 It turns out there are several different ways to define Bayesian games. This video is going to give the first definition. 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 payoffs would result given a complete action profile or action vector by everybody. So let me give you a chance to stop and think about why this is still true in 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 what 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. Now we want to think about games where these assumptions aren't true anymore. So 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 of these 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 strategy space for each agent. So how they're 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 strategy space 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. 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're 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 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 Bayesian updating. So they'll end up with a posterior belief which is obtained by starting 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 that's not what we assume in the case of Bayesian 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, also a common prior defined over these games, and a partition structure over the games for each agent. So in other words, a Bayesian game is defined by four elements. The first is a set of agents. The second is a set of games. Recall that a game is defined as a set of agents, a set of action profiles, sorry, a set of action sets, one for each agent, and a set of utility functions, one for each agent. 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 going to differ only in their 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 Bayesian games to model something that kind of makes some sort of sense in the world. And here, really, we're looking at something kind of artificial. But it's a small example that still lets us think about everything important about a Bayesian 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 than you might have seen before, but that doesn't matter. 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 10% chance that they will, in fact, be playing prisoner's dilemma, 20% chance of coordination, and a 40% chance of battle of the sexes. Now, I 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 these 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's in this equivalence class rather than this one. So that means he's going to know for sure that he's not playing matching 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 telling 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 truth would be that, in fact, battle of the sexes was being played, he would consider it possible the 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 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 do know everything about this setting. 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, then I also know that player two, 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 is possible even though I know it's not possible. And 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 Bayesian 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.