# Trembling hand perfect equilibrium

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(Normal form) trembling hand perfect equilibrium | |
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A solution concept in game theory | |

Relationship | |

Subset of | Nash Equilibrium |

Superset of | Proper equilibrium |

Significance | |

Proposed by | Reinhard Selten |

In game theory, **trembling hand perfect equilibrium** is a refinement of Nash equilibrium due to Reinhard Selten.^{[1]} 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.

## Definition[edit]

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* strategy (both pure and mixed) 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.

**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.^{[2]}

## Example[edit]

The game represented in the following normal form matrix has two pure strategy Nash equilibria, namely and . However, only is trembling-hand perfect.

Left | Right | |

Up | 1, 1 | 2, 0 |

Down | 0, 2 | 2, 2 |

Trembling hand perfect equilibrium |

Assume player 1 (the row player) is playing a mixed strategy , for .

Player 2's expected payoff from playing L is:

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

For small values of , player 2 maximizes his expected payoff by placing a minimal weight on R and maximal weight on L. By symmetry, player 1 should place a minimal weight on D if player 2 is playing the mixed strategy . Hence is trembling-hand perfect.

However, similar analysis fails for the strategy profile .

Assume player 2 is playing a mixed strategy . Player 1's expected payoff from playing U is:

Player 1's expected payoff from playing D is:

For all positive values of , player 1 maximizes his expected payoff by placing a minimal weight on D and maximal weight on U. Hence is not trembling-hand perfect because player 2 (and, by symmetry, player 1) maximizes his expected payoff by deviating most often to L if there is a small chance of error in the behavior of player 1.

## Trembling hand perfect equilibria of two-player games[edit]

For 2x2 games, the set of trembling-hand perfect equilibria coincides with the set of equilibria consisting of two undominated strategies. In the example above, we see that the equilibrium <Down,Right> is imperfect, as Left (weakly) dominates Right for Player 2 and Up (weakly) dominates Down for Player 1.^{[3]}

## Trembling hand perfect equilibria of extensive form games[edit]

Extensive-form trembling hand perfect equilibrium | |
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A solution concept in game theory | |

Relationship | |

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.

- 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 extensive-form game must be played with non-zero probability. This leads to the notion of a
**normal-form trembling hand perfect 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 non-zero probability. Limits of equilibria of such perturbed games as the tremble probabilities goes to zero are called
**extensive-form trembling hand perfect equilibria**.

The notions of normal-form and extensive-form trembling hand perfect equilibria are incomparable, i.e., an equilibrium of an extensive-form game may be normal-form trembling hand perfect but not extensive-form trembling hand perfect and vice versa. As an extreme example of this, Jean-François Mertens has given an example of a two-player extensive form game where no extensive-form trembling hand perfect equilibrium is admissible, i.e., the sets of extensive-form and normal-form trembling hand perfect equilibria for this game are disjoint.^{[citation needed]}

An extensive-form trembling hand perfect equilibrium is also a sequential equilibrium. A normal-form trembling hand perfect equilibrium of an extensive form game may be sequential but is not necessarily so. In fact, a normal-form trembling hand perfect equilibrium does not even have to be subgame perfect.

## Problems with perfection[edit]

Myerson (1978)^{[4]} pointed out that perfection is sensitive to the addition of a strictly dominated strategy, and instead proposed another refinement, known as proper equilibrium.

## References[edit]

**^**Selten, R. (1975). "A Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games".*International Journal of Game Theory*.**4**(1): 25–55. doi:10.1007/BF01766400.**^**Selten, R.: Reexamination of the perfectness concept for equilibrium points in extensive games. Int. J. Game Theory4, 1975, 25–55.**^**Van Damme, Eric (1987).*Stability and Perfection of Nash Equilibria*. doi:10.1007/978-3-642-96978-2. ISBN 978-3-642-96980-5.**^**Myerson, Roger B. "Refinements of the Nash equilibrium concept." International journal of game theory 7.2 (1978): 73-80.

## Further reading[edit]

- Osborne, Martin J.; Rubinstein, Ariel (1994).
*A Course in Game Theory*. MIT Press. pp. 246–254. ISBN 9780262650403.