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The classical secretary problem for selecting the best item is studied when the actual values of
the items are observed with noise. One of the main appeals of the secretary problem is that the
optimal strategy is able to find the best observation with the nontrivial probability of about
0.37, even when the number of observations is arbitrarily large. The results are strikingly
different when the quality of the secretaries are observed with noise. If there is no noise, then
the only information that is needed is whether an observation is the best among those already
observed. Since observations are assumed to be i.i.d. this is distribution free. In the case of
noisy data, the results are no longer distrubtion free. Furthermore, one needs to know the rank of
the noisy observation among those already seen. Finally, the probability of finding the best
secretary often goes to 0 as the number of obsevations, n, goes to infinity. The results depend
heavily on the behavior of pn, the probability that the observation that is best among the noisy
observations is also best among the noiseless observations. Results involving optimal strategies
if all that is available is noisy data are described and examples are given to elucidate the
results.

In today's increasingly interconnected world, deciding with whom and at what level to cooperate
becomes a matter of increasing importance as societies become more globalized and large-scale
cooperation becomes a viable means of addressing global issues. This tension can play out via
competition between local (e.g. within a group) and global (e.g., between groups) interests. Despite
research highlighting factors influencing cooperation in such multi-layered situations, their
biological basis is not well understood. In a double-blind placebo controlled study, we investigated
the influence of intranasally administered oxytocin and arginine vasopressin on cooperative
behavior at local and global levels. We find that oxytocin causes an increase in both the
willingness to cooperate and the expectation that others will cooperate at both levels. In contrast,
participants receiving vasopressin did not differ from those receiving placebo in their cooperative
behavior. Our results highlight the selective role of oxytocin in intergroup cooperative behavior.

In their seminal works, Arrow (1965) and Pratt (1964) defined two aspects of risk aversion:
absolute risk aversion and relative risk aversion. Based on their definitions, we define two
aspects of risk: absolute risk and relative risk. We consider situations in which, by making an
investment, an agent exchanges a certain amount of wealth w by a random distributed level of
wealth W. In such situations, we define absolute risk as the riskiness of a gamble that is
distributed as W-w, and relative risk as the riskiness of a security that is distributed as W/w.
We measure absolute risk by the Aumann and Serrano (2008) index of riskiness and relative risk by
an equivalent index that we develop in this paper. The two concepts of risk do not necessarily
agree on which one of two investments is riskier, and hence they capture two different aspects of
risk.

We present an example of a discounted stochastic game with a continuum of states,
finitely many players and actions, and deterministic transitions, that possesses no measurable
stationary equilibria, or even stationary approximate equilibria. The example is robust to
perturbations of the payoffs, the transitions, and the discount factor, and hence gives a strong
nonexistence result for stationary equilibria. The example is a game of perfect information, and
hence it also does not possess stationary extensive-form correlated equilibrium. Markovian
equilibria are also shown not to exist in appropriate perturbations of our example.

The paper analyzes a perturbation on the players' knowledge of the game in the traveler's dilemma, by introducing some uncertainty about the range of admissible actions. The ratio between changes
in the outcomes and the size of perturbation is shown to grow exponentially in the range of the
given game. This is consistent with the intuition that a wider range makes the outcome of the
traveler's dilemma more paradoxical. We compare this with the growth of the elasticity index
(Bavly (2011)) of this game.

We provide a new characterization of implementability of reduced form mechanisms in terms of straightforward second-order stochastic dominance. In addition, we present a simple proof of Matthews' (1984) conjecture, proved by Border (1991), on implementability.

Reinforcement learning in complex natural environments is a challenging task because the agent should generalize from the outcomes of actions taken in one state of the world to future actions in different states of the world. The extent to which human experts find the proper level of generalization is unclear. Here we show, using the sequences of field goal attempts made by professional basketball players, that the outcome of even a single field goal attempt has a considerable effect on the rate of subsequent 3 point shot attempts, in line with standard models of reinforcement learning. However, this change in behaviour is associated with negative correlations between the outcomes of successive field goal attempts. These results indicate that despite years of experience and high motivation, professional players overgeneralize from the outcomes of their most recent actions, which leads to decreased performance.

We develop an elasticity index of a strategic game. The index measures the robustness of the set of rational outcomes of a game. The elasticity index of a game is the maximal ratio between the change of the rational outcomes and the size of an infinitesimal perturbation. The perturbation is on the players' knowledge of the game.
The elasticity of a strategic game is a nonnegative number. A small elasticity is indicative of the robustness of the rational outcomes (for example, if there is only one player the elasticity is 0), and a large elasticity is indicative of non-robustness. For example, the elasticity of the (normalized) n-stage finitely repeated prisoner's dilemma is at least exponential in n, as is the elasticity of the n-stage centipede game and the n-ranged traveler's dilemma.
The concept of elasticity enables us to look from a different perspective at Neyman's (1999) repeated games when the number of repetitions is not commonly known, and Aumann's (1992) demonstration of the effect of irrationality perturbations.

Maskin and Riley (2003) and Lebrun (2006) prove that the Bayes-Nash equilibrium of first-price auctions is unique. This uniqueness requires the assumption that a buyer never bids above his value. We demonstrate that, in asymmetric first-price auctions (with or without a minimum bid), the relaxation of this assumption results in additional equilibria that are "substantial." Although in each of these additional equilibria no buyer wins with a bids above his value, the allocation of the object and the selling price may vary among the equilibria. Furthermore, we show that such phenomena can only occur under asymmetry in the distributions of values.

This paper investigates the effect of compensation of corporate personnel on their investment in
new technologies. We focus on a specific corporate activity, namely corporate venture capital
(CVC), describing minority equity investment by established-firms in entrepreneurial ventures.
The setting offers an opportunity to compare corporate investors to investment experts, the
independent venture capitalists (IVCs). On average, we observe a performance gap between
corporate investors and their independent counterparts. Interestingly, the performance gap is
sensitive to CVCs' compensation scheme: it is the largest when CVC personnel are awarded
performance pay. Not only do we study the association between incentives and performance
but we also document a direct relationship between incentives and the actions managers
undertake. For example, we observe disparity between the number of participants in venture
capital syndicates that involve a corporate investor, and those that consist solely of IVCs. The
disparity shrinks substantially, however, for a subset of CVCs that compensate their personnel
using performance pay. We find a parallel pattern when analyzing the relationship between
compensation and another investment practice, staging of investment. To conclude, the paper
investigates the three elements of the principal-agent framework, thus providing direct evidence
that compensation schemes (incentives) shape investment practices (managerial action), and
ultimately investors¡¦ outcome (performance).

The ability to detect a change, to accurately assess the magnitude of the change, and to react to
that change in a commensurate fashion are of critical importance in many decision domains. Thus, it is important to understand the factors that systematically affect people's reactions to change. In this article we document a novel effect: Decision makers' reactions to a change (e.g., a visual change, a technology change) were systematically affected by the type of categorizations they encountered in an unrelated prior task (e.g., the response categories associated with a survey question). We found that prior exposure to narrow, as opposed to broad, categorizations improved decision makers' ability to detect change and led to stronger reactions to a given change. These differential reactions occurred because the prior categorizations, even though unrelated, altered the extent to which the subsequently presented change was perceived as either a relatively large change or a relatively small one.

Do participants bring their own priors to an experiment? If so, do they share the same priors as
the researchers who design the experiment? In this article, we examine the extent to which
self-generated priors conform to experimenters' expectations by explicitly asking participants to
indicate their own priors in estimating the probability of a variety of events. We find in Study 1 that despite being instructed to follow a uniform distribution, participants appear to have used
their own priors, which deviated from the given instructions. Using subjects' own priors allows us to account better for their responses rather than merely to test the accuracy of their estimates. Implications for the study of judgment and decision making are discussed.

We find a herding tendency among both amateur and professional investors and conclude that the
propensity to herd is lower in the professionals. These results are obtained both when we consider herding into individual stocks and herding into stocks in general. Herding depends on the firm's systematic risk and size, and the professionals are less sensitive to these variables. The differences between the amateurs and the professionals may be attributable to the latter's superior financial training. Most of the results are consistent with the theory that herding is
information-based. We also find that the herding behavior of the two groups is a persistent
phenomenon, and that it is positively and significantly correlated with stock market returns'
volatility. Finally, herding, mainly by amateurs, causes market volatility in the Granger
causality sense.

croupier from the gambling data, as well as infer bounds on the dimensions of the roulette table.

המאמר הזה עתיד להופיע כפרק בספר "הגישה הכלכלית למשפט" בעריכת אוריאל פרוקצ'יה. מכון סאקר

למחקרי חקיקה ומשפט השוואתי, האוניברסיטה העברית, 2012 . הוא בנוי סביב כמה תופעות

פסיכולוגיות מוכרות (תורת הערך; אפקט הניסוח; אפקט הבעלות; בטחון יתר; חוכמה לאחר מעשה;

הטיית העיגון), ולגבי כל אחת מציג את התופעה, את הממצאים האמפיריים המגבים אותה, ואת

ההשתמעויות שלה למשפט. עד שיצא הנוסח הסופי של הספר נבקש לא לצטט את המאמר

"Very small but cumulated decreases in food intake may be sufficient to have significant effects, even erasing obesity over a period of years" (Rozin et al., 2011). In two studies, one a lab study and the other a real-world study, we examine the effect of manipulating the position of different foods on a restaurant menu. Items placed at the beginning or the end of the list of their category options were up to twice as popular as when they were placed in the center of the list. Given this effect, placing healthier menu items at the top or bottom of item lists and less healthy ones in their center (e.g., sugared drinks vs. calorie-free drinks) should result in some increase in favor of healthier food choices.

Since its inception, psychology has studied position effects. But the position was a temporal one in sequential presentation, and the dependent variables related to memory and learning. This paper attempts to survey position effects when position is spatial (namely, position=location), all stimuli are presented simultaneously, and the dependent variable is choice. Unlike the ubiquitous "serial position curve", position effects in simultaneous choice are not consistent. A middle bias (advantage to being away from the edges) is the most common, but advantages to being first, last, or both, have also been recorded.

We develop a measure for quantifying rank order of visitation in complex sequences of male-phase versus female-phase flowers. The measure shows whether female flowers are visited before male flowers which enhances plant fitness. We apply the new method to bumble bee visitation in Digitalus purpurea and Echium vulgare and discuss our results in relation to the evolution of protandry in insect pollinated plant species.

We provide an axiomatic characterization of the measure of riskiness of gambles (risky assets) introduced by Foster and Hart (2009). The axioms are based on the concept of “wealth requirement”.

The modal view in the cognitive sciences holds that consciousness is necessary for abstract, symbolic and rule-following computations. Hence, mathematical thinking in general, and doing arithmetic more specifically, are widely believed to require consciousness. In the current paper we use continuous flash suppression to expose participants to extremely long-duration (up to 2000 milliseconds) subliminal arithmetic equations. The results of three experiments show that the equations were solved without ever reaching consciousness. In other words, they show that arithmetic can be done unconsciously. These findings imply that the modal view of the unconscious needs to be significantly updated, to include symbolic processes that were heretofore considered to be uniquely conscious.

In matching markets the number of blocking pairs is often used as a criterion to compare matchings. We argue that this criterion is lacking an economic interpretation: In many circumstances it will neither reflect the expected extent of partner changes, nor will it capture the satisfaction of the players with the matching. As an alternative, we set up two principles which single out a particularly "disruptive" subcollection of blocking pairs. We propose to take the cardinality of that subset as a measure to compare matchings. This cardinality has an economic interpretation: The subset is a justified objection against the given matching according to a bargaining set characterization of the set of stable matchings. We prove multiple properties relevant for a workable measure of comparison.

In this paper we analyze judgement aggregation problems in which a group of agents independently votes on a set of complex propositions that has some interdependency constraint between them(e.g., transitivity when describing preferences). We consider the issue of judgement aggregation from the perspective of approximation. That is, we generalize the previous results by studying approximate judgement aggregation. We relax the main two constraints assumed in the current literature, Consistency and Independence and consider mechanisms that only approximately satisfy these constraints, that is, satisfy them up to a small portion of the inputs. The main question we raise is whether the relaxation of these notions significantly alters the class of satisfying aggregation mechanisms. The recent works for preference aggregation of Kalai, Mossel, and Keller fit into this framework. The main result of this paper is that, as in the case of preference aggregation, in the case of a subclass of a natural class of aggregation problems termed `truth-functional agendas', the set of satisfying aggregation mechanisms does not extend non-trivially when relaxing the constraints. Our proof techniques involve Boolean Fourier transform and analysis of voter influences for voting protocols.

The question we raise for Approximate Aggregation can be stated in terms of Property Testing. For instance, as a corollary from our result we get a generalization of the classic result for property testing of linearity of Boolean functions.

The term “social preference” refers to decision makers’ satisfaction with their own outcomes and those attained by comparable others. The present research was inspired by what appears to be a discrepancy in the literature on social preferences – specifically, between a class of studies demonstrating people’s concern with inequality and others documenting their motivation to increase social welfare. We propose a theoretical framework to account for this puzzling difference. In particular, we argue that a characteristic of the decision setting – an individual’s role in creating the outcomes, referred to as agency – critically affects decision makers’ weighting of opposing social motives. Namely, in settings where people can merely judge the outcomes, but cannot affect them (“low agency”), their concern with inequality figures prominently. In contrast, in settings where people determine the outcomes for themselves and others (“high agency”), their concern with the welfare of others is prominent. Three studies employing a new salary-allocation paradigm document a robust effect of agency. In the high-agency condition participants had to assign salaries, while in the low-agency condition they indicated their satisfaction with equivalent predetermined salaries. We found that compared with low-agency participants, high-agency participants were less concerned with disadvantageous salary allocations and were even willing to sacrifice a portion of their pay to better others’ outcomes. The effects of agency are discussed in connection to inequality aversion, social comparison, prosocial behavior, and preference construction.

Efficient foragers avoid returning to food sources that they had previously depleted. Bombus terrestris bumblebees use a counting-like strategy to leave Alcea setosa flowers just after visiting all of their five nectaries. We tested whether a similar strategy is employed by solitary Eucera sp. bees that also forage on A. setosa. Analyses of 261 video-recorded flower visits showed that the bees most commonly probed five nectaries, but occasionally (in 7.8% of visits) continued to a nectary they had already visited. Probing durations that preceded flower departures were generally shorter than probings that were followed by an additional nectary visit in the same flower. Assuming that probing durations correlate with nectar volumes, this suggests that flower departure frequencies increased after probing of low-rewarding nectaries. The flowers' spatial attributes were not used as departure cues, but the bees may have left flowers in response to scent marks on previously visited nectaries. We conclude that Eucera females do not exhibit numerical competence as a mechanism for efficient patch use, but rather a combination of a reward-based leaving rule and scent-marking. The bees' foraging pattern is compatible with Waage's (1979, Journal of Animal Ecology, 48, 353-371) patch departure rule, which states that the tendency to leave a foraging patch increases with time, and decreases when food items are encountered. Thus, Eucera resemble bumblebees in avoiding most revisits to already-visited nectaries, but use a different foraging strategy to do so. This difference may reflect lower learning capabilities of solitary bee species compared to social ones.

Based on a model introduced by Kaminsky, Luks, and Nelson (1984), we consider a zero-sum allocation game called the Gladiator Game, where two teams of gladiators engage in a sequence of one-to-one fights in which the probability of winning is a function of the gladiators' strengths. Each team's strategy consist the allocation of its total strength among its gladiators. We find the Nash equilibria of the game and compute its value. To do this, we study interesting majorization-type probability inequalities concerning linear combinations of Gamma random variables.

