Bourgeois Simulation Answers

Discussion of Some the Questions Dealing with the

Bourgeois vs. Hawk vs. Dove Simulation

Could you use the rules presented earlier to determine a pure ESS with all three strategies at once?

No, but you can consider all the possible combinations (Hawk vs. Dove, Hawk vs. Bourgeois, Dove vs. Bourgeois) or you can use the method we learned for examining the columns of the payoff matrix.

In any case, if one strategy is stable against several other known strategies in pairwise competition, then by definition it cannot be invaded by any of them and it can invade all of them. Thus, it is a pure ESS with respect to these strategies. You should satisfy yourself that Bourgeois beats both Hawk and Dove according to the standard criteria, at least with the payoffs that we have described, and then try playing the simulation all possible ways.

A brief discussion of simulations using initially different frequencies of H, D and B.

For any reasonable set of payoffs, Bourgeois is a pure ESS vs. Hawk and Dove. The initial frequencies have nothing to do with whether or not a strategy is a pure (or for that matter a mixed) ESS -- when dealing with an ESS the initial frequencies only dictate how long it might take to get to equilibrium. They also might cause some interesting strategy fluctuations in getting to the ESS.

For instance, you should have noted situations where Bourgeois and one of the other strategies (which one depends on the payoff matrix you are using) initially both increase as the other decreases. Sometimes the changes in frequency vary over time. An especially interested example of this occurs with the default matrix starting with f(Dove) = 0.9 and f(Hawk) = 0.09. However, eventually in every case Bourgeois wins out -- after all , that is the definition of a pure ESS.

Run the simulation using default values for the payoff matrix and f(H) = 0.9, f(D) = 0.09 and f(B) = 0.01. Then look between generations 10 and 50. What were the approximate frequencies of H and D? Then try it again after reversing the initial frequencies of Hawk and Dove. Were the frequencies between generations 10 and 50 the same regardless of whether or not you started with H or D at 0.9? Have you observed these frequencies before?

You should notice that if you used the default payoff matrix, Hawk and Dove quickly come to frequencies that are near those of a mixed Hawk/Dove ESS that we studied earlier.

What is going on here?

Regardless of whether you start with a high frequency of H or D, the same approximate equilibrium is reached. This shouldn't surpise you -- B is at such a low frequency initially that it is simply not a player -- thus the values of f(B) * E{H,B} and f(B) * E{D,B} are nearly negliable (H and D hardly ever encounter the rare B) and therefore a pseudo mixed ESS is reached between those two. However, note that B is still more fit than either (see fitness curve) and it continued to increase at a steady rate, eventually removing both H and D with D disappearing first.

Carefully examine the fitness changes that occur in the first few generations and how they happen when you start with either Hawk high or Dove high.

If B is a pure ESS why does it take so long for it to fix?

There are three parts to the answer to this question -- (i) Bourgeois is very rare initially and even if it is doubling each generation, doubling something extremely rare it still remains rare; (ii) since Bourgeois is so rare, Bourgeois's relative fitness is not initally very much greater than the other two strategies, mainly because after a few generations Bourgeois still hardly encounters itself (a good payoff) while Doves (also a good payoff) quickly become rarer than Hawks (a bad payoff) and (iii) since B is a hybrid of Hawk and Dove, its payoffs are not very different than theirs.

What would you need to do to make B fix faster, (given a starting frequency)?

There is little you can do -- B is a hybrid stategy and anything that helps it will help one of its competitors (try it!!). But it still always wins out!

Which strategy could you monitor in this game to tell when the ESS is reached?

Anytime one strategy is known to be a pure ESS, its the only one you need to monitor. So in this case, monitoring B will tell you when equilibrium is reached, whilst monitoring H or D could be deceptive -- one might go extinct while the other continues and therefore never reach equilibrium.

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