 Hi, folks. This is Matt here to tell you a little bit more about Bayesian games. So we're going to take a look at a quick example just to illustrate some of the concepts and to see, at least in a simple example, how you might think about solving one of these games. And later in the course, we'll be talking a little bit more about auctions. So you'll have a chance to look at some auctions as well. Okay. Bayesian Nash equilibrium. What did it do? We have a plan of action for every player. So we have what they're going to do is a function of their information, their types. And it's maximizing their expected utility, expecting over what they think other players are going to be doing, and expecting over the types of other players which might affect their payoff. So let's look at a very simple example. So this is sort of a Hollywood style example. So let's call it the Sheriff's Dilemma. So this is a very simple setting where you've got a sheriff and they're faced with an armed suspect. Imagine they've both pulled guns and they're standing there staring at each other with their guns in hand. And they have to decide whether to shoot at the other or not. And we could do this in the wild west. We could do it in a cop thriller, et cetera. But the idea is that you're faced with this dilemma. Do you shoot or not when you're faced with the armed suspect? And in this case, let's suppose that the suspect is either a criminal with probability P or not with probability 1 minus P. So either they're guilty of some crime or they're innocent. And in particular, when we think about this, the sheriff would rather shoot if the suspect is going to shoot. So if you're going to get shot at, you want to defend yourself. But you would rather not shoot if the suspect is not. Even if it's a criminal or not, you don't want to shoot the person if they're not going to shoot you. If it's a criminal, you'd rather take them to jail. If it's an innocent person, you'd rather not shoot them at all. So the sheriff would rather not shoot if the suspect doesn't, but will defend themselves if shot at. And the criminal would rather shoot even if the sheriff does not. So this is a situation where they'll realize they're going to be caught if they don't shoot. And so they're going to want to shoot. And the innocent suspect would rather not shoot, even if the sheriff shoots at them, because they realize if the sheriff ends up shooting, they're going to die. Maybe they'd rather not shoot and be remembered for shooting the sheriff. So that's the setting of the game, very simply. Let's take a look at possible payoffs and the structure of this. So let's have the sheriff be the column player so they can shoot or not. And here, in terms of the representation, we can think of there's two different types of the player. There's this theta for the innocent suspect and theta for the guilty suspect. So they could either be the bad or good suspect. And this is happening with, so the innocent is happening with a probability 1 minus p and the guilty is happening with a probability p. So this probability p, you've got this guilty 1 minus p on the innocent. And the sheriff doesn't know what the type of the individual is, the suspect is. Okay, so then we've got payoffs in here. And the payoffs reflect the basic structure that we talked about before. So in particular, if you're going to be shot at, if the sheriff is going to be shot at, they're going to get a better payoff from shooting than not. In either case, they'd rather, it's a negative payoff here. So actually, if you don't shoot and they're shooting you, that's a bad payoff. You're going to get killed. If you shoot back, you might end up hurting a person. In this case, they're getting a negative payoff because they're actually shooting an innocent individual and so forth. So not here is the best payoff for these individuals. For the, when they're looking at a criminal, the guilty per individual. Again, they'd rather shoot if the criminal's going to shoot. In the case where the criminal does not, they would get a payoff of one from actually capturing the criminal and taking them away and so forth. So we've got payoffs that we can look at and you can study this in a little more detail. And then the question is what's actually going to happen in terms of the play of this game, okay? So what we can do is begin to analyze, okay, if we're faced with the good suspect, the innocent one, then what are they going to do? So let's first try and calculate what the suspect is going to do. And what we see here is that the suspect in this particular situation, conditional once they see their type of being good, then they should end up, here they get a payoff of minus one if they shoot, zero if they don't. So they'd rather not shoot. Here they get a minus three if they shoot, a minus two if they don't. So we end up with a strictly dominant strategy of not shooting if you're good. So essentially what that tells us is that if we're looking at for a Bayesian equilibrium, the good player regardless of what they think the sheriff should do should not shoot, right? So we can cross this out and say that the only possible strategy for a good player is that they're not going to shoot. Okay, now we go to the bad player and we do a similar kind of calculation. And basically the criminal is going to shoot in this case, right? So we look the zero versus minus two, two versus one. This shoot strictly dominates not for the bad player once they know their type. So that tells us that in terms of either an interim plan or even if we go back ex ante and try and figure out what these players should do. Basically the good one should not shoot and the bad one should shoot. And so now we've got a probability P down here, one minus P here. And we want to ask what's the sheriff's best reply? Okay, well basically what happens if they shoot, what are they going to get? If they get zero down here, the sheriff gets a minus one up here. So they're getting minus one times one minus P if they shoot. If they don't shoot, what do they get? If they don't shoot, well they get zero up here and minus two down here, so they get a minus two times P. And so we can think of the situations, when is it better to shoot? When is it better not to? And you can check here that if P is greater than 1 third, right? So if you find that point where these two are exactly equal to each other, that's going to be the point where P is equal to a third. If P is bigger than a third, then you're more likely to be down here, you're more likely to want to shoot. And if P is less than a third, then you would want to not. So depending on what P is, you're going to have Bayesian equilibria. So the Bayesian equilibria of this game are going to be for the good type, or sorry, the innocent type, I guess. Innocent type here should always not shoot. The guilty type should always end up shooting. And then the sheriff, if P is greater than 1 third, the sheriff shoots, P is less than a third, they do not. And for P equals a third, any mixture for the sheriff. The sheriff can just flip a coin. They're completely indifferent between shooting and not when P is exactly a third. So we have a Bayesian equilibrium. In this case, the Bayesian equilibrium is going to be generically unique. It's going to be unique as long as P is not a third. And whether or not they decide to shoot depends on what their payoffs are. And so one thing that this example illustrates, it's a fairly simple example, but it still captures the basic elements of Bayesian equilibrium. How so? Well, there's several things going on. First is that the payoffs of both players depend on what the type is. Okay? So whether the sheriff is getting a higher low payoff from shooting or not, exactly how it works depends on whether they're facing a good or bad suspect. And also that determines the strategy of the other player. And so there's both strategic uncertainty about what the other player is going to do, which depends on the state. And there's payoff uncertainty about what the best thing to do is for the sheriff based on the state. And putting those two things together, we solve, we get a Bayesian equilibrium, and we end up making a prediction. So this is a simple game, but it's going to capture a lot of things in terms of how players are going to make decisions in uncertain environments. And Bayesian equilibrium moves us one step closer to applications, where in many games in the actual world, you have uncertainty in terms of what the payoffs are going to be and what other players are going to do. So summary of Bayesian Nash equilibria, what have we got? It's a model that explicitly captures uncertain environments. And players choose strategies, again, equilibrium notions. So you're maximizing your payoffs in response to uncertainty about both how other individuals are going to play and what the payoffs are from different actions. So it's a very powerful tool and one that has many applications, some of which we'll see in some of the added lectures. Take care.