The traditional premise of criminal law is that criminals who are convicted of similar crimes under similar circumstances ought to be subject to identical sentences. This article provides an efficiency-based rationale for discriminatory sentencing, i.e., establishes circumstances under which identical crimes ought to be subject to differential sentencing. We also establish the relevance of this finding to the practices of sentencing and, in particular, to the Sentencing Guidelines. Most significantly, we establish that the model can explain why celebrities, leaders, or recidivists ought to be subject to harsher sanctions than others. Discriminatory sentencing is optimal when criminals confer positive externalities on each other. If a criminal A who imposes (non-reciprocal) large positive externalities on criminal B is punished sufficiently harshly, B would expect A not to commit the crime and consequently, he would expect not to benefit from the positive externalities conferred on him by A. Given that B's expected benefits are lower, the sanctions sufficient to deter B are also lower than the ones imposed on A. The result can be easily extended to the case of reciprocal externalities. Assume that a criminal A imposes positive externalities on B and B imposes identical positive externalities on A. If A is subject to a sufficiently harsh sanction and B knows this, B would expect A not to perform the crime and therefore would expect not to benefit from the positive externalities otherwise conferred on B. Consequently, a more lenient sanction than the sanction imposed on A would be sufficient to deter B.

We consider the classical problem of selecting the best of two treatments in clinical trials with binary response. The target is to find the design that maximizes the power of the relevant test. Many papers use a normal approximation to the power function and claim that Neyman allocation that assigns subjects to treatment groups according to the ratio of the responses' standard deviations, should be used. As the standard deviations are unknown, an adaptive design is often recommended. The asymptotic justification of this approach is arguable, since it uses the normal approximation in tails where the error in the approximation is larger than the estimated quantity. We consider two different approaches for optimality of designs that are related to Pitman and Bahadur definitions of relative efficiency of tests. We prove that the optimal allocation according to the Pitman criterion is the balanced allocation and that the optimal allocation according to the Bahadur approach depends on the unknown parameters. Exact calculations reveal that the optimal allocation according to Bahadur is often close to the balanced design, and the powers of both are comparable to the Neyman allocation for small sample sizes and are generally better for large experiments. Our findings have important implications to the design of experiments, as the balanced design is proved to be optimal or close to optimal and the need for the complications involved in following an adaptive design for the purpose of increasing the power of tests is therefore questionable.

We propose a simple adaptive procedure for playing strategic games: average testing. In this procedure each player sticks to her current strategy if it yields a payoff that exceeds her
average payoff by at least some fixed \epsilon > 0; otherwise she chooses a strategy at random. We consider generic two-person games where both players play according to the average testing procedure on blocks of k-periods. We demonstrate that for all k large enough, the pair of time-average payoffs converges (almost surely) to the 3\epsilon-Pareto efficient boundary.

This research illustrates the power of reputation, such as that embodied in brand names,
demonstrating that names can enhance objective product efficacy. Study participants facing a
glaring light were asked to read printed words as accurately and as quickly as they could,
receiving compensation proportional to their performance. Those wearing sunglasses tagged Ray-Ban
made fewer errors, yet read more quickly, than those wearing the identical pair of sunglasses when
tagged Mango (a less prestigious brand). Similarly, ear-muffs blocked noise more effectively, and
chamomile tea improved mental focus more, when otherwise identical target products carried more
reputable names.

We consider uncoupled dynamics (i.e., dynamics where each player knows only his own payoff function) that reach Pareto efficient and individually rational outcomes. We prove that the number of periods it takes is in the worst case exponential in the number of players.

On several classes of n-person NTU games that have at least one Shapley NTU value, Aumann characterized this solution by six axioms: Non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives (IIA). Each of the first five axioms is logically independent of the remaining axioms, and the logical independence of IIA is an open problem. We show that for n = 2 the first five axioms already characterize the Shapley NTU value, provided that the class of games is not further restricted. Moreover, we present an example of a solution that satisffies the first 5 axioms and violates IIA for 2-person NTU games (N;V) with uniformly p-smooth V(N).

What happens when priors are not common? We show that for each type profile τ over a knowledge space (Ω, Π), where the state space Ω is connected with respect to the partition profile Π, we can associate a value 0 ≤ ε ≤ 1 that we term the prior distance of τ , where ε = 0 if and only if the profile has a common prior. If τ has ε prior distance, then for any bet f amongst the players, it cannot be common knowledge that each player expects a positive gain of ε‖f‖∞, thus extending no betting results under common priors. Furthermore, as more information is obtained and partitions are refined, the prior distance, and thus the extent of common knowledge disagreement, decreases.

Phase I clinical trials are conducted in order to find the maximum tolerated dose (MTD) of a given drug from a finite set of doses. For ethical reasons, these studies are usually sequential, treating patients or group of patients with the best available dose according to the current knowledge. However, it is proved here that such designs, and, more generally, designs that concentrate on one dose from some time on, cannot provide consistent estimators for the MTD unless very strong parametric assumptions hold. We describe a family of sequential designs that treat individuals with one of the two closest doses to the estimated MTD, and prove that such designs, under general conditions, concentrate eventually on the two closest doses to the MTD and estimate the MTD consistently. It is shown that this family contains randomized designs that assign the MTD with probability that approaches 1 as the size of the experiment goes to infinity. We compare several designs by simulations, studying their performances in terms of correct estimation of the MTD and the proportion of individuals treated with the MTD.

The Fighter problem with discrete ammunition is studied. An aircraft (fighter) equipped with n anti-aircraft missiles is intercepted by enemy airplanes, the appearance of which follows a homogeneous Poisson process with known intensity. If j of the n missiles are spent at an encounter they destroy an enemy plane with probability a(j), where a(0)=0 and {a(j)} is a known, strictly increasing concave sequence, e.g., a(j)=1 - q^j, 0 < 1. If the enemy is not destroyed, the enemy shoots the fighter down with known probability 1 - u, where 0 ≤ u ≤ 1. The goal of the fighter is to shoot down as many enemy airplanes as possible during a given time period [0,T ]. Let K(n, t) be an optimal number of missiles to be used at a present encounter, when the fighter has flying time t remaining and n missiles remaining. Three seemingly obvious properties of K(n, t) have been conjectured: [A] The closer to the destination, the more of the n missiles one should use, [B] the more missiles one has, the more one should use, and [C] the more missiles one has, the more one should save for possible future encounters. We show that [C] holds for all 0 ≤ u ≤ 1, that [A] and [B] hold for the "Invincible Fighter" (u = 1), and that [A] holds but [B] fails for the "Frail Fighter" (u = 0).

Studies of the detection of change have commonly been concerned with individuals inspecting a system or a process, whose characteristics were fully determined by the researcher. We, instead, study the detection of change in the preferences - and hence the behavior - of others with whom an individual interacts. More specifically, we study situations in which one's benefits are the result of the joint actions of one and one's partner when at times the preferred combination is the same for both and at times it is not. In other words, what we change is the payoffs associated with the different combinations of interactive choices and then look at choice behavior following such a change. We find that players are extremely quick to respond to a change in the preferences of their counterparts. This responsiveness can be explained by the players' impulsive reaction to regret - if one was due - at their most recent decision.

We compare estimators of the (essential) supremum and the integral of a function f defined on a measurable space when f may be observed at a sample of points in its domain, possibly with error. The estimators compared vary in their levels of stratification of the domain, with the result that more refined stratification is better with respect to different criteria. The emphasis is on criteria related to stochastic orders. For example, rather than compare estimators of the integral of f by their variances (for unbiased estimators), or mean square error, we attempt the stronger comparison of convex order when possible. For the supremum the criterion is based on the stochastic order of estimators.
For some of the results no regularity assumptions for f are needed, while
for others we assume that f is monotone on an appropriate domain.

The ability of invertebrates to perform complex cognitive tasks is widely debated. Bees utilize the number of landmarks en-route to their destination as cues for navigation, but their use of numerical information in other contexts has not been studied. Numerical regularity in the spatial distribution of food occurs naturally in some flowers, which contain a fixed number of nectaries. Bees that collect nectar from such flowers are expected to increase their foraging efficiency by avoiding return visits to empty nectaries. This can be achieved if bees base their flowerdeparture decisions on the number of nectaries they had already visited, or on other sources of information that co-vary with this number.
We tested, through field observations and laboratory experiments, whether bees adapt their departure behavior to the number of available food resources. Videorecorded observations of bumblebees that visited Alcea setosa flowers with five nectaries revealed that the conditional probability of flower departure after five probings was 93%. Visit duration, the spatial attributes of the flowers and scent marks could be excluded as flower-leaving cues, while the volume of nectar collected may have guided part of the departure decisions. In the laboratory the bees foraged on two patches, each with three computer-controlled feeders, but could receive only up to two sucrose-solution rewards in each patch visit. The foragers gradually increased their tendency to leave the patches after the second reward, while the frequency of patch departure after the first reward remained constant. Patch-visit duration, nectar volume, scent marks and recurring visit sequences in a patch were ruled out as possible sources of patch-leaving information.
We conclude that bumblebees distinguish among otherwise identical stimuli by their serial position in a sequence, and use this capability to increase foraging efficiency. Our findings support an adaptive role for a complicated cognitive skill in a seemingly small and simple invertebrate.

Individuals behave with choice probabilities defined by a multinomial logit (MNL) probability distribution over a finite number of alternatives which includes utilities as parameters. The salient feature of the model is that probabilities depend on the choice-set, or domain. Expanding the choice-set decreases the probabilities of alternatives included in the original set, providing positive probabilities to the added alternatives. The wider probability 'spread' causes some individuals to fur- ther deviate from their higher valued alternatives, while others find the added alternatives highly valuable. For a population with diverse preferences, there ex- ists a subset of alternatives, called the optimum choice-set, which balances these considerations to maximize social welfare. The paper analyses the dependence of the optimum choice-set on a parameter which specifies the precision of individuals' choice ('degree of rationality'). It is proved that for high values of this parame- ter the optimum choice-set includes all alternatives, while for low values it is a singleton. Numerical examples demonstrate that for intermediate values, the size and possible nesting of the optimum choice-sets is complex. Governments have various means (defaults, tax/subsidy) to directly a¤ect choice probabilities. This is modelled by 'probability weight'parameters. The paper analyses the structure of the optimum weights, focusing on the possible exclusion of alternatives. A binary example explores the level of 'type one'and 'type two'errors which justify the imposition of early eligibility for retirement benefits, common to social security systems. Finally, the e¤ects of heterogeneous degrees of rationality among individuals are briefly discussed.

We study estimation of finite population quantiles, with emphasis on estimators that are invariant under monotone transformations of the data, and suitable invariant loss functions. We discuss non-randomized and randomized estimators, best invariant and minimax estimators and sampling strategies relative to different classes. The combination of natural invariance of the kind discussed here, and finite population sampling appears to be novel, and leads to interesting statistical and combinatorial aspects.

This article explores the extent to which decision behavior is shaped by short-lived reactions to the outcome of the most recent decision. We inspected repeated decision-making behavior in two versions of each of two decision-making tasks, an individual task and a strategic one. By regressing behavior onto the outcomes of recent decisions, we found that the upcoming decision was well predicted by the most recent outcome alone, with the tendency to repeat a previous action being affected both by its actual outcome and by the outcomes of actions not taken. Because the goodness of predictions based on the most recent outcome did not diminish as participants gained experience with the task, we conclude that repeated decisions are continuously affected by impulsive reactions.

Rutherford (2010) criticizes the way some people have analyzed the 2-children problem, claiming (correctly) that slight nuances in the problem's formulation can change the correct answer. However, his own data demonstrate that even when there is a unique correct answer, participants give intuitive answers that differ from it systematically -- replicating the data reported by those he criticizes. Thus, his critique reduces to an admonition to use care in formulating and analyzing this brainteaser -- which is always a good idea -- but contributes little to what is known, analytically or empirically, about the 2-children problem.

"Waste not want not" expresses our culture's aversion to waste. "I could have gotten the same thing for less" is a sentiment that can diminish pleasure in a transaction. We study people's willingness to "pay" to avoid this spoiler. In one scenario, participants imagined they were looking for a rental apartment, and had bought a subscription to an apartment listing. If a cheaper subscription had been declined, respondents preferred not to discover post hoc that it would have sufficed. Specifically, they preferred ending their quest for the ideal apartment after seeing more, rather than fewer, apartments. Other scenarios produced similar results. We conclude that people may sometimes prefer to be wasteful in order to avoid feeling wasteful.

Evidence, both anecdotal and scientific, suggests that people treat (or are affected by) products of prestigious sources differently than those of less prestigious or anonymous sources. The "products" which are the focus of the present study are poems, and the "sources" are the poets. We explore the manner in which the poet's name affects the experience of reading a poem. Study 1 shows that a poet's reputation has a major effect on the evaluation of a poem, whereas the poem's quality is hardly discernible to lay readers. Study 2 asks whether the poet's name affects only the reader's reported evaluation (as in The Emperor's New Clothes) or is sincere. Since we conclude it is, Study 3 explores how a poet's name alters the experience of the poem. In the absence of objective criteria for measuring "true poetic experience", we propose some indirect methodological paradigms for addressing this question.

Scientists try to find out the truth about our world. Judges in a court of law try to find out the truth about the target events in the indictment. What are the similarities, and what are the differences, in the procedures that govern the search for truth in these two systems? In particular, why are quantitative tools the hallmark of science, whereas in courts they are rarely used, and when used, are prone to error?

We study the space-and-time automaton-complexity of the CYCLE-LENGTH problem. The input is a periodic stream of bits whose cycle length is bounded by a known number n. The output, a number between 1 and n, is the exact cycle length. We also study a related problem, CYCLE-DIVISOR. In the latter problem the output is a large number that divides the cycle length, that is, a number k >> 1 that divides the cycle length, or (in case the cycle length is small) the cycle length itself. The complexity is measured in terms of the SPACE, the logarithm of the number of states in an automaton that solves the problem, and the TIME required to reach a terminal state. We analyze the worst input against a deterministic (pure) automaton, and against a probabilistic (mixed) automaton. In the probabilistic case we require that the probability of computing a correct output is arbitrarily close to one.
We establish the following results:
o CYCLE-DIVISOR can be solved in deterministic SPACE o(n), and TIME O(n).
o CYCLE-LENGTH cannot be solved in deterministic SPACE X TIME smaller than (n^2).
o CYCLE-LENGTH can be solved in probabilistic SPACE o(n), and TIME O(n).
o CYCLE-LENGTH can be solved in deterministic SPACE O(nL), and TIME O(n/L), for any positive L < 1.

Standard theory assumes that voters' preferences over actions (voting) are induced by their preferences over electoral outcomes (policies, candidates). But voters may also have non-consequentialist (NC) motivations: they may care about how they vote even if it does not a¤ect the outcome. When the likelihood of being pivotal is small, NC motivations can dominate voting behavior. To examine the prevalence of NC motivations, we design an experiment that exogenously varies the probability of being pivotal yet holds constant other features of the decision environment. We find a significant e¤ect, consistent with at least 12.5% of subjects being motivated by NC concerns.

We analyze maximization of revenue in the dynamic and stochastic knapsack problem where a given capacity needs to be allocated by a given deadline to sequentially arriving agents. Each agent is described by a two-dimensional type that reflects his capacity requirement and his willingness to pay per unit of capacity. Types are private information. We first characterize implementable policies. Then we solve the revenue maximization problem for the special case where there is private information about per-unit values, but capacity needs are observable. After that we derive two sets of additional conditions on the joint distribution of values and weights under which the revenue maximizing policy for the case with observable weights is implementable, and thus optimal also for the case with two-dimensional private information. In particular, we investigate the role of concave continuation revenues for implementation. We also construct a simple policy for which per-unit prices vary with requested weight but not with time, and prove that it is asymptotically revenue maximizing when available capacity/ time to the deadline both go to infinity. This highlights the importance of nonlinear as opposed to dynamic pricing.

