A Short Introduction to Game Theory

A Short Introduction to Game Theory Heiko Hotz CONTENTS 1 Contents 1 Introduction 2 1.1 Game Theory – What is it? . 2 1.2 Game Theory – Where is it applied? . 2 2 Definitions 3 2.1 Normal Form Games . 3 2.2 Extensive Form Games . 3 2.3 Nash Equilibrium . 4 2.3.1 Best Response . 4 2.3.2 Localizing a Nash Equilibrium in a Payoff-matrix . 4 2.4 Mixed Strategies . 5 3 Games 5 3.1 Prisoner’s Dilemma (PD) . 5 3.1.1 Other Interesting Two-person Games . 6 3.2 The Ultimatum Game . 7 3.3 Public Good Game . 7 3.4 Rock, Paper, Scissors . 8 4 Evolutionary Game Theory 8 4.1 Why EGT? . 8 4.2 Evolutionary Stable Strategies . 9 4.2.1 ESS and Nash Equilibrium . 9 4.2.2 The Hawk-Dove Game . 9 4.2.3 ESS of the Hawk-Dove Game . 10 4.3 The Replicator Dynamics . 11 4.4 ESS and Replicator Dynamics . 11 5 Applications 12 5.1 Evolution of cooperation . 12 5.2 Biodiversity . 12 A Mathematical Derivation 14 A.1 Normal form games . 14 A.2 Nash equilibrium and best answer . 14 A.3 Evolutionary stable strategies (ESS) . 14 A.4 Replicator equation . 15 A.5 Evolutionary stable state of the Hawk-Dove game . 15 B Pogram Code 17 B.1 Spatial PD . 17 B.2 Hawk-Dove game . 19 1 INTRODUCTION 2 1 Introduction This description leads to a widespred definition of game theory: This paper gives a brief overview of game theory. Therefore in the first section I The subject of game theory want to outline what game theory gener- are situations, where the really is and where it is applied. sult for a player does not only In the next section, I introduce some depend on his own decisions, of the most important terms used in game but also on the behaviour of theory, such as normal form games and the other players. Nash equilibrium as well as some of the Game theory has its historical origin most popular games, e.g. the Prisoner’s in 1928. By analysing parlour games, John Dilemma or the Ultimatum Game. von Neumann realised very quickly the I then present the basic concepts of practicability of his approaches for the evolutionary game theory (EGT), a more analysis of economic problems. specialized branch of game theory. We In his book Theory of Games and Eco- will see how EGT uses new concepts such nomic Behavior, which he wrote together as evolutionary stable strategies (ESS) and with Oskar Morgenstern in 1944, he al- replicator dynamics, and its importance ready applied his mathematical theory to to sciences like biology and physics. economic applications. At last I present two examples, pro- The publication of this book is gen- grammed in the language NetLogo, which erally seen as the initial point of modern will demonstrate the applications of EGT game theory. and the similarities to condensed matter physics. 1.2 Game Theory – Where is it applied? 1.1 Game Theory – What is it? As we have seen in the previous section, The concepts of game theory provide a game theory is a branch of mathemat- common language to formulate, structure, ics. Mathematics provide a common lan- analyse and eventually understand differ- guage to describe these games. We have ent strategical scenarios. Generally, game also seen that game theory was already theory investigates conflict situations, the applied to economics by von Neumann. interaction between the agents and their When there is competition for a resource decisions. to be analysed, game theory can be used A game in the sense of game theory either to explain existing behaviour or to is given by a (mostly finite) number of improve strategies. players, who interact according to given The first is especially applied by sci- rules. Those players might be individu- ences which analyse long-term situations, als, groups, companies, associations and like biology or sociology. In animality, for so on. Their interactions will have an im- example, one can find situations, where pact on each of the players and on the cooperation has developed for the sake of whole group of players, i.e. they are in- mutual benefits. terdependent. The latter is a crucial tool in sciences To be more precise: A game is de- like economics. Companies use game the- scribed by a set of players and their possi- ory to improve their strategical situation bilities to play the game according to the in the market. rules, i.e. their set of strategies. 