Trembling hand perfect equilibrium

(Normal form) trembling hand perfect equilibrium
A solution concept in game theory
Relationship
Subset of	Nash Equilibrium
Superset of	Proper equilibrium
Significance
Proposed by	Reinhard Selten

Extensive-form trembling hand perfect equilibrium
A solution concept in game theory
Relationship
Subset of	Subgame perfect equilibrium, Perfect Bayesian equilibrium, Sequential equilibrium
Significance
Proposed by	Reinhard Selten
Used for	Extensive form games

In game theory, 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 assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability.

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, other 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.

The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely ${\displaystyle \langle {\text{Up}},{\text{Left}}\rangle }$ and ${\displaystyle \langle {\text{Down}},{\text{Right}}\rangle }$. However, only ${\displaystyle \langle {\text{U}},{\text{L}}\rangle }$ is trembling-hand perfect.

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