Mixed strategy equilibrium ■M× ed strategy equilibrium a probability distribution for each player The distributions are mutual best responses to one another in the sense of expected payoffs > It is a stochastic steady state 16
Mixed strategy equilibrium ◼ Mixed strategy equilibrium ➢ A probability distribution for each player ➢ The distributions are mutual best responses to one another in the sense of expected payoffs ➢ It is a stochastic steady state 16
Mixed strategy vS Correlated strategy
Mixed strategy vs. Correlated strategy 17
Mixed strategy equilibrium: 2-player each with two pure strategies Player 2 S21(q) S2(1-q) Player 11(r) l1(S1,S21),a2(S1,21) l1(S1,S2,2(1,S2 2(1-r)a(s12s2l,a2(s12s2)|a1(s1s2),u2(2,S2 Mixed strategy Nash equilibrium A pair of mixed strategies (r,1-r),(q,1-q2) is a Nash equilibrium if(r, I-r%)is a best response to(g*, 1-g*), and(q*, 1-g*)is a best response to(r*, I-r%). That is v1(P,1-r),(qx,1-q3)2v1(r,1-r),(q,1-q2), for al0≤r≤1 v2(P,1-r),(qx,1-q3)≥v2(P,1-P3),(q,1-q), for all0≤q≤1
Mixed strategy equilibrium: 2-player each with two pure strategies Player 2 s21 ( q ) s22 ( 1- q ) Player 1 s11 ( r ) u1 (s11, s21), u2 (s11, s21) u1 (s11, s22), u2 (s11, s22) s12 (1- r ) u1 (s12, s21), u2 (s12, s21) u1 (s12, s22), u2 (s12, s22) 18 ◼ Mixed strategy Nash equilibrium: ◼ A pair of mixed strategies ((r*, 1-r*), (q*, 1-q*)) is a Nash equilibrium if (r*,1-r*) is a best response to (q*, 1-q*), and (q*, 1-q*) is a best response to (r*,1-r*). That is, v1 ((r*, 1-r*),(q*, 1-q*)) v1 ((r, 1-r),(q*, 1-q*)), for all 0 r 1 v2 ((r*, 1-r*),(q*, 1-q*)) v2 ((r*, 1-r*),(q, 1-q)), for all 0 q 1
Find mixed strategy equilibrium in 2 player each with two pure strategies find the best response correspondence for player 1, given player 2's mixed strategy Find the best response correspondence for player 2, given player 1s mixed strategy Use the best response correspondences to determine mixed strategy Nash equilibria 19
Find mixed strategy equilibrium in 2- player each with two pure strategies ◼ Find the best response correspondence for player 1, given player 2’s mixed strategy ◼ Find the best response correspondence for player 2, given player 1’s mixed strategy ◼ Use the best response correspondences to determine mixed strategy Nash equilibria. 19
Employee Monitoring Employees can work hard or shirk Salary: $100K unless caught shirking Cost of effort $50K Managers can monitor or not Value of employee output: $200K Profit if employee doesnt work: sO Cost of monitoring: $10K
Employee Monitoring ◼ Employees can work hard or shirk ◼ Salary: $100K unless caught shirking ◼ Cost of effort: $50K ◼ Managers can monitor or not ◼ Value of employee output: $200K ◼ Profit if employee doesn’t work: $0 ◼ Cost of monitoring: $10K 20