scholarly journals Best Reply Player Against Mixed Evolutionarily Stable Strategy User

2021 ◽  
Vol 84 (1) ◽  
Author(s):  
József Garay ◽  
Tamás F. Móri
Keyword(s):  
Evolutionarily Stable Strategy ◽  
Payoff Matrix ◽  
Matrix Game ◽  
Stable Strategy ◽  
Matrix Games ◽  
Basic Assumptions ◽  
Population Games ◽  
Previous Round ◽  
The Mean ◽  
Evolutionarily Stable

AbstractWe consider matrix games with two phenotypes (players): one following a mixed evolutionarily stable strategy and another one that always plays a best reply against the action played by its opponent in the previous round (best reply player, BR). We focus on iterated games and well-mixed games with repetition (that is, the mean number of repetitions is positive, but not infinite). In both interaction schemes, there are conditions on the payoff matrix guaranteeing that the best reply player can replace the mixed ESS player. This is possible because best reply players in pairs, individually following their own selfish strategies, develop cycles where the bigger payoff can compensate their disadvantage compared with the ESS players. Well-mixed interaction is one of the basic assumptions of classical evolutionary matrix game theory. However, if the players repeat the game with certain probability, then they can react to their opponents’ behavior. Our main result is that the classical mixed ESS loses its general stability in the well-mixed population games with repetition in the sense that it can happen to be overrun by the BR player.

On the local stability of an evolutionarily stable strategy in a diploid population

1984 ◽  
Vol 21 (02) ◽  
pp. 215-224 ◽  
Author(s):  
W. G. S. Hines ◽  
D. T. Bishop
Keyword(s):  
Local Stability ◽  
Evolutionarily Stable Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Diploid Population ◽  
Haploid Population ◽  
The Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations

The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.

On the local stability of an evolutionarily stable strategy in a diploid population

1984 ◽  
Vol 21 (2) ◽  
pp. 215-224 ◽  
Author(s):  
W. G. S. Hines ◽  
D. T. Bishop
Keyword(s):  
Local Stability ◽  
Evolutionarily Stable Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Diploid Population ◽  
Haploid Population ◽  
The Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations

The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (01) ◽  
pp. 5-7
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I 1, I 2, ···, I n where I j is the strategy ‘play value m j ′ and m 1 < m 2 < ··· < m n . The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[I i , I j ] is the expected payoff for playing I i against I j . We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

ESS modelling of diploid populations II: stability analysis of possible equilibria

1994 ◽  
Vol 26 (2) ◽  
pp. 361-376 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Stability Analysis ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Stable Strategy ◽  
Local Attractors ◽  
The Mean ◽  
Arbitrary Frequency ◽  
Probability Simplex ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, a companion paper explored relevant features of the probability simplex of allelic frequencies for a population which is genetically homogeneous except possibly at a single locus.The Shahshahani metric is modified in this paper to produce a measure of distance near an arbitrary frequency F in the allelic simplex which can be used when some alleles are given zero weight by F. The equation of evolution for the modified metric can then be used to show that certain sets of frequencies (corresponding to equilibrium mean strategies) act as local attractors, as long as the mean strategies corresponding to those sets are non-singular or even, in most cases, singular. We identify conditions under which the measure of distance from an initial frequency to a nearby set of equilibrium frequencies corresponding to exceptional mean strategies might increase, either temporarily or for a protracted length of time.

ESS modelling of diploid populations I: anatomy of one-locus allelic frequency simplices

1994 ◽  
Vol 26 (02) ◽  
pp. 341-360 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Genetic Variability ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Allelic Frequency ◽  
Stable Strategy ◽  
The Mean ◽  
Probability Simplex ◽  
Weaker Conditions ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, relevant features of the probability simplex of allelic frequencies for a population with genetic variability at a single locus are explored. Singularities and related properties of mappings from the space of allele frequencies to the space of strategy frequencies are examined, and related to a certain covariance measure of variability present in the population.A companion paper builds on this characterization to establish that previous claims of stability in fact hold under slightly weaker conditions than initially indicated. The pair of papers also determines conditions under which unstable equilibria can occur, and establishes that these conditions are exceptional in practice.

Evolutionarily stable strategies in diploid populations with general inheritance patterns

1983 ◽  
Vol 20 (2) ◽  
pp. 395-399 ◽  
Author(s):  
W. G. S. Hines ◽  
D. T. Bishop
Keyword(s):  
Sexual Selection ◽  
Evolutionarily Stable Strategy ◽  
Evolutionarily Stable Strategies ◽  
Stable Strategy ◽  
Simple Argument ◽  
Sexual Population ◽  
Inheritance Patterns ◽  
The Mean ◽  
Evolutionarily Stable ◽  
Diploid Populations

A simple argument demonstrates that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (ESS) in the formal sense. This result follows under a wide set of models of genetic inheritance of strategy (including sexual selection) provided that the ESS is both attainable and maintainable.

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (1) ◽  
pp. 5-7 ◽  
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I1, I2, ···, In where Ij is the strategy ‘play value mj′ and m1 < m2 < ··· < mn. The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[Ii, Ij] is the expected payoff for playing Ii against Ij. We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

ESS modelling of diploid populations I: anatomy of one-locus allelic frequency simplices

1994 ◽  
Vol 26 (2) ◽  
pp. 341-360 ◽  
Author(s):  
W. G. S. Hines
Keyword(s):  
Genetic Variability ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Allelic Frequency ◽  
Stable Strategy ◽  
The Mean ◽  
Probability Simplex ◽  
Weaker Conditions ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, relevant features of the probability simplex of allelic frequencies for a population with genetic variability at a single locus are explored. Singularities and related properties of mappings from the space of allele frequencies to the space of strategy frequencies are examined, and related to a certain covariance measure of variability present in the population.A companion paper builds on this characterization to establish that previous claims of stability in fact hold under slightly weaker conditions than initially indicated. The pair of papers also determines conditions under which unstable equilibria can occur, and establishes that these conditions are exceptional in practice.

ESS modelling of diploid populations II: stability analysis of possible equilibria

1994 ◽  
Vol 26 (02) ◽  
pp. 361-376
Author(s):  
W. G. S. Hines
Keyword(s):  
Stability Analysis ◽  
Evolutionarily Stable Strategy ◽  
Companion Paper ◽  
Stable Strategy ◽  
Local Attractors ◽  
The Mean ◽  
Arbitrary Frequency ◽  
Probability Simplex ◽  
Evolutionarily Stable ◽  
Diploid Populations

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, a companion paper explored relevant features of the probability simplex of allelic frequencies for a population which is genetically homogeneous except possibly at a single locus. The Shahshahani metric is modified in this paper to produce a measure of distance near an arbitrary frequency F in the allelic simplex which can be used when some alleles are given zero weight by F. The equation of evolution for the modified metric can then be used to show that certain sets of frequencies (corresponding to equilibrium mean strategies) act as local attractors, as long as the mean strategies corresponding to those sets are non-singular or even, in most cases, singular. We identify conditions under which the measure of distance from an initial frequency to a nearby set of equilibrium frequencies corresponding to exceptional mean strategies might increase, either temporarily or for a protracted length of time.

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