Games - Trembling hand perfect equilibrium

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	Games - Trembling hand perfect equilibrium

The trembling hand perfection is a notion that eliminates actions of players that are unsafe because they were chosen through a slip of the hand. Players could choose actions that are not the intended ones due (trembling) that lead to unintended outcomes. Off-the-equilibrium plays are due to trembling or mistakes in choosing the action among the action set of players.

Definition

First we will define a perturbed game. A perturbed game is a copy of a base game, with the restriction that no player has a pure strategy available to them. This is the players "trembling hand", they sometimes play a different strategy. Then we can define a strategy set S (in a base game) as being trembling hand perfect if there is a sequence of perturbed games that converge to the base game in which there is a series of Nash equilibria that converge to S.

Example

The game represented in the following normal form matrix has two Nash equilibria, namely and ( and ). However, only is trembling-hand perfect.

:

Assume player 1 is playing a mixed strategy

(1-\epsilon, \epsilon)

. Player 2's expected payoff from playing L is:

:

1(1-\epsilon) + 2\epsilon = 1+\epsilon

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

:

0(1-\epsilon) + 2\epsilon = 2\epsilon

For small values of ε, player 2 maximizes his expected payoff by placing a minimal weight on R. By symmetry, player 1 should place a minimal weight on B if player 2 is playing the mixed strategy

(1-\epsilon, \epsilon)

. Hence is trembling-hand perfect.

However, similar analysis fails for the strategy profile .

Assume player 1 is playing a mixed strategy

(\epsilon, 1-\epsilon)

. Player 2's expected payoff from playing L is:

:

1\epsilon + 2(1-\epsilon) = 2-\epsilon

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

:

0(\epsilon) + 2(1-\epsilon) = 2-2\epsilon

For all positive values of ε, player 2 maximizes his expected payoff by placing a minimal weight on R. Hence is not trembling-hand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating with a small chance of error.

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