Lecture Notes: GAME THEORY School of Economics, Shandong University Contents 1 INTRODUCTION 1 1.1 What is Game Theory? . 1 1.2 Elements of a `Game' . 1 1.3 Types of Games: The Time Aspect . 3 1.4 Cooperative and Non-Cooperative Games . 3 1.5 Reading . 4 2 STATIC GAMES OF COMPLETE INFORMATION 5 2.1 THE NORMAL FORM AND TWO EQUILIBRIUM CONCEPTS . 5 2.1.1 Example: The Prisoners' Dilemma . 5 2.1.2 Equilibrium Concept 1: Iterated Elimination of Dominated Strategies . 6 2.1.3 Equilibrium Concept 2: The Nash Equilibrium . 7 2.1.4 Mixed Strategies . 9 2.1.5 Examples of Prisoners' Dilemmas in Economics . 10 2.2 APPLICATIONS TO OLIGOPOLY . 11 2.2.1 The Cournot Duopoly Game . 11 2.2.2 The Cournot-Nash Equilibrium to the Duopoly Game . 12 2.2.3 Reaction Functions . 13 2.2.4 The Bertrand Model of Duopoly . 14 2.2.5 Collusion . 15 2.3 MACROECONOMIC POLICY GAMES . 17 2.3.1 The Expectations Augmented Phillips Curve (EAPC) . 17 2.3.2 The Barro-Gordon (1983) Monetary Policy Game . 18 2.3.3 The Macroeconomic Policy Coordination Game . 23 3 DYNAMIC GAMES OF COMPLETE INFORMATION 27 3.1 Introduction . 27 3.2 A General Two-Stage Game with Observed Actions . 27 3.2.1 Solution by Backwards Induction . 28 3.2.2 Subgames and Credibility . 29 i 3.3 The Stackelberg Duopoly Model . 30 3.4 The Monopoly Union Model . 32 3.5 Bargaining Theory: Rubinstein's Sequential Bargaining Model. 33 3.6 Repeated Games. 35 3.6.1 The General Idea . 35 3.6.2 The Infinitely repeated Prisoners' Dilemma . 35 3.6.3 Collusion Between Two Duopolists . 38 3.6.4 The Barro-Gordon (1983) Monetary Policy Game . 39 4 GAMES OF INCOMPLETE INFORMATION 43 4.1 Static Games of Incomplete Information . 43 4.2 The Principal-Agent Problem and the Market for Lemons . 45 4.2.1 Adverse Selection and the Market for `Lemons' . 45 4.3 Baysian Learning using Bayes' Rule . 46 4.4 Dynamic Games of Incomplete Information with Bayesian Learning . 48 4.4.1 Introduction to Perfect Bayesian Equilibria . 49 4.4.2 Spence's Model of Job-Market Signalling . 51 4.4.3 A Reputational Model of Monetary Policy . 55 5 BOUNDED RATIONALITY: THEORY AND EXPERIMENTS 59 5.1 Introduction . 59 5.2 Less than Rational Adjustment Processes . 59 5.2.1 The Cournot Adjustment Process . 59 5.2.2 A More Rational Adjustment Process . 62 5.3 The Real World: Experimental Game Theory . 66 ii Chapter 1 INTRODUCTION 1.1 What is Game Theory? Traditional Game Theory is concerned with the strategic interaction of rational individuals or economic agents such as firms, governments, central banks and trade unions. Chapters 1 to 3 of the course will be traditional in this sense and will emphasize applications taken from both microeconomics and macroeconomics. It will be largely structured around the four main chapters of Gibbons. Rasmusen is an important additional text for this part of the course. More recently Game Theory has taken a more critical look at the treatment of rationality assumption. Casual observation would suggest that people are not rational in the sense used in game theory and this is confirmed by experimentation. Chapter 4 looks at recent developments in Game Theory which assumes people learn by trial and error and players are boundedly rational. The books particularly useful for this part of the course (but also relevant for the first four topics) are: Rasmusen, Ch 4; Binmore, and Hargreaves Heap and Varoufakis. In addition to these books you will be referred to important papers which apply game theory to different areas of economics. If as a result of this course (or in spite of it) you become `hooked' on Game Theory the book that you need to turn to, sooner or later, is Game Theory by Fudenberg and Tirole, MIT Press. 1.2 Elements of a `Game' The essential elements of a game are: rationality, players, actions (or moves), strategies, information, payoffs, equilibria and outcomes. Consider these in turn: • Rational agents do the best they can given constraints and available infor- mation. In economic theory this is formalised as utility maximisation. Un- 1 der uncertainty this becomes expected utility maximisation. In addition the 'rationality' assumption includes the assumption of common knowledge that players are rational ie, all players know that all players are rational, and that all players know that all players know that all players are rational, etc. Experimental game theory provides only limited support for the assumption that people are generally 'rational' in the economists' sense. A 'minimalist defence' of the assumption would go as follows: studying rational behaviour is a useful 'thought experiment' even if we believe that people are not ra- tional. As Myerson (1990) argues, the goal of social science is not just to predict human behavour, but to analyze social institutions (the market econ- omy, private