Multistage Game Models and Delay Supergames

The order of stages in a multistage game is often interpreted by looking at earlier stages as involving more long term decisions. For the purpose of making this interpretation precise, the notion of a delay supergame of a bounded multistage game is introduced. A multistage game is bounded if the length of play has an upper bound. A delay supergame is played over many periods. Decisions on all stages are made simultaneously, but with different delays until they become effective. The earlier the stage the longer the delay. A subgame perfect equilibrium of a bounded multistage game generates a subgame perfect equilibrium in every one of its delay supergames. This is the first main conclusion of the paper. A subgame perfect equilibrium set is a set of subgame perfect equilibria all of which yield the same payoffs, not only in the game as a whole, but also in each of its subgames. The second xmain conclusion concerns multistage games with a unique subgame perfect equilibrium set and their delay supergames which are bounded in the sense that the number of periods is finite. If a bounded multistage game has a unique subgame perfect equilibrium set, then the same is true for every one of its bounded delay supergames. Finally the descriptive relevance of multistage game models and their subgame perfect equilibria is discussed in the light of the results obtained.

[1]  W. Güth,et al.  Ultimatum bargaining behavior : a survey and comparison of experimental results , 1990 .

[2]  R. Selten,et al.  End behavior in sequences of finite prisoner's dilemma supergames , 1986 .

[3]  Reinhard Selten,et al.  A simple model of imperfect competition, where 4 are few and 6 are many , 1973 .

[4]  Robert J. Aumann,et al.  16. Acceptable Points in General Cooperative n-Person Games , 1959 .

[5]  F. Adickes,et al.  Zeitschrift für die gesamte Staatswissenschaft , 1908 .

[6]  W. Güth,et al.  An experimental analysis of ultimatum bargaining , 1982 .

[7]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[8]  R. Selten,et al.  Restabilizing Responses, Inertia Supergames, and Oligopolistic Equilibria , 1978 .

[9]  A. Rubinstein.  Equilibrium in supergames with the overtaking criterion , 1979 .

[10]  R. Selten.  Are cartel laws bad for business , 1984 .

[11]  Vernon L. Smith,et al.  Effect of Market Organization on Competitive Equilibrium , 1964 .

[12]  David M. Kreps,et al.  Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes , 1983 .

[13]  Ariel Rubinstein,et al.  Strong perfect equilibrium in supergames , 1980 .

[14]  V. Smith.  An Experimental Study of Competitive Market Behavior , 1962, Journal of Political Economy.

[15]  E. Fehr,et al.  Does Fairness Prevent Market Clearing? An Experimental Investigation , 1993 .

[16]  J. Kagel,et al.  On the Existence of Predatory Pricing: An Experimental Study of Reputation and Entry Deterrence in the Chain-Store Game , 1994 .

[17]  James W. Friedman,et al.  Oligopoly and the theory of games , 1977 .

[18]  Vijay Krishna,et al.  Finitely Repeated Games , 1985 .