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

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

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

Â What actions are available to every player.

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

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

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

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

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

Â you the answer. So the reason why imperfect information

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

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

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

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

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

Â 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.

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

Â that. So what we're relaxing is that all

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

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

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

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

Â strategies based for each agent. So.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â updating it based on their private information.

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

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

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

Â 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.

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

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

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

Â set of all possible probability distributions over games.

Â This is going to be our prior distribution.

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

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

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

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

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

Â So this example is a little bit contrived.

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

Â kind of makes some sort of sense in the world.

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

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

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

Â there are four possible games that might be played.

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

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

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

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

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

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

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

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

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

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

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

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

Â action in each game. So.

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

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

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

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

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

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

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

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

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

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

Â either coordination or battle of the sexes.

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

Â Now player two has different equivalence classes.

Â So player two. Considers these two games to be

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

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

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

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

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

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

Â played. He would consider it possible that

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

Â would consider it possible The Coordination was being played.

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

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

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

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

Â to think. They will, they do know everything about

Â the setup. So this whole kind of picture is

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â probability distribution over multiple different normal form games.

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

Â