Optimal Models
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© 1997
David H.A. Fitch
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Lecture notes

Adaptation:  Optimal models for predicting adaptive strategies

I.  Aims and assumptions of optimal models

A.  Aim:  to find which of two or more phenotypes (or some tradeoff between the two) would be optimal in terms of individual fitness

B.  Major assumptions of all optimal models:
1.  There is heritable variation for a trait, but the details of inheritance (i.e., the genetics) are ignored
2.  Populations are treated as asexually reproducing
3.  The models are based on game theory, developed to arrive at best solutions; however, natural selection arrives at the most easily attainable solution, which may not be the best according to a priori design considerations (see previous examples)

II.  Frequency-independent models

A.  Major assumption (in addition to those listed above):  The optimal strategy is not dependent on the frequency of the trait in the population (e.g., an optimal feeding strategy is not dependent on how other members of the population feed, as in the case where a particular food supply is infinitely ample)

B.  Construction of the model:  The optimal strategy is the one that maximizes the benefit (via fitness) of a strategy in relation to the cost of carrying out the strategy (e.g., a diet that maximizes caloric intake relative to the caloric cost of collecting and handling the food)

C.  Sample prediction of the model:  If an optimum type of food is sufficiently abundant, the optimum  strategy is to specialize on the type of food that yields the most calories per cost (energy spent collecting and handling the food); specialization in this case only arises from the quality and abundance of a food source, not from competition (one result of which might be (divergent) specialization on different resources)

II.  Frequency-dependent models

A.  Major assumption (in addition to those listed in I.A.2.):  Fitness (wi) contributed by a trait (such as a strategy), i, is determined by pairwise interactions among individuals that have trait i or an alternative trait (or strategy) j:

wi = w0 + [p Eii] + [q Eij] ,

where p and q are the frequencies of individuals in a population with traits i and j (note that q = 1 - p if i and j are the only alternatives),
w0 is the fitness of an individual in the absence of any interaction,
Eii is the change in fitness to an individual expressing trait i when he/she interacts with another individual also carrying trait i, and
Eij is the change in fitness to an i individual due to an interaction with a j individual.

B.  Construction of the model for particular situations is performed by making a "payoff matrix" that presents relative benefits to particular phenotypes due to particular interactions:

Due to an interaction with:
i j
Payoff to: i Eii Eij
j Eji Ejj


C.  Finding the optimal trait (called the ESS, or Evolutionarily Stable Strategy) involves finding the trait that, when possessed by nearly all individuals in the population, cannot be replaced by an alternative trait.  That is, find the trait with the highest fitness.

  • Note:  This algorithm is the same as that used in the "Prisoner's Dilemma" game.  In the game, the matrix represents the "payoff" to a prisoner who uses one of two strategies:  either he cooperates with another prisoner to escape some punishment (but runs the risk of his partner defecting) or he is a defector himself (but fails to get the benefit of cooperation).  See the Simulations links for more information about this highly useful algorithm.

However, the ESS may be neither i nor j but some mixture (or intermediate) of the two.  For example, the ESS may be to cooperate most of the time, but defect some of the time.  So let's designate the ESS as trait x.  From the definition of an ESS, in a population of nearly all x's, the payoff to an x individual must be the same, regardless of whether the individual he/she interacts with is i-like or j-like.  That is, Eix = Ejx = Exx.

If P is the probability of an x individual adopting an i-like strategy and 1-P is the probability of adopting a j-like strategy, then:

Eix = PEii + (1-P)Eij  

     =   Ejx = (1-P)Ejj + PEji

     =   Exx = P2Eii + P(1-P)Eij + P(1-P)Eji + (1-P)2Ejj  .

Solving for P will provide the ESS strategy in terms of the fraction of the time that an individual is predicted to express trait (strategy) i.  (Alternatively, this may represent the fraction of the population expected to express this trait at equilibrium.)

D.  Often, the ESS trait involves some happy medium, or tradeoff, between the two extreme traits.  Optimal models have been used, for example, to explain ritualized aggression between males in the competition for mates.

  • Optional reading:  For additional discussion on ESS models consult John Maynard Smith, 1989, Evolutionary Genetics (Oxford Univ. Press, Oxford), pp. 125-137 [on reserve at Bobst Library].

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Exercises

  1. Assume that a population contains two kinds of variants with regard to their food-getting strategy, a "hawk" (H) and a "dove" (D).  Assume that, for each interaction there are 2 pieces of food.  When H meets D, H takes both pieces of food, leaving D with none.  When D meets D, they "cooperate" and each take one piece.  When H meets H, they spend their time fighting instead of collecting the food, and each essentially loses 2 pieces of food.  The payoff matrix is thus:
Due to an interaction with:
i j
Payoff to: i -2 2
j 0 1

What is the ESS?  (What is the "optimal strategy"; i.e., what is the probability P that an individual in such a population will play the "hawk" strategy?)
 

  1. What are four problems with applying optimal models to evolving organisms in an attempt to predict adaptive strategies?

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Simulations and other offsite links

Two sites that demonstrate the frequency-dependent optimal model are:

  1. Serendip, where you play one of two strategies (in a typical "prisoner's dilemma" scenario) in several interactions with the computer to determine empirically what a good strategy might be.
     
  2. A graphical version of the "prisoner's dilemma" model, in which a population of colored squares with individual squares displaying several different strategies, with losers being replaced in the population by winners.  Despite the variety of different strategies represented, the equilibrium strategy might still not be the best one.  Why?

Additional information about the Prisoner's Dilemma model (which is applied to an astonishing variety of different problems besides adaptive explanation) may be found at the following sites:

  1. The classic presentation of the Prisoner's Dilemma
  2. Another site that provides additional links to simulations and information

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 Examples of
Adaptations  Defining
Adaptation  Levels of
Selection  Optimal
Models  Tradeoffs  Sexual
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[Examples of Adaptations] [Defining Adaptation] [Levels of Selection] [Optimal Models] [Tradeoffs] [Sexual Selection]

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