ESS and Well Posedness Property
Link: Evolutionary Stable Strategies and Well Posedness Property
From lectures, we know that the main concept of ESS is a strategy that is impermeable when adopted by a population in the adaptation to a specific environment, so the concept is largely applied in Biological modeling. Once all players in a game adopt the ESS, then no one will benefit by switching to play any alternative strategy, which can also be explained as a natural selection. The application of Game Theory to Evolution was discovered by Maynard Smith, who is specialized in mathematical models in biology. Often, in reality, there are no equilibria we can search, so approximate evolutionary stable equilibrium (eES-eq for short) is introduced in this article.
Definition Let e>0, (b*, b*) from B x B is an eES-equilibrium if the following two properties are valid:
- u(b, b*) <= u(b*, b*) + e, for all b from B
- If u(b, b*) >= u(b*, b*), then u(b, b) <= u(b*, b)+ e, for all b from B
This is also called the evolutionary well-posedness for symmetric games and the well-posedness property guarantees that there is only one equilibrium and each approx. one is very near or around this one. For example, the Hawk-Dove game is ES-well posed but the War of Attrition game is not, which is about a fight between two animals.
The symmetric games as mentioned above have always been considered for the ESS and eES models, but what can we say about asymmetric games. I think there’s still a lot of further work that we dive deeper into and it would be interesting to relate these definitions and the reality.