Uncorrelated asymmetry
Encyclopedia
In game theory
an uncorrelated asymmetry is an arbitrary asymmetry in a game which is otherwise symmetrical
. The name 'uncorrelated asymmetry' is due to John Maynard Smith
who called payoff relevant asymmetries in games with similar roles for each player 'correlated asymmetries' (note that any game with correlated asymmetries must also have uncorrelated asymmetries).
The explanation of an uncorrelated asymmetry usually makes reference to "informational asymmetry". Which may confuse some readers, since, games which may have uncorrelated asymmetries are still games of complete information
. What differs between the same game with and without an uncorrelated asymmetry is whether the players know which role they have been assigned. If players in a symmetric game know whether they are Player 1, Player 2, etc. (or row vs. column player in a bimatrix normal form game
) then an uncorrelated asymmetry exists. If the players do not know which player they are then no uncorrelated asymmetry exists. The information asymmetry
is that one player believes he is player 1 and the other believes he is player 2. Therefore, "informational asymmetry" does not refer to knowledge in the sense of an information set
in an extensive form game
.
The concept of uncorrelated asymmetries is important in determining which Nash equilibria
are evolutionarily stable strategies in discoordination games such as the game of chicken. In these games the mixing Nash is the ESS if there is no uncorrelated asymmetry, and the pure conditional Nash equilibria are ESSes when there is an uncorrelated asymmetry.
The usual applied example of an uncorrelated asymmetry is territory ownership in the hawk-dove game. Even if the two players ("owner" and "intruder") have the same payoffs (ie, the game is payoff symmetric), the territory owner will play Hawk, and the intruder Dove, in what is known as the 'Bourgeois strategy' (the reverse is also an ESS known as the 'anti-bourgeois strategy', but makes little biological sense).
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
an uncorrelated asymmetry is an arbitrary asymmetry in a game which is otherwise symmetrical
Symmetric game
In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff to the strategies, then a game is symmetric. ...
. The name 'uncorrelated asymmetry' is due to John Maynard Smith
John Maynard Smith
John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
who called payoff relevant asymmetries in games with similar roles for each player 'correlated asymmetries' (note that any game with correlated asymmetries must also have uncorrelated asymmetries).
The explanation of an uncorrelated asymmetry usually makes reference to "informational asymmetry". Which may confuse some readers, since, games which may have uncorrelated asymmetries are still games of complete information
Complete information
Complete information is a term used in economics and game theory to describe an economic situation or game in which knowledge about other market participants or players is available to all participants. Every player knows the payoffs and strategies available to other players.Complete information...
. What differs between the same game with and without an uncorrelated asymmetry is whether the players know which role they have been assigned. If players in a symmetric game know whether they are Player 1, Player 2, etc. (or row vs. column player in a bimatrix normal form game
Normal form game
In game theory, normal form is a way of describing a game. Unlike extensive form, normal-form representations are not graphical per se, but rather represent the game by way of a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some...
) then an uncorrelated asymmetry exists. If the players do not know which player they are then no uncorrelated asymmetry exists. The information asymmetry
Information asymmetry
In economics and contract theory, information asymmetry deals with the study of decisions in transactions where one party has more or better information than the other. This creates an imbalance of power in transactions which can sometimes cause the transactions to go awry, a kind of market failure...
is that one player believes he is player 1 and the other believes he is player 2. Therefore, "informational asymmetry" does not refer to knowledge in the sense of an information set
Information set
In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed. If the game has perfect information, every information set contains only one member, namely the point...
in an 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...
.
The concept of uncorrelated asymmetries is important in determining which Nash equilibria
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...
are evolutionarily stable strategies in discoordination games such as the game of chicken. In these games the mixing Nash is the ESS if there is no uncorrelated asymmetry, and the pure conditional Nash equilibria are ESSes when there is an uncorrelated asymmetry.
The usual applied example of an uncorrelated asymmetry is territory ownership in the hawk-dove game. Even if the two players ("owner" and "intruder") have the same payoffs (ie, the game is payoff symmetric), the territory owner will play Hawk, and the intruder Dove, in what is known as the 'Bourgeois strategy' (the reverse is also an ESS known as the 'anti-bourgeois strategy', but makes little biological sense).
See also
- The section on uncorrelated asymmetries in Game of chicken
- The section on discoordination games in Best response.