Press-Dyson Analysis of Asynchronous, Sequential Prisoners Dilemma

© R. D Young

Press-Dyson Analysis of Asynchronous, Sequential Prisoner’s Dilemma

Robert D. Young*

Department of Physics, Illinois State University, Normal, IL 61790-4560

(Dated December 11, 2017)

Two-player games have had a long and fruitful history of applications stretching across the social, biological, and physical sciences. Most applications of two-player games assume synchronous decisions or moves even when the games are iterated. But different strategies may emerge as preferred when the decisions or moves are sequential, or the games are iterated. Zero-determinant strategies developed by Press and Dyson are a new class of strategies that have been developed for synchronous two-player games, most notably the iterated prisoner’s dilemma. Here we apply the Press-Dyson analysis to sequential or asynchronous two-player games. We focus on the asynchronous prisoner’s dilemma. As a first application of the Press-Dyson analysis of the asynchronous prisoner’s dilemma, tit-for-tat is shown to be an efficient defense against extortionate zero-determinant strategies. Nice strategies like tit-for-tat are also shown to lead to Pareto optimal payoffs for both players in repeated prisoner’s dilemma.

1. Introduction

The theory of games has its modern genesis in a classic book published in 1944 by von Neumann and Morgenstern [1]. Since then significant work by many researchers further developed the theory with applications to the social, behavioral, biological, and physical sciences.
Prisoner’s dilemma (PD) is a game developed in 1950 at RAND by Flood and Dresher [2]. Since its formal inception, PD has provided much insight and sometimes bafflement in understanding the resolution, and even intensification, of conflict and the development of cooperation in the real world. Axelrod describes many of the strategies used in attempts to solve the PD and reports on “tournaments” to test various strategies developed by himself and many others [3]. Sigmund has recently given a short, accessible book that treats PD as well as other classic games focusing on an evolutionary approach [4]. Recently, Cleveland, Liao, and Austin have applied a game theory approach, including PD, to the physics of cancer propagation [5]. Finally, the work of Brams [6] develops a dynamic theory of moves that includes PD and that has been applied to real world conflicts [7]. The wide scope and long history of game theory, prisoner’s dilemma, and applications makes it surprising that a new technique and class of strategies for PD were recently discovered by Press and Dyson [8]. This discovery led to a resurgence of interest and new applications of the prisoner’s dilemma, mostly in evolutionary biology [9]. The main purpose of this article is to extend the Press-Dyson analysis to a model of PD [10] that is sequential, asynchronous, and compatible with the theory of moves [6,7].

• Email: rdyoung@ilstu.edu 2

II. Payoff Matrices for Prisoner’s Dilemma

Prisoner’s dilemma is developed around a landscape of payoffs for the players that is succinctly described in terms of an 2X2 matrix with ordered pair elements. This matrix captures both the decisions and the payoffs of the players [1-4,6]. Fig. 1 gives a standard form of the matrix.

   

, ,

( , ) ( , ) c R S d T P c d R T S P       X Y

Figure 1. See text for discussion.

In Fig. 1, the two players are labeled X and Y. Player X is blue, and Y is red. Strategies of play for the players are labeled by c (cooperate) and d (defect). The verb cooperate means a move that the moving player judges to lead to a positive result for both players while defect means a move that the moving player judges to lead to a positive result for the moving player but a negative result for the other player. The ordered pairs in the four quadrants enclosed by brackets are payoffs of the two players based on the externality of interest. This can be a preference rank or a payoff. The language of payoffs will be used below. The first entry of the ordered pair corresponds to the payoff for player X, the second entry to the payoff for player Y. If each player cooperates the payoff is R to each. If each player defects the payoff is P to each. If one player cooperates and the other player defects, then the player that cooperates gets payoff S and the player that defects get payoff T. The classic PD puts the following two conditions on the four payoffs: T R P S   

and 2R T S   . The first condition ensures that mutual defection with payoff   , P P is a Nash equilibrium [1,4,11]. The second condition ensures that mutual-cooperation with payoff   , R R
is the best outcome of the four ordered pairs, a Pareto optimal payoff for both players with the two payoff sets

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