# Unbeatable strategy

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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 ${\displaystyle 1/4,}$ is beaten by a higher ratio, ${\displaystyle x^{*}}$; the value of ${\displaystyle x}$ which gives its player the greatest possible advantage over the player playing ${\displaystyle x_{0}}$, is found to be given by the relationship ${\displaystyle x^{*}=(2x_{0})^{1/2}-x_{0}}$ and shows ${\displaystyle 1/2}$ 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]

## Related Research Articles

An evolutionarily stable strategy (ESS) is a strategy which, if adopted by a population in a given environment, is impenetrable, meaning that it cannot be invaded by any alternative strategy that are initially rare. It is relevant in game theory, behavioural ecology, and evolutionary psychology. An ESS is an equilibrium refinement of the Nash equilibrium. It is a Nash equilibrium that is "evolutionarily" stable: once it is fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from invading successfully. The theory is not intended to deal with the possibility of gross external changes to the environment that bring new selective forces to bear.

Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Charles Darwin popularised the term "natural selection", contrasting it with artificial selection, which in his view is intentional, whereas natural selection is not.

Sexual selection is a mode of natural selection in which members of one biological sex choose mates of the other sex to mate with, and compete with members of the same sex for access to members of the opposite sex. These two forms of selection mean that some individuals have greater reproductive success than others within a population, for example because they are more attractive or prefer more attractive partners to produce offspring. For instance, in the breeding season, sexual selection in frogs occurs with the males first gathering at the water's edge and making their mating calls: croaking. The females then arrive and choose the males with the deepest croaks and best territories. In general, males benefit from frequent mating and monopolizing access to a group of fertile females. Females can have a limited number of offspring and maximize the return on the energy they invest in reproduction.

Kin selection is the evolutionary strategy that favours the reproductive success of an organism's relatives, even at a cost to the organism's own survival and reproduction. Kin altruism can look like altruistic behaviour whose evolution is driven by kin selection. Kin selection is an instance of inclusive fitness, which combines the number of offspring produced with the number an individual can ensure the production of by supporting others, such as siblings.

William Donald Hamilton, FRS was an English evolutionary biologist, widely recognised as one of the most significant evolutionary theorists of the 20th century.

Fitness is the quantitative representation of natural and sexual selection within evolutionary biology. It can be defined either with respect to a genotype or to a phenotype in a given environment. In either case, it describes individual reproductive success and is equal to the average contribution to the gene pool of the next generation that is made by individuals of the specified genotype or phenotype. The fitness of a genotype is manifested through its phenotype, which is also affected by the developmental environment. The fitness of a given phenotype can also be different in different selective environments.

The Genetical Theory of Natural Selection is a book by Ronald Fisher which combines Mendelian genetics with Charles Darwin's theory of natural selection, with Fisher being the first to argue that "Mendelism therefore validates Darwinism" and stating with regard to mutations that "The vast majority of large mutations are deleterious; small mutations are both far more frequent and more likely to be useful", thus refuting orthogenesis. First published in 1930 by The Clarendon Press, it is one of the most important books of the modern synthesis, and helped define population genetics. It is commonly cited in biology books, outlining many concepts that are still considered important such as Fisherian runaway, Fisher's principle, reproductive value, Fisher's fundamental theorem of natural selection, Fisher's geometric model, the sexy son hypothesis, mimicry and the evolution of dominance. It was dictated to his wife in the evenings as he worked at Rothamsted Research in the day.

The sex ratio is the ratio of males to females in a population. In most sexually reproducing species, the ratio tends to be 1:1. This tendency is explained by Fisher's principle. For various reasons, however, many species deviate from anything like an even sex ratio, either periodically or permanently. Examples include parthenogenic species, periodically mating organisms such as aphids, some eusocial wasps such as Polistes fuscatus and Polistes exclamans, bees, ants, and termites.

George Robert Price was an American population geneticist.

Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.

Linear discriminant analysis (LDA), normal discriminant analysis (NDA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition, and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification.

The drawdown is the measure of the decline from a historical peak in some variable.

A population can be described as being in an evolutionarily stable state when that population’s “genetic composition is restored by selection after a disturbance, provided the disturbance is not too large”. This population as a whole can be either monomorphic or polymorphic.

In probability theory and intertemporal portfolio choice, the Kelly criterion, also known as the scientific gambling method, is a formula for bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive.

Sex allocation is the allocation of resources to male versus female reproduction in sexual species. In dioecious species, where individuals are male or female for their entire lifetimes, the allocation decision lies between producing male or female offspring. In sequential hermaphrodites, where individuals function as one sex early in life and then switch to the other, the allocation decisions lie in what sex to be first and when to change sex. Animals may be dioecious or sequential hermaphrodites. Sex allocation theory has also been applied to plants, which can be dioecious, simultaneous hermaphrodites, have unisexual plants and hermaphroditic plants in the same population, or have unisexual flowers and hermaphroditic flowers on the same plant.

Fisher's principle is an evolutionary model that explains why the sex ratio of most species that produce offspring through sexual reproduction is approximately 1:1 between males and females. A. W. F. Edwards has remarked that it is "probably the most celebrated argument in evolutionary biology".

Sex-limited genes are genes that are present in both sexes of sexually reproducing species but are expressed in only one sex and have no penetrance, or are simply 'turned off' in the other. In other words, sex-limited genes cause the two sexes to show different traits or phenotypes, despite having the same genotype. This term is restricted to autosomal traits, and should not be confused with sex-linked characteristics, which have to do with genetic differences on the sex chromosomes. Sex-limited genes are also distinguished from sex-influenced genes, where the same gene will show differential expression in each sex. Sex-influenced genes commonly show a dominant/recessive relationship, where the same gene will have a dominant effect in one sex and a recessive effect in the other. However, the resulting phenotypes caused by sex-limited genes are present in only one sex and can be seen prominently in various species that typically show high sexual dimorphism.

In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but they do not in general use the minimum number of colors possible.

Sexual reproduction is a type of reproduction that involves a complex life cycle in which a gamete with a single set of chromosomes (haploid) combines with another to produce an organism composed of cells with two sets of chromosomes (diploid). Sexual reproduction is the most common life cycle in multicellular eukaryotes, such as animals, fungi and plants. Sexual reproduction does not occur in prokaryotes, but they have processes with similar effects such as bacterial conjugation, transformation and transduction, which may have been precursors to sexual reproduction in early eukaryotes.

In computer science, an optimal binary search tree , sometimes called a weight-balanced binary tree, is a binary search tree which provides the smallest possible search time for a given sequence of accesses. Optimal BSTs are generally divided into two types: static and dynamic.

## References

1. 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.
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. Bibcode:1973Natur.246...15S. doi:10.1038/246015a0.