A Note on Stabilizing Cooperation in the Centipede Game

Steven J Brams,D Marc Kilgour

doi:10.3390/g11030035

Abstract

In the much-studied Centipede Game, which resembles the Iterated Prisoners’ Dilemma, two players successively choose between (1) cooperating, by continuing play, or (2) defecting and terminating play. The subgame-perfect Nash equilibrium implies that play terminates on the first move, even though continuing play can benefit both players—but not if the rival defects immediately, which it has an incentive to do. We show that, without changing the structure of the game, interchanging the payoffs of the two players provides each with an incentive to cooperate whenever its turn comes up. The Nash equilibrium in the transformed Centipede Game, called the Reciprocity Game, is unique—unlike the Centipede Game, wherein there are several Nash equilibria. The Reciprocity Game can be implemented noncooperatively by adding, at the start of the Centipede Game, a move to exchange payoffs, which it is rational for the players to choose. What this interchange signifies, and its application to transforming an arms race into an arms-control treaty, are discussed.

Highlights

Since Rosenthal [1] introduced what has come to be called the Centipede Game, there has been controversy about what constitutes rational play in it, which we discuss
In the Centipede Game, like IteratedPrisoners’ Dilemma (IPD), there is a tension between a Pareto-inferior subgame-perfect equilibrium outcome and a Pareto-optimal nonequilibrium outcome
Experiments with the Centipede Game show that the subgame-perfect equilibrium outcome is generally not the choice of subjects (McKelvey and Palfrey [3]; Nagel and Tang [4])

Summary

Introduction

Since Rosenthal [1] introduced what has come to be called the Centipede Game, there has been controversy about what constitutes rational play in it, which we discuss . Unlike IPD, in the Centipede Game, players do not make simultaneous choices, in ignorance of each other, in a stage game, which is repeated Instead, they make sequential choices in a game of perfect information, in which the players’ strategic situation may change after every move. Interchanging the two players’ payoffs at each stage of the Centipede Game transforms it into a new game we call the Reciprocity Game. The latter game makes it rational, via backward induction, for each player not to defect at any opportunity to do so but, instead, to cooperate until the end of play, whether or not the endpoint is known (the game is assumed to be finite)

Methods

Conclusion