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Hi folks, this is Matt here, to tell you a little bit more about Bayesian games so

Â we're going to take a look at a quick example just to illustrate some of the

Â concepts and, and to see at least in a simple example how you might think about

Â solving one of these games. and later in the course we'll be talking a little bit

Â more about auctions so you'll have a chance to look, look at some auctions as

Â well. okay, Bayesian Nash equilibrium what did it do again. It has a plan of action

Â for every player. So we have what, what they're going to do as a function of their

Â information their types. And it's maximizing their expected utility,

Â expecting over what they think other players are going to be doing. And

Â expecting over the types of other players, which might effect their payoff. So let's

Â look at a very simple example. So this is sort of a Hollywood style example, so

Â let's call it the sheriff's dilemma. So this is a very simple setting where you've

Â got a sheriff, and they are faced with an armed suspect. imagine they've both pulled

Â guns, and they're standing there staring at each other with their guns in hand and

Â they have to decide whether to shoot at the other or not. and we, you know, we

Â could do this in the wild west, we could do it in a cop thriller, et cetera. But

Â the idea is that you, you, you're faced with this dilemma; do you shoot or not

Â when, when you're faced with the armed suspect? And in this case, let's suppose

Â that the suspect is either a criminal with probability p, or not with probability 1

Â minus p. So either they're guilty of some crime or they're innocent. and, in

Â particular, when we think about this. The sheriff would rather not shoot, Would

Â rather shoot if the suspect's going to shoot. So, if, if you're going to get shot

Â at, you, you want to defend yourself. but you would rather not shoot if the suspect

Â is not. Even if it's a criminal or not. you don't want to shoot the person if

Â they're not going to shoot you. if it's a criminal, you'd rather take them to jail.

Â If it's, if it's, an innocent person you'd rather not shoot them at all so the

Â sheriff would rather not shoot if the, if the suspect doesn't, but will defend

Â themselves if shot at. And the criminal would rather shoot, even if the sheriff

Â does not. So this is a situation where if they, they'll realize they're going to be

Â caught if they don't shoot and, and so they're going to want to shoot and the

Â innocent suspect would rather not shoot, even if the sheriff shoots at them because

Â they realize if the sheriff ends up shooting They're going to die, maybe

Â they'd rather not shoot, and being remembered for shooting the sheriff, so.

Â So that's the setting of the game, very simply. Let's take a look at possible

Â payoffs and, and the structure of this .So let's have the sheriff be the column

Â player, so they can shoot or not. And here, in terms of the representation, we

Â can think of, there's 2 different types of the player. There's this, theta for the

Â innocent suspect in, in, theta for the guilty, suspect. So it could either be the

Â bad or, or, or good suspect. And this is happening with so the innocent is

Â happening with the probabillity of 1 minus p. And the guilty is happening with

Â probability p, so this probability p, you've got this guilty one, 1 minus p on

Â the innocent, and the sheriff doesn't know what the type of the individual is. the

Â suspect is. Okay so then we've got payoffs in here and the payoffs reflect the basic

Â structure that we talked about before. So in particular you know if, if you're if

Â you're going to be shot at, if the, the sheriff's going to be shot at they're

Â going to get a better payoff. From shooting than not. In either case they'd

Â rather, you know, it's a negative payoff here. so, so actually if you don't shoot

Â and they're shooting you, that's a bad payoff, you're going to get killed. if

Â you, you shoot back you might end up hurting a person in this case. You know

Â they're getting a negative payoff because they're actually shooting an innocent

Â individual and, and so forth. So, so you know the , the not, not here is the best

Â payoff for these individuals. for the when, when they're looking at a, a

Â criminal the guilty per, individual again they'd rather shoot if the criminal's

Â going to shoot the, the in the case where the criminal does not, they would get a

Â payoff of 1 from actually capturing the criminal and taking them away and so

Â forth. So we've got payoffs that we can look at and you can study this in a little

Â more detail. And then, then the question is, what's actually going to happen in

Â terms of the player of this game? Okay, so what we can do is begin to analyze okay,

Â if we were faced with the good suspect, the innocent 1. Then what are they going

Â to do? So let's first try and calculate what the suspect is going to do. And what

Â we see here is that the suspect in this particular situation, conditional once

Â they see their type of being good, then they should end up here they get a payoff

Â of minus 1 if they shoot 0 if they don't. So they rather not shoot. Here they get a

Â minus 3 if they shoot, a minus 2 if they don't. So we end up with a, a strictly

Â dominant strategy of not shooting if you're good. So, ess, essentially what

Â that tells us is that if you are looking at for a Beighing equilibrium the good

Â player, regardless of what they think their sheriff should do. Should not shoot.

