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| In [[biology]], the idea of an '''unbeatable strategy''' was proposed by [[W.D. Hamilton]] in his 1967 paper on [[sex ratio]]s in ''[[Science (journal)|Science]]''.<ref>{{cite journal |doi=10.1126/science.156.3774.477 |author=Hamilton, W.D. |authorlink=W. D. Hamilton |title=Extraordinary sex ratios |journal=Science |volume=156 |issue=3774 |pages=477–488 |year=1967 |pmid=6021675 |jstor=1721222|bibcode = 1967Sci...156..477H }}</ref><ref>{{cite book |author=Hamilton, W.D. |authorlink=W. D. Hamilton |title=Evolution of Social Behaviour |volume=1 |series=Narrow roads of gene land: the collected papers of W. D. Hamilton |publisher=Oxford University Press |location=Oxford |year=1996 |isbn=0-7167-4530-5 }}</ref> In this paper Hamilton discusses sex ratios as [[Strategy (game theory)|strategies]] in a [[game theory|game]], and cites Verner as using this language in his 1965 paper<ref>{{cite journal |doi=10.1086/282384 |author=Verner, J. |title=Selection for sex ratio |journal=American Naturalist |volume=99 |issue=908 |pages=419–421 |year=1965}}</ref> which "claims to show that, given factors causing fluctuations of the population's primary sex ratio, a 1:1 sex-ratio production proves the best overall genotypic strategy".
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| : "In the way in which the success of a chosen sex ratio depends on choices made by the co-parasitizing females, this problem resembles certain problems discussed in the "theory of games." In the foregoing analysis a game-like element, of a kind, was present and made necessary the use of the word ''unbeatable'' to describe the ratio finally established. This word was applied in just the same sense in which it could be applied to the "minimax" strategy of a zero-sum two-person game. Such a strategy should not, without qualification, be called optimum because it is not optimum against -although unbeaten by- any strategy differing from itself. This is exactly the case with the "unbeatable" sex ratios referred to." Hamilton (1967).
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| : "[...] But if, on the contrary, players of such a game were motivated to outscore, they would find that <math>1/4,</math> is beaten by a higher ratio, <math>x^*</math>; the value of <math>x</math> which gives its player the greatest possible advantage over the player playing <math>x_0</math>, is found to be given by the relationship <math>x^*=(2x_0)^{1/2} -x_0</math> and shows <math>1/2</math> to be the unbeatable play." Hamilton (1967). | |
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| The concept can be traced through [[Ronald Fisher|R.A. Fisher]] (1930)<ref>{{cite book |author=Fisher, R.A. |authorlink=Ronald Fisher |title=[[The Genetical Theory of Natural Selection]] |publisher=Clarendon |location=London |year=1930 |isbn=0-19-850440-3 }}</ref> to Darwin (1859);<ref>{{cite book |author=Darwin, C.R. |authorlink=Charles Darwin |title=[[The Origin of Species]] |publisher=John Murray |location=London |year=1859 |isbn=0-8014-1319-2}}</ref> see Edwards (1998).<ref>{{cite journal |doi=10.1086/286141 |author=Edwards, A.W.F. |title=Natural selection and the sex ratio: Fisher’s sources |journal=American Naturalist |volume=151 |issue=6 |pages=564–9 |year=1998 |pmid=18811377 }}</ref> Hamilton did not explicitly define the term "unbeatable strategy" or apply the concept beyond the evolution of sex-ratios, but the idea was very influential. [[George R. Price]] generalised the verbal argument, which was then formalised mathematically by [[John Maynard Smith]], into the [[evolutionarily stable strategy]] (ESS).<ref>{{cite journal |doi=10.1038/246015a0 |author1=Maynard Smith, J. |authorlink1=John Maynard Smith |author2=Price, G.R. |authorlink2=George R. Price |title=The logic of animal conflict |journal=Nature |volume=246 |issue=5427 |pages=15–8 |year=1973 |bibcode=1973Natur.246...15S}}</ref>
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| == References ==
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| == External links ==
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| * http://www.iiasa.ac.at/Publications/Documents/IR-02-019.pdf
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| {{game theory}}
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| {{DEFAULTSORT:Unbeatable Strategy}}
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| [[Category:Game theory]]
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