在限制預算下以隨機策略最大化多局競賽之勝率

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研究生：	陳建新 Chen, Chien-Hsin
論文名稱：	在限制預算下以隨機策略最大化多局競賽之勝率 Playing the Budget-Constrained Multi-Battle Contest with Randomized Strategies for Maximizing Winning Probabilities
指導教授：	韓永楷 Hon, Wing-Kai 陳柏安 Chen, Po-An
口試委員:	李雨青 Lee, Yu-Ching 林莊傑 Lin, Chuang-Chieh
學位類別：	碩士 Master
系所名稱：	電機資訊學院 - 資訊工程學系 Computer Science
論文出版年：	2024
畢業學年度：	112
語文別：	英文
論文頁數：	51
中文關鍵詞：	多局競賽、限制預算、隨機策略
外文關鍵詞：	multi-battle contests, budget constraints, randomized strategies
相關次數：	點閱：59 下載：0
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前述文獻, (Cheng-Chen-Hon, 2018) 針對預算受限的多次競賽，基本上得出了保證先出價的一方能夠戰勝對手的最佳預算比例及相應的競標策略。人們可能會想知道，當一個玩家的初始預算不足時該怎麼辦。我們在本文中的模型和結果提供了一種更為普遍的方法來討論預算相對較少的玩家應該如何應對：我們的目標是在每個平局決策規則下，在假設對手是對抗性的情況下推導出最大化整個遊戲獲勝概率的混合策略。此外，我們將這種最大獲勝概率表示為額外資金~$k$ 的數量。

特別地，當應用幾種基本和更複雜的規則時，獲勝概率不會受到玩家角色的影響；只有使用的遊戲規則和較富有玩家的額外資金數量才重要。當額外資金的倒數是整數且應用了最複雜的規則時，獲勝概率可以通過遞歸公式來表示。

The previous result of Cheng, Chen, and Hon (2018) for budget-constrained multi-battle contests is basically the optimal budget ratio that guarantees the first-mover to win against an adversarial follower and the corresponding bidding strategy. One may wonder what a player should do when their initial budget is not high enough. Our model and results in this thesis provide a more general way to discuss what a player with only a relatively moderate amount of budget would do: our goal is to derive the leader's mixed strategy that maximizes the winning probability for the whole game, under each rule for tie-breaking, assuming the follower plays adversarially. Moreover, we express such a maximum winning probability in terms of the amount of extra money k.

In particular, the winning probabilities when several basic and more complicated rules are applied will not be affected by the role of players; only which game rule is used and the amount of extra money of the rich player matters. When the reciprocal of the extra amount of money is integral and the most complicated rule is applied, the winning probability can be characterized by a recursive formula.

Abstract (Chinese)                                     I
Preface (Chinese)                                      II
Abstract                                               III
Preface                                                IV
Contents                                               V
Introduction                                         1
1 An Example                                       3
2 Related Work                                     5
Preliminaries: Our Game, Problem, and Results        8
1 Results                                          10
2 Technical Lemmas and Terminology                 12
Basic Rules                                          15
1 Poor Player is the Leader                        15
2 Rich Player is the Leader                        19
3 Brief Summary                                    22
Type I Rules                                         24
1 Rule A5                                          24
2 Rule A6 and Rule A7                              25
Type II Rules                                        26
1 Rule A4                                          26
1.1 Proof for Case 0                           27
1.2 Proof for Remaining Cases: When the Poor Player is the
Leader                                     28
1.3 Proof for Remaining Cases: When the Rich Player is the
Leader                                     33
2 Rule A8                                          34
2.1 Proof: When the Poor Player is the Leader  35
2.2 Proof: When the Rich Player is the Leader  38
Fractional Number of Rounds 43
1 Rule A2                                          44
1.1 Proof for a Specific Strategy When the Rich Player is the
Leader                                           44
1.2 Proof for a Specific Strategy When the Poor Player is the
Leader                                           45
2 Rule A1                                          48
Conclusions and Future Work                          49
Bibliography                                           50

                                

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