摘 要

在我們日常生活中所遭遇到的實際對局問題通常是複雜的，因此現有處理對局的方法不足以被用來解決它們。這促使我們在本論文中研究幾個還未被考慮過或尚未被完全解決的微分對局問題，包括n個追逐者與一個逃逸者的追逐逃逸對局、二個防衛者與二個入侵者的城堡攻防對局及指標函數可切換微分對局問題。

在本論文中，我們首先利用幾何的方法來考慮n個追逐者與一個逃逸者的追逐逃逸對局。兩個可以用來決定在某些情況下此對局解值的法則將被提出，在此我們可以發現一個追逐者與一個逃逸者的追逐逃逸對局是此問題的一個特例。

接著我們將對二個防衛者與二個入侵者的城堡攻防對局作定性與定量的分析。透過本問題之研究，一些從未在一個防衛者與一個入侵者的城堡攻防對局中出現的情況將被顯露出來。一個在本研究中有趣的發現是，在某些情況下為了得到一個較佳的結果，入侵者或許需要扮演追逐者的角色，這將使得本問題更複雜也更難解決。

最後我們首度提出一個具有不完全對局資訊的指標函數可切換微分對局問題。此問題與傳統微分對局的最主要差異在於參與者在對局進行當中可以任意切換指標函數，這使得此類問題的最佳性無法得到保證。一些根據不同方法的推理機制將被提出，用來決定某一個參與者的推理策略。我們將以一個實際的三城堡攻防對局問題來說明指標函數可切換微分對局的情況。多個電腦模擬將被用來比較不同推理策略的性能。指標函數可切換微分對局理論的提出對於處理具有不完全對局資訊的微分對局問題是一個重大的突破。

ABSTRACT

Usually, real game problems encountered in our daily lives are so complicated that the existing methods are no longer sufficient to deal with them. This motivates us to investigate several kinds of differential game problems, which have not been considered or solved yet, including a pursuit-evasion game with n pursuers and one evader, a problem of guarding a territory with two guarders and two invaders, and a payoff-switching differential game.

In this thesis, firstly the geometric method is used to consider the pursuit-evasion game with n pursuers and one evader. Two criteria used to find the solutions of the game in some cases are given. It will be shown that the one-on-one pursuit-evasion game is a special case of this game.

Secondly, the problem of guarding a territory with two guarders and two invaders is considered both qualitatively and quantitatively. The investigation of this problem reveals a variety of situations never occurring in the case with one guarder and one invader. An interesting thing found in this investigation is that some invader may play the role as a pursuer for achieving a more favorable payoff in some cases. This will make the problem more complicated and more difficult to be solved.

The payoff-switching differential game, first proposed by us, is a kind of differential game with incomplete information. The main difference between this problem and traditional differential games is that in a payoff-switching differential game, any one player at any time may have several choices of payoffs for the future. The optimality in such a problem becomes questionable. Some reasoning mechanisms based on different methods will be provided to determine a reasoning strategy for some player in a payoff-switching differential game. A practical payoff-switching differential game problem, i.e., the guarding three territories with one guarder against one invader, is presented to illustrate the situations of such a game problem. Many computer simulations of this example are given to show the performances of different reasoning strategies. The proposition of the payoff-switching differential game is an important breakthrough in dealing with some kinds of differential games with incomplete information.

CONTENTS
誌謝 …………………………………………………………………………… iii
摘要 …………………………………………………………………………… iv
ABSTRACT ………………………………………………………………………………… v
GLOSSARY OF SYMBOLS ………………………………………………………… vii
CHAPTER I INTRODUCTION ………………………………………………………… 1
Section 1.1 Motivation …………………………………………… 1
Section 1.2 Brief Sketch of the Contents…………………… 4
CHAPTER II DIFFERENTIAL GAMES ………………………………… 5
Section 2.1 General Concept ………………………………… 5
Section 2.2 Two-person Zero-sum Differential Games ……… 8
Section 2.2.1 Pursuit-evasion Game with One Pursuer and One
Evader ………………………………………………10
Section 2.2.2 Guarding a Territory with One Guarder and One
Invader ……………………………………………… 16
CHAPTER III MULTI-PERSON DIFFERENTIAL GAMES …………… 22
Section 3.1 Pursuit-evasion Game with n Pursuers and One
Evader ……………………………………………… 23
Section 3.2 Guarding a Territory with Two Guarders and
Two Invaders………………………………………… 42
Section 3.2.1 Preliminaries ………………………………… 42
Section 3.2.2 Problem Formulation and Definitions ………… 44
Section 3.2.3 Qualitative Analysis ……………………………… 48
Section 3.2.4 Quantitative Analysis ……………………… 56
CHAPTER IV PAYOFF-SWITCHING DIFFERENTIAL GAMES …………………… 77
Section 4.1 Differential Games with Incomplete
Information ………………………………………… 77
Section 4.2 Payoff-switching Differential Games ………… 81
Section 4.3 Reasoning Mechanisms for a Payoff-switching
Differential Game ………………………………… 87
Section 4.3.1 A Traditional Reasoning Mechanism by Computing
Minimum Payoff………………………………………… 88
Section 4.3.2 A General Reasoning Algorithm for a Payoff-
switching Differential Game ……………………… 90
Section 4.3.3 A Strategy for a Payoff-Switching Differential
Game Based on Fuzzy Reasoning…………………… 97
Section 4.3.4 Using Similarity Measure to Estimate the
Player’s Behavior in a Payoff-switching
Differential Game…………………………………… 108
CHAPTER V CONCLUSIONS AND DISSCUSSIONS ………………… 116
REFERENCES ………………………………………………………… 119

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