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.

{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

Market share and quality, or customer satisfaction, go hand in hand. Yet the inference that higher'' market share indicates higher quality is seldom made. The skepticism is in part fueled by elitism,'' the association of mass popularity with lower quality, and by cynicism, ascribing market'' leadership to an entrenched position. We find that though such skepticism is often justified, it is'' correct to make a Bayesian inference that the product with the higher market share has the better'' quality under rather tame assumptions.

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.

Recent research in psychology, especially that called "The New Unconscious", is discovering strange and unintuitive phenomena, some of which raise interesting challenges for the law. This paper discusses some of these challenges. For example, if much of our mental life occurs out of our awareness and control, and yet is subject to easy external manipulation, what implications does this have for holding defendants responsible for their deeds? For that matter, what implications does this have for trusting judges to judge and act as they should, and would, if their own mental processes were fully conscious and controlled? Some provocative ideas are suggested, such as how to make prison terms shorter and more deterring at the same time; assisting judges in overcoming inconsistency and biases; etc.

"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.

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 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.

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 toindicate 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.

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.

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.

Although possessing many beautiful features, the Hart and Mas-Colell bargaining model is not flawless: the concept of threat in this model may behave quite counter-intuitive, and its SP equilibrium expected payoff vector may not be the same as the min-max solution payoff vector in zero-sum games. If we postpone realizations of all threats to the end of the game, the two problems can be solved simultaneously. This is exactly the 2(a) model suggested by Hart and Mas-Colell in the last section of their paper. I show that the new model, unfortunately, can only guarantee the existence of an SP equilibrium in the two player case. For the original model, I reduce the computation of an SP equilibrium to a system of linear inequalities. Quantitative efficiency and symmetric SP equilibria are also discussed.

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.

Gambling frequencies on single numbers in real casino roulette were displayed in a contour map. This resulted not only in a confirmation that gamblers are subject to middle bias, but also to accessibility effects. The figure allowed us to infer the location of the roulette wheel and croupier from the gambling data, as well as infer bounds on the dimensions of the roulette table.

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.

A response to criticism of the paper "On the Matter of the Man with Three Wives," Moriah 22 (1999), 98- 107 (see also Rationality Centerdp 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.

Evidence, anecdotal and scientific, suggests that people treat (or are affected by) products of prestigious sources differently than those of less prestigious, or of 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 establishes the effect we wish to address: a poet s reputation enhances the evaluation of a poem. Study 2 asks whether it is only the reported evaluation of the poem that is enhanced by the poet s name (as was the case for The Emperor s New Clothes) or the enhancement is genuine and unaware. Finding for the latter, Study 3 explores whether the poet s name changes the reader s experience of it, so that in a sense one is reading a different  poem. We conclude that it is not so much that the attributed poem really differs from the unattributed poem, as that it is just ineffably better. The name of a highly regarded poet seems to prime quality, and the poem becomes somehow better. This is a more subtle bias than the deliberate one rejected in Study 2, but it is a bias nonetheless. Ethical implications of this kind of effect are discussed.

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

{What happens when priors are not common? We show that for each type pro¬le „ over a knowledge space (\copyright,  ), where the state space \copyright is connected with respect to the partition pro¬le  , we can associate a value 0 / 1 that we term the prior distance of „

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.