In game theory, trembling hand perfect equilibrium is a type of refinement of a Nash equilibrium that was first proposed by Reinhard Selten.
[1] 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.
A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played.
-player strategic game where every pure strategy is played with positive probability.
Then define a mixed strategy profile
as being trembling hand perfect if there is a sequence of perturbed games strategy profiles
Note: All completely mixed Nash equilibria are perfect.
Note 2: The mixed strategy extension of any finite normal-form game has at least one perfect equilibrium.
[2] The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely
Player 2's expected payoff from playing L is: Player 2's expected payoff from playing the strategy R is: For small values of
, 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 and maximal weight on U if player 2 is playing the mixed strategy
However, similar analysis fails for the strategy profile
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
, player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U.
For 2x2 games, the set of trembling-hand perfect equilibria coincides with the set of equilibria consisting of two undominated strategies.
In the example above, we see that the equilibrium
[3] There are two possible ways of extending the definition of trembling hand perfection to extensive form games.
The notions of normal-form and extensive-form trembling hand perfect equilibria are incomparable, i.e., an equilibrium of an extensive-form game may be normal-form trembling hand perfect but not extensive-form trembling hand perfect and vice versa.
As an extreme example of this, Jean-François Mertens has given an example of a two-player extensive form game where no extensive-form trembling hand perfect equilibrium is admissible, i.e., the sets of extensive-form and normal-form trembling hand perfect equilibria for this game are disjoint.
[citation needed] An extensive-form trembling hand perfect equilibrium is also a sequential equilibrium.
A normal-form trembling hand perfect equilibrium of an extensive form game may be sequential but is not necessarily so.
Myerson (1978)[4] pointed out that perfection is sensitive to the addition of a strictly dominated strategy, and instead proposed another refinement, known as proper equilibrium.