國外博弈論課件lecture(19)-資料下載頁
2025-10-09 12:47本頁面
【正文】 ? By theorem 4, we should have 2?p11+2? p12= 3?p11+1? p12 = 3?p11+2? p12 and p11+p12=1. ? Plugging p11=0, p12=1 into 2?p11+2? p12= 3?p11+1? p12 = 3?p11+2? p12 gives us 2=1=2, which is a contradiction. Hence, there is no Nash equilibrium in this case. Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 6, 2021 73347 Game TheoryLecture 14 22 Exercise of Osborne ? Case 13: check whether there is a mixed strategy in which p11=0, p120, p210, p220, p23=0 (Note this implies p12=1) ? By theorem 4, we should have 2?p11+2? p12= 3?p11+1? p12 ? 3?p11+2? p12 and p11+p12=1. ? Plugging p11=0, p12=1 into 2?p11+2? p12= 3?p11+1? p12 ? 3?p11+2? p12 gives us 2=1?2, which is a contradiction. Hence, there is no Nash equilibrium in this case. Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 6, 2021 73347 Game TheoryLecture 14 23 Exercise of Osborne ? Case 14: check whether there is a mixed strategy in which p11=0, p120, p210, p22=0, p230 (Note this implies p12=1) ? By theorem 4, we should have 2?p11+2? p12 = 3?p11+2? p12 ? 3?p11+1? p12 and p11+p12=1. Plugging p11=0, p12=1 into 2?p11+2? p12= 3?p11+2? p12 ? 3?p11+1? p12 gives us 2=2?1. OK ? By Theorem 4, we should also have 2?p21+0? p22+1? p23 ? 3?p21+1? p22+0? p23 and p21+ p22+ p23 = 1 Plugging p22=0 into these gives p23 ? p21 and p21+ p23 = 1. Then plugging p23 = 1 p21 into p23 ? p21 gives us 1 p21 ? p21 , which implies p21 ? . ? Hence, there are mixed strategy Nash equilibrium ( (0, 1), (p21, 0, 1 p21) ), for any ? p21 1 Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 6, 2021 73347 Game TheoryLecture 14 24 Exercise of Osborne ? Case 15: check whether there is a mixed strategy in which p11=0, p120, p21=0, p220, p230 (Note this implies p12=1) ? By theorem 4, we should have 2?p11+2? p12? 3?p11+1? p12 = 3?p11+2? p12 and p11+p12=1. ? Plugging p11=0, p12=1 into 2?p11+2? p12? 3?p11+1? p12 = 3?p11+2? p12 gives us 2 ? 1=2, which is a contradiction. Hence, there is no Nash equilibrium in this case. Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2 June 6, 2021 73347 Game TheoryLecture 14 25 Exercise of Osborne ? We conclude that all Nash equilibria of this game are ( (1, 0), (0, p22, 1 p22) ), for any 0 ? p22 ? ( (0, 1), (p21, 0, 1 p21) ), for any ? p21 ? 1 Player 2 L (p21) M (p22) R (p23) Player 1 T (p11) 2 , 2 0 , 3 1 , 3 B (p12) 3 , 2 1 , 1 0 , 2
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