A Proposal of Iterated Continuous Prisoner's Dilemma Game

Abstract

The purpose of this paper is to propose a new formalization of the prisoner's dilemma game, called the iterated continuous prisoner's dilemma game, and to show its usefulness. In recent years, the prisoner's dilemma game has been used in many researches which aim to investigate the principles of cooperation among agents. In the prisoner's dilemma game, each player chooses his action from two alternatives, Cooperate and Defect. However, this is unsatisfactory if we consider various kinds of human interactions, i. e., there are cases in which each player shows an inexact attitude. Therefore, we attempt to extend the range from which a player takes his action so as to represent such situations. First, we give a formulation of the continuous prisoner's dilemma game. It is a two-person non-zero-sum game in which each player can choose his action from the range [0, 1], where the value closer to 0 means more cooperative and the value closer to 1 means the player takes an action closer to defect. In the usual prisoner's dilemma game a payoff of each player is defined by a matorix, while it is given by payoff functions in the continuous version. The game is played repeatedly. This is called the iterated continuous prisoner's dilemma game. We can discuss usefulness of this new formalization by investigating various strategies each player can take on this game. Each strategy is evaluated by a present value of a total payoff under a given discount rate. Using this value, we can define a situation of invasion between two strategies. We try to compare various strategies by this invasion relation. Tit for Tat (TFT) is a well-known good strategy in the iterated prisoner's dilemma game. We evaluate various strategies against TFT, discrete TFT, and strategies that take inexact values. Lastly, we conclude this paper by discussing the principles of interaction between agents.