We characterize the incentive compatible, constrained efficient policy ("second-best") in a dynamic matching environment, where impatient, privately informed agents arrive over time, and where the designer gradually learns about the distribution of agents' values. We also derive conditions on the learning process ensuring that the complete-information, dynamically efficient allocation of resources ("first-best") is incentive compatible. Our analysis reveals and exploits close, formal relations between the problem of ensuring implementable allocation rules in our dynamic allocation problems with incomplete information and learning, and between the classical problem, posed by Rothschild [19], of finding optimal stopping policies for search that are characterized by a reservation price property .

We consider the problem of finding the mechanism that maximizes the revenue of a seller of multiple objects. This problem turns out to be significantly more complex than the case where there is only a single object (which was solved by Myerson [5]). The analysis is difficult even in the simplest case studied here, where there are two exclusive objects and a single buyer, with valuations uniformly distributed on triangular domains. We show that the optimal mechanisms are piecewise linear with either 2 or 3 pieces, and obtain explicit formulas for most cases of interest

We consider the problem of locating a facility on a network, represented by a graph. A set of strategic agents have different ideal locations for the facility; the cost of an agent is the distance between its ideal location and the facility. A mechanism maps the locations reported by the agents to the location of the facility. Specifically, we are interested in social choice mechanisms that do not utilize payments. We wish to design mechanisms that are strategyproof, in the sense that agents can never benefit by lying, or, even better, group strategyproof, in the sense that a coalition of agents cannot all benefit by lying. At the same time, our mechanisms must provide a small approximation ratio with respect to one of two optimization targets: the social cost or the maximum cost.
We give an almost complete characterization of the feasible truthful approximation ratio under both target functions, deterministic and randomized mechanisms, and with respect to different network topologies. Our main results are: We show that a simple randomized mechanism is group strategyproof and gives a tight approximation ratio of 3/2 for the maximum cost when the network is a circle; and we
show that no randomized SP mechanism can provide an approximation ratio better than 2-o(1) to the maximum cost even when the network is a tree, thereby matching a trivial upper bound of two.

We study auctions with additive valuations where agents have a limit on the number of items they may receive. We refer to this setting as capacitated allocation games. We seek truthful and envy free mechanisms that maximize the social welfare. I.e., where agents have no incentive to lie and no agent seeks to exchange outcomes with another.
In 1983, Leonard showed that VCG with Clarke Pivot payments (which is known to be truthful, individually rational, and have no positive transfers), is also an envy free mechanism for the special case of n items and n unit capacity agents. We elaborate upon this problem and show that VCG with Clarke Pivot payments is envy free if agent capacities are all equal. When agent capacities are not identical, we show that there is no truthful and envy free mechanism that maximizes social welfare if one disallows positive transfers.
For the case of two agents (and arbitrary capacities) we show a VCG mechanism that is truthful, envy free, and individually rational, but has positive transfers. We conclude with a host of open problems that arise from our work.

We study envy-free mechanisms for scheduling tasks on unrelated machines (agents) that approximately minimize the makespan. For indivisible tasks, we put forward an envy-free poly-time mechanism that approximates the minimal makespan to within a factor of O(logm), where m is the number of machines. We also show a lower bound of Omega(log m/log logm). This improves the recent result of Mu'alem [22] who give an upper bound of (m + 1)/2, and a lower bound of 2 - 1/m. For divisible tasks, we show that there always exists an envy-free poly-time mechanism with optimal makespan. Finally, we demonstrate how our mechanism for envy free makespan minimization can be interpreted as a market clearing problem.

We quantify the effect of Bayesian ignorance by comparing the social cost obtained in a Bayesian game by agents with local views to the expected social cost of agents having global views. Both benevolent agents, whose goal is to minimize the social cost, and selfish agents, aiming at minimizing their own individual costs, are considered. When dealing with selfish agents, we consider both best and worst equilibria outcomes. While our model is general, most of our results concern the setting of network cost sharing (NCS) games. We provide tight asymptotic results on the effect of Bayesian ignorance in directed and undirected NCS games with benevolent and selfish agents. Among our findings we expose the counter-intuitive phenomenon that "ignorance is bliss": Bayesian ignorance may substantially improve the social cost of selfish agents. We also prove that public random bits can replace the knowledge of the common prior in attempt to bound the effect of Bayesian ignorance in settings with benevolent agents. Together, our work initiates the study of the effects of local vs. global views on the social cost of agents in Bayesian contexts.

A response to criticism of the paper "On the Matter of the Man with Three Wives," Moriah 22 (1999), 98- 107 (see also Rationality Center DP 102, June 1996). The Moriah paper is a non-mathematical account, written in Hebrew for the Rabbinic public, of "Game-Theoretic Analysis of a Bankruptcy Problem from the Talmud," by R. Aumann and M. Maschler, J. Econ. Th. 36 (1985), 195- 213. The current response appeared in Hama'yan 50 (2010), 1- 11.

A lecture explaining the causes of the 2008‐9 world financial crisis in terms of ordinary economic processes. The lecture was delivered at the 39th St. Gallen Symposium, University of St. Gallen, Switzerland, 8 May 2009.

In 1995, Aumann showed that in games of perfect information, common knowledge of rationality is consistent and entails the back- ward induction (BI) outcome. That work has been criticized because it uses "counterfactual" reasoning|what a player "would" do if he reached a node that he knows he will not reach, indeed that he him- self has excluded by one of his own previous moves. This paper derives an epistemological characterization of BI that is outwardly reminiscent of Aumann's, but avoids counterfactual reason- ing. Specifically, we say that a player strongly believes a proposition at a node of the game tree if he believes the proposition unless it is logically inconsistent with that node having been reached. We then show that common strong belief of rationality is consistent and entails the BI outcome, where - as with knowledge - the word "common" signifies strong belief, strong belief of strong belief, and so on ad infinitum. Our result is related to - though not easily derivable from - one obtained by Battigalli and Sinischalchi [7]. Their proof is, however, much deeper; it uses a full-blown semantic model of probabilities, and belief is defined as attribution of probability 1. However, we work with a syntactic model, defining belief directly by a sound and complete set of axioms, and the proof is relatively direct.

This paper presents an overlapping generations model to explain why humans live in families rather than in other pair groupings. Since most non-human species are not familial, something special must be behind the family. It is shown that the two necessary features that explain the origin of the family are given by uncertain paternity and overlapping cohorts of dependent children. With such two features built into our model, and under the assumption that individuals care only for the propagation of their own genes, our analysis indicates that fidelity families dominate promiscuous pair bonding, in the sense that they can achieve greater survivorship and enhanced genetic fitness. The explanation lies in the free riding behavior that characterizes the interactions between competing fathers in the same promiscuous pair grouping. Kin ties could also be related to the emergence of the family. When we consider a kinship system in which an adult male transfers resources not just to his offspring but also to his younger siblings, we find that kin ties never emerge as an equilibrium outcome in a promiscuous environment. In a fidelity family environment, instead, kinship can occur in equilibrium and, when it does, it is efficiency enhancing in terms of greater survivorship and fitness. The model can also be used to shed light on the issue as to why virtually all major world religions are centered around the importance of the family.

Under the conventional approach in torts, liability for an accident is decided by comparing the injurer's costs of precautions with those of the victim, and, under the negligence rule, also with the expected magnitude of harm. In multiplevictim cases, the current paradigm holds that courts should determine liability by comparing the injurer's costs of precautions with the victims' aggregate costs and with their aggregate harm. This aggregative risk-utility test supposedly results in the imposition of liability on the least-cost avoiders of the accident, and, therefore, is assumed efficient. However, this paradigm neglects the importance of the normal differences between tort victims. When victims are heterogeneous with regard to their expected harm or costs of precaution, basing the liability-decision on the aggregate amounts may be incorrect, causing in some cases over-deterrence, while in other, under-deterrence and dilution of liability. A new paradigm is therefore needed. This Article demonstrates how aggregate liability may violate aggregate efficiency, and concludes that decisions based upon aggregate amounts are inappropriate when the victims are heterogeneous-as they typically are in real life. The Article then turns to an exploration of an alternative to the aggregative risk-utility test, and argues for a legal rule that would combine restitution for precaution costs, plus an added small "bonus," with the sampling of victims' claims.

To answer the question in the title we vary agents' beliefs against the background of a fixed knowledge space, that is, a state space with a partition for each agent. Beliefs are the posterior probabilities of agents, which we call type profiles. We then ask what is the topological size of the set of consistent type profiles, those that are derived from a common prior (or a common improper prior in the case of an infinite state space). The answer depends on what we term the tightness of the partition profile. A partition profile is tight if in some state it is common knowledge that any increase of any single agent's knowledge results in an increase in common knowledge. We show that for partition profiles which are tight the set of consistent type profiles is topologically large, while for partition profiles which are not tight this set is topologically small.

Stochastic dominance is a partial order on risky assets ("gambles") that is based on the uniform preference, of all decision-makers (in an appropriate class), for one gamble over another. We modify this, first, by taking into account the status quo (given by the current wealth) and the possibility of rejecting gambles, and second, by comparing rejections that are substantive (that is, uniform over wealth levels or over utilities). This yields two new stochastic orders: wealth-uniform dominance and utility-uniform dominance. Unlike stochastic dominance, these two orders are complete: any two gambles can be compared. Moreover, they are equivalent to the orders induced by, respectively, the Aumann-Serrano (2008) index of riskiness and the Foster-Hart (2009a) measure of riskiness.

We study a class of optimal allocation problems, including the well-known Bomber Problem, with the following common probabilistic structure. An aircraft equipped with an amount x of ammunition is intercepted by enemy airplanes arriving according to a homogenous Poisson process over a fixed time duration t. Upon encountering an enemy, the aircraft has the choice of spending any amount 0 <= y <= x of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of y. Two different goals have been considered in the literature concerning the optimal amount K(x,t) of ammunition spent: (i) Maximizing the probability of surviving for time t, which is the so-called Bomber Problem, and (ii) maximizing the number of enemy airplanes shot down during time t, which we call the Fighter Problem. Several authors have attempted to settle the following conjectures about the monotonicity of K(x,t): [A] K(x, t) is decreasing in t, [B] K(x,t) is increasing in x, and [C] the amount x - K(x,t) held back is increasing in x. [A] and [C] have been shown for the Bomber Problem with discrete ammunition, while [B] is still an open question. In this paper we consider both time and ammunition continuous, and for the Bomber Problem prove [A] and [C], while for the Fighter we prove [C] in general, and that [A] holds for one special case and [B] for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (TP2) in the Bomber and Fighter Problems, respectively. The TP2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves TP2 and other properties.

A completely uncoupled dynamic is a repeated play of a game, where each period every player knows only his action set and the history of his own past actions and payoffs. One main result is that there exist no completely uncoupled dynamics with finite memory that lead to pure Nash equilibria (PNE) in almost all games possessing pure Nash equilibria. By "leading to PNE" we mean that the frequency of time periods at which some PNE is played converges to 1 almost surely. Another main result is that this is not the case when PNE is replaced by "Nash epsilon-equilibria": we exhibit a completely uncoupled dynamic with finite memory such that from some time on a Nash epsion-equilibrium is played almost surely.

Explaining human cooperation in large groups of non-kin is a major challenge to both rational choice theory and the theory of evolution. Recent research suggests that group cooperation can be explained assuming that cooperators can punish non-cooperators or cheaters. The experimental evidence comes from economic games in which group members are informed about the behavior of all others and cheating occurs in full view. We demonstrate that under more realistic information conditions, where cheating is less obvious, punishment is ineffective in enforcing cooperation. Evidently, the explanatory power of punishment is constrained by the visibility of cheating.

We investigate sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT). In the Bayesian game Gn among n jurors, we allow for arbitrary distribution on the types of jurors. In particular, any kind of dependency is possible. If each juror i has a “constant strategy”, si (that is, a strategy that is independent of the size n≥i of the jury), such that s=( s1, s2, . . . , sn . . .) satisfies theCJT, then byMcLennan (1998) there exists a Bayesian-Nash equilibrium that also satisfies the CJT. We translate the CJT condition on sequences of constant strategies into the following problem:

(**) For a given sequence of binary random variables X = (X1, X2, ..., Xn, ...) with joint distribution P, does the distribution P satisfy the asymptotic part of the CJT ?

We provide sufficient conditions and two general (distinct) necessary conditions for (**). We give a complete solution to this problem when X is a sequence of exchangeable binary random variables.

 

Should an auctioneer start a rising auction from some starting price or set it as a reservation price? Under what circumstances might a bidder find it rational to raise the current offer by a substantial factor instead of making just a small increase above the highest bid? This paper aims to answer both of these questions by exploring the implications of jump bidding over the information sets available to the bidders. Our motivation is to find whether hiding the information about other players' signals might be beneficial for one of the bidders. We first show that it is better for the auctioneer to set a reservation price rather than "jump" to the starting price. We then prove that in a very general setting and when bidders are risk-neutral there exist no equilibrium with jump bidding (in non-weakly dominated strategies). Finally, we demonstrate that jump bidding might be a rational consequence of risk aversion, and analyze the different effects at work.

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader’s payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.

Infinite sequential games, in which Nature chooses a Borel winning set and reveals it to one of the players, do not necessarily have a value if Nature has 3 or more choices. The value does exist if Nature has 2 choices. The value also does not necessarily exist if Nature chooses from 2 Borel payoff functions. Similarly, if Player 1 chooses the Borel winning set and does not reveal his selection to Player 2, then the game does not necessarily have a value if there are 3 or more choices; it does have a value if there are only 2 choices. If Player 1 chooses from 2 Borel payoff functions and does not reveal his choice, the game need not have a value either.

This paper is concerned with the question of extending the definition of the bargaining set, a cooperative game solution, when cooperation takes place in a repeated setting. The focus is on situations in which the players face (finite or infinite) sequences of exogenously specified TU-games and receive sequences of imputations against those static cooperative games in each time period. Two alternative definitions of what a `sequence of coalitions' means in such a context are considered, in respect to which the concept of a repeated game bargaining set may be defined, and existence and non-existence results are studied. A solution concept we term subgame-perfect bargaining set sequences is also defined, and sufficient conditions are given for the nonemptiness of subgame-perfect solutions in the case of a finite number of time periods.

Extending to infinite state spaces that are compact metric spaces a result previously attained by D. Samet solely in the context of finite state spaces, a necessary and suficient condition for the existence of a common prior for several players is given in terms of the players' present beliefs only. A common prior exists if and only if for each random variable it is common knowledge that all Cesaro means of iterated expectations with respect to any permutation converge to the same value; this value is its expectation with respect to the common prior. It is further shown that compactness is a necessary condition for some of the results.

We introduce emotions into an equilibrium notion. In a mental equilibrium each player "selects" an emotional state which determines the player's preferences over the outcomes of the game. These preferences typically differ from the players' material preferences. The emotional states interact to play a Nash equilibrium and in addition each player's emotional state must be a best response (with respect to material preferences) to the emotional states of the others. We discuss the concept behind the definition of mental equilibrium and show that this behavioral equilibrium notion organizes quite well the results of some of the most popular experiments in the experimental economics literature. We shall demonstrate the role of mental equilibrium in incentive mechaisms and will discuss the concept of collective emotions, which is based on the idea that players can coordinate their emotional states.

It has been argued that increased life expectancy raises the rate of return on education, causing a rise in the investment in education followed by an increase in lifetime labor supply. Empirical evidence of these relations is rather weak. Building on a lifecycle model with uncertain longevity, this paper shows that increased life expectancy does not suffice to warrant the above hypotheses. We provide assumptions about the change in survival probabilities, specifically about the age dependence of hazard rates, which determine individuals' behavioral response w.r.t. education, work and age of retirement. Comparison is made between the case when individuals have access to a competitive annuity market and the case of no insurance.

Two salient features of modern economic growth are the rise in aggregate savings rates and the steady increase in life expectancy. This paper links these processes, showing that under certain conditions economic theory supports the hypothesis that increased longevity leads to higher aggregate savings in steady state. The analysis is based on a lifecycle model with uncertain longevity in which individuals choose an optimum consumption path and a retirement age. Conditions on the age-specific pattern of improvements in survival probabilities are shown to ensure that individual savings rise with longevity and that aggregation preserves this result. Population theory (Coale (1972)) is used to link the steady-state age density function and the population's growth rate to individuals' survival probabilities. The importance of a competitive annuity market in avoiding unintended bequests is underscored.