2 DEFINITIONS 3 Despite the deep insights he gained from 3. A payoff function, which assigns a game theory’s applications to economics, certain payoff to each player depend- von Neumann was mostly interested in ing on his strategy and the strat- applying his methods to politics and war- egy of the other players (e.g. in the fare, perhaps descending from his favorite Prisoner’s Dilemma the time each of childhood game, Kriegspiel, a chess-like the players has to spend in prison). military simulation. He used his methods to model the Cold War interaction be- The payoff function assigns each player tween the U.S. and the USSR, picturing a certain payoff depending on his strat- them as two players in a zero-sum game. egy and the strategy of the other play- He sketched out a mathematical model ers. If the number of players is limited of the conflict from which he deduced that to two and if their sets of strategies con- the Allies would win, applying some of sist of only a few elements, the outcome the methods of game theory to his pre- of the payoff function can be represented dictions. in a matrix, the so-called payoff matrix, There are many more applications in which shows the two players, their strate- the sciences, which have already been men- gies and their payoffs. tioned, and in many more sciences like sociology, philosophy, psychology and cul- Example: tural anthropology. It is not possible to list them all in this paper, more informa- Player1\Player2 L R tion can be obtained in the references at U 1, 3 2, 4 the end of this paper. D 1, 0 3, 3 2 Definitions In this example, player 1 (vertical) has two different strategies: Up (U) and Down I now introduce some of the basic defini- (D). Player 2 (horizontal) also has two tions of game theory. I use a non-mathematical different strategies, namely Left (L) and description as far as possible, since math- Right (R). ematics is not really required to under- The elements of the matrix are the stand the basic concepts of game theory. outcomes for the two players for playing However, a mathematical derivation is certain strategies, i.e. supposing, player 1 given in appendix A.1 and A.2. chooses strategy U and player 2 chooses strategy R, the outcome is (2, 4), i.e. the 2.1 Normal Form Games payoff for player 1 is 2 and for player 2 is 4. A game in normal form consists of: 2.2 Extensive Form Games 1. A finite number of players. Contrary to the normal form game, the 2. A strategy set assigned to each player. rules of an extensive form game are de- (e.g. in the Prisoner’s Dilemma each scribed such that the agents of the game player has the possibility to cooper- execute their moves consecutively. ate (C) and to defect (D). Thus his This game is represented by a game strategy set consists of the elements tree, where each node represents every C and D.) possible stage of the game as it is played. There is a unique node called the initial 2 DEFINITIONS 4 node that represents the start of the game. strategy choices and the corresponding pay- Any node that has only one edge con- offs constitute a Nash equilibrium. nected to it is a terminal node and rep- John Nash showed in 1950, that every resents the end of the game (and also a game with a finite number of players and strategy profile). Every non-terminal node finite number of strategies has at least one belongs to a player in the sense that it mixed strategy Nash equilibrium. represents a stage in the game in which it is that player’s move. Every edge repre- 2.3.1 Best Response sents a possible action that can be taken by a player. Every terminal node has The best response is the strategy (or strate- a payoff for every player associated with gies) which produces the most favorable it. These are the payoffs for every player immediate outcome for the current player, if the combination of actions required to taking other players’ strategies as given. reach that terminal node are actually played. With this definition , we can now determine the Nash equilibrium in a normal Example: form game very easily by using the payoff matrix. The formal proof that this procedure leads to the desired result is given in appendix A.2. 2.3.2 Localizing a Nash Equilibrium in a Payoff-matrix Let us use the payoff matrix of the Pris- oner’s Dilemma, which will be introduced in 3.1, to determine the Nash equilibrium: Figure 1: A game in extensive form Player1\Player2 C D C 3, 3 0, 5 In figure 1 the payoff for player 1 will be D 5, 0 1, 1 2 and for player 2 will be 1, provided that player 1 plays strategy U and player 2 The procedure is the following: First we plays strategy D’.

Load more