Hawks, Doves and Dice

In class today we looked at the Moran Process and the Karush-Kuhn-Tucker conditions. We did this by considering the Hawk Dove game.

You can see a recording of this here.

The Moran Process

The Moran process is a game theoretic model of evolution. One of the differences from the Replicator Dynamics equation is that the population is assumed to be finite: so we assumed there is a finite population of \(N\) individuals that can be of any of the types that correspond to actions of the underlying Norma Form Game.

In the example of the Hawk Dove game that we played in class we assumed there were \(N=3\) individuals and the question we attempted to understand was: if we introduce a Hawk in to a population of Doves, what will happen?

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The Moran process then follows the following:

1. Calculate the fitness of all individuals: everyone in the population plays the Normal Form game against everyone else. Their utility (or fitness) is given by their type and the type of the individual they play with.
2. Randomly select an individual for copying proportional to their fitness.
3. Randomly select an individual for removal (all individuals are equally likely to be removed).
4. Create a new individual of the same type as the one selected in step 2.
5. Remove the individual selected in step 3.

Repeat that process until there is a single type of individual in the population.

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In class we used dice to simulate the above and obtained a probability of 47% of the Hawk taking over.

Here is a notebook with some numeric simulations of the probabilities. If you look through the notes you can see approaches for calculating exact fixation probabilities.

KKT Conditions

After that we spoke about the Karush-Kuhn-Tucker Condition which I described as a rigorous mathematical set of rules for a simple idea of where optima might be on a constrained region.

You can find the notebook I used to draw the function we looked at here.