"Game Engineering" deals with the application of game theoretic methods to interactive situations or systems in which the rules are well defined, or where the designer can himself specify the rules. This talk, which addressed a business-school audience with no specific knowledge of game theory, describes five examples of game engineering: two dealing with auctions, two with traffic systems, and one with arbitration. At the end of the talk there was a Q & A session, which, too, is recorded here.

We introduce a simple "riskier than" order between gambles, from which the index of riskiness developed by Aumann and Serrano (2008) is directly obtained.

We analyze the nature of optimal contracts in a dynamic model of repeated (and persistent) adverse selection and moral hazard. In particular we consider the case of surgeons who diagnose patients and then decide whether to perform an operation, and if so, whether to exert a costly but unobservable effort. The probability of a successful operation is a function of the surgeon's effort, his quality, and the severity of the patient's problem, all of which are the surgeon's private information.
The principal observes only the history of successes and failures and is allowed to promise financial rewards as a function of the observed history. His goal is to provide incentives at minimum cost so that if the patient needs minor surgery he will be treated by any type of surgeon (low- or high-quality) but if he needs major surgery, only a high-quality surgeon will perform the operation.
The optimal contract-pair is characterized and is shown to reflect the practice often observed in the medical industry. Performing an operation is a gamble whose probability of success is higher, the higher the quality of the surgeon. A sequence of operations is exponentially less likely to be successful if the surgeon is not high-quality. An optimal contract for a high-quality surgeon exploits this fact by stipulating a high reward conditional on a long history of successes, while such a stipulation makes the contract much less attractive to a low-quality surgeon.

Several species of glaphyrid beetles forage and mate on Mediterranean red flowers. In red anemones and poppies in Israel, female beetles occupy only bowl-shaped a subset of the flowers, do not aggregate, and are hidden below the petals. This raises the question how males find their mates. The possibility that males and females orient to similar plant- generated cues, thereby increasing their mate encounter prospects, was investigated. Beetle attraction to red models increased with display area in previous studies. Choice tests with flowers and with models indicate that both male and female beetles prefer large displays to smaller ones. In anemones, beetles rest, feed and mate mainly on male- phase flowers, which are larger than female- phase flowers. Poppies that contain beetles are larger than the population average. These findings support the hypothesis that males and females meet by orienting to large red displays. Corolla size correlates with pollen reward in both plant species, suggesting that visits to large flowers also yield foraging benefits. Male beetles often jump rapidly among adjacent flowers. In contrast to the preference for large flowers by stationary individuals, these jumps sequences are random with respect to flower (in anemone) and size (in poppy). They may enable males to detect females at sex-phase close range. We hypothesize that males employ a mixed mate- searching strategy, combining orientation to floral signals and to female- produced cues. The glaphyrids' preference for large flowers may have selected for extraordinarily large displays within the "red anemone" pollination guild of the Levant.

Kolmogorov's setting for probability theory is given an original generalization to account for probabilities arising from Quantum Mechanics. The sample space has a central role in this presentation and random variables, i.e., observables, are defined in a natural way. The mystery presented by the algebraic equations satisfied by (non-commuting) observables that cannot be observed in the same states is elucidated

Recent studies involving intertemporal choice have prompted many economists to abandon the classical exponential discount utility function in favor of one characterized by hyperbolic discounting. Hyperbolic discounting, however, implies a reversal of preferences over time that is often described as dynamically inconsistent and ultimately irrational. We analyze hyperbolic discounting and its characteristic preference reversal in the context of rule-rationality, an evolutionary approach to rationality that proposes that people do not maximize utility in each of their acts; rather, they adopt rules of behavior that maximize utility in the aggregate, over all decisions to which an adopted rule applies. In this sense, people maximize over rules rather than acts. Rule-rationality provides a framework through which we may examine the rational basis for hyperbolic discounting in fundamental terms, and in terms of its evolutionary foundations. We conclude that although aspects of hyperbolic discounting may contain a certain destructive potential, it is likely that its evolutionary foundations are sound -- and its application may well be as justified and rational today as it was for our foraging ancestors.

It is known that the value of a zero-sum infinitely repeated game with incomplete information on both sides need not exist [Aumann Maschler 95]. It is proved that any number between the minmax and the maxmin of the zero-sum infinitely repeated game with incomplete information on both sides is the value of the long finitely repeated game where players' information about the uncertain number of repetitions is asymmetric.

This paper provides a new way to identify conditional cooperation in a real-time version of the standard voluntary contribution mechanism. Our approach avoids most drawbacks of the traditional procedures because it relies on endogenous cycle lengths, which are defined by the number of contributors a player waits before committing to a further contribution. Based on hypothetical distributions of randomly generated contribution sequences, we provide strong evidence for conditionally cooperative behavior. Moreover, notwithstanding a decline in contributions, conditional cooperation is found to be stable over time.

The variation of a martingale m[k] of k+1 probability measures p(0),...,p(k) on a finite (or countable) set X is the expectation of the sum of ||p(t)-p(t-1)|| (the L one norm of the martingale differences p(t)-p(t-1)), and is denoted V(m[k]). It is shown that V(m[k]) is less than or equal to the square root of 2kH(p(0)), where H(p) is the entropy function (the some over x in X of p(x)log p(x) and log stands for the natural logarithm). Therefore, if d is the number of elements of X, then V(m[k]) is less than or equal to the square root of 2k(log d). It is shown that the order of magnitude of this bound is tight for d less than or equal to 2 to the power k: there is C>0 such that for every k and d less than or equal to 2 to the power k there is a martingale m[k]=p(0),...,p(k) of probability measures on a set X with d elements, and with variation V(m[k]) that is greater or equal the square root of Ck(log d). It follows that the difference between the value of the k-stage repeated game with incomplete information on one side and with d states, denoted v(k), and the limit of v(k), as k goes to infinity, is bounded by the maximal absolute value of a stage payoff times the square root of 2(log d)/k, and it is shown that the order of magnitude of this bound is tight.

A problem of optimally allocating partially effective ammunition x to be used on randomly arriving enemies in order to maximize an aircraft's probability of surviving for time t, known as the Bomber Problem, was first posed by Klinger and Brown (1968). They conjectured a set of apparently obvious monotonicity properties of the optimal allocation function K(x,t). Although some of these conjectures, and versions thereof, have been proved or disproved by other authors since then, the remaining central question, that K(x,t) is nondecreasing in x, remains unsettled. After reviewing the problem and summarizing the state of these conjectures, in the setting where x is continuous we prove the existence of a "spend-it-all" region in which K(x,t) = x and find its boundary, inside of which the long-standing, unproven conjecture of monotonicity of K(.,t) holds. A new approach is then taken of directly estimating K(x,t) for small t, providing a complete small-t asymptotic description of K(x,t) and the optimal probability of survival.

 Cooperation poses theoretical problems because the behaviors of individuals can benefit others. Evolutionary and game-theory explanations that focus on maximizing one's own material outcomes are usually supported by experimental models with isolated and anonymous subjects. Cooperation in the natural world, however, is often a social act whereby familiar individuals coordinate behaviors for shared outcomes. Social cooperation is also associated with a cooperation bias expressed as a preference for cooperation even when noncooperation is immediately more beneficial. The authors report on evidence for such a bias in a captive group of bottlenose dolphins that voluntarily preferred to receive petting from human guides by using a pairwise coordinated approach, even though this was more difficult, and total petting amount was thereby reduced. To explain why this bias occurs, the authors propose an integrated behavioral-evolutionary approach whereby performance is determined by two kinds of immediate outcomes: material gains and intrinsic affective states associated with cooperating. The latter can provide reinforcement when immediate material gains are reduced, delayed, or absent. Over a lifetime, this proximate mechanism can lead to cooperative relationships whose long-term ultimate consequences can be adaptive.

In this study, coordinated breathing was studied in 13 common bottlenose dolphins because of its links with spontaneous coordinated behaviors (e.g., swimming, foraging, and playing). A strong link was shown between dyadic coordination levels and age/sex categories when both association patterns and spatial formation are considered. This is consistent with a significant influence of social relationships on cooperating and contrasts with an economic perspective based on immediate material outcomes alone. This cooperation bias is explained by linking proximate processes that evoke performance with ultimate evolutionary processes driven by long-term adaptive outcomes. Proximate processes can include 2 kinds of immediate outcomes: material reinforcements and affective states associated with acts of cooperating that can provide positive reinforcement regardless of immediate material benefits (e.g., when there is a time lag between cooperative acts and material outcomes). Affective states can then be adaptive by strengthening social relationships that lead to eventual gains in fitness.

This paper shows that workers can affect the productivity of their coworkers based on income maximization considerations, rather than relying on behavioral considerations such as peer pressure, social norms, and shame. We show that a worker's effort has a positive effect on the effort of coworkers if they are complements in production, and a negative effect if they are substitutes. The theory is tested using a panel data set of baseball players from 1970 to 2003. The results are consistent with the idea that the effort choices of workers interact in ways that are dependent on the technology of production.

By incentive reversal we refer to situations in which an increase of rewards for all agents results in fewer agents exerting effort. We show that externalities among peers may give rise to such intriguing situations even when all agents are fully rational. We provide a necessary and sufficient condition on the organizational technology in order for it to be susceptible to incentive reversal. The condition implies that some degree of complementarity is enough to allow incentive reversal.

A class of voting procedures based on repeated ballots and elimination of one candidate in each round is shown to always induce an outcome in the top cycle and is thus Condorcet consistent, when voters behave strategically. This is an important class as it covers multi-stage, sequential elimination extensions of all standard one-shot voting rules (with the exception of negative voting), the same one-shot rules that would fail Condorcet consistency. The necessity of repeated ballots and sequential elimination are demonstrated by further showing that Condorcet consistency would fail in all standard voting rules that violate one or both of these conditions.

Few branches of mathematics have been more influential in the social sciences than game theory. In recent years, it has become an essential tool for all social scientists studying the strategic behavior of competing individuals, firms, and countries. However, the mathematical complexity of game theory is often very intimidating for students who have only a basic understanding of mathematics. Insights into Game Theory addresses this problem by providing students with an understanding of the key concepts and ideas of game theory without using formal mathematical notation. The authors use four different topics (college admissions, social justice and majority voting, coalitions and cooperative games, and a bankruptcy problem from the Talmud) to investigate four areas of game theory. The result is a fascinating introduction to the world of game theory and its increasingly important role in the social sciences.

The secretary problem for selecting one item so as to minimize its expected rank, based on observing the relative ranks only, is revisited. A simple suboptimal rule, which performs almost as well as the optimal rule, is given. The rule stops with the smallest i such that Ri <= ic/(n + 1 - i) for a given constant c, where Ri is the relative rank of the ith observation, and n is the total number of items. This rule has added flexibility. i) A curtailed version thereof can be used to select an item with a given probability P, P < 1. ii) The rule can be used to select two or more items. The problem of selecting a fixed proportion, a, 0 < a < 1, of n, is also treated. Numerical results are included to illustrate the findings.

The subset agreement problem generalizes all forms of two-sided matching. Two agents need to agree on some subset of a given finite set of contracts. A solution concept - agreement - generalizes the notion of a stable subset. Its definition does not require the consideration of a preference ordering on sets of contracts, but only that of the choice function that reveals the agents' preferences by choosing the best subset of any given set of contracts. Under a suitable condition, called coherence, that requires that contracts are substitutes to one another, at least one greement always exists. A constructive proof is given that the structure of the set of agreements is a lattice.

We propose and discuss two postulates on the nature of errors in highly correlated noisy physical stochastic systems. The first postulate asserts that errors for a pair of substantially correlated elements are themselves substantially correlated. The second postulate asserts that in a noisy system with many highly correlated elements there will be a strong effect of error synchronization. These postulates appear to be damaging for quantum computers. The paper includes a self-contained description of the model of quantum computers.

Two-player zero-sum stochastic games with finite state and action spaces, as well as two-player zero-sum absorbing games with compact metric action spaces, are known to have undiscounted values. We study such games under the assumption that one or both players observe the actions of their opponent after some time-dependent delay. We develop criteria for the rate of growth of the delay such that a player subject to such an information lag can still guarantee himself in the undiscounted game as much as he could have with perfect monitoring. We also demonstrate that the player in the Big Match with the absorbing action subject to information lags which grow too rapidly, according to certain criteria, will not be able to guarantee as much as he could have in the game with perfect monitoring.
toring.

Managerial reasoning about performance targets and subsequent actions can be influenced by whether they focus their attention on expectations of future events or internal efforts to meet organizational goals. This study explores how managers think about expectations and aspirations by examining the semantic similarities and differences between these concepts for practicing managers and economists, the results suggesting subtle differences in how economists and managers reason about aspirations and expectations. For economists, the concept of expectations played a major role and influenced their subsequent thinking about goals and actions while managers conceptually separated factors that were controllable and uncontrollable, the concept of expectation not playing the central role for them. Implications for descriptive and prescriptive models of decision- making are discussed.

People's actions often deviate from rationality, i.e., self-interested behavior. We propose a paradigm called rule-rationality, according to which people do not maximize utility in each of their acts, but rather follow rules or modes of behavior that usually-but not always-maximize utility. Specifically, rather than choosing an act that maximizes utility among all possible acts in a given situation, people adopt rules that maximize average utility among all applicable rules, when the same rule is applied to many apparently similar situations. The distinction is analogous to that between Bentham's "act-utilitarianism" and the "rule-utilitarianism" of Mill, Harsanyi, and others. The genesis of such behavior is examined, and examples are given. The paradigm may provide a synthesis between rationalistic neo-classical economic theory and behavioral economics.

We consider stochastic dynamics whose expected (average) vector field is not necessarily continuous. We generalize the ordinary differential equation method for analyzing stochastic processes to this case, by introducing leading functions that “lead” the stochastic process across the discontinuities, which yields approximation results for the asymptotic behavior of the stochastic dynamic. We then apply the approximation results to the classical best-response dynamics used in game theory.

Define a continuous game to be one in which every player's strategy set is a Polish space, and the payoff function of each player is bounded and continuous. We prove that in this class of games the process of sequentially eliminating "never-best-reply" strategies does not terminate after the first uncountable ordinal, and that this bound is tight. Also, we examine the connection between this process and common belief of rationality in the universal type space of Mertens and Zamir.

Sexual selection theory states that male reproductive success is commonly limited by opportunities for fertilization, while female reproductive success is more often resource-limited. This creates higher selective pressure on males to attract mating partners as compared with females. Similar reasoning, when applied to plant reproduction, predicts higher investment in pollinator-attracting traits, such as nectar production, in male flowers than in female flowers. Contrary to this prediction, nectar production by female-phase flowers in the protandrous hermaphrodite shrub Salvia hierosolymita (Boiss.) (Lamiaceae) was significantly higher than in male-phase flowers in two populations over three years. Female-biased nectar production may reflect selection for pollinator attraction by female-phase flowers, possibly due to pollen limitation. In support of this interpretation, (a) the number of pollen grains on female-phase stigmas was substantially higher than on male-phase stigmas, suggesting that the female phase received more insect visits ; (b) the number of germinating pollen grains in female-phase styles only slightly exceeded the number of ovules per flower, therefore pollen availability may restrict female fitness. Proportions of female-phase flowers decreased from the base of the inflorescences towards their top. This creates a vertical gradient of nectar production, which may help reduce geitonogamous pollination by effecting pollinator behavior.

