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