http://stke.sciencemag.org/cgi/content/full/sci;303/5659/793

In the article, “Evolutionary Dynamics of Biological Games”, Martin A. Nowak and Karl Sigmund discuss the benefits of evolutionary game theory in modeling and understanding the evolution of phenotypes. Classical game theory, in the sense of networks attempting to approach a Nash equilibrium, does not fully explain newly observed population trends in biological studies, “where the long-term outcome is not a Nash equilibrium but endless regular or irregular oscillations.” Nowak and Sigmund suggest that previous ideologies be expanded to include concepts like the “unbeatable strategy” and “evolutionary stability”, dependent on the population dynamics.

To explain the foundation of evolutionary games, two specimen of type A and B are considered under the following setup, “Dominance holds if A can invade B and B cannot invade A (or vice versa), coexistence holds if A can invade B and B can invade A, and bistability holds if neither type can invade the other.” This scenario introduces the “core of evolutionary game theory”- invasibility. Supposing A is a resident population and B is an invading population, the outcome of the interaction between A and B will depend on the strategies, or in this case a Darwinian definition of “fitness”, of the two participating populations. While one might assume that the system will tend toward a Nash equilibrium, the concept of evolutionary stability must also be considered. An evolutionary stable outcome can be reached in two ways: “the mutant must either be dominated or else form a bistable pair with the resident strategy.” The language used in the case of the mutant being dominated suggests a Nash equilibrium; the mutant plays its best response to the resident’s even better response, thus it is a Nash equilibrium where the resident holds a dominant strategy. We see here that a Nash equilibrium can also be evolutionary stable.

However, a Nash equilibrium is not always evolutionary stable. We discussed this concept in class using a two-player, two-strategy game- the hawk-dove game. We generalized this game from two animals choosing from being aggressive or passive to Animal A and Animal B choosing from two strategies X and Y, receiving different payoffs depending on what the other animal chose. An X encountering an X received a, an X encountering a Y received b, a Y encountering an X received c, and a Y encountering a Y received d. In order for X to be evolutionarily stable, a must be greater than c, or if a is equal to c, b must be greater than d. This could easily not be the case. For example, X could be a best response to X, but Y might do as well as an X when it encounters an X and also does better than X when it meets another Y. Thus we see that while X is a best response to itself, it is not the only best response, therefore it is not evolutionarily stable.

It is evident that understanding the relationship between Nash equilibria and evolutionary stability is essential in “emerging fields as diverse as metabolic control networks within cells and evolutionary psychology.” Awareness of the applications of game theory to the biological world will facilitate the development of evolutionary models. Evolutionary game theory expands the Darwinian notion of equilibrium to include the chaos and variability inherent in life.