(Normal form) trembling hand perfect equilibrium  

A solution concept in game theory  
Relationships  
Subset of  Nash Equilibrium 
Superset of  Proper equilibrium 
Significance  
Proposed by  Reinhard Selten 
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.
Contents 
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.
The game represented in the following normal form matrix has two pure strategy 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 (1 − ε,ε), for 0 < ε < 1. 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 (1 − ε,ε). Hence <U,L> is tremblinghand perfect.
However, similar analysis fails for the strategy profile <D,R>.
Assume player 1 is playing a mixed strategy (ε,1 − ε). Player 2's expected payoff from playing L is:
Player 2's expected payoff from playing 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.
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.
Extensiveform trembling hand perfect equilibrium  

A solution concept in game theory  
Relationships  
Subset of  Subgame perfect equilibrium, Perfect Bayesian equilibrium, Sequential equilibrium 
Significance  
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 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.

