Predictability in civil and criminal sanctions is generally understood as desirable. Conversely, unpredictability is condemned as a violation of the rule of law. This paper explores predictability in sanctioning from the point of view of efficiency. It is argued that, given a constant expected sanction, deterrence is increased when either the size of the sanction or the probability that it will be imposed is uncertain. This conclusion follows from earlier findings in behavioral decision research and the results of an experiment conducted specifically to examine this hypothesis. The findings suggest that, within an efficiency framework, there are virtues to uncertainty that may cast doubt on the premise that law should always strive to be as predictable as possible.

Let Xi be nonnegative independent random variables with finite expectations and Xn* = max X1,..., Xn. The value Xn* is what can be obtained by a "prophet". A "mortal" onthe other hand, may use k '¥ 1 stopping rules t1,...,tk yielding a return E[max i = 1,...,k X ti]. For n '¥ k the optimal return is Vkn (X1,...,Xn) = sup E[max i = 1,...,k X ti] where the supremum is over all stopping rules which stop by time n. The well known "prophet inequality" states that for all such Xi's and one choice EXn* < 2 V1n (X1,...,Xn) and the constant "2" cannot be improved on for any n '¥ 2. In contrast we show that for k=2 the best constant d satisfying EXn* < d V2n (X1,...,Xn) for all such Xi's depends on n. On the way we obtain constants ck such that EXk+1* < ck Vkk+1 (X1,...,Xk+1).

We formulate a way to study whether the asymmetry of buyers (in the sense of having different prior probability distributions of valuations) is helpful to the seller in private-value auctions (asked first by Cantillon [2001]). In our proposed formulation, this question corresponds to two important questions previously asked: Does a first-price auction have higher revenue than a second-price auction when buyers have asymmetric distributions (asked by Maskin and Riley[2000])? And does a seller enhance revenue by releasing information (asked by Milgrom and Weber[1982])? This is shown by constructing two Harsanyi games of incomplete information each having the same ex-ante distribution of valuations but in one beliefs are symmetric while in the other beliefs are sometimes asymmetric. Our main result is that answers to all three questions coincide when values are independent and are related when values are affiliated.

We consider a permutation method for testing whether observations given in their natural pairing exhibit an unusual level of similarity in situations where any two observations may be similar at some unknown baseline level. Under a hypothesis where there is no distinguished pairing of the observations, a normal approximation, with explicit bounds and rates, is presented for determining approximate critical test levels.

We consider the following situation. A risk-neutral plaintiff sues a risk-neutral defendant for damages that are normalized to one. The defendant knows whether she is liable or not, but the plaintiff does not. We ask what are the settlement procedure and fee-shifting rule (which, together, we call a mechanism) that minimize the rate of litigation subject to maintaining deterrence. Two main results are presented. The first is a characterization of an upper bound on the rate of settlement that is consistent with maintaining deterrence. This upper bound is shown to be independent of the litigants' litigation cost. It is further shown that any mechanism that attains this upper bound must employ the English fee-shifting rule according to which all litigation costs are shifted to the loser in trial. The second result describes a simple practicable mechanism that attains this upper bound. We discuss our results in the context of recent legal reforms in the U.S. and U.K.

We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the others is returned to the agent. If the rules of rationality are followed one obtains the peculiarities of quantum probability, the uncertainty relations and the EPR paradox among others. The consequences of this approach for hidden variables and quantum logic are analyzed.

Acceptable game forms were introduced in Hurwicz and Schmeidler (1978). Dutta (1984) considered effectivity functions of acceptable game forms. This paper unifies and extends the foregoing two papers. We obtain the following characterization of the effectivity functions of acceptable game forms: An effectivity function belongs to some acceptable game form if (i) it belongs to some Nash consistent game forms; and (ii) it satisfies an extra simple condition (our (3.1) or (4.2)). (Nash consistent game forms have already been characterized by their effectivity functions in Peleg et al. (2001).) As a corollary of our characterization we show that every acceptable game form violates minimal liberalism.

We studied conflict resolution in two types of intergroup conflicts modeled as team games, a game of Assurance where the groups incentive to compete is purely fear, and a game of Chicken where the groups incentive to compete is purely greed. The games were operationalized as competitions between two groups with three players in each group. The players discussed the game with other ingroup members, after which they met with the members of the outgroup for a between group discussion, and finally had a within-group discussion before deciding individually whether to participate in their group s collective effort vis-a-vis the other group. We found that all groups playing the Assurance game managed to achieve the collectively efficient outcome of zero participation, whereas groups playing the Chicken game maintained a highly inefficient participation rate of 78%. We conclude that communication between groups is very effective in bringing about a peaceful resolution if the conflict is motivated by mutual fear and practically useless if the conflict is motivated by mutual greed.

The forecasting problem for a stationary and ergodic binary time series Xn is to estimate the probability that Xn+1 = 1 based on the observations Xi, 0 i n without prior knowledge of the distribution of the process Xn. It is known that this is not possible if one estimates at all values of n. We present a simple procedure which will attempt to make such a prediction infinitely often at carefully selected stopping times chosen by the algorithm. We show that the proposed procedure is consistent under certain conditions, and we estimate the growth rate of the stopping times.

The Presidential Address at the First International Congress of the Game Theory Society, held in Bilbao, Spain, in July of 2000. The address contains a discussion of the Congress, of the functions and activities of the Society, of the Logo of the Society, of past accomplishments of the discipline, and of some future directions for research. The address is preceded by an introduction by David Kreps.

