The Economics of Food and Agricultural Markets - 2nd Edition Chapter 6 Game Theory

Chapter . Game Theory Game Theory Introduction Game theory was introduced in the previous chapter to better understand oligopoly . Recall the definition of game theory . Game Theory A framework to study strategic interactions between players , firms , or nations . Game theory is the study of strategic interactions between players . The key to understanding strategic decision making is to understand your opponent point of view , and to deduce his or her likely responses to your actions . A game is defined as Game A situation in which firms make strategic decisions that take into account each other actions and responses . A payoff is the outcome of a game that depends of the selected strategies of the players . Payoff The value associated with a possible outcome of a game . Strategy A rule or plan of action for playing a game . An optimal strategy is one that provides the best payoff for a player in a game . Optimal Strategy A strategy that maximizes a player expected payoff . Games are of two types cooperative and games . Cooperative Game A game in which participants can negotiate binding contracts that allow them to plan joint strategies . Game A game in which negotiation and enforcement of binding contracts are not possible . In games , individual players take actions , and the outcome of the game is described by the action taken by each player , along with the payoff that each player achieves . Cooperative games are different . The outcome of a cooperative game will be specified by which group of players become a cooperative group , and the joint action that the group takes . The groups of players are called , Examples of games include checkers , the prisoners dilemma , and most business situations where there is competition for a payoff . An example of a cooperative game is a joint venture of several companies who band together to form a group ( Chapter . Game Theory 175

The discussion of the prisoner dilemma led to one solution to games the equilibrium in dominant strategies . There are several different strategies and solutions for games , including ( Dominant strategy ( Nash equilibrium ( strategy ( safety first , or secure strategy ) Cooperative strategy ( collusion ) Equilibrium in Dominant Strategies The dominant strategy was introduced in the previous chapter . Dominant Strategy A strategy that results in the highest payoff to a player regardless of the opponents action . Equilibrium in Dominant Strategies An outcome of a game in which each firm is doing the best that it can regardless of what its competitor is doing Recall the prisoners dilemma from Chapter ' CONFESS NOT CON FESS CONFESS ( 15 ) NOT CONFESS ( Figure Prisoner Dilemma Prisoner Dilemma Dominant Strategy ( If CONF , A should CONF ( 15 ) If NOT , A should CONF ( A has the same strategy ( CONF ) no matter what does . 176 Andrew The Economics of Food and Agricultural Markets

( CONF , should CONF ( 15 ) NOT , should CONF ( has the same strategy ( CONF ) no matter what A does . Thus , the equilibrium in dominant strategies for this game is ( CONF , CONF ) Nash Equilibrium A second solution to games is a Nash Equilibrium . Nash Equilibrium A set of strategies in which each player has chosen its best strategy given the strategy of its rivals . To solve for a Nash Equilibrium ( Check each outcome of a game to see if any player wants to change strategies , given the strategy of its rival . a ) If no player wants to change , the outcome is a Nash Equilibrium . If one or more player wants to change , the outcome is not a Nash Equilibrium . A game may have zero , one , or more than one Nash Equilibria . The Prisoner Dilemma is shown in Figure . We will determine if this game has any Nash Equilibria . Prisoner Dilemma Nash Equilibrium ( Outcome ( CONF , CONF ) a ) Is CONF best for A given CONF ?

Yes . Is CONF best for given A CONF ?

Yes . CONF , CONF ) is a Nash Equilibrium . Outcome ( CONF , NOT ) a ) Is CONF best for A given NOT ?

Yes . Is NOT best for given A CONF ?

No . CONF , NOT ) is not a Nash Equilibrium . Outcome ( NOT , CONF ) Chapter . Game Theory ( 177 ( a ) Is NOT best for A given CONF ?

No . Is CONF best for given A NOT ?

Yes . NOT , CONF ) is not a Nash Equilibrium . Outcome ( NOT , NOT ) a ) Is NOT best for A given NOT ?

No . Is NOT best for given A NOT ?

