An effectivity function assigns to each coalition of individuals in a society a family of subsets of alternatives such that the coalition can force the outcome of society's choice to be a member of each of the subsets separately. A representation of an effectivity function is a game form with the same power structure as that specified by the effectivity function. In the present paper we investigate the continuity properties of the outcome functions of such representation. It is shown that while it is not in general possible to find continuous representations, there are important subfamilies of effectivity functions for which continuous representations exist. Moreover, it is found that in the study of continuous representations one may practically restrict attention to effectivity functions on the Cantor set. Here it is found that general effectivity functions have representations with lower or upper semicontinuous outcome function.

The minmax in repeated games with imperfect monitoring can differ from the minmax of those games with perfect monitoring. This can happen when two or more players are able to gain common information known only to themselves, and utilize this information at a later stage. Gossner and Tomala [1] showed that in a class of such games, the minmax is given by a weighted average of the payoffs of two main strategies: one in which the information is gained, and the other in which the information is utilized. While this result is implicit, all examples analyzed to date require a single main strategy in which information is created and utilized simultaneously. We show that two strategies are indeed needed by providing and solving a concrete example of a three-player game.

We studied asymmetric competition between two (three-person) groups. Each group member received an initial endowment and had to decide whether or not to contribute it. The group with more  contributions won the competition and each of its members received a reward. The members of the losing group received nothing. The asymmetry was created by randomly and publicly selecting one group beforehand to be the winning group in the case of a tie. A theoretical analysis of this  asymmetric game generates two qualitatively different solutions, one in which members of the group that wins in the case of a tie are somewhat more likely to contribute than members of the group that loses, and another in which members of the group that loses in the case of a tie are much more likely to contribute than members of the group that wins. The experimental results are clearly in line with the first solution.

In a repeated game with perfect monitoring, correlation among a group of players may evolve in the common course of play (online correlation). Such a correlation may be concealed from a boundedly rational player. The feasibility of such "online concealed correlation" is quantified by the individually rational payoff of the boundedly rational player.We show that "strong" players, i.e., players whose strategic complexity is less stringently bounded, can orchestrate online correlation of the actions of "weak" players, in a manner that is concealed from an opponent of "intermediate" strength. The result is illustrated in two models, each captures another aspect of bounded rationality. In the first, players use bounded recall strategies. In the second, players use strategies that are implementable by finite automata.

We study a repeated game in which one player, the prophet, acquires more information than another player, the follower, about the play that is going to be played. We characterize the optimal amount of information that can be transmitted online by the prophet to the follower, and provide applications to repeated games played by finite automata, and by players with bounded recall.

When information on longevity (survival functions) is unknown early in life, individuals have an interest to insure themselves against future 'risk-class' classification. Accordingly, the First-Best typically involves transfers across states of nature. Competitive equilibrium cannot provide such transfers if insurance firms are unable to precommit their customers. On the other hand, public insurance plans that do not distinguish between 'risk-class' realizations are also inefficient. It is impossible, a-priori, to rank these alternatives from a welfare point of view.

A central question for pension design is how benefits should vary with the age of retirement beyond early eligibility age. It is often argued that in order to be neutral with respect to individual retirement decisions benefits should be actuarially fair, that is, the present value of additional contributions and benefits ('Delayed Retirement Credit' - DRC) due to postponed retirement should be equal. We show that in a self-selection, asymmetric information model, because individual decisions are suboptimal, the socially optimal benefit structure should be less than actuarially fair.

Bar-Hillel and Neter (1996) found that although people are willing to trade identical objects, they are reluctant to trade identical lottery tickets. Is this simply due to the fact that these are gambles? It was found that if the value of the tickets is guaranteed to be ex-post, not just ex ante, identical, people are more willing to exchange them. Indeed, just the possibility of ex-post difference between the lottery tickets induces as much reluctance to exchange them as when ex-post difference is guaranteed. In addition, this study examines how the vividness of lottery tickets influences the willingness to trade them. Specifically, it examines whether people are equally reluctant to exchange lottery tickets (when given a bonus for doing so) when they cannot even distinguish between them (e.g., when the tickets are concealed in envelopes). When one cannot see the ticket, it is less vivid and it is harder to imagine it winning. Indeed, it was found that people are more willing to exchange when they cannot distinguish between the tickets than when they can.In 2011, a revised version of this paper was published under the title To trade or not to trade:  The moderating role of vividness when exchanging gambles in Judgment and Decision Making, 6, 147-155.  In the link todp341, it follows the original manuscript. 

Regret-based dynamics have been introduced and studied in the context of discrete-time repeated play. Here we carry out the corresponding analysis in continuous time. We observe that, in contrast to (smooth) fictitious play or to evolutionary models, the appropriate state space for this analysis is the space of distributions on the product of the players' pure action spaces (rather than the product of their mixed action spaces). We obtain relatively simple proofs for some results known in the discrete case (related to "no-regret" and correlated equilibria), and also a new result on two-person potential games (for this result we also provide a discrete-time proof).

In this note we propose to use the mathematical formalism of Quantum Mechanics to capture the idea that agents preferences, in addition to being typically uncertain, can also be indeterminate. They are determined (realized, and not merely revealed) only when the action takes place. An agent is described by a state which is asuperposition of potential types (or preferences or behaviors). This superposed state is projected (or collapses ) onto one of the possible behaviors at the time of the interaction. In addition to the main goal of modelling uncertainty of preferences which is not due to lack of information, this formalism, seems to be adequate to describe widely observed phenomena like framing and instances of noncommutativityin patterns of behavior. We propose two experiments to test the theory.

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.