Trembling Hand Perfect Equilibrium

Equilibrium
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# Trembling hand perfect equilibrium

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Description:
Trembling hand perfect equilibrium is a refinement of Nash Equilibrium due to Reinhard Selten. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of off-the-equilibrium play into account by assumingthat the players, through a "slip of the hand" or tremble, may chooseunintended strategies, albeit with negligible probability.

## Definition

First we define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played.A totally mixed strategy is a mixed strategy where every pure strategy is played with non-zero probability.This is the "trembling hands" of the players; they sometimes play a different strategy than the one they intended to play. Then we 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 pure strategy Nash equilibria, namely <nowiki><Up, Left></nowiki> and <nowiki><Down, Right></nowiki>. However, only <nowiki><U,L></nowiki> is trembling-hand perfect.

Assume player 1 is playing a mixed strategy [itex](1-epsilon, epsilon)[/itex], for [itex] 0<epsilon <1[/itex]. Player 2's expected payoff from playing L...... ...

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