Game theory Chapter 28 and 29 Niklas Jakobsson Click to add notes

Introduction Game theory is the study of how rational agents behave in strategic situations where each agent must know the decisions of the other agents before knowing which decision is best for herself. Nonstrategic decisions can be made without taking the decisions that others make into account, as opposed to strategic decisions. Game theory is used to study e.g. economic behavior, warfare, political negotiation, and the markets for dating and marriage. 2009

A game is described by The number of players Their strategies and their turn Their payoffs (profits, utilities etc) at the outcomes of the game 2009

Payoff matrix Normal- or strategic form Player B Left Right Top 3, 0 0, -4 Bottom 2, 4 -1, 3 Player A 2009

How to solve a situation like this? The most simple case is where there is a optimal choice of strategy no matter what the other players do; dominant strategies. Explanation: For Player A it is always better to choose Top, for Player B it is always better to choose left. A dominant strategy is a strategy that is best no matter what the other player does. 2009

Nash equilibrium If Player A’s choice is optimal given Player B’s choice, and B’s choice is optimal given A’s choice, a pair of strategies is a Nash equilibrium. When the other players’ choice is revealed neither player like to change her behavior. If a set of strategies are best responses to each other, the strategy set is a Nash equilibrium. 2009

Payoff matrix Normal- or strategic form Player B Left Right Top 1, 1 2, 3 Bottom 1, 2 Player A 2009

Solution Here you can find a Nash equilibrium; Top is the best response to Right and Right is the best response to Top. Hence, (Top, Right) is a Nash equilibrium. But there are two problems with this solution concept. 2009

Problems A game can have several Nash equilibriums. In this case also (Bottom, Left). There may not be a Nash equilibrium (in pure strategies). 2009

Payoff matrix Normal- or strategic form Player B Left Right Top 1, -1 -1, 1 Bottom Player A 2009

Nash equilibrium in mixed strategies Here it is not possible to find strategies that are best responses to each other. If players are allowed to randomize their strategies we can find s solution; a Nash equilibrium in mixed strategies. An equilibrium in which each player chooses the optimal frequency with which to play her strategies given the frequency choices of the other agents. 2009

Examples I shall now consider the famous prisoner’s dilemma, repeated games, and sequential games. The Kyoto protocol will then be discussed in light of the prisoners dilemma in a one shot setting, as well as in a repeated setting. This will shed light on the fact that conclusions may differ a lot when time or dynamics are considered. 2009

The prisoner’s dilemma Two persons have committed a crime, they are held in separate rooms. If they both confess they will serve two years in jail. If only one confess she will be free and the other will get the double time in jail. If both deny they will be hold for one year. 2009

Prisoner’s dilemma Normal- or strategic form Prisoner B Confess Deny -2, -2 0, -4 -4, 0 -1, -1 Prisoner A 2009

Solution Confess is a dominant strategy for both. If both Deny they would be better off. This is the dilemma. This can describe a lot of situations. E.g. cartels, arms control, competition between municipalities (Child care fees and fiscal competition). 2009

Repeated games The game can be played repeatedly by the same players; repeated games. Player’s has the opportunity to establish a reputation for cooperation and thereby encourage other players to do the same. Thus, the prisoner’s dilemma may disappear. For egoists to play cooperatively the game has to be played an infinite number of times; there may not be a known last round of the game. 2009

Repeated games If there is a last round in the game, Confess is a dominant strategy in that round. This will also be true in the next to the last round, and so on. Players cooperate because they hope that cooperation will induce further cooperation in the future, but this requires a possibility of future play. 2009

Example Climate change and the Kyoto protocol in light of the prisoner’s dilemma. All countries will enjoy the benefits of a stable climate whether they have helped bring it about or not USA and Australia chose not to put a limit on their emissions, also many poor countries Everyone is counting on others to act; so no one will This is a prisoner’s dilemma! Rational leaders will neglect the problem and global warming can not be stopped Is this true? 2009

Example The implications are very different if we see it from the one shot game perspective or the repeated game perspective If the game is repeated the players have an incentive to cooperate so to not get punished by the opponents in the following rounds Rational countries will not be deterred by free riders but continue to curb emissions and devise sanctions for those who do not Needed is more sanctions and negotiations more frequently 2009

Example But are the dynamics this easy? Governments can negotiate Are governments rational? Power relations? 2009

Sequential games In sequential games all players do not move simultaneously. These games are best illustrated in extensive form (as opposed to normal- or strategic form) so that the time pattern become evident. Player A starts the game and can choose Top or Bottom. Player B moves second and can choose Left or Right. What is the solution of the game? (Top, Left) 2009

Summary We have had a brief look into the realm of game theory. We have used games to assess how rational agents behave in strategic situations. We have covered chapter 28. You should also read Assurance games, Chicken, How to coordinate, The kindly kidnapper, When strength is a weakness, and Savings and social security in chapter 29. These are examples of what you have seen today. Homework. If there is time, go through some problems. 2009

One minute paper What is the most important thing you learned today? What is the muddiest point still remaining at the conclusion of today’s class? 2009