No . NOT , NOT ) is not a Nash Equilibrium . Therefore , CONF , CONF ) is a Nash Equilibrium , and the only one Nash Equilibrium in the Prisoner Dilemma game . Note that in the Prisoner Dilemma game , the Equilibrium in Dominant Strategies is also a Nash Equilibrium . Advertising Game In this advertising game , two computer software firms ( and Apple ) decide whether to advertise or not . The outcomes depend on their own selected strategy and the strategy of the rival firm , as shown in Figure . ADVERTISE NOT ADVERTISE ADVERTISE ( 20 , 20 ) 10 , NOT ADVERTISE ( 10 ) 14 , 14 ) Figure 62 Advertising Two Software Firms . Outcomes in million . Advertising Dominant Strategy ( If APP AD , MIC should AD ( 20 ) If APP NOT , MIC should NOT ( 14 10 ) different strategies , so no dominant strategy for . 178 Andrew The Economics of Food and Agricultural Markets

( If MIC AD , APP should AD ( 20 ) If MIC NOT , APP should NOT ( 14 10 ) different strategies , so no dominant strategy for Apple . Thus , there are no dominant strategies , and no equilibrium in dominant strategies for this game . Advertising Nash Equilibria , Outcome ( AD , AD ) a ) Is AD best for MIC given APP AD ?

Yes . Is AD best for APP given MIC AD ?

Yes . AD ) is a Nash Equilibrium . Outcome ( AD , NOT ) a ) Is AD best for MIC given APP NOT ?

No . Is NOT best for APP given MIC AD ?

No . NOT ) is not a Nash Equilibrium . Outcome ( NOT , AD ) a ) Is NOT best for MIC given APP AD ?

No . Is AD best for APP given MIC NOT ?

No . NOT , AD ) is not a Nash Equilibrium . Outcome ( NOT , NOT ) a ) Is NOT best for MIC given APP NOT ?

Yes . Is NOT best for APP given MIC NOT ?

Yes . NOT , NOT ) is a Nash Equilibrium . There are two Nash Equilibria in the Advertising game ( AD , AD ) and ( NOT , NOT ) Therefore , in the Advertising game , there are two Nash Equilibria , and no Equilibrium in Dominant Strategies . It can be proven that in game theory , every Equilibrium in Dominant Strategies is a Nash Equilibrium . However , a Nash Equilibrium may or may not be an Equilibrium in Dominant Strategies . Chapter . Game Theory 179

Strategy ( Safety First Secure Strategy ) A strategy that allows players to avoid the largest losses is the Strategy . Strategy A strategy that maximizes the minimum payoff for one player . The , or safety first , strategy can be found by identifying the worst possible outcome for each strategy . Then , choose the strategy where the lowest payoff is the highest . Prisoner Dilemma Strategy ( Safety First ) We use Figure to find the Strategy for the Prisoner Dilemma . Player A ( a ) If CONF , worst payoff years . If NOT , worst payoff 15 years . A Strategy is CONF ( 15 ) Player ( a ) If CONF , worst payoff years . If NOT , worst payoff 15 years . Strategy is CONF ( 15 ) Therefore , the Equilibrium for the Prisoner Dilemma is ( CONF , CONF ) This outcome is also an Equilibrium in Dominant Strategies , and a Nash Equilibrium . Advertising Game Strategy ( Safety First ) a ) If AD , worst payoff 10 . If NOT , worst payoff . Strategy is AD ( 10 ) APPLE ( a ) If AD , worst payoff 10 . If NOT , worst payoff . 180 Andrew The Economics of Food and Agricultural Markets

