Aumann and Myerson (1988) defined a linking game leading to the formation of cooperation structures. They asked whether it is possible for a simple game to have a stable structure in which no coalition forms, i.e., in which the cooperation graph is not internally complete. We answer this question affirmatively; specifically, we present a simple proper weighted majority game with a connected incomplete structure, and prove it to be stable.

We present and analyze a model of non-cooperative bargaining among n participants, applied to situations describable as games in coalitional form. This leads to a unified theory that has as special cases the Shapley value in the transferable utility case, the Nash bargaining solution in the pure bargaining case, and the recently introduced Maschler-Owen consistent value solution in the general (non-transferable utility) case.

It has often been observed that cooperative behavior emerges in actual play of the repeated prisoners' dilemma. This observation seems to be in conflict with the fact that, in any finite repetition of the prisoners' dilemma, all Nash equilibria (and even all correlated equilibria) lead to the non-cooperative outcome in each stage. In this paper we show that a very small departure from the common knowledge assumption on the number, T, of repetitions already enables cooperation. More generally, with such a departure, any feasible individually-rational outcome of any one-shot game can be approximated by a Nash equilibrium of a finitely-repeated version of that game. The sense in which the departure from common knowledge is "small" is as follows: (i) With probability one, the players know T with precision +- 1. (ii) With probability 1 - %, the players know T precisely; moreover, this knowledge is mutual to degree %T. (iii) the deviation of T from its expectation is extremely small.

This chapter studies the implications of bounding the complexity of players' strategies in long term interactions. 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 its next state is a function of its current state and the actions of the other players. The size of an automaton is its number of states. The results study the equilibrium payoffs per stage of the repeated games when players' strategies are restricted to those implementable by automata of bounded size.

We report the results of two experiments designed to study tacit coordination in a class of market entry games with linear payoff functions, binary decisions, and zero entry costs, in which each of n = 20 players must decide on each trial whether or not to enter a market whose capacity is public knowledge. The results show that although the subjects differ considerably from one another in their decision policies, tacit coordination emerges quickly on the aggregate level and is accounted for most successfully by the Nash equilibrium solution for noncooperative n-person games.

Any correlated equilibrium of a strategic game with bounded payoffs and convex strategy sets which has a smooth concave potential, is a mixture of pure strategy Nash equilibrium. If moreover, the strategy sets are compact and the potential is strictly concave, then the game has a unique correlated equilibrium.

Every two person game of incomplete information in which the information to both players is identical and deterministic has an equilibrium.

Every two person repeated game of symmetric incomplete information in which the signals sent at each stage to both players are identical and generated by a state and moves dependent probability distribution on a given finite alphabet has an equilibrium payoff.

Coordination behavior is studies experimentally in a class of market entry games featuring symmetric players, complete information, zero entry costs, and several randomly presented values of the market capacity. Once the market capacity, c, becomes common knowledge, each player must decide privately whether to enter the market and receive a payoff which increases in the difference between c and the number of entrants, m, or stay out. Payoffs forstaying out are either positive, giving rise to the domain of gains, or negative, giving rise to the domain of losses. The major findings are substantial individual differences in decision policies, which do not diminish with practice, and aggregate group behavior which is organized extremely well in both the domains of gains and loses by the Nash equilibrium solution.

{The paper studies the implication 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 is an automated machine that implements a strategy; it 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 states 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 equilibrium payoffs of a sequence of such games, G(n)

This article establishes that, in a one-dimensional diffusion world, any contingent claim's delta is bounded by its delta at maturity and, if its payoff is convex, its current value is convex in the underlying's value. A decline in the present value of the exercise price can be associated with a decline in a call's price. Bounds on call prices and deltas are derived for the case when the underlying's volatility is bounded. If the underlying follows a multi-dimensional diffusion (a stochastic volatility world), or a discontinuous or non-Markovian process, call prices can be decreasing, concave function of the underlying's value.

We consider a repeated two-person zero-sum game in which the payoffs in the stage game are given by a 2 % 2 matrix. This is chosen (once) by chance, at the beginning of the game, to be either G1 or G², with probabilities p and 1 - p respectively. The maximizer is informed of the actual payoff matrix chosen but the minimizer is not. Denote by vn(p) the value of the n -times repeated game (with the payoff function defined as the average payoff per stage), and by v%(p) the value of the infinitely repeated game. It is proved that vn(p)=v%(p) + %(p)%(p)/%n + %%1/%n% , where %(p) is on appropriately scaled normal distributiondensity function evaluated at its p-quantile, and the coefficient K(p) is either 0 or the absolute value of a linear function in p.

