Game of chicken

The game of Chicken, also known as the Hawk-Dove game, is an influential model of conflict for two players in game theory. The principle of the game is that while each player prefers not to yield to the opponent, the outcome where neither player yields is the worst possible one for both players. Mathematically, Chicken and Hawk-Dove are identical; the different names stem from parallel development of the basic principles in different research areas. "Chicken", is most prevalent in the political science and economics literatures, while "Hawk-Dove" is most commonly used in the biological and evolutionary game theory literatures. The game is similar to the prisoner's dilemma in that an "agreeable" mutual solution is unstable since both players are individually tempted to stray from it.

Popular versions
The game of Chicken models 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. It is presumed that the best thing for each driver is to stay straight while the other swerves (since the other is the "chicken" while a crash is avoided). Additionally, a crash is presumed to be the worst outcome for both players. This yields a situation where each player, in attempting to secure his best outcome, risks the worse. A similar version, under the name of "chickie run", is a central plot element in the movie Rebel Without a Cause where James Dean and Corey Allen race their cars towards a cliff instead of each other.

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."

As Dixit and Nalebuff point out in Thinking Strategically, Brinkmanship involves the introduction of an element of uncontrollable risk. Even if all players act rationally in the face of risk, uncontrollable events can still trigger the catastrophic outcome. In the "chickie run" scene this happens when Corey Allen's character cannot detach himself from the car and dies in the crash. The basic game-theoretic formulation of Chicken has no element of risk, and is also the contraction of a dynamic situation into a one-shot interaction.

The Hawk-Dove version of the game imagines two players (animals) contesting an indivisible resource who can choose between two strategies, one more escalated than the other. 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.

Chicken
A formal version of the game of Chicken has been the subject of serious research in game theory. Two versions of the payoff matrix for this game are presented here (Figures 1 and 2). In Figure 1 the outcomes are represented in words, where each player would prefer to win over tying, prefer to tie over losing, and prefer to lose over crashing. Figure 2 presents numerical payoffs which conform to this situation. Here the benefit of winning is 1, the cost of losing is -1, and the cost of crashing is -10.

Both Chicken and Hawk-Dove are anti-coordination games, games in which it is mutually beneficial for the players to play different strategies. In this way it can be thought of as 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 externalities. In anti-coordination games the resource is rivalrous but non-excludable and sharing comes at a cost (or negative externality).

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 situation 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 pure strategy equilibria are the two situations wherein one player swerves while the other does not.)

Hawk-Dove
In the biological literature, this game is referred to as Hawk-Dove. 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". The traditional payoff matrix for the Hawk-Dove game is given in Figure 3, 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. C &gt; V &gt; 0. If C ≤ V, the resulting game is not a game of Chicken.

The exact value of the Dove vs. Dove playoff varies between model formulations. Sometimes the players are assumed to split the payoff equally (V/2 each), other times the payoff is assumed to be zero (since this is the expected payoff to a war of attrition game, which is the presumed models for a contest decided by display duration).

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 payoffs in Figure 4, where W &gt; T &gt; L &gt; X.

Hawk-Dove variants
Biologists have explored modified versions of classic Hawk-Dove game to investigate a number of biologically relevant factors. These include adding variation in resource holding potential, and differences in the value of winning to the different players, allowing the players to threaten each other before choosing moves in the game, and extending the interaction to two plays of the game.

Pre-commitment
One tactic in the game is for one party to signal their intentions convincingly before the game begins. For example, if one party were to ostentatiously disable their steering wheel just before the match, the other party would be compelled to swerve. 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.

Players may also make non-binding threats to not swerve. This has been modeled explicitly in the Hawk-Dove game. Such threats work, but must be wastefully costly if the threat is one of two possible signals ("I will not swerve"/"I will swerve"), or they will be costless if there are three or more signals (in which case the signals will function as a game of "Rock, Paper, Scissors".

