Two Papers on Repeated Games

Original article (link) posted: 31/10/2005

Sorin (1986) “On Repeated Games with Complete Information” Math. Of Operations Research, 11-1

Several properties of the sets of feasible payoffs for repeated games are shown. Particularly, the condition that the set of feasible payoffs are convex hull of the feasible payoffs in pure strategies is given. Namely, it is necessary and sufficient that a discount factor is larger than or equal to 1-1/N, where N is the number of the players.

Dal-Bo (2001) “Tacit Collusion under Interest Rate Fluctuations” Job Market Paper, UCLA

The paper examines the optimal tacit collusion equilibrium when the discount factor changes over time. It is shown that collusive prices and profits depend not only on the level of the discount factor but also on its volatility; they increase with a higher discount factor level and decrease with its volatility. The model is a variant of Rotemberg-Saloner model, where, instead of demand fluctuation, the discount factor is assumed to fluctuate.


Independence and Implementation

Continued from the previous post on bargaining theory, I am going to quote other parts from the Introduction. This time, my focus is on editor's (Prof. Thomson) view on a controversial axiom of Nash solution, contraction independence(*), as well as his brief survey on implementation of bargaining solutions.

(*) Contraction independence is often called the axiom of independence of irrelevant alternatives (IIA), which is defined as follows:

Contraction independence If, keeping the disagreement point constant, the feasible set contracts but the alternative chosen as solution outcome remains feasible, then it should remain the solution outcome.

On contraction independence
Contraction independence has been the object of the sharpest criticisms. Nash himself expressed some misgivings about it and Luce and Raiffa's (1957) objections are well known. In evaluating a bargaining situation, it is unavoidable and probably desirable that it be simplified and summarized, that it be reduced to its essential features. The issue is how much and what information should be discarded in the process, and one can make a convincing case that contraction independence ignores too much. Indeed, it covers contractions that affect the shape of the feasible in ways that seem relevant, for instance, the elimination of the only alternatives at which a particular agent's payoff is higher than at the initial solution outcome and the other agent's payoff lower than at the initial solution outcome; contraction independence prevents solutions from responding to such eliminations.
The main counterargument to this criticism was made by Nash himself: in practice, information is usually lacking about which alternatives are truly available, and a compromise under evaluation only competes against others that are not too different from itself. Modeling this lack or information explicitly is what an investigator should perhaps do, but there are also advantages to keeping the model simple; contraction independence is a formal way to express the idea.

On implementation
Whether a solution is implementable depends on the type of game forms that are used, and on the behavioral assumptions made about how agents confronted with such games behave. For implementation by normal form games and when agents calculate best responses taking as given the choices made by the other agents, a critical property for what is then called Nash-implementability is Maskin-monotonicity (actually an invariance property with respect to enlargements of the lower contour set at the chosen alternative). Most solutions are not Maskin-monotonic and therefore not Nash-implementable by normal form games. However, an implementation of the Kalai-Smorodinsky solution by a sequential game is possible (Moulin, 1984). Later contributions delivered the Nash solution (Howard, 1992), the egalitarian solution (Bossert and Tan, 1995)./// For subgame perfect implementation, a general theorem is offered by Miyagawa (2002). It covers all solutions obtained, after normalizing problems so that the ideal point has equal coordinates, by maximizing a monotone and quasi-concave function of the agents' payoffs. Implementation is by means of a stage game and the equilibrium notion is subgame perfection.

Bossert and Tan (1995) "An arbitration game and the egalitarian bargaining solution", Social Choice and Welfare, Volume 12, Number 1, 29-41.
Luce and Raiffa (1957), Games and Decisions: Introduction and Critical Survey, Wiley.
Miyagawa (2002) "Subgame-perfect implementation of bargaining solutions", Games and Economic Behavior, Volume 41, Issue 2, 292-308.
Moulin (1984) "Implementing the Kalai-Smorodinsky bargaining solution", Journal of Economic Theory, Volume 33, Issue 1, 32-45.


