|(Normal form) trembling hand perfect equilibrium|
|A solution concept in game theory|
|Subset of||Nash Equilibrium|
|Superset of||Proper equilibrium|
|Proposed by||Reinhard Selten|
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 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 strategy (both pure and mixed) 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 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.
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.
The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely and . However, only is trembling-hand perfect.
|Up||1, 1||2, 0|
|Down||0, 2||2, 2|
|Trembling hand perfect equilibrium|
Assume player 1 (the row player) is playing a mixed strategy , for .
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 if player 2 is playing the mixed strategy . Hence is trembling-hand perfect.
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. Hence is not trembling-hand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating most often to L if there is a small chance of error in the behavior of player 1.
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 <Down,Right> is imperfect, as Left (weakly) dominates Right for Player 2 and Up (weakly) dominates Down for Player 1.
|Extensive-form trembling hand perfect equilibrium|
|A solution concept in game theory|
|Subset of||Subgame perfect equilibrium, Perfect Bayesian equilibrium, Sequential equilibrium|
|Proposed by||Reinhard Selten|
|Used for||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.[ 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. In fact, a normal-form trembling hand perfect equilibrium does not even have to be subgame perfect.
Myerson (1978)pointed out that perfection is sensitive to the addition of a strictly dominated strategy, and instead proposed another refinement, known as proper equilibrium.
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