We have obtained experimental results on the ultimatum bargaining game that support an evolutionary explanation for subjects' behavior in the game. In these experiments we have created enviornments in which subjects interact with each other in addition to interacting with virtual players, i.e. computer programs with pre-specified stategies. Some of these virtual players were designed to play the equitable allocation, while others exhibited behavior closer to the subgame-perfect equilibrium, in which the proposer's share is much larger than that of the responder. We have observed significant differences in the behavior of real subjects depending on the type of "mutants" (virtual players) that were present in their enviornment.

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The literary source of the main ideas in Aumann's article "Backward Induction and Common Knowledge of Rationality" is exposed and analyzed. The primordial archetypal images that underlie both this literary source and Aumann's work are delineated and are used to explain the great emotive impact that this work had on the community of game theorists.

This article concerns the interactive model propositional calculus, using the multi-agent epistemic logic S5. With regard to the space % of maximally consistent sets of formulas, the knowledge of an agent is defined by its knowledge of a set of formulas. Common knowledge can be defined in at least two ways, as the common knowledge of a set of formulas or according to the meet partition generated by the knowledge partitions of the agents. With at least two agents, this meet partition is a much finer partition of % than that generated by the common knowledge of sets of formulas, yet for some points of % the two partition members coincide. Whether the two partition members coincide has radical implications for the structure of the meet partition members.

Gul (1989) introduces a non-cooperative bargaining procedure and claims that the payoffs of the resulting efficient stationary subgame perfect equilibria are close to the Shapley value of the underlying transferable utility game (when the discount factor is close to 1). We exhibit here an example showing that efficiency, even for strictly super-additive games, does not imply that all meetings end in agreement. Thus efficiency does not suffice to get Gul's result.

We study Bayesian mechanism design in the context of multidimensional types and quasi-linear preferences. We first show that any two incentive compatible mechanisms which implement the same allocation rule must be payoff equivalent up to an additive constant. This result is then applied to study multiple object auctions. We show that the Vickrey-Clarke-Groves auction maximizes the seller's expected revenue among all efficient auctions.

We analyze behavior of an experiment on the repeated centipede game played in the reduced normal form. In this game 2 players decide simultaneously when to split a cake. The longer both players wait, the higher the total gain for both. The player who is less patient to wait obtains the larger share of the pie while the other obtains the lower share of the pie. In all standard game theoretic predictions the outcome is that the pie is split immediately. We compare several static models and ouantative learning models, among them quantal response, reinforcement models and fictitious play. Furthermore, we structure behavior from period to period according to a simple cognitive process, called learning direction theory. It is shown that there is a significant difference in behavior whether a player has observed that he got the larger share of the pie or whether he got the smaller share of the pie.

This paper explores the relationships between noncooperative bargaining games and the consistent value for non-transferable utility (NTU) cooperative games. A dynamic approach to the consistent value for NTU games is introduced: the consistent vector field. The main contribution of the paper is to show that the consistent field is intimately related to the concept of subgame perfection for finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. The solutions of the dynamic system associated to the consistent field characterize the subgame perfect equilibrium payoffs of the noncooperative bargaining games. We show the for transferable utility, hyperplane and pure bargaining games, the dynamics of the consistent field converge globally to the unique consistent value. However, in the general NTU case, the dynamics of the consistent field can be complex. An example is constructed where the consistent field has cyclic solutions; moreover, the finite horizon subgame perfect equilibria do not approach the consistent value.

{The paper studies the implications of bounding the complexity of the strategies players may select, on the set of equilibrium payoffs in repeated games. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. A finite automaton has a finite number of states and an initial state. It prescribes the action to be taken as a function of the current state and a transition function changing the state of the automaton as a function of its current state and the present actions of the other players. The size of an automaton is its number of states. The main results imply in particular that in two person repeated games, the set of equilibrium payoffs of a sequence of such games. G(n)

This ariticle concerns the interactive modal propositional calculus, using the multi-agent epistemic logic S5. With regard to the space of maximally consistent sets of formulas, the relations between three aspects of common knowledge are investigated: 1) whether common knowledge defined semantically is determined by the set of formulas held in common knowledge, 2) the partial order by inclusion of the sets of formulas that can be held in common knowledge, and 3) the cardinality of a generating set of formulas for those held in common knowledge. Additionally, assuming at least two agents, it is shown that the number of connected components of holding only the tautologies in common knowledge has the cardinality of the continuum.

We consider an inspection game played on a finite time interval. The inspector wishes to detect a violation as soon as possible after it has been made by the operator. The loss to the inspector is assumed to be linear in the duration of the time elapsed between the violation and its detection. The inspection is not observed by the operator unless the inspector calls an alarm. The inspection is imperfect; it has a Type One Error which means that the inspector may call a false alarm (with probability alfa), and a Type Two Error which means that inspection may fail to detect (with probability beta) a violation which did occur. We first solve the game when alfa and beta are fixed and given. Then we consider the more general model in which the error probability alfa is chosen strategically by the inspector and may depend on the time of inspection. This yields two equilibria; one with constant alfa (and beta) and one with alfa increasing in time. The latter cannot be solved analytically. Consequently we solve a numerical example in which the inspction consist of obsderving a normally distributed signal.

