Қайырсушылар ойыны: Теориялық модельдеу және қақтығыс стратегиясы

Model of conflict for two players in game theory

The game of chicken, also known as the hawk dove game or snowdrift game, The game has also been used to describe the mutual assured destruction of nuclear warfare, especially the sort of brinkmanship involved in the Cuban Missile Crisis.

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 their best outcome, risks the worst. 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:<blockquote>Since the nuclear stalemate became apparent, the governments of East and West have adopted the policy that Mr. Dulles calls 'brinkmanship'. This is a policy adapted from a sport that, I am told, is practiced 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 toward each other from opposite ends. Each car is expected to keep the wheels on one side of 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 they pass, 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 comes, the statesmen of both sides will plunge the world into destruction. In the "chickie run" scene from the film Rebel Without a Cause, this happens when Buzz cannot escape from the car and dies in the crash. The opposite scenario occurs in Footloose where Ren McCormack is stuck in his tractor and hence wins the game as they cannot play "chicken". A similar event happens in two different games in the film The Heavenly Kid, when first Bobby, and then later Lenny become stuck in their cars and drive off a cliff. The basic game theoretic formulation of Chicken has no element of variable, potentially catastrophic, 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.

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 arbitrarily set numerical payoffs which theoretically conform to this situation. Here, the benefit of winning is 1, the cost of losing is 1, and the cost of crashing is 1000. Both Chicken and Hawk–Dove are anti coordination 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 the opponent will be reasonable and decide to swerve, leaving the first 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 their 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.)

In the biological literature, this game is known as Hawk–Dove. The earliest presentation of a form of the Hawk–Dove game was by John Maynard Smith and George Price in their 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, i. e., C > V > 0. If C ≤ V, the resulting game is not a game of Chicken but is instead a Prisoner's Dilemma. The exact value of the Dove vs. Dove payoff 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 > T > L > X. allowing the players to threaten each other before choosing moves in the game, and extending the interaction to two plays of the game.

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 themselves to an object, so that no threat can be made which would compel them to move (since they cannot move). Another example, taken from fiction, is found in Stanley Kubrick's Dr. Strangelove. In that film, the Russians sought to deter American attack by building a "doomsday machine", a device that would trigger world annihilation if Russia was hit by nuclear weapons or if any attempt were made to disarm it. However, the Russians had planned to signal the deployment of the machine a few days after having set it up, which, because of an unfortunate course of events, turned out to be too late. 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" or "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").

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.

Consider the version of "Chicken" pictured in Figure 6. Like all forms of the game, 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. It results in expected payoffs of 14/3 = 4.667 for each player. Now consider a third party (or some natural event) that draws one of three cards labeled: (C, C), (D, C), and (C, D). This exogenous draw event is assumed to be uniformly at random over the 3 outcomes. 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, they would not want to deviate supposing the other player played their assigned strategy since they will get 7 (the highest payoff possible). Suppose a player is assigned C. Then the other player has been assigned C with probability 1/2 and D with probability 1/2 (due to the nature of the exogenous draw). 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 from the drawn assignments, 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.

Although there are three Nash equilibria in the Hawk–Dove game, the one which emerges as the evolutionarily stable strategy (ESS) depends upon the existence of any uncorrelated asymmetry in the game (in the sense of anti coordination games). 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.

"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 responds irrationally (usually by giving up control over the response, so that there is sufficient risk that player 2 responds with A) player 1 will retract and agree on the compromise.

Like "Chicken", the "War of attrition" game models escalation of conflict, but they differ in the form in which the conflict can escalate. Chicken models a situation in which the catastrophic outcome differs in kind from the agreeable outcome, e. g., if the conflict is over life and death. War of attrition models a situation in which the outcomes differ only in degrees, such as a boxing match in which the contestants have to decide whether the ultimate prize of victory is worth the ongoing cost of deteriorating health and stamina.

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 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. PD is about the impossibility of cooperation while Chicken is about the inevitability of conflict. Iterated play can solve PD but not Chicken. Both games have a desirable 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 temptation away from this sensible outcome is toward a Straight move in Chicken and a Defect move in the prisoner's dilemma (generating the Temptation payoff, should the other player use the less escalated move). The essential difference between these two games is that in the prisoner's dilemma, the Cooperate strategy is dominated, whereas in Chicken the equivalent move is not dominated since the outcome payoffs when the opponent plays the more escalated move (Straight in place of Defect) are reversed.

The term "schedule chicken" is used in project management and software development circles. The condition occurs when two or more areas of a product team claim they can deliver features at an unrealistically early date because each assumes the other teams are stretching the predictions even more than they are. This pretense continually moves forward past one project checkpoint to the next until feature integration begins or just before the functionality is actually due. The practice of "schedule chicken" often results in contagious schedule slips due to the inter team dependencies and is difficult to identify and resolve, as it is in the best interest of each team not to be the first bearer of bad news. The psychological drivers underlining the "schedule chicken" behavior in many ways mimic the hawk–dove or snowdrift model of conflict.