Evolutionary game theory

A recent development in the field of theoretical biology is the study of evolutionary processes , the spread and distribution of behavioral patterns in animal populations through natural selection , spread of infections , using methods and models from game theory . The evolutionary game theory examines the temporal and / or spatial development of different phenotypes in a population. The phenotypes act on each other in constant change and use different strategies, e.g. B. when foraging or territorial fights. The strategies used decide whether the fitness of the individual phenotypes will improve or worsen over time. The change in fitness of the individual phenotypes in turn influences their distribution within the population, their frequency.

The trajectories of the temporal development of the individual phenotypes can have different attractors . Phenotypes can become extinct, but coexistence is also possible.

Pioneering work was done by John Maynard Smith and George R. Price , as well as Robert Axelrod .

This approach was motivated u. a. through the observation made in many animal populations that even heavily armed animals in turf and mating battles only rarely use their weapons to fatally injure a rival. In older biological research (e.g. Thomas Henry Huxley , Konrad Lorenz ) these phenomena were explained by the principle of preserving one's own species, which postulated peaceful behavior towards rivals of one's own species. This explanation has been abandoned in recent years.

It has been shown in several studies that game theory considerations can help to explain the phenomena mentioned above. The transfer of game theory concepts to the explanation of biological phenomena is not without controversy, since game theory concepts were initially developed for the interaction of consciously acting individuals. For this reason, some game theorists refer the theory of evolutionary games from the field of game theory in the narrower sense.

Fields of investigation

The population dynamics of strategies and behavioral patterns
The appearance of evolutionarily stable strategies (ESS)

Models

The evolutionary theory models differ significantly from 'classical' game theory in several basic assumptions. The classic approach assumes z. B. rational selection of strategies and complete information of the players. These assumptions are not made in evolutionary game theory.

In evolutionary game theory, players are not looking for a solution. Those who have a less successful strategy simply drop out of the population over time. This means that this theory starts with an extremely limited, rational decision-making behavior. However, it remains rational in that the proportion of players with a successful strategy is growing. The rationality solution does not result from the foresight of rational actors, but is the result of a selection mechanism . This can be interpreted as learning . This learning then takes place at the level of the entire population of players. However, no statements are made about the learning of a specific player.

The conscious choice of a strategy in the knowledge of interaction with other players is also completely in the background in evolutionary game theory. Instead, in an evolutionary context, the players have certain behavioral patterns like automatons and the central question aims at which behavioral patterns 'survive' to what extent in the game and which new behavioral patterns (strategies) can successfully penetrate the game.

Description of the dynamics with replicator equations

In the basic model of evolutionary game theory, the temporal development of n phenotypes (replicator species) is described with replicator equations.

{\ displaystyle {\ dot {x_ {i}}} = x_ {i} (f_ {i} - \ Phi), 1 \ leq i \ leq n}

.

The growth rate of phenotype i depends on: ${\ displaystyle {\ dot {x_ {i}}}}$

the relative frequency of the phenotype within a population, ${\ displaystyle x_ {i}}$ ${\ displaystyle i}$
the fitness of the phenotype and the ${\ displaystyle f_ {i}}$ ${\ displaystyle i}$
mean fitness of the population ${\ displaystyle \ Phi}$

Is the fitness of a phenotype i ${\ displaystyle f_ {i}}$

greater than mean fitness, the relative frequency of the phenotype in the population increases; ${\ displaystyle f_ {i}> \ Phi}$
as large as the mean fitness, the relative frequency does not change; ${\ displaystyle (f_ {i} - \ Phi) = 0}$
less than the mean fitness, the relative frequency decreases. ${\ displaystyle f_ {i} <\ Phi}$

The population is described by the vector of the relative frequencies of the individual phenotypes ; ${\ displaystyle x = (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle x_ {i} \ in [0,1]}$
the population is constant, so it is true: . ${\ displaystyle \ sum _ {i = 1} ^ {n} x_ {i} = 1}$

The interaction of the phenotypes in the model, each phenotype representing a strategy, is modeled by a payout matrix. ${\ displaystyle A = {\ begin {bmatrix} a_ {11} & \ cdots & a_ {1n} \\\ vdots & \ ddots & \ vdots \\ a_ {n1} & \ cdots & a_ {nn} \ end {bmatrix} }}$

If two phenotypes meet, the phenotype changes its fitness by the value and the phenotype by the value . ${\ displaystyle 1 \ leq i, j \ leq n}$ ${\ displaystyle i}$ ${\ displaystyle a_ {ij}}$ ${\ displaystyle j}$ ${\ displaystyle a_ {ji}}$

The overall fitness of the phenotype is then . ${\ displaystyle f_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle f_ {i} = \ sum _ {j = 1} ^ {n} x_ {j} a_ {ij}}$

The selection of a phenotype, expressed as its relative frequency, is therefore dependent on the frequency of all other phenotypes within the population.

The mean fitness is . ${\ displaystyle \ Phi = \ sum _ {i = 1} ^ {n} x_ {i} f_ {i}}$

meaning

Recently, arguments from evolutionary game theory have played an increasingly important role in modeling learning in games. Here it is in particular the aspect of the limited rationality of players that makes the adoption of elements of evolutionary game theory models attractive. Evolutionary game theory is by no means restricted to the description of biological phenomena; it is increasingly permeating areas of game theory that deal with consciously acting but not always completely rational players.

example

One of the most famous examples of evolutionary stable strategies is the hawk-dove game . The falcon-pigeon game models the competition for a resource ( territory , partner , nesting place ...). The name of the game, however, is misleading: It is not about the conflict between two different animal species, but instead the names ' falcon ' and ' dove ' stand for two behaviors that animals of one species can use in a competition:

- Pigeon: peaceful behavior

- Falcon: aggressive behavior

The question then is which of these behaviors will prevail in a population or whether both can coexist and coexist.

literature

Smith, John Maynard: Evolution and the Theory of Games; 1982
Axelrod, Robert: The evolution of cooperation , 1985
Sigmund, Karl: Games of Life; 1993
Sigmund, Karl: The Calculus of Selfishness; 2010
Nowak, Martin: Evolutionary Dynamics; 2006
Nowak, Martin; Highfield, Roger: Super Cooperators; 2011