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2014-01-30T20:43:36Z

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{{Infobox equilibrium|
name=(Normal form) trembling hand perfect equilibrium|
subsetof=[[Nash Equilibrium]]|
supersetof=[[Proper equilibrium]]|
discoverer=[[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.

==Definition==

First we define a '''perturbed game'''. A perturbed game is a copy of a base game, with the restriction that only [[mixed strategy|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 game|normal form matrix]] has two pure strategy [[Nash equilibrium|Nash equilibria]], namely <math>\langle \text{Up}, \text{Left}\rangle</math> and <math>\langle \text{Down}, \text{Right}\rangle</math>. However, only <math>\langle \text{U},\text{L}\rangle</math> is trembling-hand perfect.

{{Payoff matrix |
UL = 1, 1 | UR = 2, 0 |
DL = 0, 2 | DR = 2, 2 |
Float = right}}

Assume player 1 is playing a [[mixed strategy]] <math>(1-\varepsilon, \varepsilon)</math>, for <math> 0<\varepsilon <1</math>. Player 2's expected payoff from playing L is:

:<math>1(1-\varepsilon) + 2\varepsilon = 1+\varepsilon</math>

Player 2's expected payoff from playing the strategy R is:

:<math>0(1-\varepsilon) + 2\varepsilon = 2\varepsilon</math>

For small values of <math>\varepsilon</math>, 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 <math>(1-\varepsilon, \varepsilon)</math>. Hence <math>\langle \text{U},\text{L}\rangle</math> is trembling-hand perfect.

However, similar analysis fails for the strategy profile <math>\langle \text{D}, \text{R}\rangle</math>.

Assume player 2 is playing a [[mixed strategy]] <math>(\varepsilon, 1-\varepsilon)</math>. Player 1's expected payoff from playing U is:

:<math>1\varepsilon + 2(1-\varepsilon) = 2-\varepsilon</math>

Player 1's expected payoff from playing D is:

:<math>0(\varepsilon) + 2(1-\varepsilon) = 2-2\varepsilon</math>

For all positive values of <math>\varepsilon</math>, player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence <math>\langle \text{D}, \text{R}\rangle</math> 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.

== 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 decision rule|admissible]] equilibria, i.e., equilibria consisting of two undominated strategies. In the example above, we see that the imperfect equilibrium <nowiki><</nowiki>D,R<nowiki>></nowiki> is not admissible, as L (weakly) dominates R for Player 2.

== Trembling hand perfect equilibria of extensive form games ==

{{Infobox equilibrium|
name=Extensive-form trembling hand perfect equilibrium|
subsetof=[[Subgame perfect equilibrium]], [[Perfect Bayesian equilibrium]], [[Sequential equilibrium]]|
discoverer=[[Reinhard Selten]]|
usedfor=[[Extensive form game]]s
}}

There are two possible ways of extending the definition of trembling hand perfection to [[extensive form game]]s.

* One may interpret the extensive form as being merely a concise description of a normal form game and apply the concepts described above to this normal form game. In the resulting perturbed games, every [[strategy (game theory)|strategy]] of the extensive-form game must be played with non-zero probability. This leads to the notion of a '''normal-form trembling hand perfect equilibrium'''.

* Alternatively, one may recall that trembles are to be interpreted as modelling mistakes made by the players with some negligible probability when the game is played. Such a mistake would most likely consist of a player making another [[strategy (game theory)|move]] than the one intended at some point during play. It would hardly consist of the player choosing another [[strategy (game theory)|strategy]] than intended, i.e. a wrong plan for playing the entire game. To capture this, one may define the perturbed game by requiring that every [[strategy (game theory)|move]] at every [[information set]] is taken with non-zero probability. Limits of equilibria of such perturbed games as the tremble probabilities goes to zero are called '''extensive-form trembling hand perfect equilibria'''.

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 [[Quasi-perfect equilibrium|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 equilibrium|subgame perfect]].

== References ==

* [[Reinhard Selten|Selten, R.]] (1975) A reexamination of the perfectness concept for equilibrium points in extensive games. ''International Journal of Game Theory'' 4:25-55.

{{Game theory}}

[[Category:Game theory]]
[[Category:Non-cooperative games]]

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