Evolutionary Game Theory - The ESS

The ESS

The Evolutionary Stable Strategy (ESS) is perhaps the most widely known albeit most widely misunderstood concept in Evolutionary Game Theory. The ESS is basically akin to Nash Equilibrium in classical Game Theory, but with mathematically extended criteria.

Nash Equilibrium is a game equilibrium where it is not rational for any player to deviate from the present strategy they are executing. As discussed, in Evolutionary game Theory contestants are NOT behaving with rational choice, nor do they have the ability to totally alter their strategy, aside from executing a very limited “mixed strategy”. An ESS is instead a state of game dynamics where, in a very large (or infinite) population of competitors, another mutant strategy cannot successfully enter the population to disturb the existing dynamic (which in itself is population mix dependent). This leads to a situation where to be a successful strategy having an ESS, the strategy must be both effective against competitors when it is rare - to enter the previous competing population, and also successful when later in high proportion in the population - to “defend itself”. This in turn necessarily means that the strategy needs to be successful when it contends with others exactly like itself.

ESS is NOT:

• An OPTIMAL strategy – an optimal strategy would maximize Fitness, and many ESS states are far below the maximum fitness achievable in a fitness landscape. (see Hawk Dove graph above as an example of this)

• A singular solution – often several ESS conditions can exist in a competitive situation. A particular contest might stabilize into any one of these possibilities, but later a major perturbation in conditions can move the solution into one of the alternative ESS states.

• Always present - It is also possible for there to be no ESS. An example evolutionary game with no ESS is the Rock-Scissors-Paper game found in a number of species (an example the side-blotched lizard (Uta stansburiana))

• An unbeatable strategy - The ESS strategy is not necessarily an unbeatable strategy, it is only an uninvadable one.

The ESS state can be solved for mathematically by exploring either the dynamics of population change to determine any ESS.... or alternatively by solving equations for the stable stationary point conditions which fundamentally define an ESS. For example, in the Hawk Dove Game we can look for whether there is a static population mix condition where the fitness of Doves will be exactly the same as fitness of Hawks (therefore both having equivalent growth rates - a "static point").

Let chance of meeting a Hawk=p so therefore the chance of meeting a dove is (1-p)

Let WHawk equal the Payoff for Hawk.....

WHawk=Payoff in the chance of meeting a Dove + Payoff in the chance of meeting a Hawk

Taking the PAYOFF MATRIX results and plugging them into the above equation:

WHawk= V·(1-p)+(V/2-C/2)·p

Similarly for a Dove:

WDove= V/2·(1-p)+0·(p)

so....

WDove= V/2·(1-p)

Equating the two fitnesses, Hawk and Dove

V·(1-p)+(V/2-C/2)·p= V/2·(1-p)

... and solving for p

p= V/C

so for this "static point" where the Population Percent is an ESS solves to be ESS_{(percent Hawk)}=V/C

Similarly using inequalities it can be shown that an additional Hawk or Dove “mutant” entering this ESS state generates a situation leading eventually to LESS fitness for their kind – both a true Nash and an ESS equilibrium. This fairly simple example shows that when the risks of contest injury or death (the Cost C) is significantly greater than the potential reward offered (the benefit value V) then the stable population which is reached will be MIXED between the aggressors and the doves, and that the proportion of doves will exceed that of the aggressors. This then mathematically explains behaviours that are actually observed in nature.