Two experiments compared the Centipede game played either by 2 individuals or by 2 (3-person) groups. The 2 competitors alternate in deciding whether to take the larger portion of an increasing (or constant) pile of money, and as soon as one takes  the game ends. Assuming that both sides are concerned only with maximizing their own payoffs (and that this is common knowledge), the game theoretic solution, derived by backward induction, is for the first mover to exit the game at the first decision node. Both experiments found that although neither individuals nor groups fully complied with this solution, groups did exit the game significantly earlier than individuals. The study of experimental games has uncovered many instances in which individuals deviate systematically from the game theoretic solution. This study is in accord with other recent experiments in suggesting that game theory may provide a better description of group behavior.

Inequalities often persist because both the advantaged and the disadvantaged stand to lose from change. Despite the probability of loss, moral indignation can lead the disadvantaged to seek to alter the status quo, by encouraging them to sacrifice their material self-interest for the sake of equality. Experimental research shows that moral indignation, understood as a willingness to suffer in order to punish unfair treatment by others, is widespread. It also indicates that a propensity to apparently self-defeating moral indignation can turn out to promote people's material self-interest, if and because others will anticipate their actions. But potential rebels face collective action problems. Some of these can be reduced through the acts of "indignation entrepreneurs," giving appropriate signals, organizing discussions by like-minded people, and engaging in acts of self-sacrifice. Law is relevant as well. By legitimating moral indignation and dissipating pluralistic ignorance, law can intensify and spread that indignation, thus increasing its expression. Alternatively, law can delegitimate moral indignation, or at least raise the cost of its expression, thus stabilizing a status quo of inequality. But the effects of law are unpredictable, in part because it will have moral authority for some but not for others; here too heterogeneity is an issue both for indignation entrepreneurs and their opponents. Examples are given from a range of areas, including labor-management relations, sexual harassment, civil rights, and domestic violence.

Intergroup conflicts generally involve conflicts of interests within the competing groups as well. This paper outlines a taxonomy of games, called team games, which incorporate the intragroup and intergroup levels of conflict. Its aims are to provide a coherent framework for analyzing the prototypical problems of cooperation and competition that arise within and between groups, and to review an extensive research program which has utilized this framework to study individual and group behavior in the laboratory. Depending on the game's payoff structure, contradictions or conflicts were created between the rational choices at the individual, group, and collective levels – a generalization of the contradiction between individual and collective rationality occurring in the traditional mixed-motive games. These contradictions were studied so as to identify the theoretical and behavioral conditions that determine which level of rationality prevails.

With cheap talk, more can be achieved by long conversations than by a single message - even when one side is strictly better informed than the other.

We show in a general setup that if data Y are grouped by a covariate X in a certain way, then under a condition of monotone regression of Y on X, a Simpson s type paradox is natural rather than surprising. This model was motivated by an observation on recent SAT data which are presented.

A stationary renewal process based on iid random variables Xi is observed at a given time. The excess time, that is, the residual time until the next renewal event, is of course smaller than the total current X which consists of the residual time plus the current age. Nevertheless in certain types of data the distribution of the excess times is stochastically larger than that of Xi's. We find necessary and sufficient conditions that explain this phenomenon, and related results on stochastic orderings arising from observations on renewal processes.

We establish the existence of pure strategy equilibria in monotone bidding functions in first-price auctions with asymmetric bidders, interdependent values and affiliated one-dimensional signals. By extending a monotonicity result due to Milgrom and Weber (1982), we show that single crossing can fail only when ties occur at winning bids or when bids are individually irrational. We avoid these problems by considering limits of ever finer finite bid sets such that no two bidders have a common serious bid, and by recalling that single crossing is needed only at individually rational bids. Two examples suggest that our results cannot be extended to multidimensional signals or to second-price auctions.

Ever since the days of Francis Bacon it has been claimed that people perceive the world as less variable and more regular than it actually is. Such misperception, if shown to exist, could explain a host of perplexing behaviors. However, the only evidence supporting the claim is indirect, and there is no explanation of its cause. As a possible cause, we suggest the use of sample variability as an estimate of population variability. This is so since the sampling distribution of sample variance is downward attenuated, the attenuation being substantial for sample sizes that people are likely to consider. The results of five experiments show that people use sample variability, uncorrected for sample size, in tasks in which a correction is normatively called for, and indeed perceive variability as smaller than it actually is.

Extending earlier work of Chichilnisky and Heal, we show that any connected space of the homotopy type of a finite complex admitting a continuous symmetric choice function respeting unanimity is contractible for any fixed finite number (>1) of agents. On the other hand, removing the finiteness condition on the homotopy type, we show that there are a number of non-contractible spaces that do admit such choice functions, for any number of agents, and, characterize precisely those spaces.

The parental investment conflict considers the question of how much each sex should invest in each brood, thereby characterizing different animal groups. Each such group usually adopts a certain parental care pattern: female-care only, male-care only, biparental care, or even no parental care at all. The differences in care patterns are usually explained by the different costs and benefits arising from caring for the offspring in each animal group. This paper proposes a game-theoretical model to the parental investment conflict based on the parental behavior in Cichlid fish. Cichlid fish exhibit different parental care patterns, allowing the examination of the factors which determine the particular behavior in each mating. We present a continuous time, two-stage, asymmetric game, with two types of players: male and female. According to the model s results, three parental care patterns: male-only care, female-only care and biparental care, are possible Evolutionarily Stable Strategies. Fixation depends on the investment costs and benefits, and on the initial conditions of the game. These results may explain the different parental care patterns observed in di erent animal groups as well as in Cichlid fish.