In a tree enterprise, users reside at the nodes of the tree and their aim is to connect themselves, directly or indirectly, to the root of the tree. The construction costs of arcs of the tree are given by means of the arc-cost-function associated with the tree. Face to face with this tree enterprise, the bankruptcy venture is described in terms of the estate of the bankrupt concern and the claims of the various creditors. The objective of the paper is to provide conditions (on the claims and the surplus of the claims in the bankruptcy venture) which are sufficient and necessary for the bankruptcy venture to agree with some tree enterprise. It is established that the bankruptcy venture agrees with some tree enterprise if and only if the surplus of claims in the bankruptcy venture is at most the size of the second smallest claim (in the weak sense). For that purpose, both the tree enterprise as well as the bankruptcy venture are modelled as a cooperative game with transferable utility. Within the framework of cooperative game theory, the proof of the equivalence theorem concerning the tree enterprise game and the bankruptcy game, under the given circumstances, is based on graph theoretic tools in a tree structure.

Naive bumblebees were allowed to forage on 30 color-polymorphic artificial flowers, which were identical in morphology and reward schedule, but were marked by either a human-blue, human-green or a human-white landing surface. The probability of nectar rewards in the artificial flowers, and their spatial distribution, were manipulated experimentally. The bees' color choices in the different experimental treatments were compared. The proportions of visits to the three color morphs deviated significantly from the expected random choice (1/3-1/3-1/3) for more than 50% of the bees. Out of these bees, 38%, 32% and 30% formed a preference for human-blue, human-green and human-white, respectively. The frequency of non-random color choice, and the strength of the deviation from random choice, were highest when the different morphs were placed in separate clusters, lower when they were placed in adjacent clusters, and lowest when they were randomly intermingled. Non-random color choice was also more pronounced when the bees were rewarded according to a constant schedule, rather than probabilistically. A statistically significant preference for human-blue was found during the bees' first three visits. The bees' tendency for "runs" of consecutive visits to the same flower color can partially account for their non-random color choices. The specific color preferences of individuals could not be related to their early foraging experiences.

In Aumann (1976) - "Agreeing to Disagree" - it is shown that if there is a common prior then common knowledge of disagreement is impossible. This paper studies the converse proposition. The lack of a common prior is shown to yield common knowledge of disagreement in a variety of cases. However, an example demonstrates that this result cannot be generalized.

We look at a set % of states of the world which are defined as maximal consistent lists of formulae formed in a language which contains a knowledge operator ki for each agent i. The states of the world induce an information partition for each agent i such that all those % % % are included in the same information cell which contain the same range of knowledge for this agent (this range of knowledge we will call the agent's ken). We can then ask what it means that some agent i forgets which one of a variety of kens he has. This question can be answered easily if we use states of the world as primitives: then the answer is just to take a union over information cells. This does not make sense any more when states of the world are lists of formulae. We find a solution to this question for the case of one agent and show why the same solution cannot be used for the case of more than one agent. In an appendix, we apply results obtained before to analyze the j-operator (the knowing-whether operator). The larger context in which our question arose was to prove the Bachrach-Cave Agreement-Theorem in a model where states of the world are not primitives.

The environmental need to control the quality of the air is represented by a multi-players sequential game. One model is a two period game with n + 1 players, of which n are operators and one is an inspector. The game is analyzed via the solution concept of the strategies equilibrium (Nash equilibrium). The second model assumes that all operators are identical,i.e. the payoffs are the same to all operators. The equations that describe the Nash equilibrium, are solved analytically under this assumption, and enable us to compare games with a different number of operators (n). Numerical solutions are included. A discussion of the advantages and disadvantages of individual punishment vs. collective punishment appears in the last section. The model includes a parameter which varies from full individual punishment when the inspector raises an alarm (i.e. only the operators that acted illegally are fined), to full collective punishment (i.e. all operators are fined regardless their actions). Numerical results are added.

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.