firms, government, the world trading system, central banks, etc) and evaluate proposals for their reform. It is useful to analyze such institu- tions under the assumptions that agents are rational so as to identify the flaws in their design rather than the people within them. If institutions perform badly when agents are rational then this is a argument for better design. If well-designed institutions still perform badly because of irrational people, then this is an argument for informing and educating people better. For example, general equilibrium theory tells us that criticisms of the market economy as an institution should concentrate on its distributional consequences, market failure arising from small numbers, externalities, the existence of public goods and departures from complete information. Markets are not fundamentally anarchistic or inefficient. • The players are individuals or other economic agents who make decisions. They choose actions (or moves) to maximise their utility. • Information available to each player is central to game theory. The infor- mation set is the knowledge available to each player at each point in time. This may include observations of key variables affecting their utility such as price, the history of moves of other players up to that point and the nature of other players' utilities. Then: • A player's strategy is a rule that tells her which move to choose at each instant of the game, given her information set. Let Si be the set of strategies available to player i (or i's strategy space) and let siϵSi denote a member of this set. Strategies can be mixed strategies where each player randomizes over pure strategies si 2 Si; for example if there are 2 elements in Si, si1; si2, choosing si1 with probability p and si2 with probability 1 − p. Consider an n-player game. Let (s1; : : : ; sn) be a vector denoting an arbitrary combination of strategies on for each player. In general each player's utility will dependent on what all players are doing. Denote player i's utility by ui. The corresponding to each combination of strategies by all players is a payoff function ui(s1; : : : ; sn). 2 • The normal form of a game is defined as G = fS1;S2; ···;Sn; u1; u2; ···; ung • ∗ ∗ ∗ An equilibrium s = (s1; : : : ; sn) is a strategy combination consisting of the 'best' strategy for all players. Different types of games have their own equilibrium concept. • Corresponding to each equilibrium is an equilibrium outcome for the de- scribing all the variables of the model. 1.3 Types of Games: The Time Aspect First there is a time aspect of games. Games can be played once all moves being made simultaneously (`one-shot' or static games) or played over time moves being played sequentially and possibly many times (dynamic games). Dynamic games which continue for ever are infinite games. The Information Aspect Second, games are defined according to the information structure assumed. In the course we distinguish games of complete or incomplete information, and games of perfect or imperfect information. In games of complete information each player's payoff function is common knowledge among all the players. In games of perfect information the player with the move knows the full history of the game up to that point. Using these definitions and concepts the structure of the `traditional' part of the course (and the Gibbon's book) can be summarised as follows: Information Structure Time Aspect Equilibrium Concept Complete Static Nash Complete Dynamic Subgame-Perfect Incomplete Static Bayesian Incomplete Dynamic Perfect Bayesian 1.4 Cooperative and Non-Cooperative Games We can distinguish between cooperative and non-cooperative games. Cooper- ative games require some `precommitment mechanism' that forces players to pre- commit to a particular course of action over time. Then the issue is how the gains from cooperation can be split among the players. Non-cooperative games lack such a mechanism and the focus of the course is on this type of game. Note that coopera- tion is still studied but the emphasis is on how cooperation can be enforced without 3 an external precommitment mechanism. In effect cooperation is then a possible non-cooperative equilibrium. Zero-Sum Games Our final category is the idea of a zero-sum game. For such a game the sum of the payoffs is zero whatever the strategy combination of the players. Then there are clearly no gains from cooperation. 1.5 Reading Gibbons (G) Preface Hargreaves Heap and Varoufakis (HHV) Chs 1,2. Myerson, R. B. (1999), 'Nash Equilibrium and the History of Economic Theory', Journal of Economic Literature, 37, no. 3, 1067-1082. Nasar, S. (2001), `The Life of Mathematical Genius and Nobel Laureate John Nash', Simon & Schuster Rasmusen (R) Preface, Introduction, Chs 1 part 1.1.