Publications / Click on each table title in order to sort, click it again to reverse the sort.

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 <q< 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 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?

Much of human psychology takes place out of awareness. In the past few decades, psychologists have begun to study the working of the unconscious using new methodologies. The unconscious terrain is very different from the terrain that people have conscious access to and control over. This paper affords a peek into some of these strange processes, and explores some possible legal implications thereof (e.g., to discretion in sentencing; criminal responsibility; lie detection; judicial access to extra evidential information; etc.)

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”.