博弈论第二讲 Mixed Strategies 复旦大学 王永钦.ppt

Mixed strategy vs. Correlated strategy Theorem (Nash, 1950) Fall, 2007, Fudan Exercise 138.1 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? p12and p11+p12=1. We should have2?p21+0? p22+1? p23 ?3?p21+1? p22+0? p23  and p21+ p22+ p23 = 1 Solve these. If we can get a solution that satisfies p11=0, p120, p210, p220, p23=0 then we have a mixed strategy Nash equilibrium. Otherwise, if we can not find a solution or we find a solution that does not satisfies p11=0, p120, p210, p220, p23=0, then we do not have such a mixed strategy Nash equilibrium. 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 Fall, 2007, Fudan Exercise 138.1 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+1? p12 ? 3?p11+2? p12and p11+p12=1. We should have2?p21+0? p22+1? p23 ? 3?p21+1? p22+0? p23  and p21+ p22+ p23 = 1 Solve these. If we can get a solution that satisfies p11=0, p120, p210, p22=0, p230 then we have a mixed strategy Nash equilibrium. Otherwise, if we can not find a solution or we find a solution that does not satisfies p11=0, p120, p210, p22=0, p230, then we do not have such a mixed strategy Nash equilibrium. 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 Fall, 2007, Fudan Exercise 138.1 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? p12and p11+p12=1. We should have2?p21+0? p22+1? p23 ? 3?p21+1? p22+0? p23  and p21+ p22+ p23 = 1 Solve these. If we can get a solution that satisfies p11=0, p120, p21=0, p220, p230 then we have a mixed st