 This video is about strategies, best response, and Nash equilibrium in the perfect information extensive form. Let's look, again, at the sharing game, which you see here. Now, instead of just thinking about what this game models, I want to think about it in terms of strategies. So in a normal form game, it was easy to talk about pure strategies. Pure strategies were just actions. But you can see in an extensive form game, there are multiple places where players get to choose different actions. And so strategies are going to need to be something a little bit more complicated. So I want to ask you to think about how many pure strategies each player has in this game. You may like to pause the video at this point and work this out for yourself before I tell you the answer. We'll start by thinking about player one. Player one's kind of easy because there's only one choice note for player one here. And so player one has the three pure strategies that you would expect, taking this action, this action, and this action. Things are a little bit more complicated for player two because player two has three choice notes. So the number of pure strategies for player one is actually eight rather than six. Player one has six different actions. But the reason that there are eight pure strategies is that a pure strategy really is every combination of choices that player two could make kind of in the aggregate. So if player two were to choose like this, so player two was to make the decision that in this situation, he would decline the offer. In this situation, he would accept it. And then again, in this situation, he would decline it. That's one pure strategy. And that's different from this pure strategy here. And so the number of such pure strategies is eight. So now let's try to make that a bit more general. Generally speaking, a pure strategy for a player in a perfect information game completely specifies how that player would play the game for anything that could happen in the game. Specifically, it says what actions to take at every choice node that that player gets to make a decision on. So intuitively, the way I like to think about pure strategies in an extensive form game is in terms of giving instructions to another person to play the game for you. Imagine that player two wants to send her friend off to play the game. Then she would need to tell everything to her friend that her friend might ever need to know in order to play the game properly. And specifically, she'd have to say for every choice node that her friend might encounter what her friend would need to do. So think of a pure strategy as proxy instructions that you give to someone else to play the game for you. Then when we're counting the number of pure strategies, we really are asking how many different sets of proxy instructions is it possible to have. If we say this formally in math, the pure strategies of a player in a given perfect information extensive form game are the cross product of the action sets for that player. So if we look at the sets of actions that are available at every choice node for the player, the set of pure strategies is then the cross product of those sets across all of the choice nodes at which that player gets to make a decision. Let's do an example that is a little bit more complicated than we saw in the sharing game. So first of all, I want to ask you here to think, what are the pure strategies for player two? Here, I'm not asking you to count them, but I'm asking you actually to say what they are. And again, you may like to pause the video and think about this for both players before I give you the answer. So we'll start with player two. Well, player two has two choice nodes, this one here and this one here. And so the pure strategies for player two are going to be the cross products of the action sets at each of those different choice nodes. So here they are written out. So for example, the pure strategy CF is saying that at this choice node, player two will play C and at this choice node, player two will play F. And because they're two sets of size two, there's a total of four pure strategies. Now for player one, things are a little bit more interesting. What are the pure strategies for player one? Well, again, player one has two choice nodes, this one and this one. And so again, the pure strategies for player one are the cross products of those two sets. So again, there are four pure strategies for player one. Why is this interesting? Well, if player one takes this action, then player one knows that he will never reach this choice node because his own action has made it impossible to reach that choice node. Nevertheless, our definition of pure strategies says that the pure strategy AG is different from the pure strategy AH. So there are still four pure strategies for player one rather than three pure strategies. So pure strategies are a bit different in extensive form games than they were in normal form games. However, there's something great. Once we've got this new definition of pure strategies, we can actually leverage it in order to use all of our old definitions of all kinds of other concepts. So in normal form games, we defined mixed strategies as probability distributions over pure strategies. And in an extensive form game, we can use exactly the same definition word for word. A mixed strategy in an extensive form game is a probability distribution over mixed strategies. All the changes is that the underlying pure strategies themselves are different. They're now policies of what to do at every choice node in the game. Likewise, a best response in an extensive form game is a mixed strategy that maximizes expected utility given a mixed strategy profile of the other agents. So again, that's exactly the same definition we had in the normal form. Finally, Nash equilibrium is again, a strategy profile in which every agent is best responding to every other agent. So all three of these concepts are really just the same as they were before. Now, something we might wonder is whether Nash equilibria exists, how we can reason about them. Just having the definition doesn't of course give us that. But there's an even tighter connection to the normal form that gives us more. And that is that we can convert an extensive form game into the normal