In the interest of improving their decision-making, individuals revise their opinions on the basis of samples of opinions obtained from others. However, such a revision process may lead decision-makers to experience greater confidence in their less accurate judgments. We theorize that people tend to underestimate the informative value of independently drawn opinions, if these appear to conflict with one another, yet place some confidence even in the "spurious consensus" which may arise when opinions are sampled interdependently. The experimental task involved people’s revision of their opinions (caloric estimates of foods) on the basis of advice. The method of sampling the advisory opinions (independent or interdependent) was the main factor. The results reveal a dissociation between confidence and accuracy. A theoretical underlying mechanism is suggested whereby people attend to consensus (consistency) cues at the expense of information on interdependence. Implications for belief-updating and for individual and group decisions are discussed.

Large red bowl-shaped flowers characterize the Mediterranean “poppy guild” plants, and
were suggested to reflect convergence for beetle pollination. However, the earliest-blooming
species in this guild, Anemone coronaria (L.), starts flowering about a month before beetle
emergence. Early flowering can be adaptive if the plant receives sufficient pollination by other
means during this period. We investigated A. coronaria’s pollination prospects throughout its
flowering season by monitoring its flowering phenology, the composition of the surrounding
insect community, and insect visitors. Clear protogyny precluded self pollination, and anthesis
occurred gradually over several days. Released pollen was quickly collected by insects,
suggesting no major role for wind pollination. Beetles, flies and bees were trapped at the study
site throughout the flowering period. Honeybees were the main anemone visitors during the first
seven weeks of flowering, and were joined by Glaphyrid beetles in the remaining three weeks.
Early- and late-blooming flowers had similar female reproductive success. We propose that
effective pollination by honeybees may allow anemones to bloom in early spring and thereby
reduce competition for pollinators with later-blooming species. Our results support previous
evidence for pollination of red flowers by bees, and for the importance of generalization in
pollination interactions in heterogeneous environments.

John F. Nash, Jr., submitted his Ph.D. dissertation entitled Non-Cooperative Games to Princeton University in 1950. Read it 58 years later, and you will find the germs of various later developments in game theory. Some of these are presented below, followed by a discussion concerning dynamic aspects of equilibrium.

This paper defends judicial review on the grounds that judicial review is necessary for protecting “a right to a hearing.”

Judicial review is praised by its advocates on the basis of instrumentalist reasons, i.e., because of its desirable contingent consequences such as protecting rights, romoting democracy, maintaining stability, etc. We argue that instrumentalist easons for judicial review are bound to fail and that an adequate defense of udicial review requires justifying judicial review on non-instrumentalist grounds. A non-instrumentalist justification grounds judicial review in essential attributes of he judicial process.

In searching for a non-instrumental justification we establish that judicial review is designed to protect the right to a hearing. The right to a hearing consists of hree components: the opportunity to voice a grievance, the opportunity to be rovided with a justification for a decision that impinges (or may have impinged) on one’s rights and, last, the duty to reconsider the initial decision giving rise to the grievance. The right to a hearing is valued independently of the merit of the decisions generated by the judicial process. We also argue that the recent proposals to reinforce popular or democratic participation in shaping the Constitution are wrong because they are detrimental to the right to a hearing.

The high cost of searching for employers borne by prospective employees increases friction in the labor market and inhibits formation of efficient employer-employee relationships. It is conventionally agreed that mechanisms that reduce the search costs (e.g., internet portals for job search) lower unemployment and improve overall welfare. We demonstrate that a reduction of the search costs may have the converse effect. We consider a labor market in which workers can either establish a long-term relationship with an employer by being productive, or shirk and move from one employer to the next. In addition, the workers can signal to a potential employer their intention to be productive. We show that lower search costs lead to fewer employees willing to exert effort and, in a separating equilibrium, to more individuals opting to stay completely out of the job market and remain unemployed. Furthermore, we show that lower search costs not only deteriorate the market composition, but also impair efficiency by leading to more expensive signaling in a separating equilibrium.

Question: Which ecological factors favor the transition from plants with hermaphrodite flowers to monoecious plants with separate male and female flowers on the same individual?
Mathematical methods: ESS computation in sex allocation models Key assumptions: Within a flower, costs of attraction, pollen production, style/ovary and fruit with seeds are assumed fixed. Often costs of fruit with seeds outweigh other costs. Female flowers produce more seeds than hermaphrodite flowers, due to less pollen-stigma interference.
Conclusions: When sex allocation is female-biased at the flower level, plants respond by producing either male flowers or flowers without fruit. Hermaphroditism evolves to andromonoecy (male and hermaphrodite flowers on the same plant) and then to monoecy. In species with large fruits, sex allocation is female-biased at the flower level and the production of male flowers is favored. This facilitates the production of female flowers. The alternative route via gynomonoecy (female and hermaphrodite flowers on the same plant) is improbable since it requires unrealistically high levels of seed production in female flowers. Monoecious species are likely to have: (i) small, inexpensive flowers, (ii) large, costly fruits and seeds, and (iii) high fertilization rates.

An encyclopedia article on games with incomplete information.

Table of contents:

1. Definition of the subject and its importance
2. Introduction: Modeling incomplete information
3. Harsanyi’s model: The notion of type
4. Aumann’s model
5. Harsanyi’s model and the hierarchies of beliefs
6. The Universal Belief Space
7. Belief subspaces
8. Consistent beliefs and Common prior
9. Bayesian games and Bayesian equilibrium
10. Bayesian equilibrium and Correlated equilibrium
11. Concluding remarks and future directions
12. Bibliography

A review of Steven E. Landsburg's book More Sex is Safer Sex, the Unconventional Wisdom of Economics.

The surprise 2005 best seller Freakonomics by Steven Levitt and Stephen Dubner launched a small genre of books by economists applying economic reasoning to everyday life and finding counterintuitive results. Mathematician and economist Steven Landsburg, whose online Slate column ``Everyday Economics'' predates the Levitt and Dubner volume, has now collected and expanded some of those columns to form the basis of his new book.

In his book, Landsburg uses the ``weapons of evidence and logic, especially the logic of economics'' to draw surprising conclusions which run against common sense. ``If your common sense tells you otherwise,'' says Landsburg, ``remember that common sense also tells you the Earth is flat.''

In this review, scheduled to appear in the June/July 2008 issue of the Notices of the American Mathematical Society, we describe and discuss some of the issues and claims raised in Landsburg's book. For further discussion see the May 29 post in http://gilkalai.wordpress.com/ .

In this paper we view bargaining and cooperation as an interaction superimposed on a strategic form game. A multistage bargaining procedure for N players, the "proposer commitment" procedure, is presented. It is inspired by Nash's two-player variable-threat model; a key feature is the commitment to "threats." We establish links to classical cooperative game theory solutions, such as the Shapley value in the transferable utility case. However, we show that even in standard pure exchange economies the traditional coalitional function may not be adequate when utilities are not transferable.

We consider a multitype population model with unobservable types, in which players are engaged in the `mutual help' game: each player can increase her partner's fitness at a cost to oneself. All individuals prefer free riding to cooperation, but some of them, helpers, can establish reciprocal cooperation in a long-term relationship. Such heterogeneity can drive cooperation through a partner selection mechanism under which helpers choose to interact with one another and shun non-helpers. However, in contrast to the existing literature, we assume that each individual is matched with an anonymous partner, and therefore, stable cooperation cannot be achieved by partner selection per se. We suggest that helpers can signal their type to one another in order to establish long-term relationships, and we show that a reliable signal always exists. Moreover, due to the difference in future benefits of a long-term relationship for helpers and non-helpers, the signal need not be a handicap, in the sense that the cost of the signal need not be correlated with type.

Similarity-Projection structures abstract the numerical properties of real scalar product of rays and projections in Hilbert spaces to provide a more general framework for Quantum Physics. They are characterized by properties that possess direct physical meaning. They provide a formal framework that subsumes both classical boolean logic concerned with sets and subsets and quantum logic concerned with Hilbert space, closed subspaces and projections. They shed light on the role of the phase factors that are central to Quantum Physics. The generalization of the notion of a self-adjoint operator to SP-structures provides a novel notion that is free of linear algebra.

The Gibbard-Satterthwaite theorem states that every non-trivial voting method between at least 3 alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard-Satterthwaite theorem: a random manipulation by a single random voter will succeed with non-negligible probability for every neutral voting method between 3 alternatives that is far from being
a dictatorship.

Last summer, a member of the Rationality Center at the Hebrew University of Jerusalem installed a closed-circuit TV camera in the Center's kitchen. An email explained that the camera was installed in an effort to keep the kitchen clean. By the time the camera was removed, a week later, the members of the Center exchanged close to 120 emails among themselves, expressing their opinions for and against the camera and discussing related issues.

Taking off from this exchange, I explore the surprisingly rich set of normative concerns touched upon by the kitchen-camera incident. These include a host of issues regarding people's polarized attitudes toward public surveillance, the problem of the invasive gaze and the argument that "if you have nothing to hide you have nothing to worry," the efficacy of disciplining behavior through sanctions along with the problems related to shaming sanctions, the notion of privacy and its arguable relevance to the kitchen case, and more. Special attention is given to the notion of
cleanness and to its related norms.

In an epilogue, I offer some reflections in the wake of the incident. I find that it is precisely the smallness, concreteness and seeming triviality of this incident that helps bring a large set of interconnected, vexing normative concerns into sharper relief.

In the role of judges in bankruptcy problems, people may prescribe various just divisions of the available goods to claimants who have rights for them. Two widespread division rules are equality and proportionality. A less known rule is the Talmudic "Shnaim Ohazin" principle, whose basic rationale is applying an equal division only to that part of the goods which is genuinely under dispute. This paper demonstrates that the ratio of subjects that prefer the "Shnaim Ohazin" principle over equality and proportionality can be increased by a simple framing manipulation. These results suggest that framing effects might be a prevalent factor in the realm of distributive justice.

The present paper studies the limiting behavior of the average score of a sequentially selected group of items or individuals, the underlying distribution of which, F, belongs to the Gumbel domain of attraction of extreme value distribution. This class contains the Normal, log Normal, Gamma, Weibull and many other distributions. The selection rules are the “better than average” (β = 1) and the “β-better than average” rule, defined as follows. After the first item is selected, another item is admitted into the group if and only if its score is greater than β times the average score of those already selected. Denote by Y_k the average of the k first selected items, and by T_k the time it takes to amass them. Some of the key results obtained are: Under mild conditions, for the better than average rule, Y_k less a suitable chosen function of log k converges almost surely to a finite random variable. When 1 − F(x) = exp(-[x^α +h(x)]) , α>0 and h(x)/x^α→0 as x→∞, then T_k is of approximate order k² . When β > 1, the asymptotic results for Y_k are of a completely different order of magnitude. Interestingly, for a class of distributions, T_k, suitably normalized, asymptotically approaches 1, almost surely for relatively small β > 1, in probability for moderate sized β and in distribution when β is large.

We provide an extension of the Condorcet Theorem. Our model includes both the Nitzan-Paroush framework of “unequal competencies” and Ladha’s model of “correlated voting by the jurors”. We assume that the jurors behave “informatively”, that is, they do not make a strategic use of their information in voting. Formally, we consider a sequence of binary random variables X = (X₁,X₂, ...,X_n, ...) with range in {0,1} and a joint probability distribution P. The pair (X,P) is said to satisfy the Condorcet Jury Theorem (CJT) if lim_n→∞P(∑X_i>n/2)=1. For a general (dependent) distribution P we provide necessary as well as sufficient conditions for the CJT. Let p_i = E(X_i), p_n = (p₁ + p₂, ...+ p_n)/n and X_n = (X₁ +X₂, ...+X_n)/n. A consequence of our results is that the CJT is satisfied if lim√n(p_n-1/2)=∞ and ∑_i∑_j≠iCov(X_i,X_j) ≤ 0 for n > N₀. The importance of this result is that it establishes the validity of the CJT for a domain which strictly (and naturally) includes the domain of independent jurors. Given (X,P), let p = liminf p_n, and p= limsupp_n. Let y= liminf E(X_n - p_n)², y^*= liminf E|X_n - p_n| and y^*= limsup E|X_n - p_n|. Necessary conditions for the CJT are that p ≥1/2 + 1/2y^∗,p ≥ 1/2 + y, and also p ≥ 1/2 + y^∗. We exhibit a large family of distributions P with liminf 1/n(n-1) ∑_i∑_j≠iCov(X_i,X_j) > 0 which satisfy the CJT. We do that by ‘interlacing’ carefully selected pairs (X,P) and (X′,P′). We then proceed to project the distributions P on the planes (p,y^∗) and (p,y), and determine all feasible points in each of these planes. Quite surprisingly, many important results on the possibility of the CJT are obtained by analyzing various regions of the feasible set in these planes.

R. J. Aumann and J. H. Drèze (2008) define a rational expectation of a player i in a game G as the expected payo of some type of i in some belief system for G in which common knowledge of rationality
and common priors obtain. Our goal is to characterize the set of rational expectations in terms of the game's payoff matrix. We provide such a characterization for a specific class of strategic games, called
semi-elementary, which includes Myerson's "elementary" games.

What motivates individual self-sacrificial behavior in intergroup conflicts? Is it the altruistic desire to help the ingroup or the aggressive drive to hurt the outgroup? This paper introduces a new game paradigm, the Intergroup Prisoner’s Dilemma – Maximizing Difference (IPD-MD) game, designed specifically to distinguish between these two motives. The game involves two groups. Each group member is given a monetary endowment and can decide how much of it to contribute. Contribution can be made to either of two pools, one which benefits the ingroup at a personal cost, and another which, in addition, harms the outgroup. An experiment demonstrated that contributions in the IPD-MD game are made almost exclusively to the cooperative within-group pool. Moreover, pre-play intragroup communication increases intragroup cooperation but not intergroup competition. These results are compared with those observed in the Intergroup Prisoner's Dilemma (IPD) game, where group members' contributions are restricted to the competitive between-group pool.

This note revisits the problem of selection bias, using a simple binomial example. It focuses on selection that is introduced by observing the data and making decisions prior to formal statistical analysis. Decision rules and interpretation of confidence measure and results must then be taken relative to the point of view of the decision maker, i.e., before selection or after it. Such a distinction is important since inference can be considerably altered when the decision maker's point of view changes. This note demonstrates the issue, using both the frequentist and the Bayesian paradigms.

Harsh sanctions are conventionally assumed to primarily benefit vulnerable targets. Contrary to this perception, this article shows that augmented sanctions often serve the less vulnerable targets. While
decreasing crime, harsher sanctions also induce the police to shift enforcement efforts from more to less vulnerable victims. When this shift is substantial, augmented sanctions exacerbate--rather than reduce--the risk to vulnerable victims. Based on this insight, this article suggests several normative implications concerning the efficacy of enhanced sanctions, the importance of victims' funds,and the connection between police operations and apprehension rates.

Under the conventional tort law paradigm, a tortfeasor behaves unreasonably when two conditions are met: the tortfeasor could have averted the harm by investing in cost-effective precautions and failed to do so, and other, more cost-effective precautions were not available to the victim. Torts scholarship has long argued that making such a tortfeasor responsible for the ensuing harm induces optimal care. This Article shows that by applying the conventional analysis, courts create incentives for opportunistic investments in prevention. In order to shift liability to others, parties might deliberately invest in precautions even where such investments are inefficient. The Article presents two possible solutions to the problem. By instituting a combination of (1) broader restitution rules and (2) an extended risk-utility standard, legislators and judges can reform tort law to discourage opportunistic precautions and maximize social welfare.

This work studies the value of two-person zero-sum repeated games in which at least one of the players is restricted to (mixtures of) bounded recall strategies. A (pure) k-recall strategy is a strategy that relies only on the last k periods of history. This work improves previous results [Lehrer, Neyman and Okada] on repeated games with bounded recall. We provide an explicit formula for the asymptotic value of the repeated game as a function of the stage game, the duration of the repeated game, and the recall of the agents.

