 Hi, folks. So it's Matt again. And we are now going to talk a little bit about solving extensive form games with incomplete information and moving a little bit beyond sub-game perfection. And this is just to give you some impression of stuff that's out there in game theory. We're not going to spend too much time on this, but I'll give you a flavor of it. And the idea of solving these kinds of games that makes things difficult is sub-game perfection and backward induction had a lot of bite in games with complete information because we could analyze parts of the tree. There were a lot of sub-games, figure out what's going on in that. And that would tell us then we could simplify that, and that gives us an idea of what's going to happen in other parts of the game. Within complete information, there's no proper sub-game. So players don't really know exactly which node they're out in the game. And that can be difficult. So there may not be many proper sub-games. So the basic reasoning doesn't apply. Sub-game perfection does not apply directly in a lot of games. It doesn't have much bite. But there are ways of extending the reasoning. So there are ways to take the same kind of credibility ideas that are behind sub-game perfection and apply them in these kinds of games. So we'll just take a peek at that and just give you a taste of it, but we're not going to go into it in too much depth. OK, so let's look at a simple game. And this game is one where it's an entry decision by one, say one firm, player one. So they have a decision of either E or N. So think of E as enter, N is not. And player two is another firm, say in a market. So they're already in a marketplace selling a particular good. And firm one is deciding, should I enter into this market and compete with the other firm? So I've offered a coffee shop open on a particular corner. There's somebody else thinking, OK, should I enter right across the street and have a competing coffee shop? So firm one is now thinking about entering. Firm two is already there. And the question is, what happens if firm one entered? So in terms of payoffs here, if firm one does not enter, if this player does not enter, player one gets zero, and player two ends up getting two. So the payoff for player two here is two if firm one does not enter. And that's true either way it happens, if firm one doesn't enter. And then if firm one enters, then the payoffs depend on whether the incumbent coffee shop, say, is one that's going to fight. So F stands for fight, or A for acquiesce. So basically they can either say, OK, look, live and let live, we'll have two coffee shops, we'll lose some of our business, or we can go toe to toe by offering special coupons, discounts. We're going to make this miserable for the other company. And so the payoffs actually depend on whether firm two fights or not. And moreover, the incomplete information here is about the strength, how good player one is. So they could be a strong player, probability a half, or they could be a weak player. So the note up here is a move by nature. So nature moves first, randomly picks whether player one is strong or not, strong or weak. So with probability a half, they pick a strong player. With probability a half, they pick a weak player. And player one gets to see the outcome of that. So player one, this new coffee shop, I know whether I've got really good coffee or not. Player two doesn't know what the quality of firm one is when firm one enters. So if firm one is a strong one, or firm one is the weak one, player two cannot distinguish between those two different situations. And that's why we have these information set connected here. So that's the structure of the game. And basically where is a strong and weak manifest itself in terms of payoffs, it manifests itself in terms of, for instance, what happens if firm two fights? So if firm two fights a strong, firm one, they both get minus one. So they both lose. If firm one is strong, firm two fights, that's gonna be costly for both of them. If firm two fights a weak entrant, then firm two gets zero and firm one gets minus two. So weakness means that they do less well in fighting. We can also, in this particular game, have a situation here where the firm one, the weak version of firm one, even if firm two is accommodating, is eventually gonna go out of business. They've got really lousy coffee. They're not gonna make it. Okay, so let's try and analyze this game using sub game perfection. Well, with sub game perfection, there's actually many equilibria of this game. And part of the problem is that when we're trying to look at sub games, we can't just chop off this part and say it's a sub game because it's not. This node is connected to this node for player two. They're not sure whether they're over here or over here. So we can't chop off the small pieces and essentially the only sub game is the whole game. So the only sub game in this game is the whole game. And so sub game perfection is just the same as Nash equilibrium in this game. So if we're looking at Nash equilibrium, let's look for a couple of them. Let's take a peek at one where firm one does not enter, right? So no matter what, firm one does not enter, whether they're strong or weak and firm two plans on fighting, okay? So firm two says, I'm gonna fight you if you enter and firm one says, ooh, that's bad. I'm gonna get negative payoffs. Therefore, they don't enter, okay? So that's one Nash equilibrium. A Nash equilibrium is one, if they're strong, they don't enter. If they're weak, they don't enter. And firm two only has one information set and they fight, right? So that's a