Evolutionary stable strategy

An evolutionarily stable strategy (abbreviated ESS , . English evolutionarily stable strategy ) is a strategy that - if enough members of a population they turn to - can be improved by any alternative strategy. It is a game theory concept formulated in Theoretical Biology in 1973 by John Maynard Smith and George R. Price . Evolutionary game theory developed around them . In sociobiology and evolutionary psychology there are more or less controversial studies which attempt to interpret absurd or anti-social human behavior as ESS. For example, suitable examinations could be used to determine whether a certain proportion of criminals in the total population would correspond to an ESS.

An ESS guarantees superiority or immunity to an intruder or " mutant " in a population . If an intruder succeeds in asserting himself with his own foreign strategy, one speaks of an invasion . It is important to add that an evolutionarily stable strategy does not just refer to the behavior of an individual. In a broader sense, the hereditary endowment of a living being could also be understood as a “strategy” - for example, a plant species that is poorly adapted to acidic soils could be displaced by another species which, for genetic reasons, shows better adaptation. The evolutionarily stable state is called equilibrium in evolutionarily stable strategies .

definition

${\ displaystyle x}$ and are two (mixed) strategies ${\ displaystyle y}$
${\ displaystyle \ Pi (x, y)}$ the payout, or the reproductive success of an individual with strategy in an environment in which all other strategy run ${\ displaystyle x}$ ${\ displaystyle y}$

${\ displaystyle x}$ is only an ESS if the following applies to all : ${\ displaystyle y \ neq x}$

{\ displaystyle \ Pi (x, x)> \ Pi (y, x)}

, or

{\ displaystyle \ Pi (x, x) = \ Pi (y, x)}

and

{\ displaystyle \ Pi (x, y)> \ Pi (y, y)}

The first part of the definition is equivalent to the so-called strict equilibrium , i.e. That means: x is the best answer to itself. No other strategy does better than x than x itself. The second part provides that if strategy y is not eliminated in a pure x environment, since it is immediately successful, x can prevail over y in a pure y environment - so the population cannot tilt from x to y.

An equivalent definition is as follows: Strategy x is an ESS, if for all and sufficiently small ε> 0 applies: ${\ displaystyle y \ neq x}$

{\ displaystyle (1- \ varepsilon) \ Pi (x, x) + \ varepsilon \ Pi (x, y)> (1- \ varepsilon) \ Pi (y, x) + \ varepsilon \ Pi (y, y) }

Both sides of the inequality represent the payoff for the reproductive success in a population in which, with the exception of a small proportion ε of "mutants", all follow strategy x; on the left is this expected success for the x strategy, while on the right is that of the mutants who are pursuing strategy y.

If the defining condition is met, there can be no successful invasion by mutants in a population in which everyone is pursuing strategy x - provided that mutants can initially only appear in small numbers.

Examples

In a poker club all players play strategy X. Now a strange player comes to visit. If his other strategy Y is unsuccessful and he is not making any money, then it is likely that he will switch to strategy X (ESS is in effect). But if his Y strategy is profitable, then all members switch from strategy X to Y. So Y is the ESS; a second new member with strategy Y would do no better than the current club members.
Suppose that in Europe there was a bird that every autumn moves to the south. If an individual appears who decides to stay in Europe in mild winters , then it has the advantage that it can tap into the (food) resources in spring before the other birds return. With better fat reserves in the body, the individual is now able to raise better offspring than any other. Those birds that always migrate south will die out in the long term because they reproduce more poorly. It is also a species of disadvantage always to stay in Europe; the winter could one day be so cold that many of them die and then the birds returning from the south have a much better starting position. The ESS of the bird species consists in leaving parts of the population in Europe that survive more or less successfully depending on the mildness of the winter.
The textbook example of the ESS - the hawk-pigeon game : Individuals of the same population are classified as 'hawks' (aggressive, strong) and 'pigeons' (peaceful, evasive). If a pigeon meets a pure pigeon population, nothing changes. The same is true when a hawk joins other hawks. But there are four special cases:
- A pigeon encounters hawks: Since the pigeon evades conflicts - for example over food - and thus saves strength and physical injuries, it is pursuing a successful strategy. To do this, she can use threatening gestures to simulate aggressiveness and steal resources from hawks while saving energy.
- A hawk meets pigeons: The pigeons make room for the newcomer and give him all resources without a fight. The hawk is successful.
- A pigeon or a hawk joins a mixed population, in which pigeons and hawks occur in the correct numerical ratio (corresponds to the ESS!). For the newcomer it is now important whether he will encounter a pigeon or a hawk more often. Once the population has leveled off to the right mix, it doesn't matter whether the intruder behaves as a pigeon or a hawk.
- The so-called “citizen” strategy (English / French bourgeois ) is developing into the ESS and it is largely immune to an unbalanced hawk-pigeon population. As a “citizen”: if you defend yourself, you are a hawk; if you attack someone, you behave like a dove.
In the repeated prisoner's dilemma , a population of tit for tat strategies is not evolutionarily stable. While it is resistant to "Defect Always" strategies, it can be undermined by "Cooperate Always" strategies, as can be easily checked using the definition with x = Tit-For-Tat and y = "Cooperate Always": TFT versus “Cooperate Always” results in the same payouts for both players, so condition 1 ( ) does not apply. However, part 2 of condition 2 ( ) does not apply either, since in this case both strategies lead to the same payout for both players. A population from “cooperate always” strategies can in turn be infiltrated by “defect always”. ${\ displaystyle \ Pi (x, x)> \ Pi (y, x)}$ ${\ displaystyle \ Pi (x, y)> \ Pi (y, y)}$

literature

Kenneth G. Binmore and Larry Samuelson: Evolutionary Stability in Repeated Games Played by Finite Automata. In: Journal of Economic Theory 1992, pp. 278-305.
Maynard Smith, John: Evolution and the Theory of Games . Cambridge [u. a.]: Cambridge Univ. Press, 1982.
Hofbauer, Sigmund: Evolutionary Games and Population Dynamics , Cambridge Univ. Press, ISBN 0-521-62570-X
Krebs, John R .: Introduction to Behavioral Ecology . Berlin [etc.]: Blackwell Wissenschafts-Verlag, 1996.

Individual evidence

↑ David McFarland: Biology of Behavior. Evolution, physiology, psychobiology . 2nd Edition. Spectrum academic publishing house, Heidelberg 1999, ISBN 3-8274-0925-X .
↑ John Maynard Smith, George R. Price: The Logic of Animal Conflict. Nature , Volume 246, 1973, pp. 15-18, doi : 10.1038 / 246015a0

Web links

The evolutionary stable strategy

[1] David McFarland: Biology of Behavior. Evolution, physiology, psychobiology . 2nd Edition. Spectrum academic publishing house, Heidelberg 1999, ISBN 3-8274-0925-X .

[2] John Maynard Smith, George R. Price: The Logic of Animal Conflict. Nature , Volume 246, 1973, pp. 15-18, doi : 10.1038 / 246015a0