 In this video, I'm going to tell you about how to define formally the imperfect information extensive form and how to reason about strategies in these games. So in the perfect information extensive form, we have a player take an action at every single choice node in the game. And a consequence of that definition is that we're seeing that players know what node they're in all the time. And that means they know the whole history of all of the moves that have happened before in the game. This is reasonable for some games like chess where you really do get to see what your opponent does in every different move. It's not reasonable for games like battleship where your opponent can do something and you don't get to see what it is. And it matters to what happens to you later in the game. It can matter to your payoff. It can matter to whether the game stops or continues. So in order to model this richer situation where players aren't able to observe everything that their opponents do, we're going to add something new to the game, to the game representation. So we're going to call it the imperfect information extensive form. And the way this is going to work is we're going to take the old definition that we had before, but we're going to say that players consider some choice nodes to be equivalent to each other. So there are some choice nodes that a player isn't able to tell apart. And that's going to mean they're not able to completely figure out the history of where they are in the tree because they're not going to know which of several choice nodes they're at when they have to take a choice. And we're going to do this by taking the set of choice nodes for a given player and putting them into equivalence classes. So what that means is, kind of schematically, if these are some different choice nodes in the game that all of course belong to the same player, we might say these are one equivalence class, these are another equivalence class, and these are a third equivalence class. And what that would mean is the player wouldn't know which of these two choice nodes he was at when he was asked to make a choice. But he would know that he was at one of these two and not one of these two because they're in different equivalence classes. So let's say that a little bit more formally. So to formally define an imperfect information extensive form game, we start with a perfect information extensive form game, which we've already learned how to define before. And then we're going to add this ingredient of equivalence classes. So we're going to add this element I, which is a set of equivalent sets of equivalence classes one for every player. So for player one, we have this set of equivalence classes or let's say for player I, we have a set of equivalence classes numbered from one to K sub I. So these are all different equivalence classes and each one of these classes is some number of different choice nodes one or more choice nodes. And these are going to be different choice nodes that that player isn't able to tell apart. So if every one of these equivalence classes contains only one choice node, we're back with the perfect information case. And if any of these equivalence classes contains more than one thing, then we have something new. We have a game where the player doesn't quite know what's going on all the time. Now, in order to make this definition work, we need to add a couple of restrictions so that it makes sense. So we want to say that if two different choice nodes are part of the same equivalence class, then first of all, they have to belong to the same player. They have to be labeled with the same player because if they're not, you'd be able to tell them apart because different players would be acting. So the player kind of when he shows up, he would know which player he was. He would really not be confused about all of the nodes in the equivalence class. And the second restriction we have is that the two choice nodes have to have the same set of available actions because if the player can't tell them apart, he has to know what to do. And that's the only restrictions. So let's look at an example game here and see what the equivalence classes are. So in this game, player one has two different equivalence classes. This is one equivalence class and this is another equivalence class. So in other words, we're going to use a dotted line here to connect together choice nodes that belong to the same equivalence class. And what we want to say in this game is that player one moves and if he goes right, then the game is just going to end and they're just going to get each a payoff of one. If he goes left, then player two is going to get to make a choice and player two is going to go either A or B. And then player one is going to get to move a second time, but player one isn't going to get to observe the move that player two took. And so he's going to have to take the same action regardless of whether he's at this choice node or this choice node. And indeed you can see we've labeled it the same way. So if he says he wants to go left, he would have to go left from both of these nodes. And just to complete the answer to my question, what are the equivalence classes for player two? Well, player two only has one choice node. Actually, there's a typo here. There shouldn't be a two. Player two has only one choice node and so he has only one equivalence class. So how should we define the pure strategies for each player in this game? What's going to make sense as a definition of pure strategies? Well, intuitively, remember before what we said was that we had the cross product of all of the different action sets for each player. We don't want that here because we don't want for it to be possible for player one to do something different in this choice node and in this choice node. So instead, what we're going to use as a definition is that the pure strategies for player I are the cross product of the action sets in every different equivalence class that he has. And so player one's pure strategies here are going to be a choice here and independently a choice here. So player one could take the action LL or he could take the action RL or he could take the action LR or RR. So player one here has four different pure strategies rather than eight as he would have if we didn't have this equivalence class here. So in the imperfect information normal form, we have a more powerful representation than we did in the perfect information case. And one way we can see that is we can represent any normal form game in this representation, which you may recall we couldn't do with perfect information games. So here I'm showing you how to represent the TCP back off game or in other words the prisoner's dilemma game in imperfect information extensive form. So how does this work? Well, first of all, player one gets to decide whether to cooperate or defect. And after that player two gets to decide whether to cooperate or defect. Now, of course, in the prisoner's dilemma you don't get to see what the other person chose to do when you take your own action. So it might sound like a problem that player two gets to move second, but player two isn't able to tell which action player one took. And so although our game representation says that he goes second, it doesn't really make a difference that he goes second because he's not informed of what player one did. And then once both of them take their actions we end up with some payoff. So if they go CD and they end up with this payoff here and these are the same payoffs that we have in the game matrix. Notice that we could have things work the same way if we put player two at the root node and player one down here below because time isn't really playing a role in this game. So what I told you on the previous slide was how to start with a normal form game and make an extensive form game out of it. I can also still do the thing that we talked about with the perfect information extensive form, which is to start with an extensive form game and make a normal form game out of it. And the way this works is exactly the way I did before. So I take all of the pure strategies for player one and I make them into rows. I take all of the pure strategies for player two and I make them into columns. And then this gives me my matrix. And for every cell of the matrix, I say, well, if player one took this pure strategy and player two took this pure strategy, what payoff would result? And I put that into the cell of the matrix and I fill in the whole matrix that way. And once I have the matrix like that, then I'm kind of done. My definition of mixed strategies is exactly what it was before. It's just the mixed strategies in the induced normal form. The definition of best response in Nash equilibrium for imperfect information extensive form games, again, just kind of leverage the induced normal form. And so all of those concepts that you already understand from normal form games carry over directly to imperfect information games. And so for example, we know from Nash's theorem that a Nash equilibrium always exists for every imperfect information extensive form game because I can make a finite normal form game out of it. Now it's going to be the case that this transformation can make the game exponentially bigger as it could before, even with the perfect information case. But for example, for existence of equilibrium, that doesn't matter. Now lastly, you might wonder what happens if I take both of these transformations that I've shown you and I apply them together. So for example, I could start with an imperfect information extensive form game, make it into a normal form game, and make that back into an imperfect information extensive form game again. So you might wonder, do I end up with the same game? And the answer is no, I won't because I might have a game tree which is pretty deep. It could have all kinds of different equivalence classes. It could have all kinds of different sequential moves by the different players. And when I make it into a normal form game, I'm going to get this flat table. And then when I take a flat table and turn that into an extensive form game, it's going to be an extensive form game that only has two levels with all of this stuff in one big equivalence class. And so it's going to be an extensive form game that looks different from the one that I started with. But what is important is that it's going to have the same strategy spaces, the same sets of pure strategies for both agents, and it's going to have the same set of Nash equilibria. So although these games might look different from the perspective of how they explicitly talk about time, they're going to be strategically equivalent games. And that's it for this video.