We consider a sequential rule, where an item is chosen into the group, such as a university faculty member, only if his score is better than the average score of those already belonging to the group. We study four variables: The average score of the members of the group after k items have been selected, the time it takes (in terms of number of observed items) to assemble a group of k items, the average score of the group after n items have been observed, and the number of items kept after the first n items have been observed. We develop the relationships between these variables, and obtain their asymptotic behavior as k (respectively, n) tends to infinity. The assumption throughout is that the items are independent, identically distributed, with a continuous distribution. Though knowledge of this distribution is not needed to implement the selection rule, the asymptotic behavior does depend on the distribution. We study in some detail the Exponential, Pareto and Beta distributions. Generalizations of the "better than average" rule to the β better than average rules are also considered. These are rules where an item is admitted to the group only if its score is better than β times the present average of the group, where β > 0.

The vertical inflorescences of the Mediterranean annual Salvia viridis carry many small, colorful flowers, and are frequently terminated by a conspicuous tuft of colorful leaves ("flags") that attracts insect pollinators. Insects may use the flags as indicators of the food reward in the inflorescences, as long-distance cues for locating and choosing flowering patches, or both. Clipping of flags from patches of inflorescences in the field significantly reduced the number of pollinators that arrived at the patches, but not the total number of inflorescences and flowers visited by them. The number of flowers visited per inflorescence significantly increased with inflorescence size, however. Inflorescence and flower visits rates signific antly increased with patch size when flags were present, but not after flag removal. 6% of the plants in the study population did not develop any flag during blooming, yet suffered no reduction in seed set as compared to flag-bearing neighboring individuals. These results suggest that flags signal long-distance information to pollinators (perhaps indicating patch location or size), while flower-related cues may indicate inflorescence quality.

Plants that do not develop flags probably benefit from the flag signals displayed by their neighbors, without bearing the costs of flag production. Thus, flagproducing plants can be viewed as altruists that enhance their neighbors' fitness. Greenhouse-grown S. viridis plants allocated ≤ 0.5% of their biomass to flag production, and plants grown under water stress did not reduce their biomass
allocation to flags as compared to irrigated controls. These findings suggest that the expenses of flag production are modest, perhaps reducing the cost of altruism. We discuss additional potential evolutionary mechanisms that may select for the maintenance of flag production.

Foreign firms to break into a new market commonly undercut domestic prices and, hence, subsidise the consumer's costs of switching in order to get a positive market share. However, this may constitute the act of dumping as drawn in Article VI of the General Agreement on Tariffs and Trade (GATT). Consequently, domestic firms trying to protect themselves against potential competitors often demand an anti-dumping (AD) investigation. In a two-period model of market entry with horizontally differentiated products and exogenous switching costs, it is demonstrated that the mere existence of switching costs and AD-rules may result in an anti-competition effect: the administratively set minimum-price rule protects the domestic firm and yields larger prices. Therefore, there are some consumers who will not buy either product in both periods although they would have done so in absence of AD. Consequently, competition policy should reassess the AD-regulation.

The voting rules proposed by Dodgson and Young are both designed to find the candidate closest to being a Condorcet winner, according to two different notions of proximity; the score of a given candidate is known to be hard to compute under both rules. In this paper, we put forward an LP-based
randomized rounding algorithm which yields an O(log m) approximation ratio for the Dodgson score, where m is the number of candidates. Surprisingly, we show that the seemingly simpler Young score is NP-hard to approximate by any factor.

When two agents of unequal strength compete, the stronger one is expected to always win the competition. This expectation is based on the assumption that evaluation of performance is flawless. If, however, the agents are evaluated on the basis of only a small sample of their performance, the weaker agent still stands a chance of winning occasionally. A theoretical analysis indicates that for this to happen, the weaker agent must introduce variability into the effort he or she invests in the behavior, such that on some occasions the weaker agent's level of performance is as high as that of the stronger agent, whereas on others it is null. This, in turn, would drive the stronger agent to introduce variability into his or her behavior. We model this situation in a game, present its game-theoretic solution, and report an experiment, involving 144 individuals, in which we tested whether players are actually sensitive to their relative strengths and know how to allocate their resources given those relative strengths. Our results indicate that they do.

The problem of extracting as much information as possible from a sequence of observations of a stationary stochastic process X₀,X₁,…,X_n has been considered by many authors from different points of view. It has long been known through the work of D. Bailey that no universal estimator for P(X_n+1|X₀,X₁, ...X_n) can be found which converges to the true estimator almost surely. Despite this result, for restricted classes of processes, or for sequences of estimators along stopping times, universal estimators can be found. We present here a survey of some of the recent work that has been done along these lines.

Implementation theory tackles the following problem: given a social choice correspondence, find a decentralized mechanism such that for every constellation of the individuals' preferences, the set of outcomes in equilibrium is exactly the set of socially optimal alternatives (as specified by the correspondence). In this paper we are concerned with implementation by mediated equilibrium; under such an equilibrium, a mediator coordinates the players' strategies in a way that discourages deviation. Our main result is a complete characterization of social choice correspondences which are implementable by mediated strong equilibrium. This characterization, in addition to being strikingly concise, implies that some important social choice correspondences which are not implementable by strong equilibrium are in fact implementable by mediated strong equilibrium.

The paper analyzes an economy with asymmetric information in which agents trade in contingent assets. The new feature in the model is that each agent may have any prior belief on the states of nature and thus the posterior belief of an agent maybe any probability distribution that is consistent with his private information. We study two solution concepts: Equilibrium, which assumes rationality and market clearing, and common knowledge equilibrium (CKE) which makes the stronger assumption that rationality, market clearing, and the parameters which de…ne the economy are common knowledge. The two main results characterize the set of equilibrium prices and the set of CKE prices in terms of parameters which specify for each state s and event E the amount of money in the hands of agents who know the event E at the state s. The characterizations that are obtained apply to a broad class
of preferences which include all preferences that can be represented by the expectation of a state dependent monotone utility function. One implication of these results is a characterization of the information that is revealed in a CKE.

Consider an exchange economy with asymmetric information. What is the set of outcomes that are consistent with common knowledge of rationality and market clearing?

We propose the concept of CKRMC as an answer to this question. The set of price functions that are CKRMC is the maximal set F with the property that every f∈F defi…nes prices that clear the markets for demands that can be rationalized by some profile of subjective beliefs on F. Thus, the difference between CKRMC and Rational Expectations Equilibrium (REE) is that CKRMC allows for a situation where the agents do not know the true price function and furthermore may have different beliefs about it. We characterize CKRMC, study its properties, and apply it to a general class of economies with two commodities. CKRMC manifests intuitive properties that stand in contrast to the full revelation property of REE. In particular, we obtain that for a broad class of economies: (1) There is a whole range of prices that are CKRMC in every state. (2) The set of CKRMC outcomes is monotonic with the amount of information in the economy.

One of the most intriguing questions in insurance is the preference of consumers for low or zero deductible insurance policies. This stands in sharp contrast to a theorem proved by Mossin, 1968, that under quite common assumptions when the price of insurance is higher than its actuarial value, then full coverage is not optimal.

We show in a series of experiments that amateur subjects tend to underestimate the value of a policy with a deductible and that the degree of underestimation increases with the size of the deductible. We hypothesize that this tendency is caused by the anchoring heuristic. In particular, in pricing a policy with a deductible subjects first consider the price of a full coverage policy. Then they anchor on the size of the deductible and subtract it from the price of the full coverage policy. However, they do not adjust the price enough upward to take into account the fact that there is only a small chance that the deductible will be applied toward their payments. We also show that professionals in the field of insurance are less prone to such a bias. This implies that a policy with a deductible priced according to the true expected payments may seem “overpriced” to the insured and therefore may not be purchased. Since the values of full coverage policies are not underestimated the insured may find them as relatively better “deals”.

We study the problem of reaching Nash equilibria in multi-person games that are repeatedly played, under the assumption of uncoupledness: every player knows only his own payoff function. We consider strategies that can be implemented by …finite-state automata, and characterize the minimal number of states needed in order to guarantee that a pure Nash equilibrium is reached in every game where such an equilibrium exists.

Nectar yields (standing crops) in flowers within an individual plant are often highly variable. This variability may be a by-product of the foraging activity of insect pollinators. Alternatively, plants may be selected to produce highly variable rewards to reduce consecutive visitation by risk-averse pollinators, thus diminishing within-plant pollen transfer. This study evaluated the roles of pollinator control vs. plant control over nectar variability in the bee-pollinated shrub Rosmarinus officinalis L. We sampled nectar production, yield and pollinator visits in three shrubs of one population over 17 days during one blooming season. Nectar production rates were highly variable (CV=1.48), and increased after rainy days. Nectar yields were even more variable (CV=2.16), and decreased with increasing temperatures. Pollinator visit rates decreased with variability in nectar yields, increased with flower number per shrub, and were unaffected by variability in nectar production rates. Repeated sampling of marked flowers revealed no correlation between their nectar yields and production rates. These findings support the role of reward variance in reducing pollinator visits, but suggest that plants are not in complete control of this variability. Rather, plant-generated variability can be modified by intensive foraging activity of pollinators. Such pollinator control over nectar variability is likely to reduce the selective advantage of plant-generated reward variation. Plant-controlled variability may provide evolutionary advantage when pollinator activity is insufficient to generate reward variation.

Human altruism is a widespread phenomenon that puzzled evolutionary biologists since Darwin. Economic games illustrate human altruism by demonstrating that behavior deviates from economic predictions of profit maximization. A game that most plainly demonstrates this altruistic tendency is the Dictator Game. We hypothesized that human altruistic behavior is to some extent hardwired and that a likely candidate that may contribute to individual differences in altruistic behavior is the arginine vasopressin 1a (AVPR1a) receptor that in some mammals such as the vole has a profound impact on affiliative behaviors.

In the current investigation, 203 male and female university students played an online version of the Dictator Game, for real money payoffs. All subjects and their parents were genotyped for AVPR1a RS1 & RS3 promoter-region repeat polymorphisms. Parents did not participate in online game playing.

Since variation in the length of a repetitive element in the vole AVPR1a promoter region is associated with differences in social behavior we examined the relationship between RS1 and RS3 repeat length (base pairs) and allocation sums. Participants with short versions (308-325 bp) of the AVPR1a RS3 repeat allocated significantly (Likelihood ratio=14.75, p=0.001, DF=2) fewer shekels to the 'other' than participants with long versions (327-343 bp).

We also implemented a family-based association test, UNPHASED, to confirm and validate the correlation between the AVPR1a RS3 repeat and monetary allocations in the Dictator Game. Dictator Game allocations were significantly associated with the RS3 repeat (global p value: Likelihood ratio chi-sq = 11.73, DF= 4, p-value = 0.019).

The association between the AVPR1a RS3 repeat and altruism was also confirmed using two self-report scales (the Bardi-Schwartz Universalism and Benevolence Value-expressive Behavior Scales). RS3 long alleles were associated with higher scores on both measures.

Finally, long AVPR1a RS3 repeats were associated with higher AVPR1a human postmortem hippocampal mRNA levels than short RS3 repeats (One way-ANOVA: F=15.04, p=0.001, DF= 14) suggesting a functional molecular genetic basis for the observation that participants with the long RS3 repeats allocate more money than participants with the short repeats.

This is the first investigation showing that a common human polymorphism, with antecedents in lower mammals, contributes to decision making in an economic game. The finding that the same gene contributing to social bonding in lower animals also appears to operate similarly in human behavior suggests a common evolutionary mechanism.

We prove that games with absorbing states with compact action sets have a value.

Regularities in the environment are used to decide what course of action to take and how to prepare for future events. Here we focus on the utilization of regularities for prediction and argue that the commonly considered measure of regularity - the strength of the contingency between antecedent and outcome events - does not fully capture the goodness of a regularity for predictions. We propose, instead, a new measure - the level of expected prediction accuracy (ExpPA) - which takes into account the fact that, at times, maximal prediction accuracy can be achieved by always predicting the same, most prevalent outcome, and in others, by predicting one outcome for one antecedent and another for the other. Two experiments, testing the ExpPA measure in explaining participants' behavior, found that participants are sensitive to the twin facets of ExpPA and that prediction behavior is best explained by this new measure.

We define the riskiness of a gamble g as that unique number R(g) such that no-bankruptcy is guaranteed if and only if one never accepts gambles whose riskiness exceeds the current wealth.

We study an economic setting in which a principal motivates a team of strategic agents to exert costly effort toward the success of a joint project. The action taken by each agent is hidden and affects the (binary) outcome of the agent's individual task stochastically. A Boolean function, called technology, maps the individual tasks' outcomes into the outcome of the whole project. The principal induces a Nash equilibrium on the agents' actions through payments that are conditioned on the project's outcome (rather than the agents' actual actions) and the main challenge is that of determining the Nash equilibrium that maximizes the principal's net utility, referred to as the optimal contract.

Babaioff, Feldman and Nisan [1] suggest and study a basic combinatorial agency model for this setting. Here, we concentrate mainly on two extreme cases: the AND and OR technologies. Our analysis of the OR technology resolves an open question and disproves a conjecture raised in [1]. In particular, we show that while the AND case admits a polynomial-time algorithm, computing the optimal contract in the OR case is NP-hard. On the positive side, we devise an FPTAS for the OR case, which also sheds some light on optimal contract approximation of general technologies.

The Gibbard-Satterthwaite Theorem asserts the impossibility of designing a non-dictatorial voting rule in which truth-telling always constitutes a Nash equilibrium. We show that in voting games of complete information where a mediator is on hand, this troubling impossibility result can be alleviated. Indeed, we characterize families of voting rules where, given a mediator, truthful preference revelation is always in strong equilibrium. In particular, we observe that the family of feasible elimination procedures
has the foregoing property.

What makes a choice difficult, beyond being complex or difficult to calculate? Characterizing difficult choices as posing a special challenge to the agent, and as typically involving consequences of significant moment as well as clashes of values, the article proceeds to compare the way difficult choices are handled by rational choice theory and by the theory that preceded it, Kurt Lewin's "conflict theory." The argument is put forward that within rational choice theory no choice is in principle difficult: if the object is to maximize some value, the difficulty can be at most calculative. Several prototypes of choices that challenge this argument are surveyed and discussed (picking, multidimensionality, "big decisions" and dilemmas); special attention is given to difficult choices faced by doctors and layers. The last section discusses a number of devices people employ in their attempt to cope with difficult choices: escape, "reduction" to non-difficult choices, and second-order strategies.

A decision maker (an agent) is engaged in a repeated interaction with Nature. The objective of the agent is to guarantee to himself the long-run average payoff as large as the best-reply payoff to Nature’s empirical distribution of play, no matter what Nature does. An agent with perfect recall can achieve this objective by a simple better-reply strategy. In this paper we demonstrate that the relationship between perfect recall and bounded recall is not straightforward: An agent with bounded recall may fail to achieve this objective, no matter how long recall he has and no matter what better-reply strategy he employs.

In a series of experiments, Bar-Hillel and Budescu (1995) failed to find a desirability bias in probability estimation. The World Cup soccer tournament (of 2002 and 2006) provided an opportunity to revisit the phenomenon, in a context where wishful thinking and desirability bias are notoriously rampant (e.g., Babad, 1991). Participants estimated the probabilities of various teams to win their upcoming games. They were promised money if one particular team, randomly designated by the experimenter, would win its upcoming game. Participants judged their target team more likely to win than other participants, whose promised monetary reward was contingent on the victory of its rival team. Prima facie this seems to be a desirability bias. However, in a follow-up study we made one team salient, without promising monetary rewards, by simply stating that it is "of special interest". Again participants judged their target team more likely to win than other participants, whose "team of special interest" was the rival team. Moreover, the magnitude of the two effects was very similar. On grounds of parsimony, we conclude that what seemed like a desirability bias may just be a salience/marking effect, and -- though optimism is a robust and ubiquitous human phenomenon -- wishful thinking still remains elusive.

