Abstract
We 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.
Highlights
In game theory, the Nash equilibrium is an optimal situation where neither player can benefit by changing strategy while her opponent keeps hers unchanged
Remark 1 Maynard Smith and Price (1973) considered well-mixed interaction in (2), their idea can be used for iterated games too. (By iterated game, we mean that the same opponents play the same matrix game in a huge number of times.) if each individual is paired randomly with another individual from the population (i.e., Page 3 of 21 23 the pair formation is well mixed), each pair plays a large but the same number of games with each other, and each player can only use a genetically fixed mixed or pure strategy in all rounds, the definition of evolutionarily stable strategist (ESS) (2) remains valid
We focus on the dynamical player as a special case of the reactive player, and our general question arises: Can a mutant dynamic player invade a population of mixed ESS users?
Summary
The Nash equilibrium is an optimal situation where neither player can benefit by changing strategy while her opponent keeps hers unchanged. Consider a sufficiently large asexual population with nonoverlapping generations, where the interaction is well mixed (i.e., in each round, the probability of interactions is proportional to the relative frequency of the phenotypes) For this selection situation, based on the Darwinian tenet, Maynard Smith and Price (1973) introduced the intuitive definition of monomorphic evolutionary stability: a phenotype is evolutionarily stable if a rare enough mutant cannot invade the resident monomorphic population displaying this phenotype. (By iterated game, we mean that the same opponents play the same matrix game in a huge number of times.) if each individual is paired randomly with another individual from the population (i.e., Page 3 of 21 23 the pair formation is well mixed), each pair plays a large but the same number of games with each other (i.e., the payoffs are defined as the limit of the average payoffs per round), and each player can only use a genetically fixed mixed or pure strategy in all rounds, the definition of ESS (2) remains valid. According to our knowledge, our selection situation, where best reply players compete with classical mixed ESS players, has not been investigated so far
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