Uncorrelated asymmetry

In game theory, an uncorrelated asymmetry is an arbitrary distinguishing feature between players in an otherwise symmetric game. This concept refers to asymmetries that are unrelated to the payoffs or strategic structure of the game itself, but instead arise from players' ability to distinguish their roles or identities within the game. It is opposed to correlated asymmetries, where the asymmetry directly affects payoffs or strategic considerations. The term was introduced by John Maynard Smith in 1973.[1]

For example, consider two drivers approaching each other on a narrow road where only one can pass at a time. The payoffs are symmetric—both prefer that one yields while the other proceeds rather than both attempting to proceed simultaneously. However, if one driver arrived first or is driving on the "right" side according to local convention, this creates an uncorrelated asymmetry that can guide their strategies without changing the underlying payoff structure.

The key feature of an uncorrelated asymmetry lies in players' knowledge of their assigned roles. In a symmetric game, if players know whether they are Player 1, Player 2, or more generally whether they are the row player versus column player in a bimatrix game, then an uncorrelated asymmetry exists. Conversely, if players cannot distinguish their roles, no uncorrelated asymmetry is present.

This creates what is sometimes called an information asymmetry, though this terminology can be misleading. Games with uncorrelated asymmetries remain games of complete information in the technical sense—all players know the full game structure and payoffs. The asymmetry refers specifically to each player's knowledge of their own role: one player knows they are Player 1 while the other knows they are Player 2. This differs from information sets in extensive form games, which concern knowledge about the history of play or opponents' private information.

## Applications

Uncorrelated asymmetries play a crucial role in determining which Nash equilibria qualify as evolutionarily stable strategies (ESS) in coordination games and discoordination games. In games like the game of chicken or battle of the sexes:

* Without uncorrelated asymmetry: the mixed strategy Nash equilibrium typically serves as the ESS
* With uncorrelated asymmetry: pure strategy conditional equilibria become evolutionarily stable, where each player's strategy depends on their assigned role

The most widely cited example of uncorrelated asymmetry is territory ownership in the hawk-dove game.[2] Even when both players (the "owner" and "intruder") face identical payoffs, making the game payoff-symmetric, their roles create an uncorrelated asymmetry.[3] This enables stable strategies such as:

* Bourgeois strategy: The territory owner plays Hawk (aggressive), while the intruder plays Dove (submissive)[4]
* Anti-bourgeois strategy: The reverse pattern, though this is considered less biologically plausible

Other examples include conventions based on arbitrary physical or social markers, such as age, size, or arrival time, which can serve as coordination devices without affecting the underlying strategic incentives.

In evolutionary game theory, uncorrelated asymmetries help explain how populations can maintain stable behavioral patterns in otherwise symmetric interactions. They provide a mechanism for coordination that doesn't require communication or repeated interaction, making them particularly relevant for understanding animal behavior and the evolution of conventions in human societies.

## See also

* The section on uncorrelated asymmetries in Game of chicken
* The section on discoordination games in Best response.

## References

1. ↑ Smith, John Maynard; Parker, G. A. (1976-02-01). "The logic of asymmetric contests". Animal Behaviour 24 (1): 159–175. doi:10.1016/S0003-3472(76)80110-8. ISSN 0003-3472. https://linkinghub.elsevier.com/retrieve/pii/S0003347276801108.
2. ↑ "Uncorrelated asymmetry". https://science.umd.edu/faculty/wilkinson/bsci338/L17gametheory/tsld014.htm.
3. ↑ Kokko, Hanna; López-Sepulcre, Andrés; Morrell, Lesley J. (2006). "From Hawks and Doves to Self-Consistent Games of Territorial Behavior". The American Naturalist 167 (6): 901–912. doi:10.1086/504604. ISSN 0003-0147. PMID 16685641. Bibcode: 2006ANat..167..901K. https://www.jstor.org/stable/10.1086/504604.
4. ↑ "Ownership: the Bourgeois Strategy". https://college.holycross.edu/faculty/kprestwi/behavior/ESS/bourgeois.html.

Maynard Smith, J (1982) Evolution and the Theory of Games Cambridge University Press. ISBN 0-521-28884-3

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Topics in game theory

Definitions|

* Cooperative game
* Determinacy
* Escalation of commitment
* Extensive-form game
* First-player and second-player win
* Game complexity
* Graphical game
* Hierarchy of beliefs
* Information set
* Normal-form game
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* Sequential game
* Simultaneous game
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Equilibrium
concepts|

* Nash equilibrium
* Subgame perfection
* Mertens-stable equilibrium
* Bayesian Nash equilibrium
* Perfect Bayesian equilibrium
* Trembling hand
* Proper equilibrium
* Epsilon-equilibrium
* Correlated equilibrium
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* Quasi-perfect equilibrium
* Evolutionarily stable strategy
* Risk dominance
* Core
* Shapley value
* Pareto efficiency
* Gibbs equilibrium
* Quantal response equilibrium
* Self-confirming equilibrium
* Strong Nash equilibrium
* Markov perfect equilibrium

Strategies|

* Dominant strategies
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* Strategy-stealing argument
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Classes
of games|

* Symmetric game
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* Go
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* Kuhn poker
* Nash bargaining game
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* Trust game
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Theorems|

* Arrow's impossibility theorem
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* Folk theorem
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Key
figures|

* Albert W. Tucker
* Amos Tversky
* Ariel Rubinstein
* Claude Shannon
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* David M. Kreps
* Donald B. Gillies
* Drew Fudenberg
* Eric Maskin
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* Antoine Augustin Cournot
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* Kenneth Arrow
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* Lloyd Shapley
* Melvin Dresher
* Merrill M. Flood
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* Oskar Morgenstern
* Paul Milgrom
* Peyton Young
* Reinhard Selten
* Robert Axelrod
* Robert Aumann
* Robert B. Wilson
* Roger Myerson
* Samuel Bowles
* Suzanne Scotchmer
* Thomas Schelling
* William Vickrey

See also|

* All-pay auction
* Alpha–beta pruning
* Bertrand paradox
* Bounded rationality
* Combinatorial game theory
* Confrontation analysis
* Coopetition
* First-move advantage in chess
* Game mechanics
* Glossary of game theory
* List of game theorists
* List of games in game theory
* No-win situation
* Solving chess
* Topological game
* Tragedy of the commons
* Tyranny of small decisions

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