Lecture Notes on Game Theory

Home , Cooperative bargaining, Revelation principle

Cesaltina Pires

Fall 2000 Contents

1 Introduction 4 1.1 What is Game Theory? ...... 4 1.2 Modelling Strategic Interaction ...... 5 1.2.1 Normal Form versus Extensive Form Representation ...... 5 1.2.2 Complete/Incomplete vs Perfect/Imperfect Information ...... 11

2 Static Games of Complete Information 13 2.1 Mixed Strategies ...... 13 2.2 Iterated Deletion of Strictly Dominated Strategies ...... 14 2.2.1 Dominated Strategies Considering only Pure Strategies ...... 14 2.2.2 Dominated Strategies Considering Mixed Strategies ...... 16 2.2.3 Problems with Iterated Deletion of Dominated Strategies ...... 18 2.3 Nash Equilibrium ...... 18 2.3.1 Pure Strategy Nash Equilibrium ...... 18 2.3.2 Mixed Strategy Nash Equilibrium ...... 21 2.3.3 Comments on Mixed Strategies ...... 23 2.3.4 Discussion on Nash Equilibria ...... 24 2.3.5 Existence Theorems ...... 26

3 Dynamic Games of Complete Information 31 3.1 Games of Perfect Information and Backward Induction ...... 31 3.1.1 Backward Induction ...... 31 3.1.2 Economic Applications ...... 34

1 3.1.3 Some Problems with Backward Induction ...... 37 3.2 Games of Imperfect Information and Subgame Perfection (Selten, 65) ...... 37 3.2.1 Subgame Perfect Nash Equilibrium ...... 37 3.2.2 General Procedure to …nd SPNE in Finite Games ...... 39 3.2.3 Economic Application - Tari¤s ...... 40 3.3 Formal De…nition of Strategy and SPNE ...... 41 3.3.1 Game Tree Representation, Strategy ...... 41 3.3.2 Game Tree Representation ...... 42 3.3.3 Strategies in Extensive Form Games ...... 43 3.3.4 Equilibria in Extensive Form Games ...... 46 3.4 Comments on Backward Induction and Subgame Perfection ...... 47 3.4.1 Backward Induction ...... 47 3.4.2 Subgame Perfection ...... 50 3.5 Multistage Games with Observed Actions ...... 51 3.5.1 Repeated Games ...... 52 3.5.2 In…nitely Repeated Games ...... 57 3.5.3 The Rubinstein-Stahl Bargaining Model ...... 64

4 Static Games of Incomplete Information 70 4.1 Bayesian Game and Bayesian Equilibrium ...... 72 4.1.1 Example - Duopoly with Incomplete Information ...... 73 4.1.2 Example - Public Good with Incomplete Information ...... 75 4.2 Puri…cation of Mixed Strategies ...... 76 4.3 Mechanism Design and Revelation Principle ...... 78

5 Dynamic Games of Incomplete Information 81 5.1 Signalling Games and PBE ...... 83 5.1.1 Example - The Beer-Quiche Game ...... 84 5.1.2 Example 2 - Two Period Reputation Game ...... 86 5.2 More on Perfect Bayesian Equilibrium ...... 90 5.3 Equilibrium Re…nements ...... 91 5.3.1 Conditionally Dominated Actions ...... 91

2 5.3.2 Equilibrium Domination ...... 93 5.3.3 Intuitive Criterion (Cho-Kreps) ...... 93

6 Cooperative Game Theory 97 6.1 Introduction ...... 97 6.1.1 Basic Issues in Cooperative Game Theory ...... 97 6.1.2 Games in Coalitional Form ...... 98 6.1.3 Games in Partition Form ...... 100 6.2 Core ...... 100 6.3 Shapley Value ...... 102 6.4 Cooperative Bargaining Theory ...... 105 6.4.1 The Edgeworth Trade Example ...... 105 6.4.2 Bargaining Problem ...... 107 6.4.3 Axiomatic Bargaining Approach ...... 108