A Probabilistic Analysis of Prisoner ’ s Dilemma with an Adaptive Population

A probabilistic approach of analyzing successive round prisoner’s dilemma in a large populations. We introduce a unique model capable of handling multiple adaptive player strategies and how to compute expected distributions after several rounds from an initial distribution. We examine three cases studies, and their convergence properties. I. I NTRODUCTION Prisoner’s Dilemma is not only a popular topic in daily life, but also very important in numerical analysis. This problem was a thought experiment originally proposed by economists in the 1950’s and has been the subject of countless papers and applied across several fields [1]. presenting it as follows: Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The polic e admit they don’t have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police off r each prisoner a Faustian bargain. Each prisoner is given the opportunity either to betray the other, by testifying that t he other committed the crime, or to cooperate with the other by remaining silent. Here’s how it goes [1]: 1) If A and B both betray the other, each of them serves 2 years in prison 2) If A betrays but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa) 3) If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge) The prisoner’s dilemma is a well know game that has been extensively studied in economics, political science, mach ine learning [2], [3] and evolutionary biology [4]. In this pape r, in order to analyze this problem better, we make this problem as a mathematics model: 1) two individuals A, B are presented with $1000, and each individually given the choice to split or steal the money. 2) If both choose split the money will be evenly divided. If one chooses split and the other steal the one who choose steal gets all the money. 3) If both choose steal neither receives anything. (See Tabl e I) As we know, A single round of prisoner’s dilemma isn’t too interesting for study, but when successive rounds are playe d and where players adapt based on previous results, the probl em TABLE I THE RULES OF GAME. HHHH A B Split Steal Split 500,500 0,1000 Steal 1000,0 0,0 is worth analyzing further. In this paper we will put forth a mathematical model to help analyze the prisoner’s dilemma, and analyze the effect two different two different adaptive playing strategies. The experiments and MATLAB code are showed in Section 3, and section 4 we present the results. The main contributions in this paper are as follows. 1) This paper presents and further investigates the prisone r’ dilemma, and formulate this problem by mathematics, make it easy to further analyze, 2) Apply a probabilistic approach to analysis, 3) Analyze whether given distributions will converge. II. T HEORY A. Overview Our experiment will be played out over a series of rounds between a population of N people. Each person has two important traits: 1) a probability of choosing to Split or to Steal and 2) a playing strategy which will alter their probability bas ed on interactions with other players.we will use players with two different playing strategies in this paper, type A and type B. In a round each person in the population will be paired-with equal probability-with exactly one other person. These pai rs then play a game of the prisoner’s dilemma with each other, afterwards a player’sp value can change depending on the results of the game. B. Definitions Let the vectorXt represent a population of N people where Xt = X1t, X2t, X3t, · · · , XNt, t is some time index, and each Xit represents an ith individual person in t round. Let Split , 1 andSteal , 0. For each personXit,