Nash equilibrium, okay? It's also sub game perfection given that sub game, there's only one sub game in this. What's strange about that equilibrium? What's strange about that equilibrium is if you look at the fight decision of player two, the fight decision is essentially a dominated strategy in the sense that it gives minus one if the player's strong compared to one if they were acquiescing and zero if it's against a weak whereas one if they acquiesced. So no matter what the type of the firm, two should really acquiesce, right? They get a higher payoff from that. So this is somehow not credible. So we're losing credibility, but it's still consistent with Nash. If player one really believes firm two is gonna fight, then that's fine. And if player one really never enters, well, player two can say they're gonna fight and they never have to. So that following that strategy doesn't hurt them in the sense that they're gonna get the two no matter what and so they don't need to deviate away from F if they're never called on to move. So that's a Nash, but the what if here, the off the equilibrium path behavior of player two claiming they're gonna fight is not really credible in this game. So what if firm two was gonna acquiesce, right? So there's another strategy where for two, for two, we imagine them acquiescing. So what should one do? Well, if one then is strong, they should enter. They get a payoff of one here, zero if they don't. If they're weak, what should they do? If they're weak, well, they shouldn't enter, right? Because they get a minus one here, a zero here, so weak should not enter, okay? This is another Nash equilibrium and in some sense it's a more credible Nash equilibrium because in this situation, firm two is called on to move. They're actually doing a best response. So they're following a best response of acquiescing and firm one is doing the best it can. If it's strong, it's entering. If it's weak, it's not. And this whole thing hangs together as another Nash equilibrium. So here there's a couple of Nash equilibria. There's actually more where you have firm two doing some mixing and then firm one staying out in some circumstances and not in others. It depends on the particular mixtures you work out. So there's actually a lot of Nash equilibria to this game. And so when we wanna analyze this, sub game perfection, corresponds with Nash, it doesn't give us much bite in terms of picking out one or another. But one idea behind doing this and analyzing these games is to try and build in the idea behind sub game perfection in terms of sequential rationality. And so there are equilibrium concepts that explicitly model players beliefs about where they are in a tree for every information set. And there's two solution concepts in particular known as sequential equilibrium and perfect Bayesian equilibrium that have key features where they have players, as part of the equilibrium, you specify what the beliefs of the players are. And it should be that the beliefs are not contradicted by the actual play of the game and players best respond to those beliefs. So you have best respond and so forth, but you also make a requirement that the beliefs aren't contradicted by the actual play of the game and players have to best respond to their beliefs even off the equilibrium path. And that's going to have bite in this game. So if we look at this game again and we require that players have beliefs at different information sets. So here, what we would have to have is now player two has to say, what's the probability that I'm here? What's the probability that I'm here? So they have some beliefs. But notice in this game, no matter what those beliefs are, they should always acquiesce, right? So once we give player two beliefs here and say they have to best respond to their beliefs in any node where they have beliefs, then that ties down and says, okay, player two has to acquiesce. Then for player one, if player two is acquiescing, player one is strong, they should definitely enter. If player two is weak, they should definitely not enter. So we end up with a unique prediction in this game, whereas with Subgame Perfection there were many. So the idea here is we have these extra impositions that players have beliefs. First of all, they're not contradicted. So it has to be that what they're believing is consistent with the way that other players are playing. And players should best respond to their beliefs, which is imposing credibility at every information set in the game. Okay, so this ends up making a lot of predictions in these kinds of games. And the challenges here we see with incomplete information there may not be proper subgames. The ideas of sequential rationality can be extended, but they require extra layers of solution concepts. And once we do this, we're also layering on a lot more than we had before. We've seen Subgame Perfection already can be quite demanding of players. Here now they also have to be very good at inferring things based on where they are. But when you begin to see things like professional poker players playing, they're very much going through these kinds of calculations. So if another player raised a bet, what does that mean about what their hand is likely to be? Should I be, what should I do under different circumstances? If I have a strong hand, should I call? Should I raise there? And so forth. So what's going on in these kinds of solution concepts, nonetheless are very well suited to analyzing specific kinds of games. So there's a lot more to study, even beyond the scope of this course, but these are fascinating games to begin to wrap your head around.