Unbeatable strategy

Short description: Idea proposed by biologist W.D. Hamilton in 1967

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 x0, is found to be given by the relationship x*=(2x0)1/2−x0 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

1. ↑ Hamilton, W.D. (1967). "Extraordinary sex ratios". Science 156 (3774): 477–488. doi:10.1126/science.156.3774.477. PMID 6021675. Bibcode: 1967Sci...156..477H.
2. ↑ 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.
3. ↑ Verner, J. (1965). "Selection for sex ratio". American Naturalist 99 (908): 419–421. doi:10.1086/282384.
4. ↑ Fisher, R.A. (1930). The Genetical Theory of Natural Selection. London: Clarendon. ISBN 0-19-850440-3.
5. ↑ Darwin, C.R. (1859). The Origin of Species. London: John Murray. ISBN 0-8014-1319-2.
6. ↑ 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.
7. ↑ Maynard Smith, J.; Price, G.R. (1973). "The logic of animal conflict". Nature 246 (5427): 15–8. doi:10.1038/246015a0. Bibcode: 1973Natur.246...15S.

## External links

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

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Topics in game theory

Definitions|

* Cooperative game
* Determinacy
* Escalation of commitment
* Extensive-form game
* First-player and second-player win
* Game complexity
* Graphical game
* Hierarchy of beliefs
* Information set
* Normal-form game
* Preference
* Sequential game
* Simultaneous game
* Simultaneous action selection
* Solved game
* Succinct game

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
* Strategy-stealing argument
* Tit for tat
* Grim trigger
* Collusion
* Backward induction
* Forward induction
* Markov strategy

Classes
of games|

* Symmetric game
* Perfect information
* 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 game
* Strictly determined game
* Potential game

Games|

* Go
* Chess
* Infinite chess
* Checkers
* Tic-tac-toe
* Prisoner's dilemma
* Optional 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 game
* 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
* Rendezvous problem

Theorems|

* Arrow's impossibility theorem
* Aumann's agreement theorem
* Folk theorem
* Minimax theorem
* Nash's theorem
* Purification theorem
* Revelation principle
* Zermelo's theorem

Key
figures|

* Albert W. Tucker
* Amos Tversky
* Ariel Rubinstein
* Claude Shannon
* 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
* Jennifer Tour Chayes
* John Harsanyi
* John Maynard Smith
* Antoine Augustin Cournot
* John Nash
* John von Neumann
* Kenneth Arrow
* Kenneth Binmore
* Leonid Hurwicz
* Lloyd Shapley
* Melvin Dresher
* Merrill M. Flood
* Olga Bondareva
* Oskar Morgenstern
* Paul Milgrom
* Peyton Young
* Reinhard Selten
* Robert Axelrod
* Robert Aumann
* Robert B. Wilson
* Roger Myerson
* Samuel Bowles
* Suzanne Scotchmer
* Thomas Schelling
* William Vickrey

See also|

* All-pay auction
* Alpha–beta pruning
* Bertrand paradox
* Bounded rationality
* Combinatorial game theory
* Confrontation analysis
* Coopetition
* First-move advantage in chess
* Game mechanics
* Glossary of game theory
* List of game theorists
* List of games in game theory
* No-win situation
* Solving chess
* Topological game
* Tragedy of the commons
* Tyranny of small decisions

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