An evolutionarily stable strategy (ESS) is a

strategy Strategy (from Greek στρατηγία ''stratēgia'', "troop leadership; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the sense of the " a ...

(or set of strategies) that is ''impermeable'' when adopted by a

population Population is a set of humans or other organisms in a given region or area. Governments conduct a census to quantify the resident population size within a given jurisdiction. The term is also applied to non-human animals, microorganisms, and pl ...

in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set of strategies) which may be novel or initially rare. Introduced by

John Maynard Smith John Maynard Smith (6 January 1920 – 19 April 2004) was a British mathematical and theoretical biology, theoretical and mathematical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he ...

and

George R. Price George Robert Price (October 16, 1922 – January 6, 1975) was an American population geneticist. Price is often noted for his formulation of the Price equation in 1967. Originally a physical chemist and later a science journalist, he moved ...

in 1972/3, it is an important concept in

behavioural ecology Behavioral ecology, also spelled behavioural ecology, is the study of the evolutionary basis for animal behavior due to ecological pressures. Behavioral ecology emerged from ethology after Niko Tinbergen outlined four questions to address when ...

evolutionary psychology Evolutionary psychology is a theoretical approach in psychology that examines cognition and behavior from a modern evolutionary perspective. It seeks to identify human psychological adaptations with regard to the ancestral problems they evolved ...

, mathematical game theory and

economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...

, with applications in other fields such as

anthropology Anthropology is the scientific study of humanity, concerned with human behavior, human biology, cultures, society, societies, and linguistics, in both the present and past, including archaic humans. Social anthropology studies patterns of behav ...

philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...

and

political science Political science is the scientific study of politics. It is a social science dealing with systems of governance and Power (social and political), power, and the analysis of political activities, political philosophy, political thought, polit ...

. In game-theoretical terms, an ESS is an

equilibrium refinement In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most comm ...

of the

Nash equilibrium In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...

, being a Nash equilibrium that is also "evolutionarily

stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...

." Thus, once

fixed Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'' (film), an upcoming animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * Fi ...

in a population,

natural selection 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 Heredity, heritable traits characteristic of a population over generation ...

alone is sufficient to prevent alternative (

mutant In biology, and especially in genetics, a mutant is an organism or a new genetic character arising or resulting from an instance of mutation, which is generally an alteration of the DNA sequence of the genome or chromosome of an organism. It i ...

) strategies from replacing it (although this does not preclude the possibility that a better strategy, or set of strategies, will emerge in response to selective pressures resulting from environmental change).

History

Evolutionarily stable strategies were defined and introduced by

and

in a 1973 ''

Nature Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...

'' paper. Such was the time taken in peer-reviewing the paper for ''Nature'' that this was preceded by a 1972 essay by Maynard Smith in a book of essays titled ''On Evolution''. The 1972 essay is sometimes cited instead of the 1973 paper, but university libraries are much more likely to have copies of ''Nature''. Papers in ''Nature'' are usually short; in 1974, Maynard Smith published a longer paper in the ''

Journal of Theoretical Biology The ''Journal of Theoretical Biology'' is a biweekly peer-reviewed scientific journal covering theoretical biology, as well as mathematical, computational, and statistical aspects of biology. Some research areas covered by the journal include cell ...

''. Maynard Smith explains further in his 1982 book ''

Evolution and the Theory of Games ''Evolution and the Theory of Games'' is a book by the British evolutionary biology, evolutionary biologist John Maynard Smith on evolutionary game theory. The book was initially published in December 1982 by Cambridge University Press. Overvie ...

''. Sometimes these are cited instead. In fact, the ESS has become so central to game theory that often no citation is given, as the reader is assumed to be familiar with it. Maynard Smith mathematically formalised a verbal argument made by Price, which he read while peer-reviewing Price's paper. When Maynard Smith realized that the somewhat disorganised Price was not ready to revise his article for publication, he offered to add Price as co-author. The concept was derived from R. H. MacArthur and

W. D. Hamilton William Donald Hamilton (1 August 1936 – 7 March 2000) was a British evolutionary biologist, recognised as one of the most significant evolutionary theorists of the 20th century. Hamilton became known for his theoretical work expounding a ...

's work on

sex ratio A sex ratio is the ratio of males to females in a population. As explained by Fisher's principle, for evolutionary reasons this is typically about 1:1 in species which reproduce sexually. However, many species deviate from an even sex ratio, ei ...

s, derived from

Fisher's principle 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 mo ...

, especially Hamilton's (1967) concept of an

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''. In this paper Hamilton discusses sex ratios as strategies in a game, and cites Verner as using this language in his 1965 ...

