Evolutionarily stable strategy
Encyclopedia
In game theory
and behavioural ecology, an evolutionarily stable strategy (ESS), which is sometimes also called an evolutionary stable strategy, is a strategy
which, if adopted by a population
of players, cannot be invaded by any alternative strategy that is initially rare. 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 ESS was developed to define a class of solutions to game theoretic problems, equivalent to the Nash equilibrium, which could be applied to the evolution
of social behaviour in animals. Nash equilibria may exist due to the application of rational foresight
, but that does not apply in an evolutionary context. Teleological
forces such as rational foresight cannot explain the outcomes of trial-and-error processes, such as evolution, and thus have no place in biological applications. The definition of an ESS excludes such Nash equilibria.
First developed in 1973, the ESS is widely used in behavioural ecology and economics
, and has been used in anthropology
, evolutionary psychology
, philosophy
, and political science
.
and George R. Price
in a 1973 Nature
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
. Maynard Smith explains further in his 1982 book Evolution and the Theory of Games
. 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
's work on sex ratio
s, derived from Fisher's principle
, especially Hamilton's (1967) concept of an unbeatable strategy
. Maynard Smith was jointly awarded the 1999 Crafoord Prize
for his development of the concept of evolutionarily stable strategies and the application of game theory to the evolution of behaviour.
Uses of ESS:
is the traditional solution concept
in game theory
. 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
). 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. Individuals have no control over their strategy and need not be aware of the game. They reproduce and are subject to the forces of natural selection
(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
. 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.
or modified form of a Nash equilibrium
. (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 this is true for both players and for all T≠S:
In this definition, strategy T 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
and Price
specify two conditions for a strategy S to be an ESS. Either
for all T≠S.
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 definition of ESS, which 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, we have (adapted from Thomas, 1985):
for all T≠S.
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 players keeps his strategy S. *AND* The payoff of first player when only he 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
.
In most simple games, the ESSes and Nash equilibria coincide perfectly. For instance, in the Prisoner's Dilemma
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 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 'bstrategists 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
. There are two pure strategy Nash equilibria in this game (Swerve, Stay) and (Stay, Swerve). However, in the absence of an uncorrelated asymmetry
, neither Swerve nor Stay are ESSes. There is a third Nash equilibrium, a mixed strategy which is an ESS for this game (see Hawk-dove game
and Best response
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
, and thus have fewer equilibrium points.
are closely linked but describe different situations.
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
.
and social cooperation is the Prisoner's dilemma
. 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
. The first two strategies do the same thing regardless of the other player's actions, while the later 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.
and evolutionary psychology
attempt to explain animal and human behavior and social structures, largely in terms of evolutionarily stable strategies. Sociopathy (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. As a result, they can be used to explain human behaviours that lack any genetic influences.
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
and behavioural ecology, an evolutionarily stable strategy (ESS), which is sometimes also called an evolutionary stable strategy, is a strategy
Strategy (game theory)
In game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour...
which, if adopted by a population
Population genetics
Population genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium refinement of the Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
. It is a Nash equilibrium that is "evolutionarily
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
" stable
Ecological stability
Ecological stability can refer to types of stability in a continuum ranging from resilience to constancy to persistence. The precise definition depends on the ecosystem in question, the variable or variables of interest, and the overall context...
: once it is fixed
Fixation (population genetics)
In population genetics, fixation is the change in a gene pool from a situation where there exist at least two variants of a particular gene to a situation where only one of the alleles remains...
in a population, natural selection
Natural selection
Natural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....
alone is sufficient to prevent alternative (mutant
Mutant
In biology and especially genetics, a mutant is an individual, organism, or new genetic character, arising or resulting from an instance of mutation, which is a base-pair sequence change within the DNA of a gene or chromosome of an organism resulting in the creation of a new character or trait not...
) strategies from invading successfully.
The ESS was developed to define a class of solutions to game theoretic problems, equivalent to the Nash equilibrium, which could be applied to the evolution
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
of social behaviour in animals. Nash equilibria may exist due to the application of rational foresight
Homo economicus
Homo economicus, or Economic human, is the concept in some economic theories of humans as rational and narrowly self-interested actors who have the ability to make judgments toward their subjectively defined ends...
, but that does not apply in an evolutionary context. Teleological
Teleology
A teleology is any philosophical account which holds that final causes exist in nature, meaning that design and purpose analogous to that found in human actions are inherent also in the rest of nature. The word comes from the Greek τέλος, telos; root: τελε-, "end, purpose...
forces such as rational foresight cannot explain the outcomes of trial-and-error processes, such as evolution, and thus have no place in biological applications. The definition of an ESS excludes such Nash equilibria.
