In our brief introduction to evolutionary game theory, we defined an evolutionarily stable strategy roughly as one that tends to drive a fractionally small population of mutant strategies to extinction over time. Our setting for investigating this idea was the Hawk-Dove game played between behaviorally instinctive animals; here we saw that successive generations of animals will act according to their parent’s behavior. Another setting more closely related to our topics of human social networks deals with a population of players that choose their best response in relation to the others’ actions.

A further variation that is worth merit is a topic explored by H. Peyton Young in his paper “Stochastic Adaptive Dynamics.” As the title suggests, Young discusses the notion of introducing random errors into behavioral mechanism of normally rational players. He shows that in the context of the “Stag Hunt” game (representing it as a Markov chain), that even when players choose their strategy with a small probability of a mistake that a unique stochastically stable solution can exist. That is, as the probability of mistakes tend to zero, the long-run distribution of behaviors concentrates around a single strategy, giving the stochastically stable state. More interestingly, this notion of a stochastically evolutionarily stable solution can persist for non-trivial values of the random error parameter. This conclusion reveals that even in the presence of seemingly mistake-prone players, a long-run evolutionarily stable behavior is attainable.

A seemingly nice property of this type of behavioral analysis is that it allows for random shocks in the normally deterministic choices of the players. This randomness might be interpreted as bounded rationality of the players. This is desirable because most people in the world do not calculate their optimal responses to every situation; more likely, they base their choice on informed (but imperfect) reason and sometimes the outcomes are most certainly suboptimal. Also this variation of traditional game theoretic methods allows for other analysis that simulate the long-run dynamics of these behavioral adaptive processes to test things such as time to convergence. Luckily for this particular form of stochastically played game, the mathematics of stochastic processes are well understood (especially the dynamics of Markov Chains) which detailed alterations to test various aspects of the dynamics.

Reference:

“Stochastic Adaptive Dynamics”, forthcoming in The New Palgrave Dictionary of Economics, Second Edition, Steven N. Durlauf and Lawrence E. Blume, eds. London: Macmillan.

http://www.econ.jhu.edu/people/young/index.html