page_title: Chicken (game)
section_title: Best response mapping and Nash equilibria
hierarchical_section_title: Chicken (game) / Best response mapping and Nash equilibria
context_page_description: The game of chicken, also known as the hawk–dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the outcome is ideal for one player to yield, but the individuals try to avoid it out of pride for not wanting to look like a 'chicken'. So each player taunts the other to increase the risk of shame in yielding. However, when one player yields, the conflict is avoided, and the game is for the most part over.
  The name "chicken" has its origins in a game in which two drivers drive towards each other on a collision course: one must swerve, or both may die in the crash, but if one driver swerves and the other does not, the one who swerved will be called a "chicken", meaning a coward; this terminology is most prevalent in political science and economics. The name "hawk–dove" refers to a situation in which there is a competition for a shared resource and the contestants can choose either conciliation or conflict; this terminology is most commonly used in biology and evolutionary game theory.
context_section_description: All anti-coordination games have three Nash equilibria.  Two of these are pure contingent strategy profiles, in which each player plays one of the pair of strategies, and the other player chooses the opposite strategy.  The third one is a mixed equilibrium, in which each player probabilistically chooses between the two pure strategies. Either the pure, or mixed, Nash equilibria will be evolutionarily stable strategies depending upon whether uncorrelated asymmetries exist.
  The best response mapping for all 2x2 anti-coordination games is shown in Figure 5. The variables x and y in Figure 5 are the probabilities of playing the escalated strategy ("Hawk" or "Don't swerve") for players X and Y respectively.  The line in graph on the left shows the optimum probability of playing the escalated strategy for player Y as a function of x.  The line in the second graph shows the optimum probability of playing the escalated strategy for player X as a function of y (the axes have not been rotated, so the dependent variable is plotted on the abscissa, and the independent variable is plotted on the ordinate).  The Nash equilibria are where the players' correspondences agree, i.e., cross. These are shown with points in the right hand graph.  The best response mappings agree (i.e., cross) at three points. The first two Nash equilibria are in the top left and bottom right corners, where one player chooses one strategy, the other player chooses the opposite strategy. The third Nash equilibrium is a mixed strategy which lies along the diagonal from the bottom left to top right corners. If the players do not know which one of them is which, then the mixed Nash is an evolutionarily stable strategy (ESS), as play is confined to the bottom left to top right diagonal line.  Otherwise an uncorrelated asymmetry is said to exist, and the corner Nash equilibria are ESSes.
caption: Fig.5 - Reaction correspondences for both players in a discoordination game. Compare with replicator dynamic vector fields below
exif: {}