Mixed strategy: example Player 2 C(1/3) R(2/3) T(3/4) 2 3 1 1 Player 1 M(O 4 0 4 2 3 B(1/4) 4 5 1 0 7 ■ Player1: >(3/4, 0, 74)is a mixed strategy. That is 01(T)=314,o1M)=0ando1(B)=14 ■ Player2: >(0, 1/3, 2/3)is a mixed strategy That is 2(L)=0,o2(C)=13ando2(R)=2/3
Mixed strategy: 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 11 ◼ Player 1: ➢ (3/4, 0, ¼) is a mixed strategy. That is, 1 (T)=3/4, 1 (M)=0 and 1 (B)=1/4. ◼ Player 2: ➢ (0, 1/3, 2/3) is a mixed strategy. That is, 2 (L)=0, 2 (C)=1/3 and 2 (R)=2/3
Expected payoffs: 2 players each with two pure strategies Player 2 S21(q) S2(1-q) Player 11(r) u1(S1,S21),l2(s1,S21) l1(S1,S2),2(S1,S2 12 u1(s12,S21,a2(s12S2)|a1(s12s2,2(12,S2) Player 1 plays a mixed strategy (r, 1-r). Player 2 plays a mixed strategy (a, 1-q). Player 1s expected payoff of playing s EU1(1(q,1-q)=q×u1S1,S21)+(1-q)X1(S1,2 Player 1s expected payoff of playing s12 EU1(S12(q,1-q)=qXu1(12,S21)+(1-q)X1 129422 Player 1' s expected payoff from her mixed strategy v(G,I-r),(q,1-q)=rxEU1s1(q,1-q)+(-)×EU1(s12(q,1-q)
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(q) 1-q) Plaver1 Su (r) a(,s2,a4(,s2)|a(sn,s2),a2(m,s2) 12 u1(s12S2l,a2(s12,S2)|a1(s2S2),2(s122) Player 1 plays a mixed strategy (r, 1-r). Player 2 plays a mixed strategy( q,1-41 Player 2's expected payoff of playing s21 EU2(S21,(r,-r)=rXl2(31,S21)+(-)×2(512,S21) Player 2's expected payoff of playing s22 EU2(S2,(r,-r)=rXl2(51,2)+(1-r)Xu2(S12,52 Player 2s expected payoff from her mixed strategy v2(r,1-r),(q,1-q)=q×EU2(S21,(7,1-r)+(1-q)×EU21S2(,1-r)
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(07) H(0.4) 1 1 1 Player 1 T(0.6) 1,-1 1 1 ■ Player1: >EU1(H,(0.3,0.7)=0.3×(-1)+0.7×1=0.4 EU1(,(0.3,0.7)=0.3×1+0.7×(-1)=0.4 v1(0.4,0.6),(0.3,0.7)=0.4×0.4+0.6×(0.4)=0.08 ■ Player2 EU2(H,(0.4,0.6)=0.4×1+0.6×(-1)=-02 >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 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 C(1/3) R(2/3) T(3/4) 2 3 1 1 Player 1 M(O 4 0 4 2 3 B(1/4) 4 5 1 0 7 Mixed strategies:p1=(3/4,0,%);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×(23)=4/3 EU1(B,p2)=5×(1/3)+0×(2/3)=5/3.v/(D1,P2)=5/3 Player 2: EU2(L,p1)=2×(3/4)+4×(14)=5/2,EU2(C,p)=3×(3/4)+3×(4)=5/2, EU2(R,p1)=1×(3/4)+7×(1/4)=5/2.v(1,P2)=5/2
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