Evolutionarily Stable Strategy - Nash Equilibria and ESS

Nash Equilibria and ESS

An ESS is a refined or modified form of a Nash equilibrium. (See the next section for examples which contrast the two.) In a Nash equilibrium, if all players adopt their respective parts, no player can benefit by switching to any alternative strategy. In a two player game, it is a strategy pair. Let E(S,T) represent the payoff for playing strategy S against strategy T. The strategy pair (S, S) is a Nash equilibrium in a two player game if and only if this is true for both players and for all T≠S:

E(S,S) ≥ E(T,S)

In this definition, strategy T can be a neutral alternative to S (scoring equally well, but not better). A Nash equilibrium is presumed to be stable even if T scores equally, on the assumption that there is no long-term incentive for players to adopt T instead of S. This fact represents the point of departure of the ESS.

Maynard Smith and Price specify two conditions for a strategy S to be an ESS. Either

1. E(S,S) > E(T,S), or
2. E(S,S) = E(T,S) and E(S,T) > E(T,T)

for all T≠S.

The first condition is sometimes called a strict Nash equilibrium. The second is sometimes called "Maynard Smith's second condition". The second condition means that although strategy T is neutral with respect to the payoff against strategy S, the population of players who continue to play strategy S has an advantage when playing against T.

There is also an alternative definition of ESS, which places a different emphasis on the role of the Nash equilibrium concept in the ESS concept. Following the terminology given in the first definition above, we have (adapted from Thomas, 1985):

1. E(S,S) ≥ E(T,S), and
2. E(S,T) > E(T,T)

for all T≠S.

In this formulation, the first condition specifies that the strategy is a Nash equilibrium, and the second specifies that Maynard Smith's second condition is met. Note that the two definitions are not precisely equivalent: for example, each pure strategy in the coordination game below is an ESS by the first definition but not the second.

In words, this definition looks like this: The payoff of the first player when both players play strategy S is higher than (or equal to) the payoff of the first player when he changes to another strategy T and the second players keeps his strategy S. *AND* The payoff of first player when only he changes his strategy to T is higher than his payoff in case that both of players change their strategies to T.

This formulation more clearly highlights the role of the Nash equilibrium condition in the ESS. It also allows for a natural definition of related concepts such as a weak ESS or an evolutionarily stable set.