Bargaining Solutions: Nash, Egalitarian and Kalai-Smorodinsky

A leading researcher in bargaining theory, Professor William Thomson, recently edited the notable collection of papers in axiomatic bargaining:

Bargaining and the Theory of Cooperative Games: John Nash and Beyond

The following is quoted from the Introduction written by the editor. The parts I refer below focus on three representative solutions of the bargaining problem(*) in the literature, while his Introduction covers much more materials. I strongly recommend those who are interested in bargaining theory to read this insightful survey article.

Almost sixty years ago, Nash (1950) published his seminal paper on what is now known as the "axiomatic theory of bargaining"./// He formulated a list of properties, or "axioms", that he thought a solution should satisfy, and he established the existence and the uniqueness of a solution satisfying all of the axioms; this solution is now called the "Nash solution".

Nash's model has been one of the most successful paradigms of game theory. His paper is the founding stone of a literature that now comprises several hundred theoretical papers. The Nash solution is presented in all game theory textbooks./// Together with the Shapley value (Shapley, 1953) and the core (Gillies, 1959), it constitutes the obligatory background on cooperative games in most economics graduate programs.

In spite of the large number of reasonable solutions that one can easily define, only three solutions and variants have consistently emerged: in addition to the Nash solution, they are egalitarian(**) and Kalai-Smorodinsky solutions(***)./// The dominance of these three solutions and these variants is a central conclusion to be drawn from the literature.

The Nash solution has come out somewhat more often than the other two, and the claim can perhaps be made that it is indeed special./// But the argument is a little dangerous. Earlier, we talked about the theorist's need to simplify and summarize in order to analyze, and in axiomatic analysis simplification often takes the form of independence and invariance axioms. The Nash solution satisfies many invariance conditions, thus it is not much of a surprise that it should have dome out often. On the other hand, the monotonicity axioms that have generally led  to the Kalai-Smorodinsky and egalitarian solutions are readily understood and endorsed by the man on the street.

It is mainly on the basis of monotonicity properties that the Kalai-Smorodinsky solution should e seen as an important challenger to the Nash solution, the egalitarian solution presenting another appealing choice. This latter solution enjoys even stronger monotonicity requirements and like the Nash solution, it satisfies strong independence conditions. Unlike both the Nash and Kalai-Smorodinsky solutions, it requires interpersonal comparisons of utility however, which, depending upon the context, may be seen as a desirable feature or a limitation.

(*) A bargaining problem consists of a pair (S, d) where S, the feasible set, is the subset of alternatives, and d, the disagreement point, is a point of S./// A bargaining solution defined on a class of problems is a function that associates with each problem (S, d) in the class a unique point of S, the solution outcome of (S, d).
(**) The egalitarian solution (Kalai, 1977a) selects the maximal point of S at which utility gains from d are equal. More generally, by making utility gains proportional to a fixed vector of weights, we obtain the weighted egalitarian solution relative to these weights (Kalai, 1977b)

(***) The Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975) selects the maximal point of S that is proportional to the profile of maximal payoffs that agents can separately reach among the points of S that dominate d.

Gillies (1959) "Solutions to general non-zero-sum games" in Contributions to the Theory of Games IV, Princeton University Press, 47-85.
Kalai (1977a) "Nonsymmetric Nash solutions and replications of 2-person bargaining", International Journal of Game Theory, Volume 6, Number 3, 129-133.
Kalai (1977b) "Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons", Econometrica, Vol. 45, No. 7 (Oct., 1977), 1623-1630.
Kalai and Smorodinsky (1975) "Other Solutions to Nash's Bargaining Problem", Econometrica, Vol. 43, No. 3 (May, 1975), 513-518.
Nash (1950) "The Bargaining Problem", Econometrica, Vol. 18, No. 2 (Apr., 1950), 155-162.
Shapley (1953) "A Value for n-Person Games" in Contributions to the Theory of Games II, Princeton University Press, 307-317.