. Maynard Smith was jointly awarded the 1999

Crafoord Prize The Crafoord Prize () is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord following a donation to the Royal Swedish Academy of Sciences. It is awarded jointly by the Acade ...

for his development of the concept of evolutionarily stable strategies and the application of game theory to the evolution of behaviour. Uses of ESS: * The ESS was a major element used to analyze evolution in

Richard Dawkins Richard Dawkins (born 26 March 1941) is a British evolutionary biology, evolutionary biologist, zoologist, science communicator and author. He is an Oxford fellow, emeritus fellow of New College, Oxford, and was Simonyi Professor for the Publ ...

' bestselling 1976 book ''

The Selfish Gene ''The Selfish Gene'' is a 1976 book on evolution by ethologist Richard Dawkins that promotes the gene-centred view of evolution, as opposed to views focused on the organism and the group. The book builds upon the thesis of George C. Willia ...

''. * The ESS was first used in the

social sciences Social science (often rendered in the plural as the social sciences) is one of the branches of science, devoted to the study of society, societies and the Social relation, relationships among members within those societies. The term was former ...

by Robert Axelrod in his 1984 book '' The Evolution of Cooperation''. Since then, it has been widely used in the social sciences, including

, and

. * In the social sciences, the primary interest is not in an ESS as the end of

biological Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, origin, evolution, and distribution of ...

evolution, but as an end point in

cultural evolution Cultural evolution is an evolutionary theory of social change. It follows from the definition of culture as "information capable of affecting individuals' behavior that they acquire from other members of their species through teaching, imitation ...

or individual learning. * In

, ESS is used primarily as a model for human biological evolution.

Motivation

The

is the traditional

solution concept In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most comm ...

game theory Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...

. It depends on the cognitive abilities of the players. It is assumed that players are aware of the structure of the game and consciously try to predict the moves of their opponents and to maximize their own payoffs. In addition, it is presumed that all the players know this (see

common knowledge Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, litera ...

). These assumptions are then used to explain why players choose Nash equilibrium strategies. Evolutionarily stable strategies are motivated entirely differently. Here, it is presumed that the players' strategies are biologically encoded and

heritable Heredity, also called inheritance or biological inheritance, is the passing on of Phenotypic trait, traits from parents to their offspring; either through asexual reproduction or sexual reproduction, the offspring cell (biology), cells or orga ...

. Individuals have no control over their strategy and need not be aware of the game. They reproduce and are subject to the forces of

, with the payoffs of the game representing reproductive success (biological fitness). It is imagined that alternative strategies of the game occasionally occur, via a process like

mutation In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, ...

. To be an ESS, a strategy must be resistant to these alternatives. Given the radically different motivating assumptions, it may come as a surprise that ESSes and Nash equilibria often coincide. In fact, every ESS corresponds to a Nash equilibrium, but some Nash equilibria are not ESSes.

Nash equilibrium

An ESS is a refined or modified form of a

. (See the next section for examples which contrast the two.) In a Nash equilibrium, if all players adopt their respective parts, no player can ''benefit'' by switching to any alternative strategy. In a two player game, it is a strategy pair. Let E(''S'',''T'') represent the payoff for playing strategy ''S'' against strategy ''T''. The strategy pair (''S'', ''S'') is a Nash equilibrium in a two player game if and only if for both players, for any strategy ''T'': :E(''S'',''S'') ≥ E(''T'',''S'') In this definition, a strategy ''T''≠''S'' can be a neutral alternative to ''S'' (scoring equally well, but not better). A Nash equilibrium is presumed to be stable even if ''T'' scores equally, on the assumption that there is no long-term incentive for players to adopt ''T'' instead of ''S''. This fact represents the point of departure of the ESS.

Maynard Smith John Maynard Smith (6 January 1920 – 19 April 2004) was a British theoretical and mathematical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics un ...

and

Price A price is the (usually not negative) quantity of payment or compensation expected, required, or given by one party to another in return for goods or services. In some situations, especially when the product is a service rather than a ph ...

specify two conditions for a strategy ''S'' to be an ESS. For all ''T''≠''S'', either # E(''S'',''S'') > E(''T'',''S''), or # E(''S'',''S'') = E(''T'',''S'') and E(''S'',''T'') > E(''T'',''T'') The first condition is sometimes called a ''strict'' Nash equilibrium. The second is sometimes called "Maynard Smith's second condition". The second condition means that although strategy ''T'' is neutral with respect to the payoff against strategy ''S'', the population of players who continue to play strategy ''S'' has an advantage when playing against ''T''. There is also an alternative, stronger definition of ESS, due to Thomas. This places a different emphasis on the role of the Nash equilibrium concept in the ESS concept. Following the terminology given in the first definition above, this definition requires that for all ''T''≠''S'' # E(''S'',''S'') ≥ E(''T'',''S''), and # E(''S'',''T'') > E(''T'',''T'') In this formulation, the first condition specifies that the strategy is a Nash equilibrium, and the second specifies that Maynard Smith's second condition is met. Note that the two definitions are not precisely equivalent: for example, each pure strategy in the coordination game below is an ESS by the first definition but not the second. In words, this definition looks like this: The payoff of the first player when both players play strategy S is higher than (or equal to) the payoff of the first player when he changes to another strategy T and the second player keeps his strategy S ''and'' the payoff of the first player when only his opponent changes his strategy to T is higher than his payoff in case that both of players change their strategies to T. This formulation more clearly highlights the role of the Nash equilibrium condition in the ESS. It also allows for a natural definition of related concepts such as a weak ESS or an evolutionarily stable set.

