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【正文】 3, 2/3)) 1/3 May 29, 2021 73347 Game TheoryLecture 8 8 Expected payoffs: 2 players each with two pure strategies ? 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, 1q))=q u1(s11, s21)+(1q) u1(s11, s22) ? Player 1’s expected payoff of playing s12: EU1(s12, (q, 1q))= q u1(s12, s21)+(1q) u1(s12, s22) ? Player 1’s expected payoff from her mixed strategy: v1((r, 1r), (q, 1q))=r?EU1(s11, (q, 1q))+(1r)?EU1(s12, (q, 1q)) 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) May 29, 2021 73347 Game TheoryLecture 8 9 Expected payoffs: 2 players each with two pure strategies ? 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, 1r))=r u2(s11, s21)+(1r) u2(s12, s21) ? Player 2’s expected payoff of playing s22: EU2(s22, (r, 1r))= r u2(s11, s22)+(1r) u2(s12, s22) ? Player 2’s expected payoff from her mixed strategy: v2((r, 1r),(q, 1q))=q?EU2(s21, (r, 1r))+(1q)?EU2(s22, (r, 1r)) 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) May 29, 2021 73347 Game TheoryLecture 8 10 Mixed strategy equilibrium: 2player each with two pure strategies ? Mixed strategy Nash equilibrium: ? A pair of mixed strategies ((r*, 1r*), (q*, 1q*)) is a Nash equilibrium if (r*,1r*) is a best response to (q*, 1q*), and (q*, 1q*) is a best response to (r*,1r*). That is, v1((r*, 1r*), (q*, 1q*)) ? v1((r, 1r), (q*, 1q*)), for all 0? r ?1 v2((r*, 1r*), (q*, 1q*)) ? v2((r*, 1r*), (q, 1q)), for all 0? q ?1 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) May 29, 2021 73347 Game TheoryLecture 8 11 2player each with two strategies ? Theorem 1 (property of mixed Nash equilibrium) ? A pair of
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