 It turns out that extensive foreign games with imperfect information give rise to subtly different classes of strategies. And in particular, one can make a distinction between behavioral and mixed strategies. And they're fundamentally very simple to explain. A mixed strategy is what is defined in a completely standard way. We have a notion of a pure strategy that is what each agent needs to do in all of their information sets. And that is a unique action in each of those information sets. And a mixed strategy is simply a distribution over such pure strategies. A behavioral strategy is subtly different. It says rather than start with pure strategies, it simply says in each information set, how should you should you randomize? It may seem like the same thing, but it really isn't. And let's look at an example. So take this take this tree. And here's an example of a behavioral strategy. Player one can do take action A with probability 0.5 and G with probability 0.3. What does that mean? That over here, they'll randomize 0.5, 0.5. And over here, they'll randomize 0.3, 0.7. That's a behavioral strategy. Similarly, a mixed strategy here would be something like the following. It says let's look at two pure strategies. For example, A, G, there'll be one pure strategy, and BH, BH would be another pure strategy. And let's look at some convex combinations, some mixture of the two, 0.6 of the one and 0.4 the other. That would be a mixed strategy. Now, although they are defined quite differently, one looking at the example might think that, well, one could really do the job of the other. And in fact, you'd be correct in this case, in a very famous result, a paper by Kuhn from 1953, it was shown that in certainly all these games of perfect information, mixed strategies and behavioral strategies can emulate each other. There's no, the equilibrium in mixed strategies are outcome equivalent to the equilibria in behavioral strategies. In fact, it's not true only for games of perfect information, it's true for games of imperfect information, that is games with information sets where agents don't have full knowledge of where they are, so long as those games have what's called perfect recall. Game of imperfect information has perfect recall, if intuitively speaking, the agents have full recollection of their experience in the game. That means that wherever they are in each information set, they know all the information that they visited previously and all the actions they've taken. To see an example of a game without perfect recall, consider the following game. So player one has here two nodes, this node and this node, and he cannot tell them apart. And you can think of it as basically sending two agents on your behalf to play and neither agents know which of the two places it landed in, and particularly what the other agent did. Regardless of the interpretation, it's the case that in a behavioral, so first of all, what are the pure strategies in this case? Well, the pure strategies for agent one is simply L, and one can in this information set either do L or R. So either do L, which means you'd go here depending on where you were, or do R and go here depending on where you were. This is for agent one. And for agent two, there are again two pure strategies. So what would be a mixed strategy equilibrium in this game? Well, that turns out to be fairly easy to analyze. And we start with the observation that player two has a dominant strategy, play down. And so no matter what the other player does, player two is no worse off and in general better off by playing D rather than U. And so a best response for player one to play to playing D is playing R because they would get a payoff of two rather than a payoff of one if they played L. And so LD is in fact an equilibrium in this game. Notice the sort of the ironic or disconcerting fact that you have a very high payoff that is actually not accessible under mixed strategies. And that will be a hint about what's going to happen with pure strategy, with behavioral strategies. So what would an equilibrium in behavioral strategies look like here? Well, to start with, note that nothing has changed for player two. They still have a dominant strategy of D. And let's assume they play that. What about player one? Now, player one has the opportunity to randomize a fresh every time they find themselves in this information set. So let's assume they randomize somehow going left with probability P and right with probability one minus P. What's assuming player two plays D, what is their expected payoff given the parameter P? Well, with probability P times P, they'll end up here and get a payoff of one. So that's P squared times one. With probability P times one minus P, they'll end up here and get a payoff of 100. So that's 100 times P times one minus P. With probability one minus P times one, they will end up, because this, player one is not random, player two is not randomizing here. With probability one minus P, they will end up here and get a payoff of two. That's two times one minus P. So this is their overall payoff, assuming they randomize P and we simplify it to this expression. And we simply look at the maximum here of this equation and the and the maximum is arrived at this value. So with probability slightly less than half, they go left and slightly more than half, they go right. That is player one. And so we end up with this equilibrium where the players, the player one randomizes this way and player two plays down with probability one. So we see that the equilibria with behavioral strategies is when we have imperfect recall as we have here can be dramatically different than the equilibria with mixed strategies.