The concept of a superposition is a revolutionary novelty introduced by Quantum Mechanics. If a system may be in any one of two pure states x and y, we must consider that it may also be in any one of many superpositions of x and y. This paper proposes an in-depth analysis of superpositions. It claims that superpositions must be considered when one cannot distinguish between possible paths, i.e., histories, leading to the current state of the system. In such a case the resulting state is some compound of the states that result from each of the possible paths. It claims that states can be compounded, i.e., superposed in such a way only if they are not orthogonal. Since different classical states are orthogonal, the claim implies no non-trivial superpositions can be observed in classical systems. It studies the parameters that define such compounds and finds two: a proportion defining the mix of the different states entering the compound and a phase difference describing the interference between the different paths. Both quantities are geometrical in nature: relating one-dimensional subspaces in complex Hilbert spaces. It proposes a formal definition of superpositions in geometrical terms. It studies the properties of superpositions.

Define the riskiness of a gamble as the reciprocal of the absolute risk aversion (ARA) of an individual with constant ARA who is indifferent between taking and not taking that gamble. We characterize this index by axioms, chief among them a "duality" axiom which, roughly speaking, asserts that less risk-averse individuals accept riskier gambles. The index is homogeneous of degree 1, monotonic with respect to first and second order stochastic dominance, and for gambles with normal distributions, is half of variance/mean. Examples are calculated, additional properties derived, and the index is compared with others in the literature.

An evolutionarily stable strategy (ESS) is an equilibrium strategy that is immune to invasions by rare alternative ("mutant") strategies. Unlike Nash equilibria, ESS do not always exist in finite games. In this paper, we address the question of what happens when the size of the game increases: does an ESS exist for "almost every large" game? Letting the entries in the n x n game matrix be randomly chosen according to an underlying distribution F, we study the number of ESS with support of size 2. In particular, we show that, as n goes to infinity, the probability of having such an ESS: (i) converges to 1 for distributions F with "exponential and faster decreasing tails" (e.g., uniform, normal, exponential); and (ii) it converges to 1 - 1/sqrt(e) for distributions F with "slower than exponential decreasing tails" (e.g., lognormal, Pareto, Cauchy).
Our results also imply that the expected number of vertices of the convex hull of n random points in the plane converges to infinity for the distributions in (i), and to 4 for the distributions in (ii).

In this paper, I present a Bayesian decision theory and define choice-based subjective probabilities that faithfully represent Bayesian decision makers’ prior and posterior beliefs regarding the likelihood of the possible effects contingent on his actions. I argue that no equivalent results can be obtained in Savage’s (1954) subjective expected utility theory and give an example illustrating the potential harm caused by ascribing to a decision maker subjective probabilities that do not represent his beliefs.

In this paper I classify situations of interdependent decision-making, or games based on the type of decision-makers, or players involved. The classification builds on a distinction between three basic types of decision-making agents: individuals, cooperative or unitary groups -- groups whose members can reach a binding (and costless) agreement on a joint strategy -- and non-cooperative groups -- groups whose members act independently without being able to make a binding agreement. Pitting individuals, unitary groups, and non-cooperative groups against one another, and adding Nature as a potential “opponent”, generates a 3 (type of agent) X 4 (type of opponent) matrix of social situations. This framework is used to review the experimental decision-making literature and point out the gaps that still exist in it.

When a physicist performs a quantic measurement, new information about the system at hand is gathered. This paper studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the and then operation that combines old and new information. The and then connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the and then connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch in [3] satisfies all the properties studied in this paper. All properties satisfied by Finch's operation in modular lattices are valid in Quantum Logic. It is not known whether all properties of Quantum Logic are satisfied by Finch's operation in modular lattices.

In the context of survival analysis, we define a covariate X as protective (detrimental) for the failure time T if the conditional distribution of [T | X = x] is stochastically increasing (decreasing) as a function of x. In the presence of another covariate Y, there exist situations where [T | X = x, Y = y] is stochastically decreasing in x for each fixed y, but [T | X = x] is stochastically increasing. When studying causal effects and influence of covariates on a failure time, this state of affairs appears paradoxical and raises the question of whether X should be considered protective or detrimental. In a biomedical framework, for instance when X is a treatment dose, such a question has obvious practical importance. Situations of this kind may be seen as a version of Simpson’s paradox. In this paper we study this phenomenon in terms of the well-known Cox model. More specifically, we analyze conditions on the parameters of the model and the type of dependence between X and Y required for the paradox to hold. Among other things, we show that the paradox may hold for residual failure times conditioned on T > t even when the covariates X and Y are independent. This is due to the fact that independent covariates may become dependent when conditioned on the failure time being larger than t.

Game theory is usually applied to biology through evolutionary games. However, many competitive processes in biology may be better understood by analyzing them on a shorter time-scale than the time-course considered in evolutionary dynamics. Instead of the change in the "fitness" of a player, which is the traditional payoff in evolutionary games, we define the payoff function, tailored to the specific questions addressed. In this work we analyze the developmental competition that arises between motoneurons innervating the same muscle. The "size principle" - a fundamental principle in the organization of the motor system, stating that motoneurons with successively higher activation-threshold innervate successively larger portions of the muscle - emerges as a result of this competition. We define a game, in which motoneurons compete to innervate a maximal number of muscle-fibers. The strategies of the motoneurons are their activation-thresholds. By using a game theoretical approach we succeed to explain the emergence of the size principle and to reconcile seemingly contradictory experimental data on this issue. The evolutionary advantage of properties as the size principle, emerging as a consequence of competition rather than being genetically hardwired, is that it endows the system with adaptation capabilities, such that the outcome may be fine-tuned to fit the environment. In accordance with this idea the present study provides several experimentally-testable predictions regarding the magnitude of the size principle in different muscles.

The vertical inflorescences of several plant species are terminated by colorful bracts, which attract insect pollinators. The bracts contrast in color with the leaves below them, and are oriented perpendicular to the flowers on the inflorescence. We conducted laboratory experiments to determine the effects of color contrast and perpendicular orientation on the feeding choices of bumblebees. We first trained bees to feeders with color-contrasting perpendicular displays, composed of a horizontal and a vertical display component. We subsequently recorded the bees' choices among feeders that displayed only one of these cues. The bees preferred perpendicular displays that resembled the training model in the color of the horizontal component. None of them chose a color-contrasting display that was not perpendicular. We then evaluated the effects of the horizontal vs. vertical components of perpendicular displays on the bees' choices. After training bees to color-contrasting perpendicular displays, we allowed them to choose between displays that had either the same horizontal or the same vertical component as the training model. Foragers mostly oriented to the horizontal displays to which they had been trained. Our results suggest that (a) bumblebees can learn to associate three-dimensional perpendicular color-contrasting displays with food rewards; (b) these displays are processed hierarchically, with orientation dominating color contrast; (c) The horizontal component of perpendicular displays dominates the vertical component. We discuss possible implications of our findings for the evolution of flower signals based on extra-floral bracts.

Bracts are nonfloral showy structures associated with inflorescences. They are generally hypothesized to enhance plant reproductive success by attracting pollinating insects. We investigated whether flag-like bracts at the top of inflorescences reliably signal of floral food reward for pollinators in Salvia viridis L. Field and greenhouse data indicate incomplete synchrony between the development of flowers and bracts. Various measures of bract size, however, positively correlate with the number of open flowers on the inflorescence, and with their nectar rewards. Experimental removal of bracts from inflorescences significantly reduced honeybee visitation in the field. We compared these findings with field data on Lavandula stoechas L., another labiate species with flag-like displays. The number of open flowers in L. stoechas cannot be reliably predicted from the presence or size of the bracts. Bract clipping does not significantly reduce honeybee visits in this species. We conjecture that bees learn to orient to those bracts that reliably signal food rewards, and disregard bracts if they provide unreliable signals. Asynchronous development of bracts and floral rewards can reduce the reliability of the signals, and may explain the rarity of flag-like displays in pollination systems. We discuss additional selective forces that may favor bract displays.

We consider stochastic dynamics for the Matching Pennies game, hich behave, in expectation, like the best-response dynamics (i.e., the ontinuous fictitious play). Since the corresponding vector field is not ontinuous, we cannot apply the deterministic approximation results of Benaïm and Weibull [2003]. Nevertheless, we prove such results for our dynamics by developing the notion of a "leading coordinate."

Even with ample time and data at their disposal, people often make do with small samples, which increases their risk of making the wrong decision. A theoretical analysis indicates, however, that when the decision involves selecting among competing, adaptive agents who are eager to be selected, an error-prone evaluation may be beneficial to the decision maker. In this case, the chance of an error can motivate competitors to exert greater effort, improving their level of performance—which is the prime concern of the decision maker. This theoretical argument was tested empirically by comparing the effects of two levels of scrutiny of performance. Results show that minimal scrutiny can indeed lead to better performance than full scrutiny, and that the effect is conditional on a bridgeable difference between the competitors. We conclude by pointing out that error-prone decisions based on small samples may also maintain competition and diversity in the environment.

Let G = (I,J,g) be a two-person zero-sum game. We examine the two-person zero-sum repeated game G(k,m) in which player 1 and 2 place down finite state automata with k,m states respectively and the payoff is the average per stage payoff when the two automata face off. We are interested in the cases in which player 1 is "smart" in the sense that k is large but player 2 is "much smarter" in the sense that m>>k. Let S(g) be the value of G were the second player is clairvoyant, i.e., would know the player 1's move in advance. The threshold for clairvoyance is shown to occur for m near min(|I|, |J|)^k. For m of roughly that size, in the exponential scale, the value is close to S(g). For m significantly smaller (for some stage payoffs g) the value does not approach S(g).

A class of integer-valued allocation games -- "General Lotto games" -- is introduced and solved. The results are then applied to analyze the classical discrete "Colonel Blotto games"; in particular, optimal strategies are obtained for all symmetric Colonel Blotto games.

Regular annuities provide payment for the duration of an owner's lifetime. Period-Certain annuities provide additional payment after death to a beneficiary provided the insured dies within a certain period after annuitization. It has been argued that the bequest option offered by the latter is dominated by life insurance which provides non-random bequests. This is correct if competitive annuity and life insurance markets have full information about individual longevities. In contrast, this paper shows that when individual longevities are private information, a competitive pooling equilibrium which offers annuities at common prices to all individuals may have positive amounts of both types of annuities in addition to life insurance. In this equilibrium, individuals self-select the types of annuities that they purchase according to their longevity prospects. The break-even price of each type of annuity reflects the average longevity of its buyers. The broad conclusion that emerges from this paper is that adverse-selection due to asymmetric information is reflected not only in the amounts of insurance purchased but, importantly, also in the choice of insurance products suitable for different individual characteristics. This conclusion is supported by recent empirical work about the UK annuity market (Finkelstein and Poterba (2004)).

While auction research, including asymmetric auctions, has grown significantly in recent years, there is still little analytical solutions of first-price auctions outside the symmetric case. Even in the uniform case, Griesmer et al. (1967) and Plum (1992) find solutions only to the case where the lower bounds of the two distributions are the same. We present the general analytical solutions to asymmetric auctions in the uniform case for two bidders, both with and without a minimum bid. We show that our solution is consistent with the previously known solutions of auctions with uniform distributions. Several interesting examples are presented including a class where the two bid functions are linear. We hope this result improves our understanding of auctions and provides a useful tool for future research in auctions.

Performance sampling models of duration dependence in employee turnover and firm exit predict that hazard rates will initially be low, gradually rise to a maximum, and then fall. As we note in this paper, however, several empirical duration distributions have bimodal hazard rates. This paper shows that such bimodal hazard rates can be derived from existing models of performance sampling by small changes in the assumptions. In particular, bimodal hazard rates emerge if the mean or the variance of performances changes over time, which would occur if employees or firms face more challenging tasks over time. Using data on turnover in law firms, we show that the hazard rate predicted by these models fit data better than existing models.

I analyze a mechanism design of a tournament in which the principal can strategically enhance the probability of a tie. The principal decides on a "tie distance" and announces a rule according to which a tie is declared if the difference between the two contestants' performances is within the tie distance. I show that the contestants' equilibrium efforts do not depend on the prizes awarded in case of a tie. I find that there are cases in which the optimal mechanism has a positive tie distance.

This paper extends the standard model of optimum commodity taxation (Ramsey (1927) and Diamond-Mirrlees (1971)) to a competitive economy in which some markets are inefficient due to asymmetric information. As in most insurance markets, consumers impose varying costs on suppliers but firms cannot associate costs to customers and consequently all are charged equal prices. In a competitive pooling equilibrium, the price of each good is equal to average marginal costs weighted by equilibrium quantities. We derive modified Ramsey-Boiteux Conditions for optimum taxes in such an economy and show that they include general-equilibrium effects which reflect the initial deviations of producer prices from marginal costs, and the response of equilibrium prices to the taxes levied. It is shown that condition on the monotonicity of demand elasticities enables to sign the deviations from the standard formula. The general analysis is applied to the optimum taxation of annuities and life insurance.

Nobel Lecture.

This paper is the last part of a three-part project. The larger picture is important for the proper framing of the present paper. Here then is an abstract of the three-part paper, which is about considerateness. Focusing on two extreme poles of the spectrum of human relationships, the paper argues that considerateness is the foundation upon which relationships are to be organized in both the thin anonymous context of the public space and the thick intimate context of the family. The first part of the paper introduces the notion of considerateness among strangers and explores the idea that considerateness is the minimum that we owe to one another in the public space. By acting considerately toward strangers—for example, by holding a door open so it does not slam in the face of the next person who enters—we show respect to that which we all share as people, namely, our common humanity. The second part explores the idea that considerateness is the foundation underlying the constitution of the exemplary family. I hypothesize that each family adopts its own particular distribution of domestic burdens and benefits and I refer to it as the “family deal.” The argument is that the considerate family deal embodies a notion of fairness that is a distinct, family-oriented notion of fairness. The third part of the larger paper—which is the part I present here—takes up the notion of family fairness and contrasts it with justice. In particular, I take issue with Susan Okin’s notion of the just family and develop, instead, the notion of the not-unjust fair family. Driving a wedge between justice and fairness, I propose that family fairness is partial and sympathetic rather than impartial and empathic, and that it is particular and internal rather than universalizable. Furthermore, I claim that family fairness is based on ongoing comparisons of preferences among family members. I finally characterize the good family as a not-unjust family that is considerate and fair.

We examine incentive-compatible mechanisms for fair financing and efficient selection of a public budget (or public good). A mechanism selects the level of the public budget and imposes taxes on individuals. Individuals' preferences are quasilinear. Fairness is expressed as weak monotonicity (called scale monotonicity) of the tax imposed on an individual as a function of his benefit from an increased level of the public budget. Efficiency is expressed as selection of a Pareto-optimal level of the public budget. The budget deficit is the difference between the public budget and the total amount of taxes collected from the individuals. We show that any efficient scale-monotonic and incentive-compatible mechanism may generate a budget deficit. Moreover, it is impossible to collect taxes that always cover a fixed small fraction of the total cost.

We consider a bargaining problem where one of the players, the bureaucrat, has the power to dictate any outcome in a given set. The other players, the agents, negotiate with him which outcome to be dictated. In return, the agents transfer some part of their payoffs to the bureaucrat. We state five axioms and characterize the solutions which satisfy these axioms on a class of problems which includes as a subset all submodular bargaining problems. Every solution is characterized by a number α in the unit interval. Each agent in every bargaining problem obtains a weighted average of his individually rational level and his marginal contribution to the set of all players, where the weights are α and 1 - α, respectively. The bureaucrat obtains the remaing surplus. The solution when α = 1/2 is the nucleolus of a naturally related game in characteristic form.

We consider a multi-period auction with a seller who has a single object for sale, a large population of potential buyers, and a mediator of the trade. The seller and every buyer have independent private values of the object. The mediator designs an auction mechanism which maximizes her revenue subject to certain constraints for the traders. In each period the seller auctions the object to a set of buyers drawn at random from the population. The seller can re-auction the object (infinitely many times) if it is not sold in previous interactions. We characterize the class of mediator-optimal auction mechanisms. One of such mechanisms is a Vickrey auction with a reserve price where the seller pays to the mediator a fixed percentage from the closing price.

