A Note on the Relationship Between ESS, NE and SNE

The evolutionary stable strategy is related to Nash equilibrium in a number of ways. Technically, an ESS is also a refinement of a NE; the first condition requires that the ESS is a NE, so naturally that implies that every ESS in a game is also a NE. Thus, ESS_⊂NE. Moreover, if a NE is a strict NE (SNE), then the requirement for ESS is automatically satisfied. Therefore, we can conclude that [28]:

Strict NE_⊂ESS_⊂NE

It is crucial to point out that this claim is superficial for it boldly claims that ESS is a subset of a NE and a superset of a SNE despite the fact that NE and SNE arestrategy profiles and an ESS is a strategy. One must already assume that because ESS holds only for symmetric games, that implies a strategy profile where each player is playing the ESS (say (σ, σ)). This is also astutely noted in [17]:

... an ESS must correspond to a symmetric Nash equilibrium in that game ... I say ‘correspond’ rather than ‘is’ because an ESS is defined as a single strategy, with the understanding that it is played by all members of a monomorphic population, but a Nash equilibrium is defined as a combination of strategies, one for each player.

Given that the strategy profile corresponding to an ESS must therefore always be symmetric, the above claim fails to specify that not all strict Nash equilibria can be ESSs, but only all symmetric strict Nash equilibria are ESSs. The following game demonstrates that it is possible for a symmetric game to have a strict Nash equilibrium that is not symmetric.

A B

A 0,0 3,2

B 2,3 0,0

We must specify that an ESS is, in fact, the superset of asymmetric SNE and the subset of asymmetric NE. It is also pointed out in [17] that “it can also be shown ... that in this model any strict, symmetric Nash equilibrium corresponds to an ESS in a large population.” Therefore, the following is more accurate than the claim above:

Symmetric Strict NE_⊂ESS_⊂NE

In any case, we can not simply claim that every SNE is an ESS5_{. Under most}

circumstances the fact that NE and SNE are strategy profiles, whereas ESS is a single strategy, is relatively trivial but in a logic of evolutionary game theory, it will be crucial to acknowledge the distinction.

5_{We saw that Nash proved that every game has a Nash equilibrium: if not pure, then}

mixed. [20] claim that a “special case” of Nash’s existence theorem is “that every finite [symmeric] game has a ‘symmetric’ equilibrium.”

An evolutionary stable strategy is a special concept in evolutionary game theory. It is the one static concept on which the real dynamic character of evolution- ary game theory is built. The following section describes replicator dynamics, which demonstrates the behaviour of strategies when they are not at a stable state. The dynamics are represented by coordinate systems that visualize the changing frequency of strategies over time.

3.3

The Replicator Dynamics

Strategies are considered to bereplicators and therefore, the dynamics of strat- egy frequency can be measured over time. If a population has an evolutionary stable strategy, but is for some reason at a heterogeneous state, the replica- tion of strategies over multiple generations will always eventually result in a homogeneous population. However, a population at a homogeneous or hetero- geneous state that does not have an ESS also changes over many generations of replication. This results in a scenario where the population never stabilizes and remains in constant flux. In both cases, elementary calculus is sufficient to measure the changing frequency of strategies. This section will introduce the basic mathematics and results that compose the replicator dynamics.

The function _N(t) expresses the population size at a time t. _N(t) “can be thought of as the actual (discrete) population size divided by some normaliza- tion constant for the borderline case where both approach infinity” [28]. Given that there arenstrategiesσ1, ..., σn,xi =Ni/N of some strategyσi. The sum of xj for all σj ∈ S is 1; i.e. it is a probability distribution. We also assume that death is a constantd.

Supposetis continuous. Assuming ∆tgoes to 0, the limit equation is:

dNi dt =Ni( n " j=1 xju(i, j)−d)

“We abbreviate the expected utility of strategyσi,!n_j=1xj(i, j), asui, and the population average of the expected utility!n_i=1xiui, asu” [28]. Therefore, the corresponding differential equations are:

d_Ni

dt =Ni(ui−d)

d_N

dt =N(u−d)

Becausexi =N/Ni (and following from the definition rule for the first deriva- tive), the equation called thereplicator dynamics:

dxi

If the differential utilitydxi

dt = 0 for alli, the frequency of the strategy is question

σ remains constant over time. On the other hand, if it does not equal 0, the strategy will either increase or decrease in frequency. When it is 0, however, the strategy is considered to be static, which reflects the first condition of ESS. The second condition, that the strategy is robust against mutations or invasions of other strategies holds if the trajectories are drawn to thex value resulting in the differential utility of 0. Thenxis called anattractor.