Order Your Paper From the most reliable Essay writing Service. 

.

 



Question: ExAMPLE 117.1 (Checking whether a mixed strategy profile is a mixed strategy Nash equilibrium) I ...
Question: ExAMPLE 117.1 (Checking whether a mixed strategy profile is a mixed strategy Nash equilibrium) I ...
Question is 117.2 part (a), please answer all steps
PRECISELY!

Show transcribed image text ExAMPLE 117.1 (Checking whether a mixed strategy profile is a mixed strategy Nash equilibrium) I claim that for the game in Figure 117.1 (in which the dots indicate irrelevant payoffs), the indicated pair of strategies, ,0,l) for player 1 and (0, t, for player 2, is a mixed strategy Nash equilibrium. To verify this claim, it suffices, by Proposition 116.2, to study each player's expected payoffs to her three pure strategies. For player 1 these payoffs are 3 M 5+ Player 1's mixed strategy assigns positive probability to Tand B and probability zero to M, so the two conditions in Proposition 116.2 are satisfied for player 1. The expected payoff to each of player 2's pure strategies is .2+ .4 3 1 1+ 7 3), so the two conditions in Proposition 116.2 are satisfied also for her. Note that the expected payoff to player 2's action L, which she uses with prob- ability zero, is the same as the expected payoff to her other two actions. This equal- ity is consistent with Proposition 116.2, the second part of which requires only that the expected payoffs to actions used with probability zero be no greater than the ex- pected payoffs to actions used with positive probability (not that they necessarily be less). Note also that the fact that player 2's expected payoff to Lis the same as her expected payoffs to Cand R does not imply that the game has a mixed strategy Nash equilibrium in which player 2 uses L with positive probability it may, or it may not, depending on the unspecified payoffs. ExERCISE 117.2 (Choosing numbers) Playersland 2 each choose a positive integer up to K. If the players choose the same number, then player 2 pays $1 to player 1; otherwise no payment is made. Each player's preferences are represented by her expected monetary payoff.

ExAMPLE 117.1 (Checking whether a mixed strategy profile is a mixed strategy Nash equilibrium) I claim that for the game in Figure 117.1 (in which the dots indicate irrelevant payoffs), the indicated pair of strategies, ,0,l) for player 1 and (0, t, for player 2, is a mixed strategy Nash equilibrium. To verify this claim, it suffices, by Proposition 116.2, to study each player's expected payoffs to her three pure strategies. For player 1 these payoffs are 3 M 5+ Player 1's mixed strategy assigns positive probability to Tand B and probability zero to M, so the two conditions in Proposition 116.2 are satisfied for player 1. The expected payoff to each of player 2's pure strategies is .2+ .4 3 1 1+ 7 3), so the two conditions in Proposition 116.2 are satisfied also for her. Note that the expected payoff to player 2's action L, which she uses with prob- ability zero, is the same as the expected payoff to her other two actions. This equal- ity is consistent with Proposition 116.2, the second part of which requires only that the expected payoffs to actions used with probability zero be no greater than the ex- pected payoffs to actions used with positive probability (not that they necessarily be less). Note also that the fact that player 2's expected payoff to Lis the same as her expected payoffs to Cand R does not imply that the game has a mixed strategy Nash equilibrium in which player 2 uses L with positive probability it may, or it may not, depending on the unspecified payoffs. ExERCISE 117.2 (Choosing numbers) Playersland 2 each choose a positive integer up to K. If the players choose the same number, then player 2 pays $1 to player 1; otherwise no payment is made. Each player's preferences are represented by her expected monetary payoff.