Quasi-perfect equilibrium is a refinement of
Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
for
extensive form games due to
Eric van Damme.
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
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
significant ...
.
Mertens' voting game
It has been argued by
Jean-François Mertens
Jean-François Mertens (11 March 1946 – 17 July 2012) was a Belgian game theorist and mathematical economist.
Mertens contributed to economic theory in regards to order-book of market games, cooperative games, noncooperative games, repeated ga ...
[Jean-François Mertens. "Two examples of strategic equilibrium." ''Games and Economic Behavior'', 8:378--388, 1995.] that quasi-perfect equilibrium is superior to
Reinhard Selten
Reinhard Justus Reginald Selten (; 5 October 1930 – 23 August 2016) was a German economist, who won the 1994 Nobel Memorial Prize in Economic Sciences (shared with John Harsanyi and John Nash). He is also well known for his work in bou ...
'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.
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.
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.
References
{{game theory
Game theory equilibrium concepts