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Game Theory: Lecture 10 Transcript

October 8, 2007

Professor Ben Polak: All right, so last time we did something I think substantially harder than anything we've done in the class so far. We looked at mixed strategies, and in particular, we looked at mixed-strategy equilibria. There was a big idea last time. The big idea was if a player is playing a mixed strategy in equilibrium, then every pure strategy in the mix — that's to say every pure strategy on which they place some positive weight — must also be a best response to what the other side is doing. Then we used that trick. We used it in this game here, to help us find Nash Equilibria and the way it allowed us to find the Nash Equilibria is we knew that if, in this case, Venus Williams is mixing between left and right, it must be this case that her payoff is equal to that of right and we use that to find Serena's mix.

Conversely, since we knew that Serena is mixing again between l and r, we knew she must be indifferent between l and r and we used that to find Venus' mix. So I want to go back to this example just for a few moments just to make one more point and then we'll move on, but we'll still be talking about mixed strategies throughout today. So this was the mix that we found before we changed the payoffs, we found that Venus' equilibrium mix was .7, .3 and Serena's equilibrium mix was .6, .4. And a reasonable question at this point would be, how do we know that's really an equilibrium? We kind of found it but we didn't kind of go back and check.

So what I want to do now is actually do that, do that missing step. We rushed it a bit last time because we wanted to get through all the material. Let's actually check that in fact P* is a best response to Q*. So what I want to do is I want to check that Venus' mix P* is a best response for Venus against Serena's mix Q*. The way I'm going to do that is I'm going to look at payoffs that Venus gets now she knows - or rather now we know she's playing against Q*. So let's look at Venus' payoffs. . I'm going to figure out her payoffs for L, her payoffs for R, and also her payoff for what she's actually doing P*.

So Venus' payoffs, if she chooses L against Q* then she gets — very similar to what we had on the board last week, but now I'm going to put in what Q* is explicitly — she gets 50 times .6. . [This is Q* and this is 1-Q*.] . So she gets 50 times .6 and 80 times 1 minus .6 which is .4, 80 times .4. We can work this out, and I worked it out at home, but if somebody has a calculator they can please check me. I think this comes to .62. Somebody should just check that. If Venus chose R — remember R here means shooting to Serena's right, to Serena's forehand — if she chose R then her payoffs are 90 Q*. So 90(.6) plus 20(1-Q*) so 20(.4), so 90(.6) plus 20(.4), and again I worked that out at home, and fortunately that also comes out at .62. So what's Venus' payoff for P*? We've got her payoff for both her pure strategies, so her payoff from actually choosing P* is what?

Well, P* is .7, so .7 of the time she will actually be playing L and when she plays L, she'll get a payoff of .62, and .3 of the time she'll be playing R, and once again, she'll be getting a payoff of .62 and — do I have a calculator? Sorry, thank you. So P* is .7, yes, you're absolutely right, so this is P* and 1-P*, So let's make that clearer. I'll show you what the equilibrium is but P* itself is .7. So when Venus plays L with probability of .7,

October 8, 2007

Professor Ben Polak: All right, so last time we did something I think substantially harder than anything we've done in the class so far. We looked at mixed strategies, and in particular, we looked at mixed-strategy equilibria. There was a big idea last time. The big idea was if a player is playing a mixed strategy in equilibrium, then every pure strategy in the mix — that's to say every pure strategy on which they place some positive weight — must also be a best response to what the other side is doing. Then we used that trick. We used it in this game here, to help us find Nash Equilibria and the way it allowed us to find the Nash Equilibria is we knew that if, in this case, Venus Williams is mixing between left and right, it must be this case that her payoff is equal to that of right and we use that to find Serena's mix.

Conversely, since we knew that Serena is mixing again between l and r, we knew she must be indifferent between l and r and we used that to find Venus' mix. So I want to go back to this example just for a few moments just to make one more point and then we'll move on, but we'll still be talking about mixed strategies throughout today. So this was the mix that we found before we changed the payoffs, we found that Venus' equilibrium mix was .7, .3 and Serena's equilibrium mix was .6, .4. And a reasonable question at this point would be, how do we know that's really an equilibrium? We kind of found it but we didn't kind of go back and check.

So what I want to do now is actually do that, do that missing step. We rushed it a bit last time because we wanted to get through all the material. Let's actually check that in fact P* is a best response to Q*. So what I want to do is I want to check that Venus' mix P* is a best response for Venus against Serena's mix Q*. The way I'm going to do that is I'm going to look at payoffs that Venus gets now she knows - or rather now we know she's playing against Q*. So let's look at Venus' payoffs. . I'm going to figure out her payoffs for L, her payoffs for R, and also her payoff for what she's actually doing P*.

So Venus' payoffs, if she chooses L against Q* then she gets — very similar to what we had on the board last week, but now I'm going to put in what Q* is explicitly — she gets 50 times .6. . [This is Q* and this is 1-Q*.] . So she gets 50 times .6 and 80 times 1 minus .6 which is .4, 80 times .4. We can work this out, and I worked it out at home, but if somebody has a calculator they can please check me. I think this comes to .62. Somebody should just check that. If Venus chose R — remember R here means shooting to Serena's right, to Serena's forehand — if she chose R then her payoffs are 90 Q*. So 90(.6) plus 20(1-Q*) so 20(.4), so 90(.6) plus 20(.4), and again I worked that out at home, and fortunately that also comes out at .62. So what's Venus' payoff for P*? We've got her payoff for both her pure strategies, so her payoff from actually choosing P* is what?

Well, P* is .7, so .7 of the time she will actually be playing L and when she plays L, she'll get a payoff of .62, and .3 of the time she'll be playing R, and once again, she'll be getting a payoff of .62 and — do I have a calculator? Sorry, thank you. So P* is .7, yes, you're absolutely right, so this is P* and 1-P*, So let's make that clearer. I'll show you what the equilibrium is but P* itself is .7. So when Venus plays L with probability of .7,

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