Hi folks. This is Matt and we're going to talk a little bit now about Bayesian equilibrium an equilibrium concept and solution concept for Bayesian Games. And the idea of,of this, it is also sometimes. Referred to as Bayesian Nash Equilibria. the concept goes back to Harsanyi, John Harsanyi in, in late 1960s, 1967, and 1968, where he developed this concept. And the idea is that, that each player now, you know, when we're talking about a Bayesian game player Players have these types which determine their payoffs. And relate to the uncertainty. And can actually tell them something about what they expect other individual's types to be. And an equilibrium, now, is going to be a plan of action for each player as a function of their types. So, it's going to, to say, okay. If I, if I observe a certain type. What am I going to do in the game? And, it should be maximizing their expect utility, so it's going to be a best reply. and what are they expecting over Well now instead of just interest, in the Nash equilibrium you fix the strategy they are player and you just maximize your, your payoff your expecting over the actions of other players. So, here we've got a situation where we have to be >> Figuring out, base on what we expect their types to be, and possibly what they might be mixing, how are they playing? Based on those types, what does that lead to, the expected action distribution your going to face? And, in terms of the types, the other players types can also enter into your payoff function. So, your utility can depend on information that other people hold. So, it might be that somebody else knows the value of a stock, and I'm trying to invest based on what information I have. >> And, I realize that other people are going to have other information, and that information could affect the value to me, as well, of a particular asset, for instance. Okay. So, given a Bayesian game, we've got our set of players actions, the type space, probability distribution over the type space, and u tility functions. And, for the definitions we're going to provide here. We're going to take these to be finite sets of players, finite sets of actions, finite sets of types, and finite sets of strategies, okay. And when you start going to infinite sets in continua, you have to be a little more careful about some of the details. Of defining these things. And in particular, measurability kinds of considerations of an, an integration of, of things. So we're going to stick with a finite set, where the basic principles and ideas will be fairly, easy to understand. And extensions to these are, are. Fairly straight forward although, there some technical details you have to worry about. Okay, Pure strategy. What's a strategy for a given player? A players strategy now is a mapping so S of I which says as a function of your type, what's the action your going to take? That would be a pure strategy in the sense that you're just picking an action for each type. a mixed strategy is in the obvious extension here, where instead of picking a pure action you're picking a probability distribution. Over actions as a function of your type. and one thing that's going to be useful then is the, i, if we have a particular type, we can then talk about what the distribution over actions is. So the mix, under a mixed strategy s of i the person i plays, what's the probability that action ai is going to be chosen by them, if they happen to be of type theta i. So we use that notation. in, in, in, in some of the calculations. Okay. Now, when we start talking about Bayesian equilibrium, now we have to talk about, what a person's expected utility is when they're making their choices. And there will be different timing that we can think of. So one is ex-ante I'm, I have to form a plan for how I'm going to behave, but I actually don't know anything about anyone's type, including my own. So we might think of this as for instance you know a company forming a long term plan for how it might say bid in a series of auctions that a re coming up. But it hasn't actually gone out and collected information yet, and it hasn't actually seen the, the values of, Of, of other players and so forth so it hasn't done any of the calculations but it's trying to form a, a strategy of how it's going to behave. second possibility interim stage. So a, a person knows something about her, his or her own type but not the types of other agents yet or other players. So this is a, a setting where you have see some parts. I've done my homework I know what I've seen and I have to form a stradegy to bid at an auction, but I don't know what the other players have seen. And, that information could be valuable not only in determining what their action is but also in determining whether or not I want to go ahead and follow a certain behavior or not, based on what my pay-offs might be contingent on what information they might have. And, the third one is ex-post, so everybody know's everything about everyone's types. Now ex-post is the relatively least interesting in, in the basic sense of the kinds of calculations we're going to be doing because in that situation if peoples are making their choices. Ex-post. Then the game is going to boil down to just the complete information games we had before. Now if people have to make their choices that ex-ante and they still want them to work ex-post then that's a different story that, that we'll talk about a little later. Okay, interim expected utility. So let's talk about the expected utility that a player has if they're at the interim stage. Well, we can say what does person i expect if they're of type theta i and the strategies s are being followed? And we end up with a calculation which looks as follows. first of all, we can look at what the possible types are. So we're going to be summing the person knows their type, then that can tell them something about what they believe the probability of other people's types will be. we're going to sum across those things, and the utilites are going to be evaluated with respect to those types. So, that's one aspect of it. the second aspect is that they also have to do the calculation of what they then believe other players will be doing, or er. Including themselves if they're mixing.Um in terms of which actions will be chosen, as a function of the types. So they have a probability distribution over types, then what are the strategies that are going to be played with what probability are we going to see different actions? And then what is the utility of those actions? So we've got the payoff function of actions, we've got probability's of actions, and we have probability's of types. Okay? And so that gives us an utility calculation, which then a player can use to evaluate what do they think a given strategy is going to lead to, in terms of payoffs. That's the interim expected utility. If we, if we move things back and then have to operate at ante stage, then we can very simply say, what does I think the probability is that they'll be of different types. And what do they think their expected utility be as a function of those types, that gives you an overall expected utility, okay. So we've got an ex ante expected utility, which isn't going to condition on types, and then an interim one, which conditions on types. 'Kay, in the x post 1 they know exactly what the types are so they can just evaluate things directly as, as we did before. Okay, so the idea behind Bayes-Nash equilibrium or Bayesian Equilibrium the concept of, from, from John Hershiney's work is that we're looking for a mixed strategy profile. you can also, you know, define pure strategy equilibrium just by restricting to be pure strategies as opposed to mixed. But what has to be true is that each individual should be choosing a best response so their strategy, s sub i, which is now mapping from types into actions. Should be maximizing their expected utility here taking at the interim stage so conditional on i and they might see and it should be true for every i and every possible type so may what type I am the strategy I have chosen should be maximizing my expected utility. given what I think other people are going to do and given the expected utility that I'm calculating based on those strategies. Okay? So, this is exactly analogous to Nash equilibrium. It's just taking explicit account of the fact that individuals will see different things at different points in the game, at the interim stage and should be maximizing wiht respect to that information. Okay the above definition is the definition that we just went through is based on an interim approach. So it's asking that every individual maximize with respect to the information that they have at the interim stage. And no matter what that information turns out to be. And if, if it happens to be true that every type occurs with positive probability then this is also equivalent. To just looking at the ex ante stage and saying, look, my strategy should maximize my overall ex ante expected utility, because if it's, if it's maximizing things for every possible theta. Then it's also going to maximize things, when I average across those thetas. And likewise, if it didn't maximize with respect to sum theta, and all the thetas are receiving positive probability then it couldn't be maximizing overall. So, you can write this Bayesian equilibrium down, either from an perspective, or from an interim perspective. As long as all types have positive probability. Okay. So what do we got from Bayesian Nash Equilibrium? we've got a, an extension of Nash equilibrium to a setting to a setting, to the Bayesian game setting. It explicitly models Behaviors in settings where we've got this uncertainty, but the concept is, is simple. Players choose strategies to maximize their payoffs in response to others, accounting for these 2 aspects of uncertainty. 1 is strategic uncertainty, what do I think other players are going to be doing, as a function of types And uncertainty, and secondly, payoff uncertainty I've got to be expecting over types, which might enter my pay offs. So it's capturing both of those elements.