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Qwag Jó, Ólwz-W Cz<Fv@¬ Autor: Paulo Sérgio da Silva Borges A Model of Strategy Games Based on the Paradigm of the Iterated Prisoner's Dilemma Employing Fuzzy Sets Esta tese foi julgada adequada para a obtenção título de “Doutor em Engenharia", especialidade Engenharia de Produção e aprovada em sua forma fin pelo Programa de Pós-graduação. Ricardo Ricardo M. B PhD - Coordenador do Programa de Pós-Graduação em Engenharia ~ Produção Banca Examinadora: Ricardo M. Bírci Ph.D, advisor , Qwag Jó, ólwz-W Suresh K. Khator, Ph.D /A¿›ÀÍraha4í~ cz<fv@¬ E ar Lanze Ph.D. , A^ÀYõ-À‹> ~ ~ 0-\J\¢>‹_§Ú ” A io _ Novaes, Ph.D. / °`“7I Luis Fernando Jacinto Maia, Dr. Eng. 11 Universidade Federal de Santa Catarina Programa de Pós-Graduação em Engenharia de Produção A model 'of strategy games based on the paradigm of the Iterated Prisoner's Dilemma employing Fuzzy Sets Doctoral Dissertation submitted as a partia! fulii ih nent o f the conditions required for the obtainance of the degree of Doctor in Production and Systems Engineering Paulo Sergio da Silva Borges ii._-_i *2 1 a IIIHI I 247.395 IIàs .Zias-*_ O March 1996 UFSC-BU I I P Universidade Federal de Santa Catarina Programa de Pós Graduaçao em Engenharia de Produçao Um modelo de jogos de estratégia com base no paradigma do Dilema do Prisioneiro Iterado, com emprego de Fuzzy Sets Tese submetida como requisito parcial para obtenção do título de Doutor em Engenharia de Produçao Orientador Prof. Titular Ricardo Miranda Barcia, Ph.D. Coordenador do Programa I Prof. Titular Ricardo Miranda Barcia, Ph. D. Florianópolis, Março de 1996 iv Agradecimentos e dedicatória Agradeço ao Professor Abraham Kandel, da University of South Florida, que de forma sempre gentil incentivou a elaboração da presente Tese, provendo valiosos comentados e sugestões para melhorá-la, e dispondo-se a participar' de minha banca de examinadores. Agradeço também a Joseph Benios, estudante graduado da University of South Florida, que incansavelmente auxiliou-me na preparação do programa computacional para as simulações do modelo aplicativo usado nesta Tese. Dedico este trabalho a meus pais, a minha esposa Hélia e aos meus filhos Márcia Beatrice e Paulo Vinicius. \Í Resumo A Teoria de Jogos pode ser definida como a análise matemática formal de situações onde se observa algum tipo de conflito de interesses. Os jogadores são assumidos, pela teoria clássica, como racionais, o que significa que buscam sempre no jogo a maximização de suas utilidades esperadas. O jogo de 2 participantes conhecido por "Dilema do Prisioneiro" (PD) é considerado como o que melhor representa a contradição entre o emprego da racionalidade individual e da coletiva. Na versão tradicional do PD, as estrategias são sempre compostas pelas ações elementares “Cooperate" (C) ou “Defect” (D). O jogo iterado (IPD) constitui-seem um modelo que traduz com bastante fidelidade, e de forma simples, inúmeras questões freqüentemente encontradas na realidade, quando pessoas, grupos ou empresas disputam parcelas de recursos escassos ou limitados. A literatura disponivel sobre o tema e muito vasta, e muitas analises do lPD, incluindo simulações atraves de torneios computacionais e uso de tecnicas de inteligência artificial já foram efetuadas. Contudo, as estrategias consideradas no jogo estão restritas, na grande maioria dos estudos, a combinações deterministicas, condicionais ou probabilislicas das ações pontuais citadas, ou seja C e D. Nesta tese, uma nova versão do IPD e desenvolvida, chamada de Fuzzy Iterated Pn`soner”s Dílemma -- FIPD. Este metodo, ao permitir que os jogadores possam implementar ações graduais, objetiva uma modelagem mais realista do conflito de interesses representado pelo jogo, Adicionalmente, no FlPD os agentes decidem com base em um raciocinio qualitativo, traduzido neste trabalho pelo uso de sistemas especialistas difusos. A tese inclui dois enfoques do FIPD. O primeiro, de carater exploratório, consiste em um torneio computacional confrontando os estrategistas assistidos pelo sistema de decisão difuso entre si e com outros tipos de jogadores tradicionalmente bem sucedidos em trabalhos anteriores (TFT e Pavlov). O segundo enfoque, bem mais elaborado e detalhado, trata de uma aplicação pratica a um problema de divisão de mercado, onde varias firmas atuam vendendo produtos ou serviços a uma população diversificada de compradores, que possuem diferentes poder de compra e preferências relativas à qualidade dos itens. Esta abordagem imprimiu uma caracteristica altamente dinâmica às interações, simuladas com auxilio de um programa computacional orientado a objetos especialmente desenvolvido. As estrategias empregadas pelos participantes e seus desempenhos são analisados e discutidos, especialmente na segunda abordagem do FIPD, obtendo-se importantes conclusões a respeito do problema. Vl