Evolutionary Stability
An evolutionarily stable
strategy (ESS) is a strategy such that, if all the members of a population
adopt it, then no mutant strategy could invade the population under the
influence of natural selection. Suppose that there are two types of strategies
in the population, A and B. Let E(B, A)
denote the payoff strategy B receives in interacting with strategy A. The
strategy A is evolutionarily stable if either
1. E(A,
A) > E(B, A), or
2. E(A,
A) = E(B, A) and E(A, B) > E(B, B)
is true for all B [MAY73]. 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. The strategy A is evolutionarily stable if both
1. E(A,
A) ≥ E(B, A), and
2. E(A,
B) > E(B, B)
for all B [THO85]. The concept of ESS
considers those situations when a single mutant invades an infinite population
of homogeneous strategies. It is not concerned with the structure of the
population, the selection scheme, and other parameters of evolutionary
dynamics. If an ESS exists for an evolutionary game, the evolutionary process
is, in theory, likely to converge to the state where ESSs are common. However,
if no ESS exists, it is difficult to forecast the result of the evolution. This
definition is so strict that no known strategy is evolutionarily stable in
infinite length or indefinite length IPD [BOY87][ LOR94].
Nowak and Sigmund (1992)
considered the size of the cluster that is needed for an invader to invade a
finite population of a particular strategy. The minimal cluster size for one
strategy to invade another can be treated as an invasion barrier. If the
invasion barrier for a strategy is low, it means that a small cluster of
invaders can successfully invade and it is difficult to maintain a homogeneous population.
On the contrary, if the invasion barrier for a strategy is high, successful
invasion requires a large cluster of invaders. Therefore, different strategies
can be compared by means of their invasion barriers [NOW92]. Strategies with a higher
invasion barrier are evolutionarily stronger than those with a lower invasion
barrier. By means of replacing the quantity of each strategy in the population
with the ratio of the quantity to the size of population, an invasion barrier
can be used in evolutionary IPDs with both finite and infinite populations.
Consider a population
consisting of two types of strategies, A and B. p_{A} and p_{B} are the frequencies of A
and B in the population respectively. The IPD between two strategies
is denoted by the payoff matrix:

A 
B 
A 
a 
b 
B 
c 
d 
Then the fitness of two
strategies, E_{A} and E_{B} ,
can be expressed as:
E_{A}
= ap_{A} + bp_{B}
E_{B}
= cp_{A} + dp_{B}
The condition for A to
invade B is E_{A} > E_{B}, or (a  c)p_{A} > (d
 b)p_{B}. On the other hand, the
condition for B to invade A is (a  c)p_{A} < (d  b)p_{B}. Therefore, (a  c)p_{A} = (d
 b)p_{B} is the transition
point for the evolution. Without loss of generality, suppose that a > c and
let λ = p_{A}_{ }/
p_{B}. The
invasion barrier can be expressed as
λ_{0} = (db) / (ac)
In the case of λ = λ_{0} , two
strategies will coexist and maintain their current frequencies in the
population. λ > λ_{0} and λ < λ_{0}
indicate contrary directions of the evolution. In one case, A tends to
become dominant and B tends to die out; in the other case, A dies out and B becomes dominant.