One way to think about Nash Equilibria is that they are self-enforcing agreements, so provided that everyone believes that everyone is going to go along with this agreement, then everyone in fact will.
I want to make another, slightly more philosophical remark associated with this and it's to do with the idea of "leadership." So leadership is kind of a big word that you see written probably too often these days, in too many newspaper articles, and it probably comes up in too many Yale classes, and I don't claim to know anything about leadership.
And I don't think Game Theory is going to contribute to anything to understanding about leadership.
But one thing we can do is tell you where leadership may help.
We could call the strategies strategic complements. So before we leave coordination games, I want to look at another one, a little bit more complicated one perhaps, that we mentioned briefly last time. There's a lot of people who probably need a lot of help with their dating strategies, right? So these are both Nash Equilibria.; That isn't so much the problem. There were two well coordinated Nash Equilibria although one was better than the other. There's at least some agreement that's going to be better for everybody than a strike, and yet, because they're conflicting interests there, basically in that case conflicting interest about health and pension payments, it could well be that you end failing to coordinate. This is a game in which there are two firms who are competing in the same market, and we'll give a bit more detail in a second.
So we'll look at and play another game, and we'll call this game, "Going to the Movies." So I always regard one purpose of this class to help hapless Yale students in their dating strategies. Okay, so the idea of this game is a couple is going to meet up at the movies. So this game has a name, and actually somebody mentioned it last time. One reason this game is interesting from the point of view of economics, is this game lies between the two extreme cases that you learn about in your Intro Economics class.
So that's a problem and let's put in the payoffs, and we'll put in the payoffs that roughly I think would correspond--so my preferences and we'll talk about it--what these preferences mean in a second. So it's not to be sexist--assume this is the she and this is the he, so she would like to go and see Matt Damon beat people up and coordinate on that, her favorite thing. So this equation I've just written, if you can imagine q, total quantity on the horizontal axis, and you could imagine price on the vertical axis, and basically what we're saying is that prices depend on total quantities as follows: where the slope of this line is -b. So payoffs for these firms are going to depend on profits, the payoffs.
This was the best response of Player 1 and this was the best response of Player 2. Student: Both players doing Bourne Ultimatum and both players doing Good Shepherd. Both people would rather be at an equilibrium than to be mal-coordinated or uncoordinated, but Player 1 wants to go to Bourne Ultimatum and Player 2 wants to go to Good Shepherd, and actually I thought Nina's strategy there was pretty good. So much for talking about coordination games and helping you with your dating strategy. So this is a classic game, perhaps it's one of the most famous games, and therefore worth studying in the class.
So we learned the very first time that in the Prisoner's Dilemma communication per se won't help, but in the coordination problem, which could be just as serious socially as a Prisoner's Dilemma, in a coordination problem it may well help.
The reason it helps is you're trying to coordinate onto a Nash Equilibrium.
And they've decided to go to the Criterion [New Haven movie theatre], or the local movie house, and there were these three movies showing, and they're all excited about going to this movie except being Economics majors and not very good at dating, they have forgotten to tell each other which movie they're actually going to go to. So basically the idea is that the more these firms produce, so the more the total quantity produced q, the lower is the price in the marketplace for this product.
They're going to meet in there on the--in the back row probably and--but they're not telling us which movie. So here are the preferences for these movies of Player 1 and Player 2, and you can see from these preferences, from these payoffs, that the best thing for Player 1 is for both players, both people to meet and go to the Bourne Ultimatum; this is the action movie. Let's just draw a picture of that; we'll come back to this in a minute. Actually, let's save myself some time and bring this down. It tells me--the other way around--to look at how prices correspond to quantities, it tells me the quantity demanded at any given price. Meanwhile, let's just finish up what we're doing here and put in payoffs.
You don't want to end up uncoordinated on down left or up right. So in this game if you just played it, it's quite likely you're going to end up uncoordinated, but if you have a little bit of leadership can say okay let's make sure this is where we coordinate, or let's make sure this is where we coordinate. And I don't want to overplay the social importance of this, but go back a couple of years to what was happening in the aftermath of Katrina and realize how important--how bad things get when things fail to be coordinated.