Â Right? So we can cross this out, and say that the only possible strategy for a, a

Â good player is that they are not going to shoot. Okay, now we go to the bad player,

Â and we do a similar kind of calculation and basically the criminal, Is going to

Â shoot in this case, right? So we look 0 versus minus 2, 2 versus 1. That shoots

Â strictly dominates not for the bad player once they know their type. So that tells

Â us that in, in terms of either an interim plan, or even if we go back X ante and try

Â and figure out what these players should do. Basically the good one should not

Â shoot, and the bad one should shoot. And so now we've got a probability p down

Â here, 1 minus p here and we want to ask what's the sheriff's best reply. Okay,

Â well basically what happens if they shoot what are they going to get. They get 0

Â down here. The sheriff gets a minus 1 up here so you're getting minus 1 times 1

Â minus p, if they shoot. If they don't shoot, what do they get? If they don't

Â shoot,well, they get zero up here and minus 2 down here. So they get a minus 2

Â times p and so We can think of the situations, when is it better to shoot,

Â when is it better not to. and you can check here that if p is greater than 1

Â 3rd, right? So if you find the point where these two are exactly equal to each other,

Â that's going to be the point where p is equal to a 3rd. If p is bigger than a 3rd,

Â then you're more likely to be down here. You're more likely don't want to shoot.

Â And if p is less than a third, then you would want to not. So depending on what p

Â is, you're going to have a Basing Equilibria. So the Basing Equilibria of

Â this game, are going to be for the, the, the good type. oh sorry, the innocent type

Â I guess, innocent type here should always not shoot. The guilty type should always

Â end up shooting. And then the sheriff if p. Is greater than one third the share of

Â shoots. P is less that a third, they do not. And for p equals a third, any

Â mixture. For the sheriff. The sheriff can just flip a coin they're completely

Â indifferent between shooting and not when p is exactly a third. So we have a

Â Bayesian equilibrium. In this case, the Bayesian equilibrium is going to be

Â generically unique. It's going to be unique as long as p is not a third, and

Â whether or not they decide to shoot. Depends on what their payoffs are. And so

Â what, one thing that this, this example illustrates for, it's a fairly simple

Â example, but it still captures the basic elements of Bayesian Equilibrium. How so?

Â Well, there's several things going on. First is that the payoffs. Of both players

Â depend on what the type is, okay? So whether the sheriff is getting a higher or

Â low payoff from shooting or not, exactly how it works depends on, on whether the,

Â they're facing a good or bad suspect and also that determines the strategy of the

Â other player and so there's both strategic uncertainty about what the other player's

Â going to do, which depends on the state and there is payoff uncertainty about what

Â the best thing to do is for the sheriff based on the state and putting those two

Â things together we saw if we get a base in equilibrium and we end up making a

Â prediction. And, so this is a simple game but you know it's going to capture a lot

Â of things in terms of how, How players are going to make decisions in uncertain

Â environments, and Bayesian Equilibrium moves us one step closer to applications

Â where in many, many games in the actual world, you have uncertainty in terms of,

Â of, of what. The payoffs are going to be and what other players are going to do.

Â So, summary of Bayesian Nash Equilibrium, what have we got? It's, it's a model that,

Â that explicitly captures uncertain environments. And players choose

Â strategies again equilibrium notions so your maximizing your payoffs in response

Â to uncertainty about both how other individuals are going to play, and, what

Â the payoffs are from, from different actions. So it's a very powerful tool and

Â one that has many applications some of which we'll see in some of the added

Â