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 assumingthat the players, through a "slip of the hand" or tremble,
may chooseunintended strategies, albeit with negligible probability.
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 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 <nowiki><Up, Left></nowiki> and <nowiki><Down, Right></nowiki>. However, only <nowiki><U,L></nowiki> is trembling-hand perfect.
Assume player 1 is playing a mixed strategy
<math>(1-epsilon, epsilon)</math>, for <math> 0<epsilon <1</math>. Player 2's expected payoff from playing L......