First developed in 1973, the ESS is widely used in behavioural ecology and economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, and has been used in anthropology
Anthropology
Anthropology is the study of humanity. It has origins in the humanities, the natural sciences, and the social sciences. The term "anthropology" is from the Greek anthrōpos , "man", understood to mean mankind or humanity, and -logia , "discourse" or "study", and was first used in 1501 by German...
, evolutionary psychology
Evolutionary psychology
Evolutionary psychology is an approach in the social and natural sciences that examines psychological traits such as memory, perception, and language from a modern evolutionary perspective. It seeks to identify which human psychological traits are evolved adaptations, that is, the functional...
, philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
, and political science
Political science
Political Science is a social science discipline concerned with the study of the state, government and politics. Aristotle defined it as the study of the state. It deals extensively with the theory and practice of politics, and the analysis of political systems and political behavior...
.
History
Evolutionarily stable strategies were defined and introduced by John Maynard SmithJohn Maynard Smith
John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
and George R. Price
George R. Price
George Robert Price was an American population geneticist. Originally a physical chemist and later a science journalist, he moved to London in 1967, where he worked in theoretical biology at the Galton Laboratory, making three important contributions: first, rederiving W.D...
in a 1973 Nature
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
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
Journal of Theoretical Biology
The Journal of Theoretical Biology is a scientific journal about theoretical biology; dealing with theoretical issues, as well as mathematical and computational aspects of biology. Some research areas covered by the papers published in the journal are population genetics, morphogenesis,...
. Maynard Smith explains further in his 1982 book Evolution and the Theory of Games
Evolution and the Theory of Games
Evolution and the Theory of Games is a 1982 book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. In it, Maynard Smith summarises work on evolutionary game theory that had developed in the 1970s, to which he made several important contributions...
. 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
Robert MacArthur
Robert Helmer MacArthur was an American ecologist who made a major impact on many areas of community and population ecology....
and W. D. Hamilton
W. D. Hamilton
William Donald Hamilton FRS was a British evolutionary biologist, widely recognised as one of the greatest evolutionary theorists of the 20th century....
's work on sex ratio
Sex ratio
Sex ratio is the ratio of males to females in a population. The primary sex ratio is the ratio at the time of conception, secondary sex ratio is the ratio at time of birth, and tertiary sex ratio is the ratio of mature organisms....
s, derived from Fisher's principle
Fisher's principle
Fisher's principle explains why the sex ratio of most species is approximately 1:1. It was famously outlined by Ronald Fisher in his 1930 book The Genetical Theory of Natural Selection . Nevertheless, A. W. F. Edwards has remarked that it is "probably the most celebrated argument in evolutionary...
, especially Hamilton's (1967) concept of an unbeatable strategy
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 paper which "claims to show that, given factors causing...
. Maynard Smith was jointly awarded the 1999 Crafoord Prize
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...
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 DawkinsRichard DawkinsClinton Richard Dawkins, FRS, FRSL , known as Richard Dawkins, is a British ethologist, evolutionary biologist and author...
' bestselling 1976 book The Selfish GeneThe Selfish GeneThe Selfish Gene is a book on evolution by Richard Dawkins, published in 1976. It builds upon the principal theory of George C. Williams's first book Adaptation and Natural Selection. Dawkins coined the term "selfish gene" as a way of expressing the gene-centred view of evolution as opposed to the...
. - The ESS was first used in the social sciencesSocial sciencesSocial science is the field of study concerned with society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences usually exclusive of the administrative or managerial sciences...
by Robert AxelrodRobert AxelrodRobert M. Axelrod is an American political scientist. He is Professor of Political Science and Public Policy at the University of Michigan where he has been since 1974. He is best known for his interdisciplinary work on the evolution of cooperation, which has been cited in numerous articles...
in his 1984 book The Evolution of CooperationThe Evolution of CooperationThe evolution of cooperation can refer to:* the study of how cooperation can emerge and persist as elucidated by application of game theory,* a 1981 paper by political scientist Robert Axelrod and evolutionary biologist W. D...
. Since then, it has been widely used in the social sciences, including anthropologyAnthropologyAnthropology is the study of humanity. It has origins in the humanities, the natural sciences, and the social sciences. The term "anthropology" is from the Greek anthrōpos , "man", understood to mean mankind or humanity, and -logia , "discourse" or "study", and was first used in 1501 by German...
, economicsEconomicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, philosophyPhilosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
, and political sciencePolitical sciencePolitical Science is a social science discipline concerned with the study of the state, government and politics. Aristotle defined it as the study of the state. It deals extensively with the theory and practice of politics, and the analysis of political systems and political behavior...
. - In the social sciences, the primary interest is not in an ESS as the end of biological evolution, but as an end point in cultural evolution or individual learning.
- In evolutionary psychologyEvolutionary psychologyEvolutionary psychology is an approach in the social and natural sciences that examines psychological traits such as memory, perception, and language from a modern evolutionary perspective. It seeks to identify which human psychological traits are evolved adaptations, that is, the functional...
, ESS is used primarily as a model for human biological evolutionHuman evolutionHuman evolution refers to the evolutionary history of the genus Homo, including the emergence of Homo sapiens as a distinct species and as a unique category of hominids and mammals...
.
Motivation
The Nash equilibriumNash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
is the traditional solution concept
Solution concept
In game theory, a solution concept is a formal rule for predicting how the game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players, therefore predicting the result of the game...
in game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
. 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 (logic)
Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.The concept was first introduced in...
). 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. Individuals have no control over their strategy and need not be aware of the game. They reproduce and are subject to the forces of natural selection
Natural selection
Natural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....
(with the payoffs of the game representing reproductive success (biological fitness
Fitness (biology)
Fitness is a central idea in evolutionary theory. It can be defined either with respect to a genotype or to a phenotype in a given environment...
). It is imagined that alternative strategies of the game occasionally occur, via a process like mutation
Mutation
In molecular biology and genetics, mutations are changes in a genomic sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a virus. They can be defined as sudden and spontaneous changes in the cell. Mutations are caused by radiation, viruses, transposons and mutagenic...
. 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 equilibria and ESS
An ESS is a refinedSolution concept
In game theory, a solution concept is a formal rule for predicting how the game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players, therefore predicting the result of the game...
or modified form of a Nash equilibrium
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
. (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 this is true for both players and for all T≠S:
- E(S,S) ≥ E(T,S)
In this definition, strategy T 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
John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
and Price
George R. Price
George Robert Price was an American population geneticist. Originally a physical chemist and later a science journalist, he moved to London in 1967, where he worked in theoretical biology at the Galton Laboratory, making three important contributions: first, rederiving W.D...
specify two conditions for a strategy S to be an ESS. Either
- E(S,S) > E(T,S), or
- E(S,S) = E(T,S) and E(S,T) > E(T,T)
for all T≠S.
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 definition of ESS, which 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, we have (adapted from Thomas, 1985):
- E(S,S) ≥ E(T,S), and
- E(S,T) > E(T,T)
for all T≠S.
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 players keeps his strategy S. *AND* The payoff of first player when only he 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
Evolutionarily stable set
In game theory an evolutionarily stable set , sometimes referred to as evolutionary stable sets, is a set of strategies, which score equally against each other, each of which would be an evolutionarily stable strategy were it not for the other members of the set. ES sets were first defined by...
.
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
Prisoner's dilemma
The prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
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 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 'bstrategists 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
Chicken (game)
The game of chicken, also known as the hawk-dove or snowdrift game, is an influential model of conflict for two players in game theory...
. There are two pure strategy Nash equilibria in this game (Swerve, Stay) and (Stay, Swerve). However, in the absence of an uncorrelated asymmetry
Uncorrelated asymmetry
In game theory an uncorrelated asymmetry is an arbitrary asymmetry in a game which is otherwise symmetrical. The name 'uncorrelated asymmetry' is due to John Maynard Smith who called payoff relevant asymmetries in games with similar roles for each player 'correlated asymmetries' .The explanation of...
, neither Swerve nor Stay are ESSes. There is a third Nash equilibrium, a mixed strategy which is an ESS for this game (see Hawk-dove game
Chicken (game)
The game of chicken, also known as the hawk-dove or snowdrift game, is an influential model of conflict for two players in game theory...
and Best response
Best response
In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given...
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
Symmetric equilibrium
In game theory, a symmetric equilibrium is an equilibrium where both players use the same strategy in the equilibrium. In the Prisoner's Dilemma game pictured to the right, the only Nash equilibrium is . Since both players use the same strategy, the equilibrium is symmetric.Symmetric equilibria...
, and thus have fewer equilibrium points.
ESS vs. Evolutionarily Stable State
In population biology, the two concepts of an evolutionarily stable strategy (ESS) and an evolutionarily stable stateEvolutionarily stable state
"A population is said to be in an evolutionarily stable state if its genetic composition is restored by selection after a disturbance, provided the disturbance is not too large...
