Production Games A best response to a best response to a best response to a.. is called a Nash If all the payoffs in a particular cell are underlined then the strategies that correspond to that cell are a Na rategies. Sometimes this can be very complicated. Chess In the prisoners dilemma the Nash equilibrium is (D, D] Player 2 BRI Player 1 In the above graph the best responses are plotted for each player. These are rea rves. The Nash equilibrium is where the wes intersect. This is a useful method for finding equilibria. Along the axis are the possible actions taken by players 1 and 2. BR, is the best reply function of player 1 and BR2 is player 2s. They intersect at ( D, DI, the equilibrium. What are the dotted lines for? Mixed Strategies Suppose a player could "toss a coin"to decide which action to take. What would their opponent de Such an action is called a mired strategy. For example. in the prisoners'dilemma, with a fair coin, the opponent would expect to see C half the time and D half the time. They calculate a best response If they choose to play C, half the time they would get 3, and half the time they would get 0. If they choose to play D half the time they would get 5, and half the time they would get 1. Their expected payoffs 32+0=2<3=57+1 2 mixes half-half BRI Therefore they would play D in response to such a mixture. In fact, they would play D in response to any mixture. Hence the dotted lines can be filled in as above. Mixed strategies are beyond the level required hereProduction — Games 7 Nash Equilibrium • A best response to a best response to a best response to a ... is called a Nash equilibrium. • If all the payoffs in a particular cell are underlined then the strategies that correspond to that cell are a Nash equilibrium. Notice equilibrium is a collection of strategies. Sometimes this can be very complicated. Chess? • In the prisoners’ dilemma the Nash equilibrium is {D, D}. ................................................................................................................................................................................................................................................................................ ............. ............. ............. ............. ............. ............. ............. ............. ............. . . . . . . . . . Player 2 Player 1 BR2 BR1 C D C D • • • • In the above graph the best responses are plotted for each player. These are reaction curves. The Nash equilibrium is where the two reaction curves intersect. This is a useful method for finding equilibria. • Along the axis are the possible actions taken by players 1 and 2. BR1 is the best reply function of player 1 and BR2 is player 2s. They intersect at {D, D}, the equilibrium. What are the dotted lines for? Production — Games 8 Mixed Strategies • Suppose a player could “toss a coin” to decide which action to take. What would their opponent do? • Such an action is called a mixed strategy. For example, in the prisoners’ dilemma, with a fair coin, the opponent would expect to see C half the time and D half the time. They calculate a best response. • If they choose to play C, half the time they would get 3, and half the time they would get 0. If they choose to play D half the time they would get 5, and half the time they would get 1. Their expected payoffs are: 3 1 2 + 0 1 2 = 3 2 < 3 = 5 1 2 + 1 1 2 ................................................................................................................................................................................................................................................................................ ......................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . 2 mixes half-half Player 2 Player 1 BR2 BR1 C D C D • • • • Therefore they would play D in response to such a mixture. In fact, they would play D in response to any mixture. Hence the dotted lines can be filled in as above. Mixed strategies are beyond the level required here