Quasi-perfect equilibrium

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Quasi-perfect Equilibrium
A solution concept in game theory
Relationship
Subset of Sequential equilibrium, normal-form trembling hand perfect equilibrium
Significance
Proposed by Eric van Damme
Used for Extensive form games
Example Mertens' voting game

Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme. [1]

Eric Eleterius Coralie van Damme is a Dutch economist and Professor of Economics at the Tilburg University, known for his contributions to game theory.

Informally, a player playing by a strategy from a quasi-perfect equilibrium takes observed as well as potential future mistakes of his opponents into account but assumes that he himself will not make a mistake in the future, even if he observes that he has done so in the past.

Quasi-perfect equilibrium is a further refinement of sequential equilibrium. It is itself refined by normal form proper equilibrium.

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.

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.

Mertens' voting game

It has been argued by Jean-François Mertens [2] that quasi-perfect equilibrium is superior to Reinhard Selten's notion of extensive-form trembling hand perfect equilibrium as a quasi-perfect equilibrium is guaranteed to describe admissible behavior. In contrast, for a certain two-player voting game no extensive-form trembling hand perfect equilibrium describes admissible behavior for both players.

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

Reinhard Justus Reginald Selten was a German economist, who won the 1994 Nobel Memorial Prize in Economic Sciences. He is also well known for his work in bounded rationality and can be considered as one of the founding fathers of experimental economics.

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.

The voting game suggested by Mertens may be described as follows:

• Two players must elect one of them to perform an effortless task. The task may be performed either correctly or incorrectly. If it is performed correctly, both players receive a payoff of 1, otherwise both players receive a payoff of 0. The election is by a secret vote. If both players vote for the same player, that player gets to perform the task. If each player votes for himself, the player to perform the task is chosen at random but is not told that he was elected this way. Finally, if each player votes for the other, the task is performed by somebody else, with no possibility of it being performed incorrectly.

In the unique quasi-perfect equilibrium for the game, each player votes for himself and, if elected, performs the task correctly. This is also the unique admissible behavior. But in any extensive-form trembling hand perfect equilibrium, at least one of the players believes that he is at least as likely as the other player to tremble and perform the task incorrectly and hence votes for the other player.

In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is no other rule that is always "better" than it, in the precise sense of "better" defined below. This concept is analogous to Pareto efficiency.

The example illustrates that being a limit of equilibria of perturbed games, an extensive-form trembling hand perfect equilibrium implicitly assumes an agreement between the players about the relative magnitudes of future trembles. It also illustrates that such an assumption may be unwarranted and undesirable.

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References

1. Eric van Damme. "A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games." International Journal of Game Theory 13:1--13, 1984.
2. Jean-François Mertens. "Two examples of strategic equilibrium." Games and Economic Behavior, 8:378--388, 1995.