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Evolutionarily stable strategy

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                                           Evolutionarily stable strategy
                                        A solution concept in game theory
                                                            Relationships
            Subset of:                                    Nash equilibrium
      Intersects with: Subgame perfect equilibrium, Trembling hand perfect
                                 equilibrium, Perfect Bayesian equilibrium
                                                             Significance
          Proposed by:              John Maynard Smith and George R. Price
             Used for:    Biological modeling and Evolutionary game theory
              Example:                     Hawk-dove (aka Game of chicken)

   In game theory, an evolutionarily stable strategy (or ESS; also
   evolutionary stable strategy) is a strategy which if adopted by a
   population cannot be invaded by any competing alternative strategy. The
   concept is an equilibrium refinement to a Nash equilibrium. The
   difference between a Nash equilibrium and an ESS is that a Nash
   equilibrium may sometimes exist due to the assumption that rational
   foresight prevents players from playing an alternative strategy with no
   short term cost, but which will eventually be beaten by a third
   strategy. An ESS is defined to exclude such equilibria, and assumes
   that natural selection is the only force which selects against using
   strategies with lower payoffs.

   The term was introduced and defined by John Maynard Smith and George R.
   Price in a 1973 Nature paper and is central to Maynard Smith's (1982)
   book Evolution and the Theory of Games. The concept was derived from
   R.H. MacArthur and W.D. Hamilton's work on sex ratios, especially
   Hamilton's (1967) concept of an unbeatable strategy. The idea can be
   traced back to Ronald Fisher (1930) and Charles Darwin (1859), (see
   Edwards, 1998).

Nash equilibria and ESS

   A Nash equilibrium is a strategy in a game such that if all players
   adopt it, no player will benefit by switching to play any alternative
   strategy. If a player choosing strategy J in a population where all
   other players play strategy I receives a payoff of E(J,I), then
   strategy I is a Nash equilibrium if,

          E(I,I) ≥ E(J,I) for any J

   This equilibrium definition allows for the possibility that strategy J
   is a neutral alternative to I (it scores equally, but not better). A
   Nash equilibrium is presumed to be stable even if J scores equally, on
   the assumption that players do not play J

   Maynard Smith and Price (1973) specify two conditions for a strategy I
   to be an ESS. Either
    1. E(I,I) > E(J,I), or
    2. E(I,I) = E(J,I) and E(I,J) > E(J,J)

   must be true for all I ≠ J, where E(I,J) is the expected payoff to
   strategy I when playing against strategy J.

   The first condition is sometimes called a 'strict Nash' equilibrium
   (Harsanyi, 1973), the second is sometimes referred to as 'Maynard
   Smith's second condition'.

   There is also an alternative definition of ESS which, though it
   maintains functional equivalence, 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(I,I) ≥ E(J,I), and
    2. E(I,J) > E(J,J)

   In this formulation, the first condition specifies that the strategy be
   a Nash equilibrium, and the second specifies that Maynard Smith's
   second condition be met. Note that despite the difference in
   formulation, the two definitions are actually equivalent.

   One advantage to this change is that the role of the Nash equilibrium
   in the ESS is more clearly highlighted. It also allows for a natural
   definition of other concepts like a weak ESS or an evolutionarily
   stable set (Thomas, 1985).

An example

   Consider the following payoff matrix, describing a coordination game:

                                 A      B
                              A 1,1    0,0
                              B 0,0    1,1
                              Coordination game

   Both strategies A and B are ESS, since a B player cannot invade a
   population of A players nor can an A player invade a population of B
   players. Here the two pure strategy Nash equilibria correspond to the
   two ESS. In this second game, which also has two pure strategy Nash
   equilibria, only one corresponds to an ESS:

                                    C   D
                                 C 1,1 0,0
                                 D 0,0 0,0
                                 Simple game

   Here (D, D) is a Nash equilibrium (since neither player will do better
   by unilaterally deviating), but it is not an ESS. Consider a C player
   introduced into a population of D players. The C player does equally
   well against the population (she scores 0), however the C player does
   better against herself (she scores 1) than the population does against
   the C player. Thus, the C player can invade the population of D
   players.

   Even if a game has pure strategy Nash equilibria, it might be the case
   that none of the strategies are ESS. Consider the following example
   (known as Chicken):

                                   E      F
                               E  0,0   -1,+1
                               F +1,-1 -20,-20
                                  Chicken

   There are two pure strategy Nash equilibria in this game (E, F) and (F,
   E). However, in the absence of an uncorrelated asymmetry, neither F nor
   E are ESSes. A third Nash equilibrium exists, a mixed strategy, which
   is an ESS for this game (see Hawk-dove game and Best response for
   explanation).

Bishop-Cannings theorem

   Just as Nash equilibria can be either a pure strategy, or probabalistic
   mixtures of pure strategies (a mixed strategy), evolutionarily stable
   strategies can be either pure of mixed.

   The Bishop-Cannings theorem (Bishop & Cannings, 1978) proves that all
   members of a mixed evolutionarily stable strategy have the same payoff,
   and that none of these can also be a pure evolutionarily stable
   strategy. The same logic also applies to Nash equilibria and so the
   same will hold true for members of a mixed Nash as for members of a
   mixed ESS.

ESS vs. Evolutionarily Stable State

          An ESS or evolutionarily stable strategy is a strategy such
          that, if all the members of a population adopt it, no mutant
          strategy can invade. -- Maynard Smith (1982).

          A population is said to be in an evolutionarily stable state if
          its genetic composition is restored by selection after a
          disturbance, provided the disturbance is not too large. Such a
          population can be genetically monomorphic or polymorphic. --
          Maynard Smith (1982).

   An ESS is a strategy with the property that, once virtually all members
   of the population use it, then no 'rational' alternative exists. An
   evolutionarily stable state is a dynamical property of a population to
   return to using a strategy, or mix of strategies, if it is perturbed
   from that strategy, or mix of strategies. The former concept fits
   within classical game theory, whereas the latter is a population
   genetics, dynamical system, or evolutionary game theory concept.

   Thomas (1984) applies the term ESS to an individual strategy which may
   be mixed, and evolutionarily stable population state to a population
   mixture of pure strategies which may be formally equivalent to the
   mixed ESS.

Prisoner's dilemma and ESS

   Consider a large population of people who, in the iterated prisoner's
   dilemma, always play Tit for Tat in transactions with each other.
   (Since almost any transaction requires trust, most transactions can be
   modelled with the prisoner's dilemma.) If the entire population plays
   the Tit-for-Tat strategy, and a group of newcomers enter the population
   who prefer the Always Defect strategy (i.e. they try to cheat everyone
   they meet), the Tit-for-Tat strategy will prove more successful, and
   the defectors will be converted or lose out. Tit for Tat is therefore
   an ESS, with respect to these two strategies. On the other hand, an
   island of Always Defect players will be stable against the invasion of
   a few Tit-for-Tat players, but not against a large number of them. (see
   Robert Axelrod's The Evolution of Cooperation).

ESS and human behaviour

   The recent, controversial sciences of sociobiology and now evolutionary
   psychology attempt to explain animal and human behaviour and social
   structures, largely in terms of evolutionarily stable strategies.
   Sociopathy (chronic antisocial/criminal behaviour) has been suggested
   to be best explained as a combination of two such strategies.

   Although ESS were originally considered as stable states for biological
   evolution, it need not be limited to such contexts. In fact, ESS are
   stable states for a large class of adaptive dynamics. As a result ESS
   are used to explain human behavior without presuming that the behaviour
   is necessarily determined by genes.

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