## 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: