Quasi-perfect equilibrium
Encyclopedia
Quasi-perfect equilibrium is a refinement of Nash Equilibrium
for extensive form game
s 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
.
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.
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. 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 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.
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a 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 his own strategy unilaterally...
for extensive form game
Extensive form game
An extensive-form game is a specification of a game in game theory, allowing explicit representation of a number of important 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 he...
s 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
Sequential 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 eachof the players but also a belief for each of the players...
. It is itself refined by normal form proper equilibrium
Proper equilibrium
Proper equilibrium is a refinement of Nash Equilibrium due to Roger B. Myerson.Proper equilibrium further refines Reinhard Selten's notion of atrembling hand perfect equilibrium by assuming that more costly trembles are made with...
.
Mertens' voting game
It has been argued by Jean-François Mertens that quasi-perfect equilibrium issuperior to Reinhard Selten
Reinhard Selten
-Life and career:Selten was born in Breslau in Lower Silesia, now in Poland, to a Jewish father, Adolf Selten, and Protestant mother, Käthe Luther. For his work in game theory, Selten won the 1994 Nobel Memorial Prize in Economic Sciences...
's notion of extensive-form trembling hand perfect equilibrium
Trembling hand perfect equilibrium
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...
as a quasi-perfect equilibrium is guaranteed to describe admissible
Admissible decision rule
In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....
behavior. In contrast, for a certain two-player voting game no extensive-form trembling hand perfect equilibrium
Trembling hand perfect equilibrium
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...
describes admissible
Admissible decision rule
In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....
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. This is also the unique admissible
Admissible decision rule
In statistical decision theory, an admissible decision rule is a rule for making a decision such that there isn't any other rule that is always "better" than it, in a specific sense defined below....
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 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.