顫抖手精煉均衡 (trembling hand perfect equilibrium)

Definitioin

“顫抖手精煉均衡”概念是澤爾騰提出的對納什均衡的一個改進。顫抖手精煉均衡的基本思想是:在任何一個博弈中，每個局中人都有一定的犯錯誤的可能性(類似一個人用手抓東西時，手一顫抖，他就抓不住他想抓的東西)。一個策略對是一個顫抖手精煉均衡時，它必須具有如下性質:各局中人ｉ要採用的策略，不僅在其他局中人不犯錯誤時是最優的，而且在其他局中人偶爾犯錯誤(概率很小，但大於0)時還是最優的。可以看出，顫抖手精煉均衡是一種較穩定的均衡。

all text is from MBALib:
http://wiki.mbalib.com/zh-tw/%E9%A2%A4%E6%8A%96%E6%89%8B%E7%B2%BE%E7%82%BC%E5%9D%87%E8%A1%A1

Example

The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely <Up, Left> and <Down, Right>. However, only <U,L> is trembling-hand perfect.

	Left	Right
Up	1, 1	2, 0
Down	0, 2	2, 2
Trembling hand perfect equilibrium

Assume player 1 is playing a mixed strategy , for . Player 2's expected payoff from playing L is:

Player 2's expected payoff from playing the strategy R is:

For small values of ε, player 2 maximizes his expected payoff by placing a minimal weight on R and maximal weight on L. By symmetry, player 1 should place a minimal weight on D if player 2 is playing the mixed strategy . Hence <U,L> is trembling-hand perfect.

However, similar analysis fails for the strategy profile <D,R>.

Assume player 2 is playing a mixed strategy . Player 1's expected payoff from playing U is:

Player 1's expected payoff from playing D is:

For all positive values of ε, player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence <D, R> is not trembling-hand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating most often to L if there is a small chance of error in the behavior of player 1.

all text is from Wikipedia
http://en.wikipedia.org/wiki/Trembling_hand_perfect_equilibrium