(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.

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.

**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] }

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.

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] }

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.

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.

In game theory, the **Nash equilibrium**, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.

In game theory, the **best response** is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.

In game theory, the **centipede game**, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.

In game theory, a player's **strategy** is any of the options which he or she chooses in a setting where the outcome depends *not only* on their own actions *but* on the actions of others. A player's strategy will determine the action which the player will take at any stage of the game.

In game theory, a **solution concept** is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

An **extensive-form game** is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature".

In game theory, a **Bayesian game** is a game in which players have incomplete information about the other players. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. These beliefs are represented by a probability distribution over the possible payoff functions.

In game theory, **folk theorems** are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.

In game theory, a **correlated equilibrium** is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.

In game theory, the **purification theorem** was contributed by Nobel laureate John Harsanyi in 1973. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.

**Sequential equilibrium** is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. A sequential equilibrium specifies not only a strategy for each of the players but also a **belief** for each of the players. A belief gives, for each information set of the game belonging to the player, a probability distribution on the nodes in the information set. A profile of strategies and beliefs is called an **assessment** for the game. Informally speaking, an assessment is a perfect Bayesian equilibrium if its strategies are sensible given its beliefs **and** its beliefs are confirmed on the outcome path given by its strategies. The definition of sequential equilibrium further requires that there be arbitrarily small perturbations of beliefs and associated strategies with the same property.

**Quasi-perfect equilibrium** is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme.

In game theory, a **subgame perfect equilibrium** is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that if the players played any smaller game that consisted of only one part of the larger game, their behavior would represent a Nash equilibrium of that smaller game. Every finite extensive game with perfect recall has a subgame perfect equilibrium.

**Quantal response equilibrium** (**QRE**) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

**Proper equilibrium** is a refinement of Nash Equilibrium due to Roger B. Myerson. Proper equilibrium further refines Reinhard Selten's notion of a trembling hand perfect equilibrium by assuming that more costly trembles are made with significantly smaller probability than less costly ones.

In game theory, an **epsilon-equilibrium**, or near-Nash equilibrium, is a strategy profile that approximately satisfies the condition of Nash equilibrium. In a Nash equilibrium, no player has an incentive to change his behavior. In an approximate Nash equilibrium, this requirement is weakened to allow the possibility that a player may have a small incentive to do something different. This may still be considered an adequate solution concept, assuming for example status quo bias. This solution concept may be preferred to Nash equilibrium due to being easier to compute, or alternatively due to the possibility that in games of more than 2 players, the probabilities involved in an exact Nash equilibrium need not be rational numbers.

In game theory, a **stochastic game**, introduced by Lloyd Shapley in the early 1950s, is a dynamic game with **probabilistic transitions** played by one or more players. The game is played in a sequence of stages. At the beginning of each stage the game is in some **state**. The players select actions and each player receives a **payoff** that depends on the current state and the chosen actions. The game then moves to a new random state whose distribution depends on the previous state and the actions chosen by the players. The procedure is repeated at the new state and play continues for a finite or infinite number of stages. The total payoff to a player is often taken to be the discounted sum of the stage payoffs or the limit inferior of the averages of the stage payoffs.

**Jean-François Mertens** was a Belgian game theorist and mathematical economist.

**Mertens stability** is a solution concept used to predict the outcome of a non-cooperative game. A tentative definition of stability was proposed by Elon Kohlberg and Jean-François Mertens for games with finite numbers of players and strategies. Later, Mertens proposed a stronger definition that was elaborated further by Srihari Govindan and Mertens. This solution concept is now called Mertens stability, or just stability.

The **Berge equilibrium** is a game theory solution concept named after the mathematician Claude Berge. It is similar to the standard Nash equilibrium, except that it aims to capture a type of altruism rather than purely non-cooperative play. Whereas a Nash equilibrium is a situation in which each player of a strategic game ensures that they personally will receive the highest payoff given other players' strategies, in a Berge equilibrium every player ensures that all other players will receive the highest payoff possible. Although Berge introduced the intuition for this equilibrium notion in 1957, it was only formally defined by Vladislav Iosifovich Zhukovskii in 1985, and it was not in widespread use until half a century after Berge originally developed it.

- ↑ 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.

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

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