Sunday 9 March 2014

A (very) brief interactive look at evolutionary game theory in class

In class last week my students and I started looking at evolutionary game theory (Chapters 11 and 12 of my class). One concept of evolutionary game theory that is important to understand is that in a sense the barrier between strategies and players becomes a bit fuzzy.

To try and illustrate this in class I brought in 2 packs of cards. I actually ended up only using the cards as binary markers (whether or not they were facing 'UP' or 'DOWN'). I then proceeded to describe the following:

"If an UP card interacts with a DOWN card in any given round the DOWN card changes to UP on the next round. Otherwise everything else stays the same."

I used this game to illustrate how a strategy $\sigma$ can induce a population vector $\chi$ and I also touched upon what we would mean by $\sigma$ being stable.

We played the following games:


The left half the class was told to play UP and the other half to play DOWN. Thus our initial value of $\chi=(.5,.5)$. This was given by the fact that half the population was playing $\sigma=(1,0)$ and the other half playing $\sigma(0,1)$.

We played a couple of rounds (which was fairly academic as the outcome is obvious) and arrived at a final population vector of $\chi=(1,0)$ (all the DOWN cards had been changed to UP cards). This is a stable population.

I asked what would happen if I nudged the population by introducing some more DOWN cards in to the population, to perhaps $\chi=(.9,1)$. Everyone realised that the population would swiftly move right back to $\chi=(1,0)$.

We also talked about what would happen if we started with $\chi=(0,1)$ (all cards started as DOWN), everyone realised that $\chi$ would not change over time as we played (since there were no UP cards to force a change). This is also a stable population.

It's obvious though that if we introduce some UP cards (say nudging the population to $\chi=(.1,.9)$) then the population would swiftly move to $\chi=(1,0)$.

The difference between these two stable populations is that one stays stable under evolutionary conditions.

That basically leads us to the definition of an evolutionary stable strategy.


The next game we played was to ask students to randomly (with equal probability) assign themselves a strategy: so everyone was playing a strategy $\sigma=(.5,.5)$.

I won't go in to the details of what we did with that (mainly re-confirming the above conclusions) but the important part is to see that any given population vector $\chi$ can be induced by a strategy vector $\sigma$. This leads to the idea of considering whether or not a strategy is evolutionary stable which corresponds to whether or not it is stable in the population that it induces.


With more time I would have liked to do more with the cards and played more games...


Tomorrow we'll be talking about pairwise contest games and the connection between normal form games and evolutionary games.

Here's a video that I put together a while ago that shows some code that allows us to investigate emergent behaviour:

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