Subset of Nash Equilibrium | ||
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.
Contents
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, 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.
Example
The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely
Assume player 1 (the row player) is playing a mixed strategy
Player 2's expected payoff from playing the strategy R is:
For small values of
However, similar analysis fails for the strategy profile
Assume player 2 is playing a mixed strategy
Player 1's expected payoff from playing D is:
For all positive values of
Trembling hand perfect equilibria of two-player games
For two-player games, the set of trembling hand perfect equilibria coincides with the set of admissible equilibria, i.e., equilibria consisting of two undominated strategies. In the example above, we see that the imperfect equilibrium <D,R> is not admissible, as L (weakly) dominates R for Player 2 and U (weakly) dominates D for Player 1.
Trembling hand perfect equilibria of extensive form games
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.
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. In fact, a normal-form trembling hand perfect equilibrium does not even have to be subgame perfect.