In game theory, trembling hand perfect equilibrium is a type of refinement of a Nash equilibrium that was first proposed by 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 in an ${\displaystyle n}$-player strategic game where every pure strategy is played with positive 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 mixed strategy profile ${\displaystyle \sigma =(\sigma _{1},\ldots ,\sigma _{n})}$ as being trembling hand perfect if there is a sequence of perturbed games strategy profiles ${\displaystyle \{\sigma ^{k}\}_{k=1,2,\ldots }}$ that converges to ${\displaystyle \sigma }$ such that for every ${\displaystyle k}$ and every player ${\displaystyle 1\leq i\leq n}$ the strategy ${\displaystyle \sigma _{i}}$ is the best reply to ${\displaystyle \sigma _{-i}^{k}}$.

Note: All completely mixed Nash equilibria are perfect.

Note 2: The mixed strategy extension of any finite normal-form game has at least one perfect equilibrium.

The game represented in the following normal form matrix has two pure strategies Nash equilibria, namely ${\displaystyle \langle {\text{Up}},{\text{Left}}\rangle }$ and ${\displaystyle \langle {\text{Down}},{\text{Right}}\rangle }$. However, only ${\displaystyle \langle {\text{U}},{\text{L}}\rangle }$ is trembling-hand perfect.

Assume player 1 (the row player) is playing a mixed strategy ${\displaystyle (1-\varepsilon ,\varepsilon )}$, for ${\displaystyle 0<\varepsilon <1}$.

Player 2's expected payoff from playing L is:

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