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Chicken (game)
Encyclopedia
The game of chicken, also known as the hawk-dove or snowdrift 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 other, the worst possible outcome occurs when both players do not yield.
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
. From a game-theoretic point of view, "chicken" and "hawk-dove" are identical; the different names stem from parallel development of the basic principles in different research areas. 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 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
:
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 from the film Rebel Without a Cause
, this happens when Corey Allen's character 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 he can't play "chicken". 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.
. 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 -10.
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 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.)
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, i.e., C > V > 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 > T > L > X.
, 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.
'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 failed to signal — they deployed their doomsday machine covertly.
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
").
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.
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.
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, 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 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.
(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.
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 7a. 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 and Chicken thus illustrate an interesting case where the qualitative results for the two different version of the replicator dynamics differ wildly.
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.
" 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 schedules 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.
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
. The principle of the game is that while each player prefers not to yield to the other, the worst possible outcome occurs when both players do not yield.
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
Political science
Political Science is a social science discipline concerned with the study of the state, government and politics. Aristotle defined it as the study of the state. It deals extensively with the theory and practice of politics, and the analysis of political systems and political behavior...
and economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
. 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
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
and evolutionary game theory
Evolutionary game theory
Evolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...
. From a game-theoretic point of view, "chicken" and "hawk-dove" are identical; the different names stem from parallel development of the basic principles in different research areas. The game has also been used to describe the mutual assured destruction
Mutual assured destruction
Mutual Assured Destruction, or mutually assured destruction , is a doctrine of military strategy and national security policy in which a full-scale use of high-yield weapons of mass destruction by two opposing sides would effectively result in the complete, utter and irrevocable annihilation of...
of nuclear warfare
Nuclear warfare
Nuclear warfare, or atomic warfare, is a military conflict or political strategy in which nuclear weaponry is detonated on an opponent. Compared to conventional warfare, nuclear warfare can be vastly more destructive in range and extent of damage...
, especially the sort of brinkmanship
Brinkmanship
Brinkmanship is the practice of pushing dangerous events to the verge of disaster in order to achieve the most advantageous outcome...
involved in the Cuban Missile Crisis
Cuban Missile Crisis
The Cuban Missile Crisis was a confrontation among the Soviet Union, Cuba and the United States in October 1962, during the Cold War...
.
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 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
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
famously compared the game of Chicken to nuclear
Nuclear warfare
Nuclear warfare, or atomic warfare, is a military conflict or political strategy in which nuclear weaponry is detonated on an opponent. Compared to conventional warfare, nuclear warfare can be vastly more destructive in range and extent of damage...
brinkmanship
Brinkmanship
Brinkmanship is the practice of pushing dangerous events to the verge of disaster in order to achieve the most advantageous outcome...
:
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.
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 from the film Rebel Without a Cause
Rebel Without a Cause
Rebel Without a Cause is a 1955 American drama film about emotionally confused suburban, middle-class teenagers. Directed by Nicholas Ray, it offered both social commentary and an alternative to previous films depicting delinquents in urban slum environments...
, this happens when Corey Allen's character 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 he can't play "chicken". 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.
Chicken
A formal version of the game of Chicken has been the subject of serious research in game theoryGame theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
. 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 -10.
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
Coordination game
In game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies...
, 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
Externality
In economics, an externality is a cost or benefit, not transmitted through prices, incurred by a party who did not agree to the action causing the cost or benefit...
. In anti-coordination games the resource is rivalrous but non-excludable
Non-excludable good
In economics, a good or service is said to be excludable when it is possible to prevent people who have not paid for it from having access to it, and non-excludable when it is not possible to do so.- Examples :...
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
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
, 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 SmithJohn Maynard Smith
John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
and George Price
George R. Price
George Robert Price was an American population geneticist. Originally a physical chemist and later a science journalist, he moved to London in 1967, where he worked in theoretical biology at the Galton Laboratory, making three important contributions: first, rederiving W.D...
in their 1973 Nature
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
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.
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
War of attrition (game)
In game theory, the war of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while constantly accumulating costs over the time t that the contest lasts. The model was originally formulated by John Maynard Smith, a mixed evolutionary stable...
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.
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 potentialResource holding potential
In biology, resource holding potential is the ability of an animal to win an all-out fight if one were to take place.The term was coined by Geoff Parker to disambiguate physical fighting ability from the motivation to persevere in a fight . Originally the term used was 'Resource Holding Power',...
, 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 ostensibly 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 KubrickStanley Kubrick
Stanley Kubrick was an American film director, writer, producer, and photographer who lived in England during most of the last four decades of his career...
'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 failed to signal — they deployed their doomsday machine covertly.
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
Handicap principle
The handicap principle is a hypothesis originally proposed in 1975 by biologist Amotz Zahavi to explain how evolution may lead to "honest" or reliable signaling between animals who have an obvious motivation to bluff or deceive each other...
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
Rock, Paper, Scissors
Rock-paper-scissors is a hand game played by two people. The game is also known as roshambo, or another ordering of the three items ....
").
