Editing Evolutionarily stable strategy (section)

===Examples of differences between Nash equilibria and ESSes===

{|align=block
|-
|{{Payoff matrix | Name =  Prisoner's Dilemma
                | 2L =  Cooperate    | 2R = Defect  |
1U =  Cooperate | UL = 3, 3       | UR = 1, 4       |
1D =  Defect   | DL = 4, 1       | DR = 2, 2       }}
|{{Payoff matrix | Name =  Harm thy neighbor
                | 2L = A     | 2R = B    |
1U = A          | UL = 2, 2  | UR = 1, 2 |
1D = B          | DL = 2, 1  | DR = 2, 2 }}
|}

In most simple games, the ESSes and Nash equilibria coincide perfectly.  For instance, in the [[prisoner's dilemma]] there is only one Nash equilibrium, and its strategy (''Defect'') is also an ESS.

Some games may have Nash equilibria that are not ESSes. For example, in harm thy neighbor (whose payoff matrix is shown here) both (''A'', ''A'') and (''B'', ''B'') are Nash equilibria, since players cannot do better by switching away from either.  However, only ''B'' is an ESS (and a strong Nash). ''A'' is not an ESS, so ''B'' can neutrally invade a population of ''A'' strategists and predominate, because ''B'' scores higher against ''B'' than ''A'' does against ''B''.  This dynamic is captured by Maynard Smith's second condition, since E(''A'', ''A'') = E(''B'', ''A''), but it is not the case that E(''A'',''B'') > E(''B'',''B'').

{|align=block style="clear: right"
|-
|{{Payoff matrix | Name = Harm everyone
                | 2L = C     | 2R = D    |
1U = C          | UL = 2, 2  | UR = 1, 2 |
1D = D          | DL = 2, 1  | DR = 0, 0 }}
|{{Payoff matrix | Name = Chicken
                | 2L = Swerve     | 2R = Stay       |
1U = Swerve     | UL = 0,0        | UR = −1,+1      |
1D = Stay       | DL = +1,−1      | DR = −20,−20    }}
|}

Nash equilibria with equally scoring alternatives can be ESSes.  For example, in the game ''Harm everyone'', ''C'' is an ESS because it satisfies Maynard Smith's second condition. ''D'' strategists may temporarily invade a population of ''C'' strategists by scoring equally well against ''C'', but they pay a price when they begin to play against each other; ''C'' scores better against ''D'' than does ''D''.  So here although E(''C'', ''C'') = E(''D'', ''C''), it is also the case that E(''C'',''D'') > E(''D'',''D'').  As a result, ''C'' is an ESS.

Even if a game has pure strategy Nash equilibria, it might be that none of those pure strategies are ESS. Consider the [[Chicken (game)|Game of chicken]].  There are two pure strategy Nash equilibria in this game (''Swerve'', ''Stay'') and (''Stay'', ''Swerve''). However, in the absence of an [[uncorrelated asymmetry]], neither ''Swerve'' nor ''Stay'' are ESSes. There is a third Nash equilibrium, a [[mixed strategy]] which is an ESS for this game (see [[Chicken (game)|Hawk-dove game]] and [[Best response]] for explanation).

This last example points to an important difference between Nash equilibria and ESS.  Nash equilibria are defined on ''strategy sets'' (a specification of a strategy for each player), while ESS are defined in terms of strategies themselves.  The equilibria defined by ESS must always be [[Symmetric equilibrium|symmetric]], and thus have fewer equilibrium points.