0:00

Hi folks. This is Matt and we're going to talk a

Â little bit now about Bayesian equilibrium an equilibrium concept and solution

Â concept for Bayesian Games. And the idea of,of this, it is also sometimes. Referred

Â to as Bayesian Nash Equilibria. the concept goes back to Harsanyi, John

Â Harsanyi in, in late 1960s, 1967, and 1968, where he developed this concept. And

Â the idea is that, that each player now, you know, when we're talking about a

Â Bayesian game player Players have these types which determine their payoffs. And

Â relate to the uncertainty. And can actually tell them something about what

Â they expect other individual's types to be. And an equilibrium, now, is going to

Â be a plan of action for each player as a function of their types. So, it's going

Â to, to say, okay. If I, if I observe a certain type. What am I going to do in the

Â game? And, it should be maximizing their expect utility, so it's going to be a best

Â reply. and what are they expecting over Well now instead of just interest, in the

Â Nash equilibrium you fix the strategy they are player and you just maximize your,

Â your payoff your expecting over the actions of other players. So, here we've

Â got a situation where we have to be >> Figuring out, base on what we expect their

Â types to be, and possibly what they might be mixing, how are they playing? Based on

Â those types, what does that lead to, the expected action distribution your going to

Â face? And, in terms of the types, the other players types can also enter into

Â your payoff function. So, your utility can depend on information that other people

Â hold. So, it might be that somebody else knows the value of a stock, and I'm trying

Â to invest based on what information I have. >> And, I realize that other people

Â are going to have other information, and that information could affect the value to

Â me, as well, of a particular asset, for instance. Okay.

Â So, given a Bayesian game, we've got our set of players actions, the type space,

Â probability distribution over the type space, and u tility functions. And, for

Â the definitions we're going to provide here. We're going to take these to be

Â finite sets of players, finite sets of actions, finite sets of types, and finite

Â sets of strategies, okay. And when you start going to infinite sets in continua,

Â you have to be a little more careful about some of the details. Of defining these

Â things. And in particular, measurability kinds of considerations of an, an

Â integration of, of things. So we're going to stick with a finite set, where the

Â basic principles and ideas will be fairly, easy to understand. And extensions to

Â these are, are. Fairly straight forward although, there some technical details you

Â have to worry about. Okay, Pure strategy. What's a strategy for a given player? A

Â players strategy now is a mapping so S of I which says as a function of your type,

Â what's the action your going to take? That would be a pure strategy in the sense that

Â you're just picking an action for each type. a mixed strategy is in the obvious

Â extension here, where instead of picking a pure action you're picking a probability

Â distribution. Over actions as a function of your type. and one thing that's going

Â to be useful then is the, i, if we have a particular type, we can then talk about

Â what the distribution over actions is. So the mix, under a mixed strategy s of i the

Â person i plays, what's the probability that action ai is going to be chosen by

Â them, if they happen to be of type theta i. So we use that notation. in, in, in, in

Â some of the calculations. Okay. Now, when we start talking about Bayesian

Â equilibrium, now we have to talk about, what a person's expected utility is when

Â they're making their choices. And there will be different timing that we can think

Â of. So one is ex-ante I'm, I have to form a plan for how I'm going to behave, but I

Â actually don't know anything about anyone's type, including my own. So we

Â might think of this as for instance you know a company forming a long term plan

Â for how it might say bid in a series of auctions that a re coming up. But it

Â hasn't actually gone out and collected information yet, and it hasn't actually

Â seen the, the values of, Of, of other players and so forth so it hasn't done any

Â of the calculations but it's trying to form a, a strategy of how it's going to

Â behave. second possibility interim stage. So a, a person knows something about her,

Â his or her own type but not the types of other agents yet or other players. So this

Â is a, a setting where you have see some parts. I've done my homework I know what

Â I've seen and I have to form a stradegy to bid at an auction, but I don't know what

Â the other players have seen. And, that information could be valuable not only in

Â determining what their action is but also in determining whether or not I want to go

Â ahead and follow a certain behavior or not, based on what my pay-offs might be

Â contingent on what information they might have. And, the third one is ex-post, so

Â everybody know's everything about everyone's types. Now ex-post is the

Â relatively least interesting in, in the basic sense of the kinds of calculations

Â we're going to be doing because in that situation if peoples are making their

Â choices. Ex-post. Then the game is going to boil down to

Â just the complete information games we had before. Now if people have to make their

Â choices that ex-ante and they still want them to work ex-post then that's a

Â different story that, that we'll talk about a little later. Okay, interim

Â expected utility. So let's talk about the expected utility that a player has if

Â they're at the interim stage. Well, we can say what does person i expect if they're

Â of type theta i and the strategies s are being followed? And we end up with a

Â calculation which looks as follows. first of all, we can look at what the possible

Â types are. So we're going to be summing the person knows their type, then that can

Â tell them something about what they believe the probability of other people's

Â types will be. we're going to sum across those things, and the utilites are going

Â to be evaluated with respect to those types. So, that's one aspect of it. the

Â second aspect is that they also have to do the calculation of what they then believe

Â other players will be doing, or er. Including themselves if they're mixing.Um

Â in terms of which actions will be chosen, as a function of the types. So they have a

Â probability distribution over types, then what are the strategies that are going to

Â be played with what probability are we going to see different actions? And then

Â what is the utility of those actions? So we've got the payoff function of actions,

Â we've got probability's of actions, and we have probability's of types. Okay? And so

Â that gives us an utility calculation, which then a player can use to evaluate

Â what do they think a given strategy is going to lead to, in terms of payoffs.

Â That's the interim expected utility. If we, if we move things back and then have

Â to operate at ante stage, then we can very simply say, what does I think the

Â probability is that they'll be of different types. And what do they think

Â their expected utility be as a function of those types, that gives you an overall

Â expected utility, okay. So we've got an ex ante expected utility, which isn't going

Â to condition on types, and then an interim one, which conditions on types. 'Kay, in

Â the x post 1 they know exactly what the types are so they can just evaluate things

Â directly as, as we did before. Okay, so the idea behind Bayes-Nash equilibrium or

Â Bayesian Equilibrium the concept of, from, from John Hershiney's work is that we're

Â looking for a mixed strategy profile. you can also, you know, define pure strategy

Â equilibrium just by restricting to be pure strategies as opposed to mixed. But what

Â has to be true is that each individual should be choosing a best response so

Â their strategy, s sub i, which is now mapping from types into actions. Should be

Â maximizing their expected utility here taking at the interim stage so conditional

Â on i and they might see and it should be true for every i and every possible type

Â so may what type I am the strategy I have chosen should be maximizing my expected

Â utility. given what I think other people are going to do and given the expected

Â utility that I'm calculating based on those strategies. Okay? So, this is

Â exactly analogous to Nash equilibrium. It's just taking explicit account of the

Â fact that individuals will see different things at different points in the game, at

Â the interim stage and should be maximizing wiht respect to that information. Okay the

Â above definition is the definition that we just went through is based on an interim

Â approach. So it's asking that every individual maximize with respect to the

Â information that they have at the interim stage. And no matter what that information

Â turns out to be. And if, if it happens to be true that every type occurs with

Â positive probability then this is also equivalent. To just looking at the ex ante

Â stage and saying, look, my strategy should maximize my overall ex ante expected

Â utility, because if it's, if it's maximizing things for every possible

Â theta. Then it's also going to maximize things, when I average across those

Â thetas. And likewise, if it didn't maximize with respect to sum theta, and

Â all the thetas are receiving positive probability then it couldn't be maximizing

Â overall. So, you can write this Bayesian equilibrium down, either from an

Â