Characterizing the Value and Effect of Perceptiveness in Various Game-Theoretic Settings

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Characterizing the Value and Effect of Perceptiveness in Various Game-Theoretic Settings, Is A Well-Researched Topic, It Is To Be Used As A Guide Or Framework For Your Research.

Abstract

This thesis investigates the value and e↵ect that perceptiveness has in three game-theoretic settings. I consider a player to be expert if they know the value of a particular payo↵-relevant parameter in the models I study. If the player does not know such value, I consider the player to
be inexpert. A player is perceptive if they know with certainty whether their opponent is expert. Otherwise, the player is imperceptive. The goal of this thesis is to provide insight regarding the potential value and e↵ect that perceptiveness has in the game-theoretic settings I study. The first model I consider emulates a two-player, one-round game of poker. The second model I investigate is a two-player market-entry game. The third model I study depicts a twoplayer market-entry game that is influenced by an information designer who aims to maximize producer surplus. In each model, I consider distinct information structures, which vary in terms of the players’ levels of expertise and perceptiveness. In the first two models, I solve for the Bayesian Nash equilibria of each game and compute each agent’s expected payo↵. Then, by comparing the equilibrium action and expected payo↵ of an agent when perceptive to that when
imperceptive, holding all else constant, I determine the agent’s value of perceptiveness and the e↵ect that perceptiveness has on the agent’s equilibrium strategy. In the third model, I solve for the information designer’s attainable decision rules, then determine which of the attainable decision rules maximizes producer surplus. Among other insights, I find that perceptiveness is generally valuable, whether that be from the perspective of a poker player, a player considering market entry, or an information designer in a market-entry game. Furthermore, under an equilibrium that treats the market-entry players as symmetrically as possible, the value of perceptiveness is positive when both players have a sufficiently high probability of being expert; whereas, the value of perceptiveness is zero when either player is inexpert with a sufficiently high probability. Also, perceptiveness is generally less beneficial to players considering market entry than it is to players playing poker.

Table of Contents

Abstract ii
Summary for Lay Audience iii
Acknowledgements iv
List of Figures x
List of Tables xii
List of Appendices xiv
1 Introduction 1
2 Perceptiveness in a Game-Theoretic Model of Poker 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Players, Actions, States . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Types, Information, & Beliefs . . . . . . . . . . . . . . . . . . . . . . 10
2.3.4 Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Strategies, Best Responses, & Equilibria . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Best Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Inexpert & Perceptive vs. Inexpert . . . . . . . . . . . . . . . . . . . . 15
Expert & Perceptive vs. Inexpert . . . . . . . . . . . . . . . . . . . . . 15
Inexpert & Perceptive vs. Expert . . . . . . . . . . . . . . . . . . . . . 17
Expert & Perceptive vs. Expert . . . . . . . . . . . . . . . . . . . . . . 17
Inexpert & Imperceptive . . . . . . . . . . . . . . . . . . . . . . . . . 18
Expert & Imperceptive . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.3 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Information Structure: (✏A, ✏B) = (0,0) . . . . . . . . . . . . . . . . . . 21
Information Structure: (✏A, ✏B) = (0,1) . . . . . . . . . . . . . . . . . . 21
Information Structure: (✏A, ✏B) = (1,1) . . . . . . . . . . . . . . . . . . 22
Information Structure: (✏A, ✏B) = (0,1/2) . . . . . . . . . . . . . . . . . . 22
Information Structure: (✏A, ✏B) = (1,1/2) . . . . . . . . . . . . . . . . . . 22
Information Structure: (✏A, ✏B) = (1/2,1/2) . . . . . . . . . . . . . . . . . 23
2.4.4 Ex-Ante Expected Payo↵s . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Value of Expertise & Perceptiveness . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Perceptiveness in a Market-Entry Game 34
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Players, Actions, States . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Microfoundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Strategies, Best Responses & Equilibria . . . . . . . . . . . . . . . . . . . . . 39
Player Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Best Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Equilibrium Refinement . . . . . . . . . . . . . . . . . . . . . . . . . 47
Ex-Ante Expected Payo↵s . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Value of Perceptiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.1 Application to Market-Entry Setting . . . . . . . . . . . . . . . . . . . 52
3.4.2 Comparing Results Between Chapters 2 & 3 . . . . . . . . . . . . . . . 56
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Perceptiveness in a Market-Entry, Information Design Setting 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 Players, Actions, States . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Equilibria & Obedience Constraints . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1 Obedience Constraints for Expert Agents . . . . . . . . . . . . . . . . 69
4.3.2 Obedience Constraints for Inexpert Agents . . . . . . . . . . . . . . . 71
4.3.3 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Maximizing Producer Surplus . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.1 Region 1: (“H,”L) 2 [1,1) ⇥ (−1,−1/9) . . . . . . . . . . . . . . . . . 80
4.4.2 Region 2: (“H,”L) 2 [14/9,1) ⇥ [−1/9, 4/9) . . . . . . . . . . . . . . . . . 81
4.4.3 Region 3: (“H,”L) 2 [1,14/9) ⇥ [−1/9, 4/9) & “H % 1 + 12
(1+✏)(49
−”L) . . . 82
4.4.4 Region 4: (“H,”L) 2 [1,14/9) ⇥ [−1/9, 4/9) & “H  1+12
(1+✏)(49
−”L) . . . . 82
Region 4.1: “H−”L % 10
9 & “H 2 [1+12
(1−✏)(49
−”L), 1+12
(1+✏)(49
−”L)) . 82
Region 4.2: “H−”L % 10
9 & “H 2 [1, 1+12
(1−✏)(49
−”L)) . . . . . . . . . . 83
Region 4.3: “H−”L  10
9 & “H 2 [ 2
(1+✏) (49
+✏)−”L, 1+12
(1+✏)(49
−”L)) . . . 84
Region 4.4: “H−”L  10
9 & “H 2 [1, 2
(1+✏) (49
+✏)−”L) . . . . . . . . . . . 85
4.5 The E↵ect of Perceptiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 Conclusion 90
A Appendices for Chapter 2 93
A.1 Payo↵ Grid: Inexpert Agent i Choosing All-In vs. Expert Agent j . . . . . . . 93
B Appendices for Chapter 3 94
B.1 Cournot Competition: Derivation of ⇡D = 4/9 . . . . . . . . . . . . . . . . . . . 94
B.2 Example: Oil Investment Decision Problem . . . . . . . . . . . . . . . . . . . 95
B.3 Proof of Corollary 3.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C Appendices for Chapter 4 100
C.1 Microfoundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Bibliography 101
Curriculum Vitae 109

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YourPastQuestions Brand

Additional information

Author

Terrence Adam Rooney

No of Chapters

4

No of Pages

123

Reference

YES

Format

PDF

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