Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 452042, 10 pages

http://dx.doi.org/10.1155/2015/452042

## A Study of Prisoner’s Dilemma Game Model with Incomplete Information

^{1}School of Applied Mathematics, Guangdong University of Technology, Guangzhou, Guangdong 510006, China^{2}Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China

Received 25 May 2014; Revised 22 September 2014; Accepted 23 September 2014

Academic Editor: Yiu-ming Cheung

Copyright © 2015 Xiuqin Deng and Jiadi Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Prisoners’ dilemma is a typical game theory issue. In our study, it is regarded as an incomplete information game with unpublicized game strategies. We solve our problem by establishing a machine learning model using Bayes formula. The model established is referred to as the Bayes model. Based on the Bayesian model, we can make the prediction of players’ choices to better complete the unknown information in the game. And we suggest the hash table to make improvement in space and time complexity. We build a game system with several types of game strategy for testing. In double- or multiplayer games, the Bayes model is more superior to other strategy models; the total income using Bayes model is higher than that of other models. Moreover, from the result of the games on the natural model with Bayes model, as well as the natural model with TFT model, it is found that Bayes model accrued more benefits than TFT model on average. This demonstrates that the Bayes model introduced in this study is feasible and effective. Therefore, it provides a novel method of solving incomplete information game problem.

#### 1. Introduction

Incomplete information games are influenced by the private information owned by at least one game player, such as current game state, mechanism of other players in decision-making, the game state of other players, and the reward/punishment mechanism of the game [1]. However, due to the absence of its optimal or relatively optimal solution (at any certain game state), such an incomplete information game is insoluble by traditional study methods. This is due to the fact that the other players restrain the strategy of incomplete information games. Harsanyi [2] analyzed an incomplete information game using Bayesian game player strategies and proposed methods for modeling and analyzing game problems. Zinkevich et al. [3] investigated incomplete information games using Nash equilibrium and minimizing regret method and proposed some game strategies for improving poker game problems.

As a branch of artificial intelligence and a cutting-edge research topic, machine learning has been paid great attention in related fields in recent years. Machine learning is defined as a research method aiming at obtaining a more desired approximate solution based on the general rules obtained by analyzing a large amount of data [4]. Statistical machine learning is a branch of machine learning. It integrates statistical theory into machine learning by combining probability theory and stochastic mathematical knowledge with machine learning to improve the efficiency and accuracy [5, 6]. The Bayesian classification algorithm is a commonly used machine learning method [7]. The simplified model of naive Bayesian classification is often used in text classification.

The prisoners’ dilemma game is a classic cooperation and selection problem based on the assumption of selfish human motives [8]. It is popular and widely applied in mathematics and economics [9]. For a long time, it has been a classical game theory problem and attracted great interest from mathematics and economics researchers around the world. Game theory was born in the mid-twentieth Century and was founded by von Neumann (a famous mathematician and founding father of computing) and Morgenstern (a famous economist). The starting point for the development of game theory was the publication of John von Neumann and Oscar Morgenstern’s seminal work* The Theory of Games and Economic Behavior* in 1944 [10]. Game theory brought radical changes to economics and provided a standard analysis tool for economists. In light of the contributions of game theory to economics, the Royal Swedish Academy of Sciences awarded Nobel Prizes for economics to Nash, Harsanyi, and Selten in 1994 and Aumann and Schelling in 2005, respectively [10]. In the famous artificial intelligence algorithm competition of prisoners’ dilemma, Axelrod concluded that a TFT (tit-for-tat) model was the optimum solution through brute force competition in participating algorithms [11, 12]. Miller [13] introduced an automaton model to simplify and analyze prisoners’ dilemma and proposed a more general prisoners’ dilemma decision analysis method. He also applied the model to solve problems arising from generalisation. And Press suggested that the prisoner’s dilemma is an ultimatum game and gave an example of strategy, which can gain an unfair share of rewards, to support his claim [14]. Using genetic algorithm, Lin and Wu [15] studied the evolution of strategies in the iterated prisoner’s dilemma on complex networks and found that the agents located on complex networks can naturally develop some self-organization mechanics of cooperation, which can not only result in the emergence of cooperation but also strengthen and sustain the persistent cooperation.

Evolutionary game theory [16, 17] extends and combines ideas from game theory and evolutionary biology to study the evolution of an interacting population of individuals. Perhaps one of the simplest games in evolutionary game theory is the so-called evolutionary spatial prisoners’ dilemma (ESPD) [18]. Cardillo et al. [19] investigated the coevolution of strategies and update rules in the evolutionary spatial prisoners’ dilemma (ESPD). The authors concluded, for a variety of underlying graph topologies, that when the dynamics coevolves with the strategies it leads to more cooperation in the weak prisoners’ dilemma in general. Du et al. discussed another evolutionary method that uses the improved weighted network to solve the problem [20]. Literature [21] proposed a model using two graphs in conjunction with the ESPD: one for determining player interaction and second for updating strategies. Moreover, Wang et al. have proposed some evolutionary algorithm to solve relevant problems [22, 23]. Game theory techniques have been widely applied to various engineering design problems in which the action of one component has impact on (and perhaps conflicts with) that of any other component.

Prisoners’ dilemma could be regarded as a game with incomplete information. It satisfies the conditions for an incomplete information game; namely, the players of each game are incapable of determining the choice of their rival in any current station. In our study, we propose a naive Bayesian classification method, which is used to establish the machine learning model for prisoners’ dilemma in an attempt to solve it through statistical machine learning. With the use of Bayesian classification, the opponents’ strategy can be presented as the possibility of choice which means the accuracy of the prediction on opponents’ strategy has been promoted. Moreover, we introduce an evaluation with multiple processes to provide the information with high precision for the final decision in our strategy. In the step of record, we suggest some efficient data structures and ensure a reasonable space and time complexity in our method. We test the proposed method during the competitions with some typical methods. The simulation experimental results show that our method outperforms four classical methods (see Figure 1 for more details). We further apply the Bayes method to multiplayer games. The simulation result indicates that the Bayes method gains the highest income in the multiplayer test (see Figure 5 for more details).