We consider the prosaic system of matching which is specified by the following two common rules: (i) Each woman (man) proposes to at most one man (woman). (ii) A man and a woman marry each other if they propose to each other. Weprove that this system implements the correspondence of stable matchings by strong Nash Equilibria. We also find a simple extensive game form which implements the same correspondence by subgame perfect equilibria.

This paper reports two ultimatum game experiments comparing the behavior of individuals with that of three-person groups. Group members conducted a short face-to-face discussion in order to decide, as a collective, on a proposed division or on whether to accept or reject a proposal. Both experiments found that groups offered significantly less than individuals. But, as indicated by the low rejection rate in both treatments, groups were also willing to accept less.

This paper traces the ideological career of the notion of the invisible hand, from the 18th century to the 20th. Two main models of invisible-hand explanations are distinguished: the aggregative and the evolutionary. The argument is made that the contemporary use of the idea of the invisible hand by conservatives as against liberals and social planners springs from not distinguishing between these two models. The latter part of the paper draws a comparison between the idea of the invisaible hand and Hegel's historically-related idea of the cunning of reason.

Two preimputations of a given TU game can be compared via the Lorenz order applied to the vectors of satisfactions. One preimputation is `socially more desirable' than the other, if its corresponding vector of satisfactions Lorenz dominates the satisfaction vector with respect to the second preimputation. It is shown that the prenucleolus, the anti-prenucleolus, and the modified nucleolus are maximal in this Lorenz order. Here the modified nucleolus is the unique preimputation which lexicographically minimizes the envies between the coalitions, i.e. the differences of excesses. Recently Sudh?lter developed this solution concept. Properties of the set of all undominated preimputations, the maximal satisfaction solution, are discussed. A function on the set of preimputations is called collective satisfaction function if it respects the Lorenz order. We prove that both classical nucleoli are unique minimizers of certain `weighted Gini inequality indices', which are derived from some collective satisfaction functions. For the (pre)nucleolus the function proposed by Kohlberg, who characterized the nucleolus as a solution of a single minimization problem, can be chosen. Finally, a collective satisfaction function is defined such that the modified nucleolus is its unique maximizer.

It is shown that the linkage principle (Milgrom and Weber(1982)) does not extend to the multi-unit auction setting. An analysis of the equilibium bidding strategies is carried out for the gneral two-agent/two-unit Vickrey auction in order to provide economic insight into the nature of the failure. In addition, an explicit counterexample is provided.

For fixed i let X(i)=(X1(i),...,Xd(i) be a d-dimensional random vector with some known joint distribution. Here i should be considered a time variable. Let X(i)=1,...,n be a sequence of n independent vectors, where n is the total horizon. In many examples Xj(i) can be thought of as the return to partner j, when there are d2 partners, and one stops with the i-th observation. If the j-th partner alone could decide on a (random) stopping rule t, his goal would be to maximize EXj(t) over all possible stopping rules tn. In the present "multivariate" setup the d partners must however cooperate and stop at the same stopping time t, so as to maximize some agreed upon function h( ) of the individual expected returns. The goal is thus to find a stopping rule t* for which h(EX1(t),...,EXd(t)=h(EX(t) is maximized. For continuous and monotone h we describe the class of optimal stopping rules t*. With some additional symmetry assumptions we show that the optimal rule is one which (also) maximizes EZt where Zi= Xj(i), and hence has a particularly simple structure. Examples are included, and the results are extended both to the infinite horizon case and to the case when X(1),..., X(n) are dependent. Asymptotic comparisons between the present problem of finding sup h(E X(t)) and the "classical" problem of finding sup Eh( X(t)) are given. Comparisons between the optimal return to the statistician and to a "prophet" are also included. In the present context a "prophet" is someone who can base his (random)choice g on the full sequence X(1),..., X(n), with corresponding return sup h(E X(g)).

P-values for hypotheses are considered as random variables. Their expected value (EPV) is expressed in a simple form. In simple examples they are directly computable, also under the alternative hypothesis, and in more complicated examples they are easily simulated. Their major advantage is that they do not depend on any significant level. It is suggested that the use of EPV can replace the use of power, which is always significance level dependent EPV can also be used for comparison of tests when more than one test is available for a given hypothesis. Examples are given, as well as tables which relate significance level and power to EPV. A comparison of the two-sample one-sided Kolmogorov-Smirnov, Mann-Whitney and t tests is included, for a variety of underlying distributions.

Whereas probabilistic calibration has been a central normative concept of accuracy in previous research on interval estimates, we suggest here that normative approaches for the evaluation of judgmental estimates should consider the communicative interaction between the individuals who produce the judgements and those who receive or use them for making decisions. We analyze precision and error in judgement and consider the role of accuracy-informativeness trade-off (Yaniv & Foster, 1995) in the communication of estimates. The results shed light on puzzling findings reported earlier in the literature concerning the calibration accuracy of subjective confidence intervals.