Editing Evolutionarily stable strategy (section)

== Nash equilibrium ==
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{{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 }}
-->

An ESS is a [[solution concept|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 for both players, for any strategy ''T'':
:E(''S'',''S'') ≥ E(''T'',''S'')

In this definition, a strategy ''T''≠''S'' can be a neutral alternative to ''S'' (scoring equally well, but not better). 
<!--For example, in ''Harm thy neighbour'', (''A'', ''A'') is a Nash equilibrium because one cannot do ''better'' by switching to ''B''. **will move this to "comparison" section, trying to avoid mixing A&B and S&T strategies in same paragraph -->
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.

[[John Maynard Smith|Maynard Smith]] and [[George R. Price|Price]]<ref name="JMSandP73"/> specify two conditions for a strategy ''S'' to be an ESS.  For all ''T''≠''S'', either
# E(''S'',''S'') > E(''T'',''S''), '''or'''
# E(''S'',''S'') = E(''T'',''S'') and E(''S'',''T'') > E(''T'',''T'')

The first condition is sometimes called a ''strict'' Nash equilibrium.<ref>{{cite journal |doi=10.1007/BF01737572 |author=Harsanyi, J |author-link=John Harsanyi |title=Oddness of the number of equilibrium points: a new proof |journal=Int. J. Game Theory |volume=2 |issue=1 |pages=235–50 |year=1973 }}</ref> 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, stronger definition of ESS, due to Thomas.<ref name="Thomas85">{{cite journal |author=Thomas, B. |title=On evolutionarily stable sets |journal=J. Math. Biology |volume=22 |pages=105–115 |year=1985 |doi=10.1007/bf00276549}}</ref>  This 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, this definition requires that for all ''T''≠''S''

# E(''S'',''S'') ≥ E(''T'',''S''), '''and'''
# E(''S'',''T'') > E(''T'',''T'')

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 player keeps his strategy S  ''and'' the payoff of the first player when only his opponent 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]].<ref name="Thomas85"/>

===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.