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 game theoryGame theoryGame theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
.
- In an evolutionarily stable state, a population's genetic composition will be 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 geneticsPopulation geneticsPopulation genetics is the study of allele frequency distribution and change under the influence of the four main evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors of recombination, population subdivision and population...
, dynamical systemDynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
, or evolutionary game theoryEvolutionary game theoryEvolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...
.
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
Polymorphism (biology)
Polymorphism in biology occurs when two or more clearly different phenotypes exist in the same population of a species — in other words, the occurrence of more than one form or morph...
.
Prisoner's dilemma and ESS
A common model of altruismAltruism
Altruism is a concern for the welfare of others. It is a traditional virtue in many cultures, and a core aspect of various religious traditions, though the concept of 'others' toward whom concern should be directed can vary among cultures and religions. Altruism is the opposite of...
and social cooperation is the Prisoner's dilemma
Prisoner's dilemma
The prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
. 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
Repeated game
In game theory, a repeated game is an extensive form game which consists in some number of repetitions of some base game . The stage game is usually one of the well-studied 2-person games...
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
Tit for tat is an English saying meaning "equivalent retaliation". It is also a highly effective strategy in game theory for the iterated prisoner's dilemma. It was first introduced by Anatol Rapoport in Robert Axelrod's two tournaments, held around 1980. An agent using this strategy will initially...
. The first two strategies do the same thing regardless of the other player's actions, while the later 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.
ESS and human behavior
The fields of sociobiologySociobiology
Sociobiology is a field of scientific study which is based on the assumption that social behavior has resulted from evolution and attempts to explain and examine social behavior within that context. Often considered a branch of biology and sociology, it also draws from ethology, anthropology,...
and evolutionary psychology
Evolutionary psychology
Evolutionary psychology is an approach in the social and natural sciences that examines psychological traits such as memory, perception, and language from a modern evolutionary perspective. It seeks to identify which human psychological traits are evolved adaptations, that is, the functional...
attempt to explain animal and human behavior and social structures, largely in terms of evolutionarily stable strategies. Sociopathy (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. As a result, they can be used to explain human behaviours that lack any genetic influences.
See also
- Antipredator adaptation
- Evolutionary game theoryEvolutionary game theoryEvolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...
- Fitness landscapeFitness landscapeIn evolutionary biology, fitness landscapes or adaptive landscapes are used to visualize the relationship between genotypes and reproductive success. It is assumed that every genotype has a well-defined replication rate . This fitness is the "height" of the landscape...
- Hawk-Dove game
- Prisoner's dilemmaPrisoner's dilemmaThe prisoner’s dilemma is a canonical example of a game, analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interest to do so. It was originally framed by Merrill Flood and Melvin Dresher working at RAND in 1950. Albert W...
- War of attrition (game)War of attrition (game)In game theory, the war of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while constantly accumulating costs over the time t that the contest lasts. The model was originally formulated by John Maynard Smith, a mixed evolutionary stable...
Further reading
. An 88-page mathematical introduction; see Section 3.8. Free online at many universities.- Parker, G.A.Geoff ParkerProfessor Geoffrey Alan Parker FRS is a Derby professor of biology at the University of Liverpool.He has a particular interest in behavioural ecology and evolutionary biology, and is most noted for introducing the concept of sperm competition in 1970.Much of his work from the 1970s onwards has...
(1984) Evolutionary stable strategies. In Behavioural Ecology: an Evolutionary Approach (2nd ed) Krebs, J.R.John KrebsJohn Richard Krebs, Baron Krebs FRS is a world leader in zoology and more specifically bird behaviour. He is currently the Principal of Jesus College, Oxford University...
& Davies N.B., eds. pp 30–61. Blackwell, Oxford.. A comprehensive reference from a computational perspective; see Section 7.7. Downloadable free online. - John Maynard SmithJohn Maynard SmithJohn Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
. (1982) Evolution and the Theory of GamesEvolution and the Theory of GamesEvolution and the Theory of Games is a 1982 book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. In it, Maynard Smith summarises work on evolutionary game theory that had developed in the 1970s, to which he made several important contributions...
. ISBN 0-521-28884-3. Classic reference.
External links
- Evolutionarily Stable Strategies at Animal Behavior: An Online Textbook by Michael D. Breed.
- Game Theory and Evolutionarily Stable Strategies, Kenneth N. Prestwich's site at College of the Holy Cross.
- Evolutionarily stable strategies knol