Best response mapping and Nash equilibria
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
chooses between the two pure strategies. Either the pure, or mixed, Nash equilibria will be evolutionarily stable strategies depending upon whether uncorrelated asymmetries
Uncorrelated asymmetry
In game theory an uncorrelated asymmetry is an arbitrary asymmetry in a game which is otherwise symmetrical. The name 'uncorrelated asymmetry' is due to John Maynard Smith who called payoff relevant asymmetries in games with similar roles for each player 'correlated asymmetries' .The explanation of...
exist.
The best response
Best response
In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given...
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
Abscissa
In mathematics, abscissa refers to that element of an ordered pair which is plotted on the horizontal axis of a two-dimensional Cartesian coordinate system, as opposed to the ordinate...
, and the independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
is plotted on the ordinate
Ordinate
In mathematics, ordinate refers to that element of an ordered pair which is plotted on the vertical axis of a two-dimensional Cartesian coordinate system, as opposed to the abscissa...
). 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
Evolutionarily stable strategy
In game theory and behavioural ecology, an evolutionarily stable strategy , which is sometimes also called an evolutionary stable strategy, is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium...
(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 for the 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 (a range of possible instinctive reactions for a single situation) or whether the population is a polymorphic mixture of players dedicated to choosing a particular pure strategy(a single reaction differing from individual to individual). 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 symmetricSymmetric equilibrium
In game theory, a symmetric equilibrium is an equilibrium where both players use the same strategy in the equilibrium. In the Prisoner's Dilemma game pictured to the right, the only Nash equilibrium is . Since both players use the same strategy, the equilibrium is symmetric.Symmetric equilibria...
Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
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 "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, 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 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
Correlated equilibrium
In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann . The idea is that each player chooses his/her action according to his/her observation of the value of the same public...
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
Although there are three Nash equilibria in the Hawk-Dove game, the one which emerges as the evolutionarily stable strategyEvolutionarily stable strategy
In game theory and behavioural ecology, an evolutionarily stable strategy , which is sometimes also called an evolutionary stable strategy, is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium...
(ESS) depends upon the existence of any uncorrelated asymmetry
Uncorrelated asymmetry
In game theory an uncorrelated asymmetry is an arbitrary asymmetry in a game which is otherwise symmetrical. The name 'uncorrelated asymmetry' is due to John Maynard Smith who called payoff relevant asymmetries in games with similar roles for each player 'correlated asymmetries' .The explanation of...
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.
Replicator dynamics
Evolutionary game theory
Evolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...
. 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
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
pictured in Figure 7a. 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.
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 moveStrategic move
A strategic move in game theory is an action taken by a player outside the defined actions of the game in order to gain a strategic advantage and increase one's payoff...
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.
War of attrition
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.Schedule Chicken & Project Management
The term "Schedule ChickenSchedule chicken
The concept of "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...
" is used in project management
Project management
Project management is the discipline of planning, organizing, securing, and managing resources to achieve specific goals. A project is a temporary endeavor with a defined beginning and end , undertaken to meet unique goals and objectives, typically to bring about beneficial change or added value...
and software development
Software development
Software development is the development of a software product...
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
Integration testing
Integration testing is the phase in software testing in which individual software modules are combined and tested as a group. It occurs after unit testing and before validation testing...
begins or just before the functionality is actually due.
The practice of "Schedule Chicken" often results in contagious schedules 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.
See also
- Coordination gameCoordination gameIn game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies...
- Matching penniesMatching penniesMatching pennies is the name for a simple example game used in game theory. It is the two strategy equivalent of Rock, Paper, Scissors. Matching pennies is used primarily to illustrate the concept of mixed strategies and a mixed strategy Nash equilibrium....
- Volunteer's dilemmaVolunteer's dilemmaThe volunteer's dilemma game models a situation in which each of N players faces the decision of either making a small sacrifice from which all will benefit, or freeriding....
- War of attritionWar of attrition (game)In game theory, the war of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while constantly accumulating costs over the time t that the contest lasts. The model was originally formulated by John Maynard Smith, a mixed evolutionary stable...
- BrinkmanshipBrinkmanshipBrinkmanship is the practice of pushing dangerous events to the verge of disaster in order to achieve the most advantageous outcome...
- Prisoner's dilemmaPrisoner's dilemmaThe prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
External links
- The game of Chicken as a metaphor for human conflict
- Game-theoretic analysis of Chicken
- Game of Chicken – Rebel Without a Cause by Elmer G. Wiens.
- David M. Dikel, David Kane, James R. Wilson (2001). Software Architecture: Organizational Principles and Patterns, University of Michigan, ISBN 9780130290328
- Michael Ficco (2001). What Every Engineer Should Know about Career Management, CRC Press, ISBN 9781420076820
- David M. Dikel, David Kane, James R. Wilson (2002). Software Craftsmanship: The New Imperative, Addison-Wesley, ISBN 9780130290328
- Online model: Expected Dynamics of an Imitation Model in the Hawk-Dove Game
- Online model: Expected Dynamics of an Intra-Population Imitation Model in the Two-Population Hawk-Dove Game