Unbeatable strategy

In biology, the idea of an unbeatable strategy was proposed by W.D. Hamilton in his 1967 paper on sex ratios in Science.^[1]^[2] In this paper Hamilton discusses sex ratios as strategies in a game, and cites Verner as using this language in his 1965 paper^[3] 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".

"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).

"[...] But if, on the contrary, players of such a game were motivated to outscore, they would find that

1/4,

is beaten by a higher ratio,

x^{*}

; the value of

x

which gives its player the greatest possible advantage over the player playing

x_{0}

, is found to be given by the relationship

x^{*}=(2x_{0})^{1/2}-x_{0}

and shows

1/2

to be the unbeatable play." Hamilton (1967).

The concept can be traced through R.A. Fisher (1930)^[4] to Darwin (1859);^[5] see Edwards (1998).^[6] 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).^[7]

References

↑ Hamilton, W.D. (1967). "Extraordinary sex ratios". Science. 156 (3774): 477–488. Bibcode:1967Sci...156..477H. doi:10.1126/science.156.3774.477. JSTOR 1721222. PMID 6021675.
↑ Hamilton, W.D. (1996). Evolution of Social Behaviour. Narrow roads of gene land: the collected papers of W. D. Hamilton. 1. Oxford: Oxford University Press. ISBN 0-7167-4530-5.
↑ Verner, J. (1965). "Selection for sex ratio". American Naturalist. 99 (908): 419–421. doi:10.1086/282384.
↑ Fisher, R.A. (1930). The Genetical Theory of Natural Selection. London: Clarendon. ISBN 0-19-850440-3.
↑ Darwin, C.R. (1859). The Origin of Species. London: John Murray. ISBN 0-8014-1319-2.
↑ Edwards, A.W.F. (1998). "Natural selection and the sex ratio: Fisher's sources". American Naturalist. 151 (6): 564–9. doi:10.1086/286141. PMID 18811377.
↑ Maynard Smith, J.; Price, G.R. (1973). "The logic of animal conflict". Nature. 246 (5427): 15–8. Bibcode:1973Natur.246...15S. doi:10.1038/246015a0.

External links

http://www.iiasa.ac.at/Publications/Documents/IR-02-019.pdf

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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