Trembling hand perfect equilibrium

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. 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 pure strategy 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. The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely ⟨ Up , Left ⟩ {displaystyle langle { ext{Up}},{ ext{Left}} angle } and ⟨ Down , Right ⟩ {displaystyle langle { ext{Down}},{ ext{Right}} angle } . However, only ⟨ U , L ⟩ {displaystyle langle { ext{U}},{ ext{L}} angle } is trembling-hand perfect. Assume player 1 (the row player) is playing a mixed strategy ( 1 − ε , ε ) {displaystyle (1-varepsilon ,varepsilon )} , for 0 < ε < 1 {displaystyle 0<varepsilon <1} .