Examples of differences between Nash equilibria and ESSes

In most simple games, the ESSes and Nash equilibria coincide perfectly. For instance, in the

prisoner's dilemma The prisoner's dilemma is a game theory thought experiment involving two rational agents, each of whom can either cooperate for mutual benefit or betray their partner ("defect") for individual gain. The dilemma arises from the fact that while def ...

there is only one Nash equilibrium, and its strategy (''Defect'') is also an ESS. Some games may have Nash equilibria that are not ESSes. For example, in harm thy neighbor (whose payoff matrix is shown here) both (''A'', ''A'') and (''B'', ''B'') are Nash equilibria, since players cannot do better by switching away from either. However, only ''B'' is an ESS (and a strong Nash). ''A'' is not an ESS, so ''B'' can neutrally invade a population of ''A'' strategists and predominate, because ''B'' scores higher against ''B'' than ''A'' does against ''B''. This dynamic is captured by Maynard Smith's second condition, since E(''A'', ''A'') = E(''B'', ''A''), but it is not the case that E(''A'',''B'') > E(''B'',''B''). Nash equilibria with equally scoring alternatives can be ESSes. For example, in the game ''Harm everyone'', ''C'' is an ESS because it satisfies Maynard Smith's second condition. ''D'' strategists may temporarily invade a population of ''C'' strategists by scoring equally well against ''C'', but they pay a price when they begin to play against each other; ''C'' scores better against ''D'' than does ''D''. So here although E(''C'', ''C'') = E(''D'', ''C''), it is also the case that E(''C'',''D'') > E(''D'',''D''). As a result, ''C'' is an ESS. Even if a game has pure strategy Nash equilibria, it might be that none of those pure strategies are ESS. Consider the

Game of chicken The game of chicken, also known as the hawk-dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the ideal outcome is for one player to yield (to avoid the worst outcome if n ...

. There are two pure strategy Nash equilibria in this game (''Swerve'', ''Stay'') and (''Stay'', ''Swerve''). However, in the absence of an

uncorrelated asymmetry In game theory, an uncorrelated asymmetry is an arbitrary distinguishing feature between players in an otherwise symmetric game. This concept refers to asymmetries that are unrelated to the Payoff matrix, payoffs or strategic structure of the game i ...

, neither ''Swerve'' nor ''Stay'' are ESSes. There is a third Nash equilibrium, a

mixed strategy In game theory, a move, action, or play is any one of the options which a player can choose in a setting where the optimal outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the actio ...

which is an ESS for this game (see

Hawk-dove game The game of chicken, also known as the hawk-dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the ideal outcome is for one player to yield (to avoid the worst outcome if n ...

and

Best response In game theory, the best response is the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution ...

for explanation). This last example points to an important difference between Nash equilibria and ESS. Nash equilibria are defined on ''strategy sets'' (a specification of a strategy for each player), while ESS are defined in terms of strategies themselves. The equilibria defined by ESS must always be

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

, and thus have fewer equilibrium points.

Vs. evolutionarily stable state

In population biology, the two concepts of an ''evolutionarily stable strategy'' (ESS) and an ''

evolutionarily stable state 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" (Maynard Smith, 1982).Maynard Smith, J.. (1 ...

'' are closely linked but describe different situations. In an evolutionarily stable ''strategy,'' if all the members of a population adopt it, no mutant strategy can invade. Once virtually all members of the population use this strategy, there is no 'rational' alternative. ESS is part of classical

. In an evolutionarily stable ''state,'' a population's genetic composition is restored by selection after a disturbance, if the disturbance is not too large. An evolutionarily stable state is a dynamic property of a population that returns to using a strategy, or mix of strategies, if it is perturbed from that initial state. It is part of

population genetics Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, s ...

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

, or

evolutionary game theory 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 Darwinism, Darwinian competition can be modelled. It originated in 1973 wi ...

. This is now called convergent stability. B. Thomas (1984) applies the term ESS to an individual strategy which may be mixed, and evolutionarily stable population state to a population mixture of pure strategies which may be formally equivalent to the mixed ESS. Whether a population is evolutionarily stable does not relate to its genetic diversity: it can be genetically monomorphic or polymorphic.

