Discoordination game
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- For other uses, see Chicken (disambiguation).
In game theory a discoordination game, also known as congestion game or crowding game is a class of games in which it is mutually beneficial for the players to play different strategies. It is the opposite of a coordination game, where playing the same strategy Pareto dominates playing different strategies. The underlying concept is that players use a shared resource. In coordination games, sharing the resource creates a benefit for all: the resource is non-rivalrous, and the shared usage creates positive network externalities. In discoordination games the resource is rivalrous and sharing comes at a cost (or negative network externality). The canonical example of a congestion game for many players is the tragedy of the commons. A hybrid form between coordination and discoordination is the matching pennies game, where sharing benefits one player and comes at a cost to the other.
The most common forms of this game are "Chicken", which is most prevalent in the political science and economics literatures, and the "Hawk-Dove game", which is most commonly used in the biological literature. The principle of the game is that while it is preferable not to yield to the opponent, if neither player yields, then outcome is the worst possible one for both players.
Discoordination 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 Nash is a mixed equilibrium, in which the each player probabilistically chooses one of the two pure strategies. Either the pure, or mixed, Nash equilibria will be evolutionarily stable strategies depending upon whether uncorrelated asymmetries exist.
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[edit] Chicken
The game of chicken models a two drivers both headed for a single lane bridge from opposite directions. The first to swerve away yields the bridge to the other. If neither player swerves, the result is a costly deadlock in the middle of the bridge, or a potentially fatal head-on collision.
A formal version of the game of chicken has been the subject of serious research in game theory[1]. Because the "loss" of swerving is so trivial compared to the crash that occurs if nobody swerves, the reasonable strategy would seem to be to swerve before a crash is likely. Yet, knowing this, if one believes one's opponent to be reasonable, one may well decide not to swerve at all, in the belief that he will be reasonable and decide to swerve, leaving the other player the winner. This unstable strategy can be formalized by saying there is more than one Nash equilibrium, which is a pair of strategies for which neither player gains by changing his own strategy while the other stays the same. (In this case, the equilibria are the two situations wherein one player swerves while the other does not.)
| Swerve | Straight | |
| Swerve | 0, 0 | −1, +1 |
| Straight | +1, −1 | −10, −10 |
| Chicken example | ||
One tactic in the game is for one party to signal their intentions convincingly before the game begins (this has been modelled explicitly in the Hawk-Dove game[2]). For example, if one party were to ostentatiously disable their steering wheel just before the match, the other party would be compelled to swerve [3]. This shows that, in some circumstances, reducing one's own options can be a good strategy. One real-world example is a protester who handcuffs himself to an object, so that no threat can be made which would compel him to move (since he cannot move). Another example, taken from fiction, is found in Stanley Kubrick's Dr. Strangelove. In that film, the Russians build a "doomsday machine," a device that would trigger world annihilation once Russia was to be hit by nuclear weapons. That way, they sought to deter any American attack. While in the movie the Russians deploy their doomsday machine covertly, thus failing to signal, the plotline does highlight the importance of signalling for understanding Chicken.
The phrase game of chicken is also used as a metaphor for a situation where two parties engage in a showdown where they have nothing to gain, and only pride stops them from backing down. Bertrand Russell famously compared the game of chicken to nuclear brinkmanship:
"Since the nuclear stalemate became apparent, the Governments of East and West have adopted the policy which Mr. Dulles calls 'brinkmanship'. This is a policy adapted from a sport which, I am told, is practised by some youthful degenerates. This sport is called 'Chicken!'. It is played by choosing a long straight road with a white line down the middle and starting two very fast cars towards each other from opposite ends. Each car is expected to keep the wheels of one side on the white line. As they approach each other, mutual destruction becomes more and more imminent. If one of them swerves from the white line before the other, the other, as he passes, shouts 'Chicken!', and the one who has swerved becomes an object of contempt. As played by irresponsible boys, this game is considered decadent and immoral, though only the lives of the players are risked. But when the game is played by eminent statesmen, who risk not only their own lives but those of many hundreds of millions of human beings, it is thought on both sides that the statesmen on one side are displaying a high degree of wisdom and courage, and only the statesmen on the other side are reprehensible. This, of course, is absurd. Both are to blame for playing such an incredibly dangerous game. The game may be played without misfortune a few times, but sooner or later it will come to be felt that loss of face is more dreadful than nuclear annihilation. The moment will come when neither side can face the derisive cry of 'Chicken!' from the other side. When that moment is come, the statesmen of both sides will plunge the world into destruction."[4]
[edit] Hawk-Dove game
The game is also known as the Hawk-Dove game in biological game theory. In this interpretation two players contesting an indivisible resource choose between two strategies, one more escalated than the other[5]. They can use threat displays (play Dove), or physically attack each other (play Hawk). If both players choose the Hawk strategy, then they fight until one is injured and the other wins. If only one player chooses Hawk, then this player defeats the Dove player. If both players play Dove, there is a tie, and each player receives a payoff lower than the profit of a hawk defeating a dove. The exact value of the Dove vs Dove playoff varies between model formulations, sometime the players are assumed to split the payoff equally (1/2V each) other times the payoff is assumed to be zero (since this is the expected payoff to a war of attrition, which is the presumed models for a contest decided by display duration).
