## Evolutionarily stable strategy

An evolutionarily stable strategy is defined as a strategy that, once it prevails in a population, can never be overcome by another (minority) strategy (Maynard Smith & Price 1973). This is the strategy, that of all the alternative strategies, is most successful in competition with its own copies. Translated into the language of biologists, the long-term numerical prevalence of bearers of an evolutionarily stable strategy in the population is not threatened by the incidental appearance of mutants or migrants, because the bearers of any alternative strategy will have lower fitness than the bearers of the majority strategy.
The best known model that can demonstrate the principle of competition of alternative strategies is the model called the dove and the hawk; it was described from mathematical point of view in another context in Sect. IV.5.1. The dove and the hawk are names for two alternative strategies asserted when two individuals compete over a certain resource, e.g. a piece of food.  If two individuals competing over a piece of food direct their behavior according to the dove strategy (for simplicity we will further talk only about two doves, two hawks, etc.), they will share the food and each gets, on an average, half of the reward. If two hawks compete, they will fight over the food and only one of them gets the whole piece; the other one will be injured more or less seriously with the negative value of the injuries usually prevailing over the positive value of the food acquired. The average reward that two hawks, the winner and the loser, get from their competition, is therefore negative. If a hawk meets a dove, the dove retreats without a fight, therefore without injury; the hawk gets all the food. An example of the pay-off matrix is given in Fig. IV.5. Analysis of the model shows that neither the dove nor the hawk represents evolutionarily stable strategies. If all the individuals in a population behave as doves, then the mutant, the hawk, wins all competitions without injury and the particular strategy will spread in population. Analogously, in a hawk population, the mutant, the dove, gets the biggest, i.e. zero reward from all competitions, because the hawks will mostly compete with other hawks so their average reward will be negative. It is obvious that finally a balance will be set up in the population entailing frequencies of both strategies where the dove’s average reward and the hawk’s average reward will be the same. If we admit the existence of mixed strategies, an evolutionarily stable strategy will be to behave with p1 probability as a hawk and with (1 – p1) probability as a dove.
Of course, the evolutionary stability of a strategy is only conditional; the given strategy is stable only under the conditions described in our idealized model. If, in a real population, a minority (mutated) new strategy occurred, one that was not included in the original reward matrix, the original winning strategy could easily lose its evolutionary stability. This limiting condition, obvious to a mathematician, is, of course, valid for any theoretical model; no model can predict the behavior of the system under conditions that were not considered while creating it.