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, 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, look at some auctions as well. okay, Bayesian Nash equilibrium what did it do again. It has a plan of action for every player. So we have what, what they're going to do as 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 effect 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 are 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, you know, we could do this in the wild west, we could do it in a cop thriller, et cetera. But the idea is that you, you, you're faced with this dilemma; do you shoot or not when, 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 not shoot, Would rather shoot if the suspect's going to shoot. So, if, if you're going to get shot at, you, 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, if it's, an innocent person you'd rather not shoot them at all so the sheriff would rather not shoot if the, 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 if they, they'll realize they're going to be caught if they don't shoot and, 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 being remembered for shooting the sheriff, so. So that's the setting of the game, very simply. Let's take a look at possible payoffs and, 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 2 different types of the player. There's this, theta for the innocent suspect in, in, theta for the guilty, suspect. So it could either be the bad or, or, or good suspect. And this is happening with so the innocent is happening with the probabillity of 1 minus p. And the guilty is happening with probability p, so this probability p, you've got this guilty one, 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 you know if, if you're if you're going to be shot at, if the, the sheriff's going to be shot at they're going to get a better payoff. From shooting than not. In either case they'd rather, you know, it's a negative payoff here. so, 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, you shoot back you might end up hurting a person in this case. You know they're getting a negative payoff because they're actually shooting an innocent individual and, and so forth. So, so you know the , the not, not here is the best payoff for these individuals. for the when, when they're looking at a, a criminal the guilty per, individual again they'd rather shoot if the criminal's going to shoot the, the in the case where the criminal does not, they would get a payoff of 1 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, then the question is, what's actually going to happen in terms of the player of this game? Okay, so what we can do is begin to analyze okay, if we were faced with the good suspect, the innocent 1. 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 1 if they shoot 0 if they don't. So they rather not shoot. Here they get a minus 3 if they shoot, a minus 2 if they don't. So we end up with a, a strictly dominant strategy of not shooting if you're good. So, ess, essentially what that tells us is that if you are looking at for a Beighing equilibrium the good player, regardless of what they think their sheriff should do. Should not shoot. Right? So we can cross this out, and say that the only possible strategy for a, a good player is that they are 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 0 versus minus 2, 2 versus 1. That shoots strictly dominates not for the bad player once they know their type. So that tells us that in, in terms of either an interim plan, or even if we go back X 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, 1 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. They get 0 down here. The sheriff gets a minus 1 up here so you're getting minus 1 times 1 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 2 down here. So they get a minus 2 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 3rd, right? So if you find the point where these two are exactly equal to each other, that's going to be the point where p is equal to a 3rd. If p is bigger than a 3rd, then you're more likely to be down here. You're more likely don't 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 a Basing Equilibria. So the Basing Equilibria of this game, are going to be for the, the, the good type. oh 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 one third the share of shoots. P is less that 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 what, one thing that this, this example illustrates for, 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 or low payoff from shooting or not, exactly how it works depends on, on whether the, 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's going to do, which depends on the state and there is payoff uncertainty about what the best thing to do is for the sheriff based on the state and putting those two things together we saw if we get a base in equilibrium and we end up making a prediction. And, so this is a simple game but you know it's going to capture a lot of things in terms of how, How players are going to make decisions in uncertain environments, and Bayesian Equilibrium moves us one step closer to applications where in many, many games in the actual world, you have uncertainty in terms of, of, of what. The payoffs are going to be and what other players are going to do. So, summary of Bayesian Nash Equilibrium, what have we got? It's, it's a model that, that explicitly captures uncertain environments. And players choose strategies again equilibrium notions so your maximizing your payoffs in response to uncertainty about both how other individuals are going to play, and, what the payoffs are from, 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
