Ehud Kalai

Other Solutions to Nash's Bargaining Problem

A two-person bargaining problem is considered. It is shown that under four axioms that describe the behavior of players there is a unique solution to such a problem. The axioms and the solution presented are different from those suggested by Nash. Also, families of solutions which satisfy a more limited set of axioms and which are continuous are discussed. WE CONSIDER a two-person bargaining problem mathematically formulated as follows. To every two-person game we associate a pair (a, S), where a is a point in the plane and S is a subset of the plane. The pair (a, S) has the following intuitive interpretation: a = (a1, a2) where ai is the level of utility that player i receives if the two players do not cooperate with each other. Every point x = (x1, x2) e S represents levels of utility for players 1 and 2 that can be reached by an outcome of the game which is feasible for the two players when they do cooperate. We are interested in finding an outcome in S which will be agreeable to both players. This problem was considered by Nash [3] and his classical result was that under certain axioms there is a unique solution. However, one of his axioms of independence of irrelevant alternatives came under criticism (see [2, p. 128]). In this paper we suggest an alternative axiom which leads to another unique solution. Also, it was called to our attention by the referee that experiments conducted by H. W. Crott [1] led to the solution implied by our axioms rather than to Nashs solution. We also consider the class of continuous solutions which are required to satisfy only the axioms of Nash which are usually accepted. We give examples of families of such solutions.

On weighted shapley values

Nonsymmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It is shown axiomatically that two families of solutions of this type are possible. These families are strongly related to each other through the duality relationship on games. While the first family lends itself to applications of nonsymmetric revenue sharing problems the second family is suitable for applications of cost allocation problems. The intersection of these two families consists essentially of the symmetric Shapley value. These families are also characterized by a probabilistic arrival time to the game approach. It is also demonstrated that lack of symmetries may arise naturally when players in a game represent nonequal size constituencies.

Observable Contracts: Strategic Delegation and Cooperation

The role of commitments in noncooperative games is well acknowledged and documented. One way to achieve commitments is by letting delegates represent the players of a game. In this paper, the authors study a delegation game in which the players can use agents strategically to play on their behalf and the contracts they sign with them are common knowledge. They show that, in such cases, every Pareto optimal outcome of the game can become the unique subgame perfect Nash equilibrium of the delegation game. The authors demonstrate this result by discussing the Cournot-type duopolistic game. Copyright 1991 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Totally Balanced Games and Games of Flow

A class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games. The proof relies on the max flow-min cut theorem of Ford and Fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation.

Generalized Network Problems Yielding Totally Balanced Games

A class of multiperson mathematical optimization problems is considered and is shown to generate cooperative games with nonempty cores. The class includes, but is not restricted to, numerous versions of network flow problems. It was shown by Owen that for games generated by linear programming optimization problems, optimal dual solutions correspond to points in the core. We identify a special class of network flow problems for which the converse is true, i.e., every point in the core corresponds to an optimal dual solution.

Persistent equilibria in strategic games

A perfect equilibrium [Selten] can be viewed as a Nash equilibrium with certain properties of local stability. Simple examples show that a stronger notion of local stability is needed to eliminate unreasonable Nash equilibria. The persistent equilibrium is such a notion. Properties of this solution are studied. In particular, it is shown that in each strategic game there exists a pesistent equilibrium which is perfect and proper.

Bounded Rationality and Strategic Complexity in Repeated Games

Publisher Summary This chapter discusses three important areas in modern decision theory, which are bounded rationality, artificial decision making, and management information systems. It discusses some new methodologies and results within the area of repeated games and with complexity measures that use notions of automata. The proposal to apply the notion of automaton to describe a player in a repeated game comes from a survey of repeated games. This notion is suggested as a way to distinguish between simple and complicated strategies based on the number of states of automata describing them. It studies the interactive behavior of bounded players by studying a game with appropriately restricted sets of strategies. The chapter also discusses the effect of complexity costs on the outcome of the game. The players are restricted to use automata of any finite size to play the game. However, their final payoffs decrease as they use automata of bigger sizes. Thus, tension in a player is created between high overall utility and increasing complexity. In the modified version of the game, the equilibrium outcomes have a nice simple structure and the set of equilibrium payoffs is dramatically reduced. This is even the case as the complexity costs approach zero and thus, their model points out a fundamental discontinuity regarding complexity costs.

Social welfare functions when preferences are convex, strictly monotonic, and continuous

The paper shows that if the class of admissible preference orderings is restricted in a manner appropriate for economic and political models, then Arrows impossibility theorem for social welfare functions continues to be valid. Specifically if the space of alternatives is R+n, n ≥ 3, where each dimension represents a different public good and if each persons preferences are restricted to be convex, continuous, and strictly monotonic, then no social welfare function exists that satisfies unanimity, independence of irrelevant alternatives, and nondictatorship.

Arbitration of Two-Party Disputes Under Ignorance

When an arbitrator lacks complete information about the dispute in question, he may have to turn to the disputants themselves to provide information. If they know how the information is to be used, they may have incentives to hide the truth. By using the players reports as checks on each other, a completely ignorant arbitrator of a dispute between two completely informed players can induce truthful revelation in the sense that the truth is a Nash equilibrium of the game which the arbitrators decision process imposes on the players. Such a scheme may be used in conjunction with any one from a class of functions which select Pareto-optimal, individually-rational outcomes in two-person normal-form games.

Barriers to Trade and Disadvantageous Middlemen : Nonmonotonicity of the Core*

Received September 14, 1977; revised April 20, 1978 Tn applying cooperative game theory to economic problems of exchange, it is standard to assume that all logically possible coalitions may form. However, because of institutional, legal, or physical barriers, it may in fact be impossible for certain sets of agents to communicate or trade with one another directly. It would seem worthwhile to recognize this and to analyze the impact of such barriers to trade. In