The earliest presentation of a form of the Hawk-Dove game was by John Maynard Smith and George Price in their 1973 Nature paper, The logic of animal conflict[6]. The traditional [5][7] payoff matrix for the Hawk-Dove game is:
-
Hawk Dove Hawk 1/2*(V-C), 1/2*(V-C) V, 0 Dove 0, V V/2, V/2
Where V is the value of the contested resource, and C is the cost of an escalated fight. It is (almost always) assumed that the value of the resource is less than the cost of a fight is, i.e. V<C. Were it not the case that V<C, then the game will not correspond to the game of chicken. A common payoff variant is to assume that the Dove vs. Dove outcome leads to a War of Attrition, for which the expected payoff is zero, and thus the V/2 payoffs are replaced with zeros.
While the Hawk Dove game is typically taught and discussed with the payoffs in terms of V and C, the solutions hold true for any matrix with the following payoffs:
-
I J I a, a b, c J c, b d, d
where a<c, and d<b. [7]
[edit] Uncorrelated asymmetries and solutions to the Hawk Dove game
The Hawk Dove game has three Nash equilibria:
- the row player chooses Hawk while the column player chooses Dove,
- the row player chooses Dove while the column player chooses Hawk, and
- both players play a mixed strategy where Hawk is played with probability p, and Dove is played with probability 1-p.
It can be demonstrated that p=V/C in the traditional payoff version, and p=(b-d)/(b+c-a-d) in the generic payoff version.
While there are three Nash equilibria, which will be evolutionarily stable strategies (ESSs) depends upon the existence of any uncorrelated asymmetry in the game (see also section on discoordination games in best response). In order for row players to choose one strategy and column players the other, the players must be able to distinguish which role (column or row player) they have. The standard biological interpretation of this uncorrelated asymmetry is that one player is the territory owner, while the other is an intruder on the territory.
If no such uncorrelated asymmetry exists then both players must choose the same strategy, and the ESS will be the mixing Nash equilibrium. If there is an uncorrelated asymmetry, then the mixing Nash is not an ESS, but the two pure, role contigent, Nash equilibria are.
[edit] Best response mapping
The best response mapping for all 2x2 discoordination games is shown in Fig.1. The variables x and y in Fig. 1 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 (not the axes have not been rotated, and so the dependent variable is plotted on the abscissa, and the independent variable is plotted on the ordinate). The Nash equilibria are where the two player's correspondences agree, ie. cross. These are shown with points in the right hand graph. The best response mappings agree (i.e. cross) at three points. These three Nash equilibria are in the top left and bottom right corners -where one player chooses one strategy, the other player chooses the opposite strategy- and 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.
[edit] See also
[edit] References
- ^ Rapoport, A. & Chammah, A. M. (1966) The game of chicken, American Behavioral Scientist 10: 10-14, 23-28.
- ^ Kim, Y-G. 1995. Status signalling games in animal contests, Journal of Theoretical Biology 176 221-231.
- ^ Kahn, H. 1965. On escalation: metaphors and scenarios Praeger Publ. Co. New York, cited in Rapoport & Chammah, 1966)
- ^ Russel, B. W. (1959) Common Sense and Nuclear Warfare London: George Allen & Unwin, p30
- ^ a b Maynard Smith, J. & Parker, G.A. 1976. The logic of asymmetric contests. Animal Behaviour 24, 159-175.
- ^ Maynard Smith, J. & Price, G.R. (1973) The logic of animal conflict, Nature 246, 15-18.
- ^ a b Maynard Smith, J. (1982) Evolution and the Theory of Games. Cambridge University Press
- Deutsch, M. The Resolution of Conflict: Constructive and Destructive Processes. Yale University Press, New Haven, CT, 1973.
- Moore, C. W. The Mediation Process: Practical Strategies for Resolving Conflict. Jossey-Bass, San Francisco, 1986.
[edit] External links
- The game of chicken as a metaphor for human conflict
- Game-theoretic analysis of the game of chicken
- Game of Chicken - Rebel Without a Cause by Elmer G. Wiens.