Abstract Game Theory can be defined as the formal mathematical analysis of situations where some kind of conflict ofinterest exists. The classical theory assumes that the players are rafional, in the sense that they always seek the maximization ot their expected utilities. The 2-person game known by the name of “Prisoner's Dilemma” (PD) is considered as the situation which best depicts the contradiction between the individual and the collective rationality. ln its traditional version, the strategies in the PD are always composed by the elementary and dichotomic actions "Cooperate“ (C) ou “Defect" (D). The iterated game (IPD) constitutes a model that mirrors, in a simple and direct manner, several problems often found in the real world, when people, groups or companies are involved in a dispute over some limited or scarce resource. The available literature regarding that theme is vast, and many comprehensive analysis of the IPD have been already performed, including computational tournaments and the employment of artificial intelligence techniques. However, the strategies considered in the game are usually confined to deterministic or probabilistic combinations of the mentioned punctual actions C and D. ln this dissertation, an innovative- version of the IPD is presented and developed, called Fuzzy lterated Pn`soner's Dilemma FlPD. That method, by allowing the players to implement gradual actions, aims al attaining a more realistic model of the conflict of interested represented by the game. Additionally, in the FlPD the agents decide based on a qualitative system of reasoning, which in this work regards the utilization of mzzy expert systems (FES). The dissertation includes two approaches of the FlPD. The first has an exploratory character, and consists of a computational contest where the FES-assisted strategists confront each other and also other well known successful rules (TFT and Pavlov). The second approach, much more elaborate and perfected, is directed at a practical application concerning a market share problem. There, several firms are present offering their products or services to a diversified population of buyers, the Consumers, which on their turn possess different buying powers and preferences relative to the quality of itens desired. That method has the advantage of imprinting a highly dynamic feature to the environment, which has been simulated by means of an C++ object-oriented program especially written. The strategies employed by the participants and their respective performances are extensively analyzed and discussed, mainly in the second approach, from which many important conclusions have been drawn. vii Contents Page Resumo v Abstract v1 Chapter 1 Introduction 1.1 Preambie ........................................................................... ._ 1.1 1.2 The Prisoner's Dilemma ..................................................... .. 1.2 1.3 Description of the Problem and Objective of the Dissertation 1.3 1.4 Methodoiogy ...................................................................... .. 1.6 1.5 Structure and Outiine of the Dissertation .......................... .. 1.8 1.6 Main Contributions of the Dissertation ................................ .. 1.10 Chapter 2 A Review of Fundamental Topics of Game Theory 2.1 introduction ....................................................................... ._ 21 Representation of 2.2 Strategic Games ................................... .. 24 2.2.1 Decision Trees ...................................................................... .. 24 2.2.2 Rule Sets .............................................................................. .. 25 2.3 Usual Methods for the Evaluation of the Sub-optima! Function ............................................................................ .. 25 The Minimax .. 2.3.1 Method ............................................................ 26 2.3.2 Alpha-Beta Cuts ................................................................... .. 27 2.3.3 Other Methods ...................................................................... .. 2.8 V111 2.4 information Sets ................................................................ .. 28 2.5 Strategies .......................................................................... .. 29 2.5.1 Saddle Points ....................................................................... ._ 2.10 25.2 Pure and Mixed Strategies ................................................... _. 211 2.5.3 The Minimax Theorem .......................................................... _. 2.14 2.6 Preferences, Utility and Rationality ..................................... .. 216 2.6.1 Fundamental Axioms of Utility Theory .................................. _. 2_17 2.6.2 Prospect Theory ..................................................................
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