The purpose of the study is to explore, in the context of the Israeli-Palestinian conflict, the influence of framing a decision task as inclusion or exclusion on Israeli-Jewish respondents' support for the concession of Jewish settlements in the West Bank and Gaza. Respondents received a list of 40 Jewish settlements. Details such as the number of residents and geographical location were provided for each settlement. The respondents were randomly assigned to one of two conditions. In the inclusion condition 55 respondents were asked to mark the settlements for which they recommended that Israeli sovereignty be conceded. In the exclusion condition 53 respondents were asked to mark the settlements for which they recommended that Israeli sovereignty not be conceded. The findings confirm the predictions tested and indicate that: (1) Framing the task in terms of inclusion or exclusion affects respondents' support for territorial compromise, so that respondents in the exclusion condition support the concession of more settlements than respondents in the inclusion condition. (2) Framing the task in terms of inclusion or exclusion has a greater effect on support for conceding options (settlements) that are perceived as ambiguous (less consensual in the climate of opinion) in comparison to options (settlements) that are perceived as more clear-cut (more consensual). The theoretical and practical implications of these findings are discussed.

In daily decision making, people often solicit one another's opinions in the hope of improving their own judgment. According to both theory and empirical results, integrating even a few opinions is beneficial, with the accuracy gains diminishing as the bias of the judges or the correlation between their opinions increases. Decision makers using intuitive policies for integrating others' opinions rely on a variety of accuracy cues in weighting the opinions they receive. They tend to discount dissenters and to give greater weight to their own opinion than to other people's opinions.

The Shapley value is an a priori evaluation of the prospects of a player in a multi-person game. Introduced by Lloyd S. Shapley in 1953, it has become a central solution concept in cooperative game theory. The Shapley value has been applied to economic, political, and other models.

Our main purpose is to present a very surprising new characterization of the Shannon entropy of stationary ergodic processes. We will use two basic concepts: isomorphism of stationary processes and a notion of finite observability, and we will see how one is led, inevitably, to Shannon's entropy. A function J with values in some metric space, defined on all finite-valued, stationary, ergodic processes is said to be finitely observable (FO) if there is a sequence of functions S_n(x₁,x₂,...,x_n) that for all processes Χ converges to J(Χ) for almost every realization x₁∞ of Χ. It is called an invariant if it returns the same value for isomorphic processes. We show that any finitely observable invariant is necessarily a continuous function of the entropy. Several extensions of this result will also be given.

We study the question of how long it takes players to reach a Nash equilibrium in &quot;uncoupled&quot; setups, where each player initially knows only his own payoff function. We derive lower bounds on the number of bits that need to be transmitted in order to reach a Nash equilibrium, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. Finally, we show that some very simple and naive procedures lead to similar exponential upper bounds.

On the occasion of Robert J. Aumann's being awarded the 2005 Nobel Prize in Economics, this paper gives a personal view on some of Aumann's contributions, and primarily on his approach to foundational issues in game theory, economics, and science as a whole. It is based on numerous discussions and e-mail exchanges we had in the 1990's, dealing with various scientific and political matters, including our long debate on the ``Bible Code'' controversy.

Behavior in competitive situations requires decision makers to evaluate their own as well as their competitors' positions. Using data from a realistic competitive risk-taking setting, Jeopardy's Tournament of Champions (TOC), we test whether players choose the strategic best response when making their betting decisions. Analyses show that the percentage of players choosing the strategic best response is very low, a surprising finding because the TOC is contested by the best and most experienced players of the game. We conjecture that performance aspiration and survival targets that guide risk-taking behavior in competitive situations may lead players to select inferior competitive strategies.

An overview of the landmark contributions of Robert J. Aumann, winner of the 2005 Nobel Memorial Prize in Economic Sciences.

A two-house legislature can often be modelled as a proper simple game whose outcome depends on whether a coalition wins, blocks or loses in two smaller proper simple games. It is shown that there are exactly five ways to combine the smaller games into a larger one. This paper focuses on one of the rules, lexicographic composition, where a coalition wins in G₁ ⇒ G₂ when it either wins in G₁, or blocks in G₁ and wins in G₂. It is the most decisive of the five. A lexicographically decomposable game is one that can be represented in this way using components whose player sets partition the whole set. Games with veto players are not decomposable, and anonymous games are decomposable if and only if they are decisive and have two or more players. If a player's benefit is assessed by any semi-value, then for two isomorphic games a player is better off from having a role in the first game than having the same role in the second. Lexicographic decomposability is sometimes compatible with equality of roles. A relaxation of it is suggested for its practical benefits.

In many tournaments investments are made over time and conducting a review only once at the end, or also at points midway through, is a strategic decision of the tournament designer. If the latter is chosen, then a rule according to which the results of the different reviews are aggregated into a ranking must also be determined. This paper takes a first step in the direction of answering how such rules should be optimally designed. A characterization of the optimal aggregation rule is provided for a two-agent two-stage tournament. In particular, we show that treating the two reviews symmetrically may result in an equilibrium effort level that is inferior to the one in which only a final review is conducted. However, treating the two reviews lexicographically by first looking at the final review, and then using the midterm review only as a tie-breaking rule, strictly dominates the option of conducting a final review only. The optimal mechanism falls somewhere in between these two extreme mechanisms. It is shown that the more effective the first-stage effort is in determining the final review's outcome, the smaller is the weight that should be assigned to the midterm review in determining the agents' ranking.

The existence of a value and optimal strategies is proved for the class of two-person repeated games where the state follows a Markov chain independently of players' actions and at the beginning of each stage only player one is informed about the state. The results apply to the case of standard signaling where players' stage actions are observable, as well as to the model with general signals provided that player one has a nonrevealing repeated game strategy. The proofs reduce the analysis of these repeated games to that of classical repeated games with incomplete information on one side.

While traditional mechanism design typically assumes isomorphism between the agents' type- and action spaces, in many situations the agents face strict restrictions on their action space due to, e.g., technical, behavioral or regulatory reasons. We devise a general framework for the study of mechanism design in single-parameter environments with restricted action spaces. Our contribution is threefold. First, we characterize sufficient conditions under which the information-theoretically optimal social-choice rule can be implemented in dominant strategies, and prove that any multilinear social-choice rule is dominant-strategy implementable with no additional cost. Second, we identify necessary conditions for the optimality of action-bounded mechanisms, and fully characterize the optimal mechanisms and strategies in games with two players and two alternatives. Finally, we prove that for any multilinear social-choice rule, the optimal mechanism with k actions incurs an expected loss of O(1/k²) compared to the optimal mechanisms with unrestricted action spaces. Our results apply to various economic and computational settings, and we demonstrate their applicability to signaling games, public-good models and routing in networks.

One way to express bounded rationality of a player in a game theoretic models is by specifying a set of feasible strategies for that player. In dynamic game models with finite automata and bounded recall strategies, for example, feasibility of strategies is determined via certain complexity measures: the number of states of automata and the length of recall. Typically in these models, a fixed finite bound on the complexity is imposed resulting in finite sets of feasible strategies. As a consequence, the number of distinct feasible strategies in any subgame is finite. Also, the number of distinct strategies induced in the first T stages is bounded by a constant that is independent of T.

In this paper, we initiate an investigation into a notion of feasibility that reflects varying degree of bounded rationality over time. Such concept must entail properties of a strategy, or a set of strategies, that depend on time. Specifically, we associate to each subset Ψ_i of the full (theoretically possible) strategy set a function y_i from the set of positive integers to itself. The value ψ_i(t) represents the number of strategies in Ψ_i that are distinguishable in the first t stages. The set Ψ_i may contain infinitely many strategies, but it can differ from the fully rational case in the way y_i grows reflecting a broad implication of bounded rationality that may be alleviated, or intensified, over time.

We examine how the growth rate of y_i affects equilibrium outcomes of repeated games. In particular, we derive an upper bound on the individually rational payoff of repeated games where player 1, with a feasible strategy set Ψ₁, plays against a fully rational player 2. We will show that the derived bound is tight in that a specific, and simple, set Ψ₁ exists that achieves the upper bound.

As a special case, we study repeated games with non-stationary bounded recall strategies where the length of recall is allowed to vary in the course of the game. We will show that a player with bounded recall can guarantee the minimax payoff of the stage game even against a player with full recall so long as he can remember, at stage t, at least K log(t) stages back for some constant K >0. Thus, in order to guarantee the minimax payoff, it suffices to remember only a vanishing fraction of the past. A version of the folk theorem is provided for this class of games.

Let A be a finite set of m alternatives, let N be a finite set of n players and let R^N be a profile of linear preference orderings on A of the players. Let u^N be a profile of utility functions for R^N. We define the NTU game V_u^N that corresponds to simple majority voting, and investigate its Aumann-Davis-Maschler and Mas-Colell bargaining sets.

The first bargaining set is nonempty for m ≤ 3 and it may be empty for m ≥ 4. However, in a simple probabilistic model, for fixed m, the probability that the Aumann-Davis-Maschler bargaining set is nonempty tends to one if n tends to infinity.

The Mas-Colell bargaining set is nonempty for m ≤ 5 and it may be empty for m ≥ 6. Furthermore, it may be empty even if we insist that n be odd, provided that m is sufficiently large. Nevertheless, we show that the Mas-Colell bargaining set of any simple majority voting game derived from the k-th replication of R^N is nonempty, provided that k ≥ n + 2.

We design two computationally-efficient incentive-compatible mechanisms for combinatorial auctions with general bidder preferences. Both mechanisms are randomized, and are incentive-compatible in the universal sense. This is in contrast to recent previous work that only addresses the weaker notion of incentive compatibility in expectation. The first mechanism obtains an O(√m)-approximation of the optimal social welfare for arbitrary bidder valuations -- this is the best approximation possible in polynomial time. The second one obtains an O(log² m)-approximation for a subclass of bidder valuations that includes all submodular bidders. This improves over the best previously obtained incentive-compatible mechanism for this class which only provides an O(√m)-approximation.

A version of a secretary problem is considered: Let X_j, j = 1,...,n be i.i.d. random variables. Like in the classical secretary problem the optimal stopper only observes Y_j = 1, if X_j is a (relative) record, and Y_j = 0, otherwise. The actual X_j-values are not revealed. The goal is to maximize the expected X-value at which one stops. We describe the structure of the optimal stopping rule, its asymptotic properties and the asymptotic expected reward. Three different families of distributions of X are considered, belonging to the three different domains of attraction of the maximum. It is shown that both the time of stopping, as well as the expected reward are strongly distribution dependent. The last section discusses an ‘inverse' of ‘Robbins' Problem'.

Recent advancement in genetics testing for late-onset diseases raises fundamental decision dilemmas. The first study surveyed people's willingness to undergo predictive testing to find out about their own predisposition for certain incurable, late-onset diseases. The second study investigated the respondents' willingness to be tested as a function of the base rate of the disease, test diagnosticity, and the availability of treatment for the disease. In addition we surveyed (in the first study) people's willingness to disclose to others personal information about their genetic predisposition. The findings show that people often prefer not to know, as if they are choosing "protective ignorance". Respondents' verbal justifications of their choices were also analyzed. Respondents offered emotional, cognitive-instrumental, and strategic reasons for their preferences. The findings are compared with other issues in behavioral decision theory, including attitudes towards uncertainty and desire for control. The implications of the findings for policies and legislation on genetic tests are also considered.

Seeking advice is a basic practice in making real life decisions. Until recently, however, little attention has been given to it in either empirical studies or theories of decision making. The studies reported here investigate the influence of advice on judgment and the consequences of advice use for judgment accuracy. Respondents were asked to provide final judgments on the basis of their initial opinions and advice presented to them. The respondents' weighting policies were inferred. Analysis of the these policies show that (a) the respondents tended to place a higher weight on their own opinion than on the advisor's opinion (the self/other effect); (b) more knowledgeable individuals discounted the advice more; (c) the weight of advice decreased as its distance from the initial opinion increased; and (d) the use of advice improved accuracy significantly, though not optimally. A theoretical framework is introduced which draws in part on insights from the study of attitude change to explain the influence of advice. Finally the usefulness of advice for improving judgment accuracy is considered.

Effectivity functions for finitely many players and alternatives are considered. It is shown that every monotonic and superadditive effectivity function can be augmented with equal chance lotteries to a finite lottery model - i.e., an effectivity function that preserves the original effectivity in terms of supports of lotteries - which has a Nash consistent representation. In other words, there exists a finite game form which represents the lottery model and which has a Nash equilibrium for any profile of utility functions, where lotteries are evaluated by their expected utility. No additional condition on the original effectivity function is needed.

For the last fifty years, countries in Asia and elsewhere witnessed a surge in aggregate savings per capita. Many empirical studies attribute this trend to the highly significant increases in life longevity of the populations of these countries. Some argue that the rise in savings is short-run, to be eventually dissipated by the dissaving of the elderly, whose proportion in the population rises along with longevity. This paper examines whether these conclusions are supported by economic theory. A model of life cycle decisions with uncertain survival is used to derive individuals'savings and chosen retirement age response to changes in longevity. Conditions on the age-profile of improvements in survival probabilities are shown to be necessary in order to predict the direction of this response (the uneven history of age specific improvements in longevity is recorded by Cutler (2004)). Population theory (e.g. Coale (1952)) is used to derive the dependence of the steady-state population age density on longevity. This, in turn, enables the explicit aggregation of individual response functions and a comparative steady-state analysis. Sufficient conditions for a sustainable positive effect of increased longevity on aggregate savings per capita are then derived. The importance of the availability of insurance markets is briefly discussed.

This work follows "Evolutionary dynamics and backward induction" (Hart [2000]) in the study of dynamic models consisting of selection and mutation, when the mutation rate is low and the populations are large. Under the assumption that there is a single backward induction (or subgame perfect) equilibrium of a perfect information game, Hart [2000] proved that this point is the only stable state. In this work, we examine the case where there are multiple backward induction equilibria.

This paper investigates the effectiveness of shaming penalties. It establishes that there may be an inverse relation between the rate of shaming penalties and their deterrent effects - the more people are shamed the less deterring shaming penalties become. This conclusion is based on a search model in which the costs of searching for law-abiding partners increase with the rate of shaming, and may lead to lower expected sanction for offenders. The inverse relation between the rate of shaming penalties and their effectiveness is later used to show that increasing the probability of detection, increasing the magnitude of shaming penalties or reducing the number of wrongful acquittals does not necessarily increase the deterrent effects of shaming penalties (and may, in fact, decrease these effects).

 

We will try to explore, primarily from the complexity-theoretic point of view, limitations of error-correction and fault-tolerant quantum computation.

We consider stochastic models of quantum computation on n qubits subject to noise operators that are obtained as products of tiny noise operators acting on a small number of qubits. We conjecture that for realistic random noise operators of this kind there will be substantial dependencies between the noise on individual qubits and, in addition, the dependence structure of the noise acting on individual qubits will necessarily depend (systematically) on the dependence structure of the qubits themselves. The main hypothesis of this paper is that these properties of noise are sufficient to reduce quantum computation to probabilistic classical computation. Some potentially relevant mathematical issues and problems will be described. Our line of thought appears to be related to that of physicists Alicki, Horodecki, Horodecki and Horodecki [AHHH].

 

In this paper we study the social preferences obtained from monotone neutral social welfare functions for random individual preferences. It turns out that there are two extreme types of behavior. On one side, there are social welfare functions, such as the majority rule, that lead to stochastic stability of the outcome in terms of perturbations of individual preferences. We identify and study a class of social welfare functions that demonstrate an extremely different type of behavior which is a completely chaotic: they lead to a uniform probability distribution on all possible social preference relations and, for every ε>0, if a small fraction ε of individuals change their preferences (randomly) the correlation between the resulting social preferences and the original ones tends to zero as the number of individuals in the society increases. This class includes natural multi-level majority rules.