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So before thinking about Bayesian Games everything we've thought about has

assumed that all of the players know what game is being played.

That is, everybody knows how many players there are in the world.

What actions are available to every player.

And what pay-offs would result given a complete action profile or action vendor

by everybody. So, let me give you a chance to stop and

think about why this is still true and imperfect information games because

intuitively it might seem to you like it isn't true.

So you might want to pause the video now. And think about that, and then I'll tell

you the answer. So the reason why imperfect information

games still have all of these things being true, is that when you don't

remember in an imperfect information game is what actions other players have played

at the moment when you're about to take your own action.

But it's still the case that you know, what actions are available to everybody

and it's still the case that if you know what everybody fully did a complete

action profile by everybody, what payoff would result.

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Now we want to think about games where these assumptions aren't true anymore.

So, what we are still going to make some new assumptions, so let me tell you about

that. So what we're relaxing is that all

players know what game is being played. So we're now going to think about the

idea that there's more than one possible game that the players can think about.

And, among all the possible games that the players will reason about.

They're all going to be games that have the same number of agents, and the same

strategies based for each agent. So.

How their going to be different is just in the utility functions.

And that's important because, it's just difficult to model a situation where

you're not quite sure what strategies base you have, it wouldn't be clear how

to act. Now, you might find it useful to reason

about games in which you're not sure what other agents there are.

And it turns out that's actually possible to capture within this framework.

In that case you would just always believe that the maximum number of agents

were present in every game, but you'd set up the utility functions in a way where

sometimes it doesn't matter that some of the agents are present because they are

not able to affect anything. The second assumption that we're going to

make, it has to do with the beliefs that agents have about these different

possible games. So, in order for this to work, it has to

be the case that agents still have well-defined beliefs about what is

possible in the world. And we're going to say that the agents

start out with a common prior, so that is everyone has the same beliefs about what

it is that's possible in the world, what what games it is that are possible.

And then, they might get individual private information about in fact what

game is being played and then, they'll do Bassion updating, so they'll end up with

a posterior belief which is obtained by studying from this common prior and

updating it based on their private information.

So, that's an assumption, we could believe that the agents had different

prior beliefs but, but that's not what we assume in the case of Bassion games.

So here's one definition of Bayesian Games in terms of information sets.

So, a Bayesian Game is a set of games that differ only in their payoffs.

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so we're going to restrict the set of games here.

To all have in common, the same set of agents.

And also the same action sets so these games are only going to differ in the

utility functions. Then P is going to be an element of the

set of all possible probability distributions over games.

This is going to be our prior distribution.

So this is going to tell us how likely each of these games from the set G is.

And finally, we're going to have a set of partitions of G, one for each agent.

So this is going to be a set of equivalence classes that will say from

the point of view of an agent, certain games are indistinguishable from each

other and others are not. Let's look at an example.

So this example is a little bit contrived.

Ordinarily we're going to use Beijing games to model something that, that

kind of makes some sort of sense in the world.

And here really we're, we're looking at something kind of artificial.

But it's a small example that still lets us think about everything important about

a Beijing game. So, the first thing to notice is that

there are four possible games that might be played.

Matching Pennies, Prisoner's Dilemma, Coordination or Battle of the Sexes.

Incidentally, Prisoner's Dilemma here is being played with different payoffs then

you might have seen before, but that doesn't matter, it's, it's strategically

the same game. And we have a common prior over the

games, so there's a 30% chance that the game that the players will in fact be

playing is matching pennies. There's a ten% chance that they will in

fact be playing presidential dilemma, twenty% chance of coordination, and a 40%

chance of battle of the sexes. Now haven't marked the actions in these

games but our assumption is that they have to be the same.

So let's say player one has two choices top or bottom.

He gets to choose the top or bottom action and it's the same in every game.

And likewise, player two can choose left or right, and he gets the left or right

action in each game. So.

What is interesting here of course is that we have this information sets.

So player one gets to find out. Which of these two sets the game is in.

What that means is. In fact, nature is going to decide which

game gets played. So, randomly it's going to be decided

which of the four games being played, according to the common prior.

Let's say the most likely thing happens, and the players end up playing Battle Of

The Sexes. In that case, what player one is going to

find out is that he is in this equivalence class than this one.

So, that means, he is going to know for sure that he is not playing Imagine

Pennies or Prisoner's dilemma, but he's going to think that he might be playing

either coordination or battle of the sexes.

He's going to have no way of turning them apart.

Now player two has different equivalence classes.

So player two. Considers these two games to be

indistinguishable, and likewise considers these two games to be indistinguishable.

And, continuing our example from before. If this was really the game that was

randomly chosen by nature, then player two would find out that he was in this

equivalence class, rather than this one. Meaning, that he would think the game

being played was either Prisoner's Dilemma or Battle of the Sexes.

And, the ground, the ground truth would be in fact Battle of the Sexes was being

played. He would consider it possible that

Prisoner's Dilemma was being played. And we've already seen that player one

would consider it possible The Coordination was being played.

And what this means is that when the players are deciding what, what action to

take, they're going to have to play an action without fully knowing what game is

going to be played. And they're going to have to reason about

what their opponent is doing without fully knowing what the opponent is going

to think. They will, they do know everything about

the setup. So this whole kind of picture is

something that the players know. They know the common prior.

They know their own equivalence classes and they know their opponent's

equivalence classes. So if I'm player one and I want to reason

about what player two is going to do and I know.

That I'm in this equivalence class. That I also know that player two, that if

the game is really coordination, which I believe is possible, then player two

thinks he's in this equivalence class. And thinks that matching pennies is

possible, even though I know it's not possible.

Or, on the other hand, if Battle of the Sexes is the real game that's being

played, then I know that player two thinks he's in this equivalence class,

which means he's going to think prisoner's dilemma's possible although I

know it's not possible. And I'll, I'll leave for a future video

actually how we reason about these games. But what we've learned here is how to

define a bastion game, by writing it as a probability distribution, a common

probability distribution over multiple different normal form games.

All of which share the same number of players and the same action sets.