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Game Theory: Lecture 12 Transcript

October 15, 2007

Professor Ben Polak: All right, I want to talk about evolution again today but just before I do, it's hard not to be a little bit happy today because I woke up this morning and heard on the radio that three game theorists won the Nobel prize this morning, which is very nice. Eric Maskin, Roger Myerson and Leo Hurwicz, all of whom worked in an area called Mechanism Design, which is using game theoryif you like, it's about designing gamesto give people incentives and to try and use information available to society or in a firm, to do as well as you canfigure out how well you can and try to achieve that. So, Mechanism Design, or Incentive Design, we'll just touch on but not really cover in the second half of this classit will goit's something you cover in more detail if you take the follow-up class, in 156, or if you take the Auction seminar later on but this is goodthis is great thing.

All three of these guys, Hurwicz, Maskin, Myersonthey're incredibly elegant guys, they're incredible intellectuals, you can talk to them about anything. In fact, Maskinhe was my teacher, so I'm getting a biasbut let me talk about Myerson for a second: when I had a job interview years ago at Northwestern, where Myerson was then, I went in there ready to be all nerdy and talk about Economics for half an hour and try and impress Roger Myerson and all he wantedhe realized I was English and had an interest in Historyand all he wanted to talk to me about was Oliver Cromwell. So we talked about Oliver Cromwell for half an hour and I thought that was it, I thought I wouldn't get the job. Well, in fact, I did get the job, so it wasn't so bad. I didn't take the job, but I did get it.

All right, so let's move on. So let's go back to evolutionary game theoryI should say one other thingEric Maskin was here two weeks ago in Yale giving a talk, talking about this, talking about evolutionary game theory.

All right, so this was the definition we saw at the end last time. I've tried to write it a bit larger, it just repeats the second of those definitions. This is the definition that connects the notion of evolutionary stability, for now in pure strategies, with Nash Equilibrium. Basically it says this, to check whether a strategy is evolutionarily stable in pure strategies, first check whether (Ŝ,Ŝ) is a symmetric Nash Equilibrium. And, if it is, if it's a strict Nash Equilibrium we're done. And if it's not strict, that means there's at least another strategy that would tie with Ŝ against Ŝ, then compare how Ŝ does against this mutation with how the mutation does against itself. And if Ŝ does better than the mutation than the mutation does against itself then we're okay. One virtue of this definition is it's very easy to check, so let's try an example to see that and also to get us back into gear and reminding ourselves what we're doing a bit.

So in this example — sort of trivial game but still — the game looks like this. And suppose we're asked the question what is evolutionarily stable in this game? So no prizes for finding the symmetric Nash Equilibrium in this game. Shout it out. What's the symmetric Nash Equilibrium in this game? (A,A). So (A,A) is a symmetric Nash Equilibrium. That's easy to check. So really the only candidate for an evolutionarily stable strategy here is A. So the second thing you would check is: is (A,A) a strict Nash Equilibrium? So what does strict Nash Equilibrium mean?

October 15, 2007

Professor Ben Polak: All right, I want to talk about evolution again today but just before I do, it's hard not to be a little bit happy today because I woke up this morning and heard on the radio that three game theorists won the Nobel prize this morning, which is very nice. Eric Maskin, Roger Myerson and Leo Hurwicz, all of whom worked in an area called Mechanism Design, which is using game theoryif you like, it's about designing gamesto give people incentives and to try and use information available to society or in a firm, to do as well as you canfigure out how well you can and try to achieve that. So, Mechanism Design, or Incentive Design, we'll just touch on but not really cover in the second half of this classit will goit's something you cover in more detail if you take the follow-up class, in 156, or if you take the Auction seminar later on but this is goodthis is great thing.

All three of these guys, Hurwicz, Maskin, Myersonthey're incredibly elegant guys, they're incredible intellectuals, you can talk to them about anything. In fact, Maskinhe was my teacher, so I'm getting a biasbut let me talk about Myerson for a second: when I had a job interview years ago at Northwestern, where Myerson was then, I went in there ready to be all nerdy and talk about Economics for half an hour and try and impress Roger Myerson and all he wantedhe realized I was English and had an interest in Historyand all he wanted to talk to me about was Oliver Cromwell. So we talked about Oliver Cromwell for half an hour and I thought that was it, I thought I wouldn't get the job. Well, in fact, I did get the job, so it wasn't so bad. I didn't take the job, but I did get it.

All right, so let's move on. So let's go back to evolutionary game theoryI should say one other thingEric Maskin was here two weeks ago in Yale giving a talk, talking about this, talking about evolutionary game theory.

All right, so this was the definition we saw at the end last time. I've tried to write it a bit larger, it just repeats the second of those definitions. This is the definition that connects the notion of evolutionary stability, for now in pure strategies, with Nash Equilibrium. Basically it says this, to check whether a strategy is evolutionarily stable in pure strategies, first check whether (Ŝ,Ŝ) is a symmetric Nash Equilibrium. And, if it is, if it's a strict Nash Equilibrium we're done. And if it's not strict, that means there's at least another strategy that would tie with Ŝ against Ŝ, then compare how Ŝ does against this mutation with how the mutation does against itself. And if Ŝ does better than the mutation than the mutation does against itself then we're okay. One virtue of this definition is it's very easy to check, so let's try an example to see that and also to get us back into gear and reminding ourselves what we're doing a bit.

So in this example — sort of trivial game but still — the game looks like this. And suppose we're asked the question what is evolutionarily stable in this game? So no prizes for finding the symmetric Nash Equilibrium in this game. Shout it out. What's the symmetric Nash Equilibrium in this game? (A,A). So (A,A) is a symmetric Nash Equilibrium. That's easy to check. So really the only candidate for an evolutionarily stable strategy here is A. So the second thing you would check is: is (A,A) a strict Nash Equilibrium? So what does strict Nash Equilibrium mean?

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