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【正文】 s. May 30, 2021 73347 Game TheoryLecture 9 13 Example 1 ? Bruce’s expected payoff of playing Head ? EU1(H, (q, 1–q)) = q 1 + (1–q) 4 = 4–3q ? Bruce’s expected payoff of playing Tail ? EU1(T, (q, 1–q)) = q 4 + (1–q) 1 = 1+3q ? Bruce is indifferent between playing Head and Tail ? EU1(H, (q, 1–q)) = EU1(T, (q, 1–q)) 4–3q = 1+3q 6q = 3 This give us q = 1/2 Sheila H ( q ) T ( 1–q ) Bruce H ( r ) 1 , 4 4 , 1 T ( 1–r ) 4 , 1 1 , 4 May 30, 2021 73347 Game TheoryLecture 9 14 Example 1 ? Sheila’s expected payoff of playing Head ? EU2(H, (r, 1–r)) = r 4+(1–r) 1 = 3r + 1 ? Sheila’s expected payoff of playing Tail ? EU2(T, (r, 1–r)) = r 1+(1–r) 4 = 4 – 3r ? Sheila is indifferent between playing Head and Tail ? EU2(H, (r, 1–r)) = EU2(T, (r, 1–r)) 3r + 1 = 4 – 3r 6r = 3 This give us r = 189。 ? ( (1/2, 1/2), (1/2, 1/2) ) is a mixed strategy Nash equilibrium. Sheila H ( q ) T ( 1–q ) Bruce H ( r ) 1 , 4 4 , 1 T ( 1–r ) 4 , 1 1 , 4 May 30, 2021 73347 Game TheoryLecture 9 15 Example 2 ? Player 1’s expected payoff of playing T ? EU1(T, (q, 1–q)) = q 6 + (1–q) 0 = 6q ? Player 1’s expected payoff of playing B ? EU1(B, (q, 1–q)) = q 3 + (1–q) 6 = 63q ? Player 1 is indifferent between playing T and B ? EU1(T, (q, 1–q)) = EU1(B, (q, 1–q)) 6q = 63q 9q = 6 This give us q = 2/3 Player 2 L ( q ) R ( 1–q ) Player 1 T ( r ) 6 , 0 0 , 6 B ( 1–r ) 3 , 2 6 , 0 May 30, 2021 73347 Game TheoryLecture 9 16 Example 2 ? Player 2’s expected payoff of playing L ? EU2(L, (r, 1–r)) = r 0+(1–r) 2 =2 2r ? Player 2’s expected payoff of playing R ? EU2(R, (r, 1–r)) = r 6+(1–r) 0 = 6r ? Player 2 is indifferent between playing L and R ? EU2(L, (r, 1–r)) = EU2(R, (r, 1–r)) 2 2r = 6r 8r = 2 This gives us r = 1
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