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100 CHAPTER 3 Moving from Static to Dynamic Games of the population. It is thus a generalization of the strong equilibrium (see Chapter 1) in the context of large population. The evolutionarily stable state or strategy (ESS) introduced by Smith and Price (1973) is a weaker concept than the unbeatable state concept. ·························································································· · · · · · · · A strategy x is an evolutionarily stable state or strategy (ESS) if, for every y = x, there exists · · some threshold fraction of mutants y > 0 such that y - x, f ( y + (1 - )x) < 0 holds for all · · · ]0, y [. · Definition 120: Evolutionarily Stable Strategy ESS For a symmetric bi-matrix game, the ESS definition is equivalent to the following: ·························································································· · · · · · · · A population profile x is an ESS if and only if y = x the following conditions hold: · · · · y - x, f (x) 0, and (3.46) · · · · · y - x, f (x) = 0 = y - x, f (y) < 0 (3.47) · · Theorem 121 A proof can be found in Weibull (1997, Proposition 2.1), and in Hofbauer and Sigmund (1998, Theorem 6.4.1, page 63). In fact, if condition (3.46) is satisfied, then the fraction of mutations in the population will tend to decrease, as it has a lower utility, meaning a lower growth rate. Thus the population profile x is then immune to mutations. If this is not case, but condition (3.47) holds, then a population using x (A) is "weakly" immune to a mutation using y. Indeed, if the mutant population grows, then there will be more frequently players with strategy x competing with mutants. In such cases, the condition x - y, f (y) > 0 ensures that the growth rate of the original population exceeds that of the mutants. Recall that a mixed strategy x (A) that satisfies (3.48) for all y = x is a Nash equilibrium state. A mixed strategy x which satisfies (3.46) for all y = x is called a strict Nash equilibrium. In conclusion, a strict Nash equilibrium is an ESS, and an ESS implies a Nash equilibrium. Note that for finite bi-matrix games, there is at least one mixed Nash equilibrium, but an ESS may not exist. However, an ESS exists in generic symmetric bi-matrix two-by-two games (i.e., two actions for each player); see Weibull (1997). ·························································································· · · · · · · · Any generic two-by-two evolutionary game has at least one ESS (in pure or mixed strategies). · Proposition 122 3.5.2.2 The Two-Population Case Consider two large populations of players which are randomly matched from differ- ent populations. Each player j plays an asymmetric bi-matrix game against some randomly selected player j . Denote by = {1, 2} the set of populations. Every