Mixed strategy vs.Correlated strategy 下
Mixed strategy vs. Correlated strategy 17
Mixed strategy equilibrium:2-player each with two pure strategies Player 2 521(9) s22(1-9) Player 1 S1(r) 411,S21,21,S21） 41(S1,S22,u2S1,522) S12(1-r） 41S12,S2i,42S12S2) u1(S12yS22),2S123S22) ■ Mixed strategy Nash equilibrium: A pair of mixed strategies (*,1-r),(q*,1-q) is a Nash equilibrium if(r*,1-r*)is a best response to(g*, 1-g*),and (g*,1-g*)is a best response to (r,1-r*).That is, y(r*,1-r),(g*,1-q)≥y(,1-r),(g*,1-q*),f0rall0≤r≤1 v2(r*,1-r),(q*,1-q*)≥v2(r*,1-r),(g,1-q),for all0q≤1 18
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 1's 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 doesn't work:$0 Cost of monitoring:$10K 20
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
Employee Monitoring ■ Employee's best response B(q): Expected >Shirk (r=0)if q<0.5 payoffs >Work (r=1)if q>0.5 50 ,Any mixed strategy(0≤r≤1)ifq=0.5100(1-qW Expected 100r-10 200r-100 payoffs Manager Monitor(q)Not Monitor(1-q) Work (r) 50, 90 50 100 Employee Shirk(1-r） 0 ,-10 100, -100 21
Employee Monitoring ◼ Employee’s best response B1(q): ➢ Shirk (r=0) if q<0.5 ➢ Work (r=1) if q>0.5 ➢ Any mixed strategy (0r1) if q=0.5 21 Manager Monitor ( q ) Not Monitor (1-q) Employee Work ( r ) 50 , 90 50 , 100 Shirk (1-r ) 0 , -10 100 , -100 50 100(1-q) Expected payoffs Expected payoffs 100r-10 200r-100