Stochastic ESS

In the classic definition of an ESS, no mutant strategy can invade. In finite populations, any mutant could in principle invade, albeit at low probability, implying that no ESS can exist. In an infinite population, an ESS can instead be defined as a strategy which, should it become invaded by a new mutant strategy with probability p, would be able to counterinvade from a single starting individual with probability >p, as illustrated by the evolution of bet-hedging.

Prisoner's dilemma

A common model of

altruism Altruism is the concern for the well-being of others, independently of personal benefit or reciprocity. The word ''altruism'' was popularised (and possibly coined) by the French philosopher Auguste Comte in French, as , for an antonym of egoi ...

and social cooperation is the

Prisoner's dilemma The prisoner's dilemma is a game theory thought experiment involving two rational agents, each of whom can either cooperate for mutual benefit or betray their partner ("defect") for individual gain. The dilemma arises from the fact that while def ...

. Here a group of players would collectively be better off if they could play ''Cooperate'', but since ''Defect'' fares better each individual player has an incentive to play ''Defect''. One solution to this problem is to introduce the possibility of retaliation by having individuals play the game repeatedly against the same player. In the so-called '' iterated'' Prisoner's dilemma, the same two individuals play the prisoner's dilemma over and over. While the Prisoner's dilemma has only two strategies (''Cooperate'' and ''Defect''), the iterated Prisoner's dilemma has a huge number of possible strategies. Since an individual can have different contingency plan for each history and the game may be repeated an indefinite number of times, there may in fact be an infinite number of such contingency plans. Three simple contingency plans which have received substantial attention are ''Always Defect'', ''Always Cooperate'', and ''

Tit for Tat Tit for tat is an English saying meaning "equivalent retaliation". It is an alternation of '' tip for tap'' "blow for blow", first recorded in 1558. It is also a highly effective strategy in game theory. An agent using this strategy will fi ...

''. The first two strategies do the same thing regardless of the other player's actions, while the latter responds on the next round by doing what was done to it on the previous round—it responds to ''Cooperate'' with ''Cooperate'' and ''Defect'' with ''Defect''. If the entire population plays ''Tit-for-Tat'' and a mutant arises who plays ''Always Defect'', ''Tit-for-Tat'' will outperform ''Always Defect''. If the population of the mutant becomes too large — the percentage of the mutant will be kept small. ''Tit for Tat'' is therefore an ESS, ''with respect to only these two strategies''. On the other hand, an island of ''Always Defect'' players will be stable against the invasion of a few ''Tit-for-Tat'' players, but not against a large number of them. If we introduce ''Always Cooperate'', a population of ''Tit-for-Tat'' is no longer an ESS. Since a population of ''Tit-for-Tat'' players always cooperates, the strategy ''Always Cooperate'' behaves identically in this population. As a result, a mutant who plays ''Always Cooperate'' will not be eliminated. However, even though a population of ''Always Cooperate'' and ''Tit-for-Tat'' can coexist, if there is a small percentage of the population that is ''Always Defect'', the selective pressure is against ''Always Cooperate'', and in favour of ''Tit-for-Tat''. This is due to the lower payoffs of cooperating than those of defecting in case the opponent defects. This demonstrates the difficulties in applying the formal definition of an ESS to games with large strategy spaces, and has motivated some to consider alternatives.

Human behavior

The fields of

sociobiology Sociobiology is a field of biology that aims to explain social behavior in terms of evolution. It draws from disciplines including psychology, ethology, anthropology, evolution, zoology, archaeology, and population genetics. Within the study of ...

and

attempt to explain animal and human behavior and social structures, largely in terms of evolutionarily stable strategies.

Sociopathy Psychopathy, or psychopathic personality, is a personality construct characterized by impaired empathy and remorse, along with bold, disinhibited, and egocentric traits. These traits are often masked by superficial charm and immunity to s ...

(chronic antisocial or criminal behavior) may be a result of a combination of two such strategies. Evolutionarily stable strategies were originally considered for biological evolution, but they can apply to other contexts. In fact, there are stable states for a large class of

adaptive dynamics Evolutionary invasion analysis, also known as adaptive dynamics, is a set of mathematical modeling techniques that use differential equations to study the long-term evolution of traits in asexually and sexually reproducing populations. It res ...

. As a result, they can be used to explain human behaviours that lack any genetic influences.

References

External links

Evolutionarily Stable Strategies
at Animal Behavior: An Online Textbook by Michael D. Breed.

Kenneth N. Prestwich's site at College of the Holy Cross.
Evolutionarily stable strategies knol
Archived: https://web.archive.org/web/20091005015811/http://knol.google.com/k/klaus-rohde/evolutionarily-stable-strategies-and/xk923bc3gp4/50# {{DEFAULTSORT:Evolutionarily Stable Strategy Game theory equilibrium concepts Evolutionary game theory