Starting with the analysis of arms control and disarmament problems in the sixties, inspection games have evolved into a special area of game theory with specific theoretical aspects, and, equally important, practical applications in various fields of human activities where inspection is mandatory. In this contribution, a survey of applications is given first. Then, the general problem of inspection is presented in a game theoretic framework as an extension of a statistical hypothesis presented in a testing problem. Using this framework, two important models are solved: material accountancy and dataverification. A second important aspect of inspection games are limited inspection resources that have to be used strategically. This is presented in the context of sequential inspection games, where many mathematically challenging models have been studies. Finally, the important concept of leadership, where the inspector becomes a leader by announcing and committing himself to his strategy, is shown to apply naturally to inspection games.

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or "player". One is interested not only in each person's knowledge about substantive matters, but also in his knowledge about the others' knowledge. These notes examine two parallel approaches to the subject. The first is the semantic approach, in which knowledge is represented by a space % of states of the world, together with partitions %i of % for each player i; the atom of %i containing a given state % of the world represents the set of those states that i cannot distinguish from %. The second is the syntactic approach, in which knowledge is represented by abstract formulas constructed according to certain syntactic rules. These notes examine the relation between the two approaches, and show that they are in a sense equivalent. In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space % and the partitions %i can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach. Other topics that are taken up include various formalizations of "common knowledge", and the "Agreement Theorem" of J. Cave and M. Bacharach. The notes end with an application of these ideas to the context of probabilistic beliefs.

It has been argued that in the presence of an `Atmosphere Externality' and competitive behavior by households, a uniform commodity tax on an externality - generating good attains the first best. It is demonstrated, however, that if income redistribution is desirable then personalized taxes are required for a second-best optimum. Each of these taxes is the sum of a uniform (across households) tax and a component, positive or negative, which depends on the household's income and demand elasticities. Second-best optimal indirect taxes and rules for investment in externality-reducing measures are also considered.

A survey of bounded rationality models and ideas in Game Theory. Topics covered include: The evolutionary approach to optimization – and specifically to game theory – and its implications for the idea of bounded rationality; evolutionary dynamics; "rule rationality" as opposed to "act rationality"; "trembles" and refinements in general; "crazy" perturbations; failure of common knowledge of rationality; limiting average payoff in infinitelyrepeated games; epsilon equilibria; players modeled as computers, finite state automata, or Turing machines; paradoxes (such as Ellsberg or Allais); laboratory experiments; and finally, an open problem.

The reactive bargaining set (Granot [1994]) is the set of outcomes for which no justified objection exists. Here, in a justified objection the objector first watches how the target tries to act (if he has such an option), and then reacts by making a profit and ruining the target's attempt to maintain his share. In this paper we explore properties of the reactive bargaining set, set up the system of inequalities that defines it, and construct a dynamic system in the sense of Stearn's transfer scheme that leads the players to this set. We also extend the definition of the reactive bargaining set to NTU games in a way that keeps it nonempty. To shed light on its nature and its relative ease of computation, we compute the reactive bargaining set for games that played important role in the game theory literature.

A reply to Margalit and Yaari's paper "Rationality and Comprehension", in which they comment on my papers "Agreeing to disagree" [1] and "Notes on Interactive Epistemology" [2]. Inter alia, we point out that contrary to Margalit and Yaari's claim, in [1] the agents need not condition on the same events; and in [2], the state space is not assumed as analytic knowledge, but is derived. In addition, a simple resolution of the "hangman's paradox" is offered.

Motivated by recent auctions of licenses for the radio-frequency spectrum, we consider situations where multiple objects are auctioned simultaneously by means of a second-price, sealed-bid auction. For some buyers, called global bidders, the value of multiple objects exceeds the sum of objects' values separately. Others, called local bidders, are interested in only one object. In a simple independent private values setting, we (a) characterize an equilibrium that is symmetric among the global bidders; (b) show that the addition of bidders often leads to less aggressive bidding; and (c) compare the revenues obtained from the simultaneous auction to those from its sequential counterpart.

For a class of nonatomic congestion games, two solution concepts, a noncooperative one and a cooperative one, are compared. Each player in the game chooses one of several common facilities. The player's payoff is the difference between the reward and the cost associated with choosing that facility. The rewards are fixed and player-specific. The costs are uniform, but variable: they strictly increase with the measure of the set of players using the facility. The noncooperative solution of the game is the (unique) Nash equilibrium outcome. The cooperative one is the Aumann-Shapley value of the cooperative game that results when players are willing to cooperate in order to minimize the total utility. Using a new result in the theory of values of nonatomic games, we derive a formula for the value. We show that there is exactly one case in which the Nash equilibrium outcome andthe value always coincide: this is the case in which the costs increase logarithmically with the measure of the set of users.