APPLE Strategy is AD ( 10 ) Therefore , the Equilibrium in the Advertising game is ( AD , AD ) Recall that this outcome is one of two Nash Equilibria in the advertising game ( AD , AD ) and ( NOT , NOT ) If both players choose , there is only one equilibrium ( AD , AD ) The relationships between the game theory strategies can be summarized ( An Equilibrium in Dominant Strategies is always a Equilibrium . A Equilibrium is NOT always an Equilibrium in Dominant Strategies . An Equilibrium in Dominant Strategies is always a Nash Nash Equilibrium is NOT always an Equilibrium in Dominant Strategies . Cooperative Strategy ( Collusion ) The cooperative strategy is defined as the best joint outcome for both players together . Cooperative Strategy A strategy that leads to the highest joint payoff for all players . Thus , the cooperative strategy is identical to collusion , where players work together to achieve the best joint outcome . In the Prisoner Dilemma ( Figure ) the cooperative outcome is found by summing the two players outcomes together , and finding the outcome that has the smallest jail sentence for the prisoners together ( NOT , NOT ) This outcome is the collusive solution , which provides the best outcome if the prisoners could make a joint decision and stick with it . Of course , there is always the temptation to cheat on the agreement , where each player does better for themselves , at the expense of the other prisoner . Similarly , the cooperative outcome in the advertising game ( Figure ) is ( AD , AD ) 20 , 20 ) This outcome provides the highest profits ( 40 million ) to both firms . Note that the advertising game is not a prisoner dilemma , since there is no incentive to cheat once the cooperative solution has been achieved . Game Theory Example Steak Pricing Game A pricing game for steaks if shown in Figure . In this game , two beef processors , Tyson and , are determining what price to charge for steaks . Suppose that these two firms are the major players in this steak market , and the outcomes depend on the strategies of both firms , since players choose which company to purchase from based on price . If both firms choose low prices , the outcome is low profits . Additional profits are Chapter . Game Theory 181

earned by choosing high prices . However , when both firms have high prices , there is an incentive to undercut the other firm with a low price , to increase profits at the expense of the other firm . LOW ( 12 , HIGH ( 12 ) 10 , 10 ) Figure 63 Steak Pricing Game Two Beef Firms . Outcomes in million . Steak Pricing Game Dominant Strategy ( If TYSON LOW , should LOW ( If TYSON HIGH , should LOW ( 12 10 ) the dominant strategy for TYSON is LOW . If LOW , TYSON should LOW ( If HIGH , TYSON should LOW ( 12 10 ) the dominant strategy for is LOW . The Equilibrium in Dominant Strategies for the Steak Pricing game is ( LOW , LOW ) This is an unexpected result , since it is a less desirable scenario than ( HIGH , HIGH ) for both firms . We have seen that an Equilibrium in Dominant Strategies is also a Nash Equilibrium and a Equilibrium . These results will be checked in what follows . Steak Pricing Game Nash Equilibrium ( Outcome ( LOW , LOW ) a ) Is LOW best for given TYSON LOW ?

Yes . Is LOW best for TYSON given LOW ?

Yes . LOW , LOW ) is a Nash Equilibrium . 182 Andrew The Economics of Food and Agricultural Markets ( Outcome ( LOW , HIGH ) a ) Is LOW best for given TYSON HIGH ?

Yes . Is HIGH best for TYSON given LOW ?

No . LOW , HIGH ) is not a Nash Equilibrium . Outcome ( HIGH , LOW ) a ) Is HIGH best for given TYSON LOW ?

No . Is LOW best for TYSON given HIGH ?

Yes . HIGH , LOW ) is not a Nash Equilibrium . Outcome ( HIGH , HIGH ) a ) Is HIGH best for given TYSON HIGH ?

No . Is HIGH best for TYSON given HIGH ?

No . HIGH , HIGH ) is not a Nash Equilibrium . Therefore , there is only one Nash Equilibrium in the Steak Pricing game ( LOW , LOW ) Steak Pricing Game ( Safety First ) a ) If LOW , worst payoff . If HIGH , worst payoff . Strategy is LOW ( TYSON ( a ) If LOW , worst payoff . If HIGH , worst payoff . TYSON Strategy is LOW ( The Equilibrium in the Steak Pricing game is ( LOW , LOW ) Interestingly , if both firms cooperated , they could achieve much higher profits . Steak Pricing Game Cooperative Equilibrium ( Collusion ) Chapter . Game Theory 183

Both and Tyson can see that if they were to cooperate , either explicitly or implicitly , profits would increase significantly . The cooperative outcome is ( HIGH , HIGH ) This is the outcome with the highest combined profits . Both firms are better off in this outcome , but each firm has an incentive to cheat on the agreement to increase profits from 10 to 12 . 184 Andrew The Economics of Food and Agricultural Markets