第2个回答 2011-12-11
棒子 ---A
虎 ---B
鸡 --- C
虫子 ---D
A B C D
A 0,0 1,-1 0,0 -1,1
B -1,1 0,0 1,-1 0,0
C 0,0 -1,1 0,0 1,-1
D 1,-1 0,0 -1,1 0,0
Let two peoples' strategy functions be f(.) and g(.)
Mixed Nash equilibrium makes each other indifferent
=>
f(a)g(a)u1(a,a)+f(a)g(b)u1(a,b)+f(a)g(c)u1(a,c)+f(a)g(d)u1(a,d)
=
f(b)g(a)u1(b,a)+f(b)g(b)u1(b,b)+f(b)g(c)u1(b,c)+f(b)g(d)u1(b,d)
=
f(c)g(a)u1(c,a)+f(c)g(b)u1(c,b)+f(c)g(c)u1(c,c)+f(c)g(d)u1(c,d)
=
f(d)g(a)u1(d,a)+f(d)g(b)u1(d,b)+f(d)g(c)u1(d,c)+f(d)g(d)u1(d,d)
==>
f(a)g(b)-f(a)g(d)=f(b)g(c)-f(b)g(a)=f(c)g(d)-f(c)g(b)=f(d)g(a)-f(d)g(c)
AND f(a)+f(b)+f(c)+f(d)=g(a)+g(b)+g(c)+g(d)=1
Since the game is symmetric, so strategies are identical, so
ab-ad = bc - ba = cd-cb = da-dc
=>
f(a)=f(b)=f(c)=f(d)=0.25
Mixed NE:
(0.25A+0.25B+0.25C+0.25D,0.25A+0.25B+0.25C+0.25D)