We study the question of how long it takes players to reach a Nash equilibrium in "uncoupled" 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.

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).

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.

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 Bena¯m and Weibull [2003]. Nevertheless, we prove such results for our dynamics by developing the notion of a "leading coordinate."

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)).

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.

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 Sn(x1,x2,...,xn) that for all processes \S converges to J(\S) for almost every realization x1ˆ\v z of \S. 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.