Editing Evolutionary game theory (section)

==Evolutionary games==
===Models===
[[File:Game Diagram AniFin.gif|thumb|400px|Evolutionary game theory analyses Darwinian mechanisms with a [[system model]] with three main components –  ''population'', ''game'', and ''replicator dynamics''. The system process has four phases:<br />
     <br />
1) The model (as evolution itself) deals with a ''population'' (Pn).  The population will exhibit [[Evolution#Sources of variation|variation]] among competing individuals.  In the model this competition is represented by the game.<br>
     <br>
2) The game tests the strategies of the individuals under the rules of the game.  These rules produce different payoffs – in units of [[Fitness (biology)|fitness]] (the production rate of offspring).  The contesting individuals meet in pairwise contests with others, normally in a highly mixed distribution of the population.  The mix of strategies in the population affects the payoff results by altering the odds that any individual may meet up in contests with various strategies. The individuals leave the game pairwise contest with a resulting fitness determined by the contest outcome, represented in a ''payoff matrix''.<br> 
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3) Based on this resulting fitness each member of the population then undergoes replication or culling determined by the exact mathematics of the ''replicator dynamics process''.  This overall process then produces a ''new generation'' P(n+1).  Each surviving individual now has a new fitness level determined by the game result.<br>
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4) The new generation then takes the place of the previous one and the cycle repeats. The population mix may converge to an ''evolutionarily stable state'' that cannot be invaded by any mutant strategy.]]

Evolutionary game theory encompasses Darwinian evolution, including competition (the game), natural selection (replicator dynamics), and heredity. Evolutionary game theory has contributed to the understanding of [[group selection]], [[sexual selection]], [[altruism]], [[parental care]], [[co-evolution]], and [[ecology|ecological]] dynamics. Many counter-intuitive situations in these areas have been put on a firm mathematical footing by the use of these models.<ref name=Hammerstein>{{cite book |author1=Hammerstein, Peter |author2-link=Reinhard Selten |author2=Selten, Reinhard |chapter=Game theory and evolutionary biology |title=Handbook of Game Theory with Economic Applications |volume=2 |editor1=Aumann, R.|editor2=Hart, S. |publisher=Elsevier |date=1994 |pages=929–993 |doi=10.1016/S1574-0005(05)80060-8 |isbn=978-0-444-89427-4}}</ref>

The common way to study the evolutionary dynamics in games is through [[replicator equation]]s. These show the growth rate of the proportion of organisms using a certain strategy and  that rate is equal to the difference between the average payoff of that strategy and the average payoff of the population as a whole.<ref name=Samuelson>{{cite journal | last1=Samuelson | first1=L. | year=2002 | title=Evolution and game theory | journal=  Journal of Economic Perspectives| volume=16 | issue=2| pages=46–66 | doi=10.1257/0895330027256 | doi-access=free }}</ref> Continuous replicator equations assume infinite populations, [[continuous time]], [[complete mixing]] and that strategies breed true. Some [[attractor]]s (all global asymptotically stable fixed points) of the equations are [[evolutionarily stable state]]s.<ref name="Zeeman">{{Citation
| last1 = Zeeman | first1 = E. C.
| contribution = Population dynamics from game theory
| title = Global theory of dynamical systems
| publisher = Springer Verlag
| place = Berlin
| pages = 471–497
| date = 1980
|contribution-url = http://www.lms.ac.uk/sites/lms.ac.uk/files/1980%20Population%20dynamics%20from%20game%20theory%20(preprint).pdf}}</ref>  A strategy which can survive all "mutant" strategies is considered evolutionarily stable. In the context of animal behavior, this usually means such strategies are programmed and heavily influenced by [[genetics]], thus making any player or organism's strategy determined by these biological factors.<ref>{{cite book |author=Weibull, J. W. |date=1995 |title=Evolutionary game theory |publisher=MIT Press }}</ref><ref>{{cite book |author1=Hofbauer, J. |author2-link=Karl Sigmund |author2=Sigmund, K.  |date=1998 |title=Evolutionary games and population dynamics |publisher=Cambridge University Press }}</ref>

Evolutionary games are mathematical objects with different rules, payoffs, and mathematical behaviours. Each "game" represents different problems that organisms have to deal with, and the strategies they might adopt to survive and reproduce. Evolutionary games are often given colourful names and cover stories which describe the general situation of a particular game. Representative games include [[Chicken (game)|hawk-dove]],<ref name=Price/> [[war of attrition (game)|war of attrition]],<ref name=SelfishGene/> [[stag hunt]], [[Cheating (biology)|producer-scrounger]], [[tragedy of the commons]], and [[prisoner's dilemma]]. Strategies for these games include hawk, dove, bourgeois, prober, defector, assessor, and retaliator. The various strategies compete under the particular game's rules, and the mathematics are used to determine the results and behaviours.