Best response mapping and Nash equilibria


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 the 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 (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, 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.

Strategy polymorphism vs strategy mixing
The ESS to the classic Hawk-Dove game is a mixed strategy. Formal game theory is indifferent to whether this mixture is due to all players in a population choosing randomly between the two pure strategies or whether the population is a polymorphic mixture of players dedicated to choosing a particular pure strategy. Biologically, these two options are strikingly different ideas. The Hawk-Dove game has been used as a basis for evolutionary simulations to explore which of these two modes of mixing ought to predominate in reality.

Symmetry breaking
In both Chicken and Hawk-Dove the only symmetric Nash equilibrium is the mixed strategy Nash equilibrium, where both individuals randomly chose between playing Hawk/Straight or Dove/Swerve. This mixed strategy equilibrium is often sub-optimal – both players would do better if they could coordinate their actions in some way. This observation has been made independently in two different contexts, with almost identical results.

Correlated equilibrium and Chicken
Consider the version of pictured in Figure 6. Like all forms of Chicken, there are three Nash equilibria. The two pure strategy Nash equilibria are (D, C) and (C, D). There is also a mixed strategy equilibrium where each player Dares with probability 1/3.

Now consider a third party (or some natural event) that draws one of three cards labeled: (C, C), (D, C), and (C, D). After drawing the card the third party informs the players of the strategy assigned to them on the card (but not the strategy assigned to their opponent). Suppose a player is assigned D, he would not want to deviate supposing the other player played their assigned strategy since he will get 7 (the highest payoff possible). Suppose a player is assigned C. Then the other player will play C with probability 1/2 and D with probability 1/2. The expected utility of Daring is 0(1/2) + 7(1/2) = 3.5 and the expected utility of chickening out is 2(1/2) + 6(1/2) = 4. So, the player would prefer to Chicken out.

Since neither player has an incentive to deviate, this probability distribution over the strategies is known as a correlated equilibrium of the game. Notably, the expected payoff for this equilibrium is 7(1/3) + 2(1/3) + 6(1/3) = 5 which is higher than the expected payoff of the mixed strategy Nash equilibrium.

Uncorrelated asymmetries and solutions to the Hawk-Dove game
While there are three Nash equilibria, which will be evolutionarily stable strategies (ESSes) depends upon the existence of any uncorrelated asymmetry in the game (see also section on anti-coordination 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. 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 contingent, Nash equilibria are.

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. In most cases, the territory owner plays Hawk while the intruder plays Dove. In this sense, the evolution of strategies in Hawk-Dove can be seen as the evolution of a sort of prototypical version of ownership. Game-theoretically, however, there is nothing special about this solution. The opposite solution – where the owner plays dove and the intruder plays Hawk – is equally stable. In fact, this solution is present in a certain species of spider; when an invader appears the occupying spider leaves. In order to explain the prevalence of property rights over "anti-property rights" one must discover a way to break this additional symmetry.

Replicator dynamics


The replicator dynamics is a simple model of strategy change commonly used in evolutionary game theory. In this model a strategy which does better than the average increases in frequency at the expense of strategies that do worse than the average. There are two versions of the replicator dynamics. In one version, there is a single population which plays against itself. In another, there are two population models where each population only plays against the other population (and not against itself).

In the one population model, the only stable state is the mixed strategy Nash equilibrium. Every initial population proportion (except all Hawk and all Dove) converge to the mixed strategy Nash Equilibrium where part of the population plays Hawk and part of the population plays Dove. (This occurs because the only ESS is the mixed strategy equilibrium.) In the two population model, this mixed point becomes unstable. In fact, the only stable states in the two population model correspond to the pure strategy equilibria, where one population is composed of all Hawks and the other of all Doves. In this model one population becomes the aggressive population while the other becomes passive. This model is illustrated by the vector field pictured in Figure 7. The one dimensional vector field of the single population model (Figure 7b) corresponds to the bottom left to top right diagonal of the two population model.



