Expected payoffs:2 players each with two pure strategies Player 2 S21（9） 522(1-4) Player 1 S1(r) 41(S11,S21,42(S11,S2） 41S1522,u2S1,S22) S12(1-r） 41(S12S21,u251252i） 41(S123S22,u2S123S22) Player 1 plays a mixed strategy(r,1-r).Player 2 plays a mixed strategy (g,1-q). Player 1's expected payoff of playing s11: EU1S11,(g,1-q)-qX1S1,S2+(1-qW)Xu1S11S22) Player 1's expected payoff of playing s12: EU1(S12,(9,1-q)=9Xu1S12,S2+(1-④)Xu1S12,S22) Player 1's expected payoff from her mixed strategy: y(m,1-r,(9,1-q)=×EU(S1,(4,1-q)+(1-)xEU1(S12,(4,1-q) 2
Expected payoffs: 2 players 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) 12 ◼ Player 1 plays a mixed strategy (r, 1- r ). Player 2 plays a mixed strategy ( q, 1- q ). ➢ Player 1’s expected payoff of playing s11: EU1 (s11, (q, 1-q))=q×u1 (s11, s21)+(1-q)×u1 (s11, s22) ➢ Player 1’s expected payoff of playing s12: EU1 (s12, (q, 1-q))= q×u1 (s12, s21)+(1-q)×u1 (s12, s22) ◼ Player 1’s expected payoff from her mixed strategy: v1 ((r, 1-r), (q, 1-q))=rEU1 (s11, (q, 1-q))+(1-r)EU1 (s12, (q, 1-q))
Expected payoffs:2 players each with two pure strategies Player 2 S21（9) 52(1-9） Player 1 Su (r) u1(S1,S21,u2S1,S2) 41(S11S22,W2(11,S22) S12(1-r) 41(S12,S21,42S12,S21） 41(S12522),u212,S22) ■ Player 1 plays a mixed strategy(r,1-r).Player 2 plays a mixed strategy (4,1-q). >Player 2's expected payoff of playing s21: EU2S21,(,1-r)=X2S11,S2it(-r)X2(S12,S2i) Player 2's expected payoff of playing s22: EU2S22,(,1-)FrXu2S11,S22H(l-）Xu2S12,S22) Player 2's expected payoff from her mixed strategy: v2(,1-r),(g,1-q0)=q×EU221,(r,1-)+(1-q)xEU2S22(,1-r) 13
Expected payoffs: 2 players 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) 13 ◼ Player 1 plays a mixed strategy (r, 1- r ). Player 2 plays a mixed strategy ( q, 1- q ). ➢ Player 2’s expected payoff of playing s21: EU2 (s21, (r, 1-r))=r×u2 (s11, s21)+(1-r)×u2 (s12, s21) ➢ Player 2’s expected payoff of playing s22: EU2 (s22, (r, 1-r))= r×u2 (s11, s22)+(1-r)×u2 (s12, s22) ◼ Player 2’s expected payoff from her mixed strategy: v2 ((r, 1-r),(q, 1-q))=qEU2 (s21, (r, 1-r))+(1-q)EU2 (s22, (r, 1-r))
Expected payoffs:example Player 2 H(0.3) T(0.7) H(0.4) -1,1 1,-1 Player 1 T(0.6) 1,-1 -1,1 ■Player1: >EU1(H,(0.3,0.7)=0.3X(-1)+0.7X1=0.4 >EU1T,(0.3,0.7)=0.3×1+0.7×(-1)=-0.4 >y(0.4,0.6),(0.3,0.7)=0.4×0.4+0.6×(-0.4)=-0.08 ■Player2: >EU2H,(0.4,0.6)=0.4×1+0.6×(-1)=-0.2 >EU2(T,0.4,0.6)=0.4×-1)+0.6×1=0.2 >y20.4,0.6,(0.3,0.7)=0.3X(-0.2)+0.7×0.2=0.08 安
Expected payoffs: example Player 2 H (0.3) T (0.7) Player 1 H (0.4) -1 , 1 1 , -1 T (0.6) 1 , -1 -1 , 1 14 ◼ Player 1: ➢ EU1 (H, (0.3, 0.7)) = 0.3×(-1) + 0.7×1=0.4 ➢ EU1 (T, (0.3, 0.7)) = 0.3×1 + 0.7×(-1)=-0.4 ➢ v1 ((0.4, 0.6),(0.3, 0.7))=0.40.4+0.6(-0.4)=-0.08 ◼ Player 2: ➢ EU2 (H, (0.4, 0.6)) = 0.4×1+0.6×(-1) = -0.2 ➢ EU2 (T, (0.4, 0.6)) = 0.4×(-1)+0.6×1 = 0.2 ➢ v2 ((0.4, 0.6), (0.3, 0.7))=0.3×(-0.2)+0.7×0.2=0.08
Expected payoffs:example Player 2 L(O) C(113) R(23) T(314) 0, 2 3， 3 1, 1 Player 1 M(0) 4 0 0,4 2, 3 B(114) 4 5, 1 0,7 ■ Mixed strategies:p1-=(3/4,0,4);p2=(0,1/3,2/3). ■Player1: >EU1(T,P2=3×(1/3)+1×(2/3)=5/3,EU1(M,p2=0×(1/3)+2×(2/3)=4/3 EU1(B,p2)=5×(1/3)+0x(2/3)=5/3.v1D1,P2)=5/3 ■Player2: >EU2(L,p1)=2×(3/4)+4×(1/4)=5/2,EU2(C,p1)=3×(3/4)+3×(1/4)F5/2, EU2R,p1)=1×(3/4)+7×(1/4)=5/2.y01,P2)=5/2 15
Expected payoffs: example Player 2 L (0) C (1/3) R (2/3) Player 1 T (3/4) 0 , 2 3 , 3 1 , 1 M (0) 4 , 0 0 , 4 2 , 3 B (1/4) 3 , 4 5 , 1 0 , 7 15 ◼ Mixed strategies: p1=( 3/4, 0, ¼ ); p2=( 0, 1/3, 2/3 ). ◼ Player 1: ➢ EU1 (T, p2 )=3(1/3)+1(2/3)=5/3, EU1 (M, p2 )=0(1/3)+2(2/3)=4/3 EU1 (B, p2 )=5(1/3)+0(2/3)=5/3. v1 (p1 , p2 ) = 5/3 ◼ Player 2: ➢ EU2 (L, p1 )=2(3/4)+4(1/4)=5/2, EU2 (C, p1 )=3(3/4)+3(1/4)=5/2, EU2 (R, p1 )=1(3/4)+7(1/4)=5/2. v1 (p1 , p2 ) = 5/2
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 6
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