===Hawk dove===
[[File:HawkDove2.jpg|thumb|300px| Solution of the [[Chicken (game)|hawk dove]] game for V=2, C=10 and fitness starting base B=4. The fitness of a hawk for different population mixes is plotted as a black line, that of dove in red. An ESS (a stationary point) will exist when hawk and dove fitness are equal: Hawks are 20% of population and doves are 80% of the population.]]
{{main|Chicken (game)}}

The first game that [[Maynard Smith]] analysed is the classic ''[[Chicken (game)|hawk dove]]''{{efn|Maynard Smith chose the name "hawk dove" from descriptions of political views current during the [[Vietnam War]].}} game. It was conceived to analyse Lorenz and Tinbergen's problem, a contest over a shareable resource. The contestants can be either  a hawk or a dove. These are two subtypes or morphs of one species with different strategies. The hawk first displays aggression, then escalates into a fight until it either wins or is injured (loses). The dove first displays aggression, but if faced with major escalation runs for safety. If not faced with such escalation, the dove attempts to share the resource.<ref name=Price/>

{| class="wikitable" style="text-align:center"
|+style="white-space:nowrap"| Payoff matrix for hawk dove game
|-
|  || ''' meets hawk ''' || ''' meets dove '''
|-
| '''if hawk''' ||V/2 − C/2||V
|-
|'''if dove''' ||0 ||V/2
|}

Given that the resource is given the value V, the damage from losing a fight is given cost C:<ref name=Price/>

*If a hawk meets a dove, the hawk gets the full resource V 
*If a hawk meets a hawk, half the time they win, half the time they lose, so the average outcome is then V/2 minus C/2
*If a dove meets a hawk, the dove will back off and get nothing – 0
*If a dove meets a dove, both share the resource and get V/2

The actual payoff, however, depends on the probability of meeting a hawk or dove, which in turn is a representation of the percentage of hawks and doves in the population when a particular contest takes place. That, in turn, is determined by the results of all of the previous contests. If the cost of losing C is greater than the value of winning V (the normal situation in the natural world) the mathematics ends in an [[evolutionarily stable strategy]] (ESS), a mix of the two strategies where the population of hawks is V/C. The population regresses to this equilibrium point if any new hawks or doves make a temporary perturbation in the population.
The solution of the hawk dove game explains why most animal contests involve only [[Ritualized aggression|ritual fighting]] behaviours in contests rather than outright battles. The result does not at all depend on "[[group selection|good of the species]]" behaviours as suggested by Lorenz, but solely on the implication of actions of so-called [[selfish genes]].<ref name=Price/>

===War of attrition===
{{main|War of attrition (game)}}

In the hawk dove game the resource is shareable, which gives payoffs to both doves meeting in a pairwise contest. Where the resource is not shareable, but an alternative resource might be available by backing off and trying elsewhere, pure hawk or dove strategies are less effective. If an unshareable resource is combined with a high cost of losing a contest (injury or possible death) both hawk and dove payoffs are further diminished.  A safer strategy of lower cost display, bluffing and waiting to win, is then viable – a bluffer strategy. The game then becomes one of accumulating costs, either the costs of displaying or the costs of prolonged unresolved engagement. It is effectively an auction; the winner is the contestant who will swallow the greater cost while the loser gets the same cost as the winner but no resource.<ref name=SelfishGene>{{harvp|Dawkins|2006|pp=76-78}}</ref> The resulting evolutionary game theory mathematics lead to an optimal strategy of timed bluffing.<ref>{{cite book |author-link=John Maynard Smith |last=Maynard Smith |first=John |date=1982 |title=Evolution and the Theory of Games |isbn=978-0-521-28884-2 |page=[https://archive.org/details/evolutiontheoryg00jmsm/page/n35 28]|title-link=Evolution and the Theory of Games |publisher=Cambridge University Press }}</ref>

[[File:Attrition graph.jpg|thumb|300px| [[War of attrition (game)|War of attrition]] for different values of resource. Note the time it takes for an accumulation of 50% of the contestants to quit vs. the value (V) of resource contested for.]]

