Yesterday started a spring school at HEC Montréal on evolutionary games (here). Jörgen Weibull (here) gave a very interesting talk about evolution games and some notion(s) of stability. And he came back on John Nash's notion of equilibrium (actually, Jörgen visited the Princeton campus to meet Nash in the 80's- as mentioned in the talk and there - so he has a lot of interesting stories to tell).

John Nash is famous for his interpretation of a fixed point theorem (see here for a nice story by Ivar Ekeland, in French). And actually, he wrote extremely interesting non-mathematical ideas, especially in in his PhD thesis (here), where he provided two interpretations of the so-called Nash-equilibrium concept (for non-cooperative games): the first one, which became the standard interpretation, with a game played only once, where participants are assumed to be “rational” (the one rationalistic interpretation). "We proceed by investigating the question: What would be a rational prediction of the behavior to be expected of rational playing the game in question? By using the principles that a rational prediction should be unique, that the players should be able to deduce and make use of it, and that such knowledge on the part of each player of what to expect the others to do should not lead him to act out of conformity with the prediction, one is led to the concept of a solution defined before".

The second one is a population-statistic (or mass-action) interpretation, that was actually given by John Nash in his PhD thesis (which is actually not too long to read, since it was only 27 pages) "It is unnecessary to assume that the participants have full knowledge of the total structure of the game, or the ability and inclination to go through any complex reasoning processes. But the participants are supposed to accumulate empirical information on the relative advantages of the various pure strategies at their disposal."

"To be more detailed, we assume that there is a population (in the sense of statistics) of participants for each position of the game. Let us also assume that the ‘average playing’ of the game involves n participants selected at random from the n populations, and that there is a stable average frequency with which each pure strategy is employed by the ‘average member’ of the appropriate population."

"Since there is to be no collaboration between individuals playing in different positions of the game, the probability that a particular n-tuple of pure strategies will be employed in a playing of the game should be the product of the probabilities indicating the chance of each of the n pure strategies to be employed in a random playing"

"Now let us consider what effects the experience of the participants will produce. To assume, as we did, that they accumulate empirical evidence on the pure strategies at their disposal is to assume that those playing in position i learn the numbers p(i). But if they know these they will employ only optimal pure strategies [...]. Consequently, since s, expresses their behavior, s, attaches positive coefficients only to optimal pure strategies, [...]. But this is simply a condition for s to be an equilibrium point. Thus the assumption we made in this ‘mass-action’ interpretation lead to the conclusion that the mixed strategies representing the average behavior in each of the populations form an equilibrium point."