Editing Evolutionary game theory (section)

==Unstable games, cyclic patterns==

===Rock paper scissors===
[[File:Rock paper scissors.jpg|thumb|Rock paper scissors]]
[[File:RockPaperMatrix.jpg|thumb|Mutant invasion for rock paper scissors payoff matrix – an endless cycle]]
{{main|Rock paper scissors}}

[[File:RPS dynamics.jpg|thumb|A computer simulation of the rock scissors paper game. The associated [[Normal-form game|RPS game payoff matrix]] is shown. Starting with an arbitrary population the percentage of the three morphs builds up into a continuously cycling pattern.]]

Rock paper scissors incorporated into an evolutionary game has been used for modelling natural processes in the study of [[ecology]].<ref>Allesina and Levine, "A competitive network theory of species diversity", Proceedings of the National Academy of Sciences, 2011</ref> 
Using [[experimental economics]] methods, scientists have used RPS games to test human social evolutionary dynamical behaviours in laboratories. The social cyclic behaviours, predicted by evolutionary game theory, have been observed in various laboratory experiments.<ref>{{cite journal | last1=Hoffman | first1=M | last2=Suetens | first2=S | last3=Gneezy | first3=U | last4=Nowak | first4=M | year=2015 | title=An experimental investigation of evolutionary dynamics in the Rock-Paper-Scissors game | journal=Scientific Reports | volume=5 |  page=8817 | doi=10.1038/srep08817  |bibcode=2015NatSR...5E8817H | pmid=25743257 | pmc=4351537}}</ref><ref>{{cite journal | last1=Cason | first1=T | last2=Friedman | first2=D | last3=Hopkins | first3=E | year=2014 | title=Cycles and Instability in a Rock–Paper–Scissors Population Game: A Continuous Time Experiment | journal=Review of Economic Studies | volume=81 | issue=1 | pages=112–136 | doi=10.1093/restud/rdt023 | citeseerx=10.1.1.261.650 }}</ref>

===Side-blotched lizard plays the RPS, and other cyclical games===
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The first example of RPS in nature was seen in the behaviours and throat colours of a small lizard of western North America. The [[side-blotched lizard]] (''Uta stansburiana'') is [[Polymorphism (biology)|polymorphic]] with three throat-colour morphs<ref>Sinervo, B. and Lively, C. M. 1996. The rock-scissors-paper game and the evolution of alternative male strategies. Nature 340:246</ref> that each pursue a different mating strategy:

[[File:SideblotchedLizard.jpg|thumb|left|The [[side-blotched lizard]] effectively uses a rock-paper-scissors mating strategy]]

* The orange throat is very aggressive and operates over a large territory – attempting to mate with numerous females 
* The unaggressive yellow throat mimics the markings and behavior of female lizards, and "sneakily" slips into the orange throat's territory to mate with the females there (thereby taking over the population)
* The blue throat mates with, and carefully guards, one female – making it impossible for the sneakers to succeed and therefore overtakes their place in a population
However the blue throats cannot overcome the more aggressive orange throats. Later work showed that the blue males are altruistic to other blue males, with three key traits: they signal with blue color, they recognize and settle next to other (unrelated) blue males, and they will even defend their partner against orange, to the death. This is the hallmark of another game of cooperation that involves a [[green-beard effect]].<ref>Sinervo, B. and Clobert, J. 2003. Morphs, dispersal, genetic similarity and the evolution of cooperation. [[Science (journal)|Science]] 300: 1949–1951</ref><ref>Sinervo, B., Chaine, A., Clobert, J., Calsbeek, R., McAdam, A., Hazard, H., Lancaster, L., Alonzo, S., Corrigan, G., and M. Hochberg. 2006a. Self-recognition, color signals and cycles of greenbeard mutualism and transient altruism. Proceedings of the National Academy of Sciences (U.S.A.). 102: 7372–7377</ref>

The females in the same population have the same throat colours, and this affects how many offspring they produce and the size of the progeny, which generates cycles in density, yet another game – the ''r-K'' game.<ref>Sinervo, B., Svensson, E. and Comendant, T. 2000. Density cycles and an offspring quantity and quality game driven by natural selection. Nature 406: 985–988</ref> Here, ''r'' is the [[Malthusian growth model|Malthusian parameter]] governing exponential growth, and ''K'' is the [[Carrying capacity|carrying capacity of the environment]]. Orange females have larger [[Clutch (eggs)|clutches]] and smaller offspring which do well at low density. Yellow & blue females have smaller clutches and larger offspring which do well at high density. This generates perpetual cycles tightly tied to population density. The idea of cycles due to density regulation of two strategies originated with rodent researcher [[Dennis Chitty]], ergo these kinds of games lead to "Chitty cycles". There are games within games within games embedded in natural populations. These drive RPS cycles in the males with a periodicity of four years and ''r-K'' cycles in females with a two year period.

The overall situation corresponds to the rock, scissors, paper game, creating a four-year population cycle. The RPS game in male side-blotched lizards does not have an ESS, but it has a Nash equilibrium (NE) with endless orbits around the [[Fixed point (mathematics)#Attracting fixed points|NE attractor]]. Following this Side-blotched lizard research, many other three-strategy polymorphisms have been discovered in lizards and some of these have RPS dynamics merging the male game and density regulation game in a single sex (males).<ref>Sinervo, B., Heulin, B., Surget-Groba, Y., Clobert, J., Corl, A., Chaine, A, and Davis, A. 2007. Models of density-dependent genic selection and a new Rock-Paper-Scissors social system. The American Naturalist, 170: 663–680.</ref> More recently, mammals have been shown to harbour the same RPS game in males and ''r-K'' game in females, with coat-colour polymorphisms and behaviours that drive cycles.<ref>Sinervo, B., A. Chaine, and D. B. Miles. Social Games and Genic Selection Drives Mammalian Mating System Evolution and Speciation. The American Naturalist 195:247–274.</ref> This game is also linked to the evolution of male care in rodents, and monogamy, and drives [[Speciation#Rates|speciation rates]]. There are ''r-K'' strategy games linked to rodent population cycles (and lizard cycles).<ref>Chitty, D. (1996). Do lemmings commit suicide?: beautiful hypotheses and ugly facts. Oxford University Press.</ref>

When he read that these lizards were essentially engaged in a game with a rock-paper-scissors structure, John Maynard Smith is said to have exclaimed "They have read my book!".<ref>Sigmund, Karl, " Interim Report IR-05-076 John Maynard Smith and Evolutionary Game Theory",  International Institute for Applied Systems Analysis, Dec. 2005</ref>