This is because in the war of attrition any strategy that is unwavering and predictable is unstable, because it will ultimately be displaced by a mutant strategy which relies on the fact that it can best the existing predictable strategy by investing an extra small delta of waiting resource to ensure that it wins. Therefore, only a random unpredictable strategy can maintain itself in a population of bluffers. The contestants in effect choose an acceptable cost to be incurred related to the value of the resource being sought, effectively making a random bid as part of a mixed strategy (a strategy where a contestant has several, or even many, possible actions in their strategy). This implements a distribution of bids for a resource of specific value V, where the bid for any specific contest is chosen at random from that distribution.  The distribution (an ESS) can be computed using the [[Bishop-Cannings theorem]], which holds true for any mixed-strategy ESS.<ref>{{cite book |author-link=John Maynard Smith |last=Maynard Smith |first=John |date=1982 |title=Evolution and the Theory of Games |isbn=978-0-521-28884-2 |page=[https://archive.org/details/evolutiontheoryg00jmsm/page/n40 33]|title-link=Evolution and the Theory of Games |publisher=Cambridge University Press }}</ref> The distribution function in these contests was determined by Parker and Thompson to be:

:<math>p(x)=\frac{e^{-x/V}}{V}.</math>
 
The result is that the cumulative population of quitters for any particular cost m in this "mixed strategy" solution is:

:<math>p(m)=1- e^{-m/V},</math>
   
as shown in the adjacent graph.  The intuitive sense that greater values of resource sought leads to greater waiting times is borne out. This is observed in nature, as in male dung flies contesting for mating sites, where the timing of disengagement in contests is as predicted by evolutionary theory mathematics.<ref>{{cite journal | last1=Parker | last2=Thompson | year=1980 | title=Dung Fly Struggle: a test of the War of Attrition | journal=Behavioral Ecology and Sociobiology | volume=7 | issue=1 | pages=37–44 | doi=10.1007/bf00302516| s2cid=44287633 }}</ref>

===Asymmetries that allow new strategies===

{{multiple image
 | width=200
 | footer=Animal strategy examples: by examining the behaviours, then determining both the costs and the values of resources attained in a contest the strategy of an organism can be verified 
 | image1=Scatophaga stercoraria.jpg
 | caption1= Dung fly (''Scatophaga stercoraria'') – a war of attrition player
 | image2=30-EastTimor-Dive2 Maubara 35 (Mantis Shrimp)-APiazza.JPG
 | caption2=The mantis shrimp guarding its home with the bourgeois strategy
 }}

In the war of attrition there must be nothing that signals the size of a bid to an opponent, otherwise the opponent can use the cue in an effective counter-strategy.  There is however a mutant strategy which can better a bluffer in the [[War of attrition (game)|war of attrition]] game if a suitable asymmetry exists, the bourgeois strategy. Bourgeois uses an asymmetry of some sort to break the deadlock. In nature one such asymmetry is possession of a resource. The strategy is to play a hawk if in possession of the resource, but to display then retreat if not in possession.  This requires greater cognitive capability than hawk, but bourgeois is common in many animal contests, such as in contests among [[mantis shrimp]]s and among [[speckled wood butterfly|speckled wood butterflies]].

===Social behaviour===
[[File:Game Theory Strategic Social Alternatives.jpg|thumb|300px|Alternatives for game theoretic social interaction]]

Games like hawk dove and war of attrition represent pure competition between individuals and have no attendant social elements.  Where social influences apply, competitors have four possible alternatives for strategic interaction.  This is shown on the adjacent figure, where a plus sign represents a benefit and a minus sign represents a cost.

* In a ''cooperative'' or ''mutualistic'' relationship both "donor" and "recipient" are almost indistinguishable as both gain a benefit in the game by co-operating, i.e. the pair are in a game-wise situation where both can gain by executing a certain strategy,  or alternatively both must act in  concert because of some encompassing  constraints that effectively puts them  "in the same boat".
* In an ''altruistic'' relationship the donor, at a cost to themself provides a benefit to the recipient.  In the general case the recipient will have a kin relationship to the donor and the donation is one-way.  Behaviours where benefits are donated alternatively (in both directions) at a cost, are often called "altruistic", but on analysis such "altruism" can be seen to arise from optimised  "selfish" strategies.
* ''Spite'' is essentially a “reversed” form of cooperation where neither party receives a tangible benefit. The general case is that the ally is kin related and the benefit is an easier competitive environment for the ally.   <small>Note: George Price, one of the early mathematical modellers of both altruism and spite, found this equivalence particularly disturbing at an emotional level.</small><ref>{{cite book |author=Harman, O. |title=The Price of Altruism |date=2010 |publisher=Bodley Head |pages=Chapter 9 |isbn=978-1-847-92062-1}}</ref>
* ''Selfishness'' is the base criteria of all strategic choice from a game theory perspective – strategies not aimed at self-survival and self-replication are not long for any game. Critically however, this situation is impacted by the fact that competition is taking place on multiple levels  – i.e. at a genetic, an individual and a group level.