计算纳什均衡
Posted MatrixCancer
tags:
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EXAMPLE1
[
[0, 2, -1, 1],
[-2, 0, 3, 1],
[1, -3, 0, 1],
]
双人零和对称博弈(
M
=
−
M
T
M = -M^T
M=−MT)的所有纳什均衡都是对称纳什均衡.
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\\max\\limits_p \\min\\limits_q pMq^T = \\min\\limits_q \\max\\limits_p pMq^T = pMp^T = -pMp^T = 0
pmaxqminpMqT=qminpmaxpMqT=pMpT=−pMpT=0
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max min 问题如下:
max p min q p M q T ⟺ max p min p M ⟺ max p , v v p M ≽ v 1 p ≽ 0 p 1 T = 1 \\beginaligned & \\max\\limits_p \\min\\limits_q pMq^T \\\\ \\iff& \\max\\limits_p \\min\\pM\\ \\\\ \\iff& \\begincases \\max\\limits_p,v v \\\\ pM \\succcurlyeq v1 \\\\ p \\succcurlyeq 0 \\\\ p1^T = 1 \\\\ \\endcases \\\\ \\endaligned ⟺⟺pmaxqminpMqTpmaxminpM⎩ ⎨ ⎧p,vmaxvpM≽v1p≽0p1T=1
⟺ min p , v − v 2 p 1 − p 2 + v ⩽ 0 − 2 p 0 + 3 p 2 + v ⩽ 0 p 0 − 3 p 1 + v ⩽ 0 − p 0 ⩽ 0 − p 1 ⩽ 0 − p 2 ⩽ 0 p 0 + p 1 + p 2 = 1 \\iff \\begincases & \\min\\limits_p,v -v \\\\ & \\beginmatrix & 2p_1 & -p_2 & +v &\\leqslant &0 \\\\ -2p_0 & & +3p_2 & +v &\\leqslant &0 \\\\ p_0 & -3p_1 & & +v &\\leqslant &0 \\\\ -p_0 & & & &\\leqslant &0 \\\\ & -p_1 & & &\\leqslant &0 \\\\ & & -p_2 & &\\leqslant &0 \\\\ p_0 & +p_1 & +p_2 & &= &1 \\\\ \\endmatrix \\\\ \\endcases ⟺⎩ ⎨ ⎧p,vmin−v−2p0p0−p0p02p1−3p1−p1+p1−p2+3p2−p2+p2+v+v+v⩽⩽⩽⩽⩽⩽=0000001
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max min 问题解得 p ∗ = [ 1 2 , 1 6 , 1 3 ] p^*=[\\frac12, \\frac16, \\frac13] p∗=[21,61,31], v ∗ = 0 v^*=0 v∗=0.
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min max 问题如下:
min q max p q M T p T ⟺ min q max q M T ⟺ min q , u u q M T ≼ u 1 q ≽ 0 q 1 T = 1 \\beginaligned & \\min\\limits_q \\max\\limits_p qM^Tp^T \\\\ \\iff& \\min\\limits_q \\max\\qM^T\\ \\\\ \\iff& \\begincases \\min\\limits_q, u u \\\\ qM^T \\preccurlyeq u1 \\\\ q \\succcurlyeq 0 \\\\ q1^T = 1 \\\\ \\endcases \\\\ \\endaligned ⟺⟺qminpmax《纳什均衡与博弈论》纳什博弈论及对自然法则的研究