Trembling hand perfect equilibrium
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Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision 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 offtheequilibrium play into account by assuming that the players, through a "slip of the hand" or tremble, may choose unintended strategies, albeit with negligible probability.
DefinitionEdit
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 nonzero 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.
ExampleEdit
The game represented in the following normal form matrix has two Nash equilibria, namely <Up, Left> and <Down, Right>. However, only <U,L> is tremblinghand perfect.
Left  Right  
Up  1, 1  2, 0 
Down  0, 2  2, 2 
Trembling hand perfect equilibrium 
Assume player 1 is playing a mixed strategy . Player 2's expected payoff from playing L is:
Player 2's expected payoff from playing the strategy R is:
For small values of ε, player 2 maximizes his expected payoff by placing a minimal weight on R. By symmetry, player 1 should place a minimal weight on D if player 2 is playing the mixed strategy . Hence <U,L> is tremblinghand perfect.
However, similar analysis fails for the strategy profile <D,R>.
Assume player 1 is playing a mixed strategy . Player 2's expected payoff from playing L is:
Player 2's expected payoff from playing the strategy R is:
For all positive values of ε, player 2 maximizes his expected payoff by placing a minimal weight on R. Hence <D, R> is not tremblinghand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating if there is a small chance of error.
Trembling hand perfect equilibria of twoplayer games Edit
For twoplayer 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.
Trembling hand perfection of equilibria of extensive form games Edit

There are two possible ways of extending the definition of trembling hand perfection to extensive form games.
 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 of the extensiveform game must be played with nonzero probability. This leads to the notion of a normalform trembling hand prefect 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 move than the one intended at some point during play. It would hardly consist of the player choosing another 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 move at every information set is taken with nonzero probability. Limits of equilibria of such perturbed games as the tremble probabilities goes to zero are called extensiveform trembling hand perfect equilibria.
The notions of normalform and extensiveform trembling hand perfect equilibria are incomparable, i.e., an equilibrium of an extensiveform game may be normalform trembling hand perfect but not extensiveform trembling hand perfect and vice versa. As an extreme example of this, JeanFrançois Mertens has given an example of a twoplayer extensive form game where no extensiveform trembling hand perfect equilibrium is admissible, i.e., the sets of extensiveform and normalform trembling hand perfect equilibria for this game are disjoint.
An extensiveform trembling hand perfect equilibrium is also a sequential equilibrium. A normalform trembling hand perfect equilibrium of an extensive form game may be sequential but is not necessarily so. In fact, a normalform trembling hand perfect equilibrium does not even have to be subgame perfect.
References Edit
 Reinhard Selten. "A reexamination of the perfectness concept for equilibrium points in extensive games". International Journal of Game Theory 4:2555, 1975.
 Selten, R (1983) Evolutionary stability in extensive twoperson games. Math. Soc. Sci. 5:269363.
 Selten, R.(1988) Evolutionary stability in extensive twoperson games  correction and further development. Math. Soc. Sci. 16:223266
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