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Game Theory: Lecture 8 Transcript

October 1, 2007

Professor Ben Polak: So last time we left things in the middle of a model, which was the candidate-voter model. What was — I don't want to go over the whole model again, but just to reiterate a little bit — what was different about that model from what we saw before — the main thing that was different was that the candidates cannot choose their positions. If you like, every voter is a potential candidate but you know the positions of the voters.

Let me just bring out two lessons that we left hanging last time. I want to just put them on the board to make sure they're in your notes. So the first lesson is — one thing we saw already last time — is there can be lots of different Nash Equilibrium this model. There are multiple possible Nash Equilibrium in this model and more to the point, not all of those equilibria have the candidates crowded at the center. We saw early on in the classic Downs or median-voter model that that model predicted crowding the center. This one doesn't, and we'll come back to that in a second. And a second thing we saw last time was that entry can — if you enter on the left one affect of entering on the left can be to make the candidate on the right more likely to win. Conversely, if you enter on the right — you're a right-wing voter candidate and you enter, that can make it more likely that the left wings are — can lead to the winner being more — being further from your ideal position.

Just to revisit this a little bit, let me go back and just illustrate those two points again. So I'll take a row further back this time and get a nice full row that we can see the whole of. This way, I'm going to take a row further back so that this time we have no confusion about what left wing and right wing is, at least almost no confusion. So let me choose this row, this row good okay. I'm sorry, the people in the balcony are going to have to imagine this. It's a penalty being in the balcony. So this row. And here's my left wing of this row (for everyone who's in front of you which is almost everybody), and here's my right wing.

Let's try and illustrate some equilibrium we saw last time. So, in particular, if I can get the guy in the blue Yale shirt to stand up a second and the guy with his computer to stand up a second. Sorry. Let's assume all the seats are filled for now so — just to help me out a little bit since I'm doing this on the fly. This, I'm going to claim is an equilibrium. Notice that there are two candidates standing and notice that they are not particularly close to the center. We have our sort of middle-of-the-democratic-party left candidates. So I'm tempted to give you a name but perhaps I will not. And here's a middle-of-the-republican-party candidate. And this is the election. They're going to split the votes equally if I've actually chosen correctly.

So, just to observe some things here. First, for this to be an equilibrium, it better be the case that they're symmetric on left and right. If they're not symmetric, then it isn't an equilibrium because one of those candidates is going to lose for sure and the way we set up the model that means that one of them will drop out: they will deviate to drop out. Is that correct? Let's also illustrate this. So, we've already illustrated that they're not particularly close to the center. Let's illustrate what I meant by somebody on the left causing the right wing to win.

October 1, 2007

Professor Ben Polak: So last time we left things in the middle of a model, which was the candidate-voter model. What was — I don't want to go over the whole model again, but just to reiterate a little bit — what was different about that model from what we saw before — the main thing that was different was that the candidates cannot choose their positions. If you like, every voter is a potential candidate but you know the positions of the voters.

Let me just bring out two lessons that we left hanging last time. I want to just put them on the board to make sure they're in your notes. So the first lesson is — one thing we saw already last time — is there can be lots of different Nash Equilibrium this model. There are multiple possible Nash Equilibrium in this model and more to the point, not all of those equilibria have the candidates crowded at the center. We saw early on in the classic Downs or median-voter model that that model predicted crowding the center. This one doesn't, and we'll come back to that in a second. And a second thing we saw last time was that entry can — if you enter on the left one affect of entering on the left can be to make the candidate on the right more likely to win. Conversely, if you enter on the right — you're a right-wing voter candidate and you enter, that can make it more likely that the left wings are — can lead to the winner being more — being further from your ideal position.

Just to revisit this a little bit, let me go back and just illustrate those two points again. So I'll take a row further back this time and get a nice full row that we can see the whole of. This way, I'm going to take a row further back so that this time we have no confusion about what left wing and right wing is, at least almost no confusion. So let me choose this row, this row good okay. I'm sorry, the people in the balcony are going to have to imagine this. It's a penalty being in the balcony. So this row. And here's my left wing of this row (for everyone who's in front of you which is almost everybody), and here's my right wing.

Let's try and illustrate some equilibrium we saw last time. So, in particular, if I can get the guy in the blue Yale shirt to stand up a second and the guy with his computer to stand up a second. Sorry. Let's assume all the seats are filled for now so — just to help me out a little bit since I'm doing this on the fly. This, I'm going to claim is an equilibrium. Notice that there are two candidates standing and notice that they are not particularly close to the center. We have our sort of middle-of-the-democratic-party left candidates. So I'm tempted to give you a name but perhaps I will not. And here's a middle-of-the-republican-party candidate. And this is the election. They're going to split the votes equally if I've actually chosen correctly.

So, just to observe some things here. First, for this to be an equilibrium, it better be the case that they're symmetric on left and right. If they're not symmetric, then it isn't an equilibrium because one of those candidates is going to lose for sure and the way we set up the model that means that one of them will drop out: they will deviate to drop out. Is that correct? Let's also illustrate this. So, we've already illustrated that they're not particularly close to the center. Let's illustrate what I meant by somebody on the left causing the right wing to win.

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