Evolutionarily Stable Strategies
by Prof. Matt Miller, Department of Mathematics, University of South Carolina email: miller@math.sc.edu
Based on John Maynard Smith, 1982. Evolution and the Theory of Games. Cambridge University Press.
> restart; with(linalg):
Warning, the protected names norm and trace have been redefined and unprotected
>
Consider a population in which there are conflicts. Each individual has fitness w prior to the conflict, and its fitness changes after the conflict. The average fitness of an individual is its baseline fitness plus the fitness change resulting froman encounter with another individual, weighted by the probability of such an encounter. For a population with 2 types of individuals, H and D, in proportion p and (1-p) in the population, the fitness wh of type H individuals is dependent upon the fitness change ehh associated with encountering another H individual ( the probability of such an encounter is p), and the fitness change ehd associated with encountering a D individual (the probability of such an encounter is 1-p):
> w[h]:=w+p*E[hh]+(1-p)*E[hd];
The average fitness of a type D individual is calculated similarly, where edh is the fitness change associated with encountering a type H individual and edd is the fitness change associated with encountering a type D individual.
> w[d]:=w+p*E[dh]+(1-p)*E[dd];
The average fitness in the population is the weighted sum of the fitness of type H and D individuals:
> wbar:=p*w[h]+(1-p)*w[d];
The proportion of type H individuals in the next generation is dependent upon the proportion in the previous generation and the ratio of fitness of type H individuals to average fitness. This is the same calculation used in the density dependent natural selection papers.
> pprime:=p*w[h]/wbar;
At equilibrium, the fitness of the two types should be equal. This is the solution on p 181 of Maynard Smith:
> equil:= solve(w[h] = w[d], p);
Notice that there may fail to be an equilibrium (if the denominator is zero). Maynard Smith suggests a possible Hawk-Dove interaction, where the pre-interaction individuals produce 5 offspring, and payoffs from interactions are as follows: E[HH] = -1, E[HD] = 2, E[DH] = 0, E[DD] = 1
> parms_E:= E[hh]= -1, E[hd]= 2, E[dh]= 0, E[dd]= 1, w = 5;
> equil1a:= subs(parms_E, equil);
Try to decide if this mixed strategy is stable to invasion by a different strategy. First we will see if a population that starts close to this mixed strategy will retain it:
> p[0]:= .52; # select an initial condition in the range 0 to 1
> plotlist:= NULL; values:= NULL;
> for t from 0 to 35 do
> plotlist:= plotlist, [t, p[t]]:
> values:= values, p[t]:
> p[t+1]:=subs(parms_E,p=p[t],pprime):
> od:
> print( [values]);
> plot( [plotlist] );
Is this equilibrium an ESS? What happens if you start much further from the equilibrium? What ESS conditions exist in this population?
Maynard Smith argues that
if E[HH] > E[DH], strategy H is an ESS
if E[DD] > E[HD] strategy D is an ESS
if both E[HH] > E[DH] and E[DD] > E[HD] then both H and D are ESS's
if both conditions fail then the mixed strategy defined by the equilbrium above is an ESS.
***Test these conclusions. Record all your results! Is it possible to have an equilibrium that is NOT an ESS?***