form. And there are a couple of reasons why this is interesting. The first is because there exists a normal form game, we can leverage results we have about the normal form like the existence of equilibrium just by virtue of the fact that there's a corresponding game. Also, if we find it easier to reason about the normal form game, we can actually construct it and look at it rather than looking at the extensive form. So I'm gonna show you how to do that conversion. Here's the extensive form game we just thought about. The conversion is actually really straightforward. Here's the corresponding normal form game. And what you'll see is we've just listed all of the pure strategies of each agent as the actions in the normal form game. So you'll remember these because we went through them before on this game already. Here are the four pure strategies for player one and here are the four pure strategies for player two. Now to fill in the payoff values, what we do is we just kind of simulate play of the game. So if for example, I wanted to figure out how to put the numbers in this cell of the game, I would look at the pure strategy BG by player one and the pure strategy CF by player two. So BG means playing like this and CF means playing like that. And then I look at what node I would actually reach in the game and I would get down here and follow the tree like that. And so I write in the number 210. And so this whole table has been filled in that way. And this is what we call the induced normal form of this extensive form game. Now one thing to notice about this induced normal form is that it has more numbers in it than there are leaf nodes in the extensive form. You'll notice there are repetitions. So for example, three eight gets repeated four times here even though it only corresponds to one payoff value. Similarly, eight three gets repeated four times even though it only corresponds to one payoff value. And that's not an accident. That's because there are four pure strategy profiles that lead to that same leaf node in the tree. So this can be a problem because this blow up is actually exponential. It doesn't look so bad here because the game we're looking at is very small. But as the size of the game tree grows, this blow up can be really profound. It can mean that in practice it might be very difficult for us to write down this induced normal form. Another thing that's important to notice is that we can't always do a transformation in reverse. So you might be curious and wonder if you give me a normal form game, can I make a perfect information extensive form game out of it? And the answer is in general, no. This kind of special structure that you see to the game where payoffs are repeated is kind of important. And general extensive form games, sorry, general normal form games can't be turned into extensive form games. An example of that is matching pennies. Intuitively in matching pennies it's really important that the two players play simultaneously. And we don't really have a way of talking about two players playing simultaneously in a perfect information game because one of the players would have to move first. The second player would then get to see that move and there's just no way of representing a simultaneous move game like matching pennies that way. So intuitively we shouldn't expect a transformation from matching pennies into a perfect information game. Seems like something would have to be lost there. Indeed, there's a theorem that says that every perfect information extensive form game always has at least one pure strategy Nash equilibrium. That's not something that's true in general of normal form games. Matching pennies, which we just talked about doesn't have a pure strategy equilibrium. It's easy to see why this theorem is true. Intuitively, randomization often serves the role of confusing the other player. And there's really no reason that that could ever work in a perfect information game. If player one randomized about this choice, player two would nevertheless get to see what player one had done. And so it can't gain us anything to have randomization in the game. That can't create the opportunity for an equilibrium that wasn't there before. So lastly, I want to look at this game and reason about what its three pure strategy equilibria are. Now, we can do this by actually looking at the game tree and trying to think about what would make sense as an equilibrium in that game. But that can be a little bit hard to do in part because we can't quite so easily read the pure strategies off the game tree. Instead, what can be more convenient for a small game like this is to construct its induced normal form here, which lists the pure strategies directly, and then to just reason about the pure strategies directly on this game. So let's do that. So I'll let you again pause the video here if you like and try to find those equilibria for yourself and then I'll tell you what they are. So the three pure strategy equilibria are A-G-C-F, A-H-C-F, and B-H-C-E. So let's talk about how we're able to see that those are equilibria. Recall always the way that we test for a pure strategy equilibrium in a normal form game is to, for each player, check and see whether there's any deviation that would give that player greater utility. So let's, for example, look at B-H-C-E. If player one was to deviate here, you can see there's no other action he could take that would give him more than five. And similarly, if player two was to deviate here, you can see there's no other action she could take that would give her more than five. In both cases, there's something that would tie, but that's okay because a best response just says that there isn't anything better. So that confirms that this is an equilibrium. In contrast, if I looked at something like this, which I've claimed isn't an equilibrium, you can see that it isn't an equilibrium by checking for each player. So player two indeed can't do anything better than 10. So this is a best response. C-F is a best response to B-G for player two. But on the other hand, player one could deviate from B-G to A-G and get a payoff of three instead of a payoff of two. So B-G is not a best response to C-F for player one, and so this is not an ash equilibrium. And that's it for this video. Thanks.