The single population model presents a situation where no uncorrelated asymmetries exist, and so the best players can do is randomize their strategies. The two population models provide such an asymmetry and the members of each population will then use that to correlate their strategies. In the two population model, one population gains at the expense of another. Hawk-Dove/Chicken thus illustrates an interesting case where the qualitative results for the two different version of the replicator dynamics differ wildly.

Prisoner's dilemma
Chicken and prisoner's dilemma both start from the same premise of a mutually agreeable, "compromise" solution (C, C) that is threatened by a Pareto dominated, "aggressive" solution (A, A). The threat comes from the fact each player is individually better off switching to A if the other player plays C, but if both switch they end up in (A, A). The games differ in their response to one player switching individually. Assuming player 1 chooses A, the best response in prisoner's dilemma for player 2 is to switch to A as well, while in Chicken player 2 is better off remaining in C: prisoner's dilemma allows player 2 to retaliate while Chicken does not. This has consequences if the game is played repeatedly: in the iterated prisoner's dilemma it is possible for (C, C) to be stable if the threat of retaliation is credible, while in iterated Chicken a stable compromise can only be achieved through brinkmanship.

Brinkmanship
Chicken and brinkmanship are often used synonymously in the context of conflict, but in the strict game-theoretic sense brinkmanship refers to a strategic move designed to avert the possibility of the opponent switching to aggressive behavior. The move involves a credible threat of the risk of irrational behavior in the face of aggression. If player 1 unilaterally moves to A a rational player 2 cannot retaliate since (A, C) is preferable to (A, A). Only if player 1 has grounds to believe that there is sufficient risk that player 2 is irrational and responds with A player 1 will retract and agree on the compromise.

War of attrition
The war of attrition game and Chicken model escalation of conflict, but they differ in the form in which the conflict can escalate. Chicken is used if the catastrophic outcome differs in kind from the agreeable outcome, e.g. if the conflict is over life and death. War of attrition is used when outcomes differ in degrees, e.g. the conflict is a boxing match and the pugilists have to decide whether the ultimate prize of victory is worth the ongoing cost of deteriorating health and stamina.

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Hawk-Dove and war of attrition
The Hawk-Dove game is the most commonly used game theoretical model of aggressive interactions in biology. The war of attrition is another very influential model of aggression in biology. The two models investigate slightly different questions. The Hawk-Dove game is a model of escalation, and addresses the question of when ought an individual escalate to dangerously costly physical combat. The war of attrition seeks to answer the question of how contests may be resolved when there is no possibility of physical combat. The war of attrition is an auction in which both players pay the lower bid (an all-pay second price auction). The bids are assumed to be the duration which the player is willing to persist in making a costly threat display. Both players accrue costs while displaying at each other, the contest ends when the individual making the lower bid quits. Both players will then have paid the lower bid.

Chicken and prisoner's dilemma
Chicken is a symmetrical 2x2 game with conflicting interests, the preferred outcome is to play Straight while the opponent plays Swerve. Similarly, the prisoner's dilemma is a symmetrical 2x2 game with conflicting interests, the preferred outcome is to Defect while the opponent plays Cooperate. Both games have a what seems a "sensible" cooperative outcome in which both players choose the less escalated strategy, Swerve-Swerve in the Chicken game, and Cooperate-Cooperate in the prisoner's dilemma, such that players receive the Coordination payoff C (see tables below). The obvious temptation away from this sensible outcome is towards the Temptation payoff, a Straight move in Cicken and a Defect move in the prisoner's dilemma. The essential difference between these two games is that in the prisoner's dilemma, the Cooperate strategy is dominated, whereas in the Hawk-Dove game the equivalent move is not dominated since the outcome preferences when the opponent plays the more escalated move (Straight/Defect) are reversed.

PD is about the impossibility of cooperation while Chicken is about the inevitability of conflict. Iterated play can solve PD but not Chicken. -->