Hi, folks. So, it's Matt again. And, we're now going to talk a little bit about solving extensive form games with incomplete and moving a little bit beyond subgame perfection. And this is just to give you some impression of, of stuff that's out there in game theory. We're not going to spend too much time on this, but it'll give you a flavor of it. And yu know, the, the idea of, of solving these kinds of games is, that, that makes things difficult is, you know, subgame perfection and backward induction had a lot of bite in games with complete information because we could analyze parts of the tree there were lots of subgames, figure out what's going on in that, and that would tell us then we could simplify that and, and that gives us an idea of, of what's going to happen in other parts of the game. With incomplete information, there's no proper subgames. So, players don't really know exactly which node they're at in the game and that can be difficult. So, they may not be many proper subgames. So, the, the basic reasoning doesn't apply, subgame perfection does not apply directly in a lot of games, 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 subgame 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. Okay, so let's look at a simple game. And this game is one where it's an entry decision by one say one firm of Player 1. so they have a decision of either E or N. So, think of E as Enter, N as Not. And Player 2 is another firm say, in a market. So, they are already in a market place selling a particular good, and Firm 1 is deciding, should I enter into this market and compete with the other firm, okay? So, I've offered you know, there's a coffee shop open on a particular corner. There's somebody else thinking, okay, should I enter right across the street and have a competing coffee shop? So, Firm 1 is now thinking about entering from 2's already there. And the question is, what happens once the, the Firm 1 answered? So if, if the terms of payoffs here, if Firm 1 does not enter, if this player does not enter, Player 1 gets zero, and Player 2 ends up getting two. so, the, the payoff for Player 2 here is 2, if Firm 1 does not enter. And that's, that's true either way it happens, if Firm 1 doesn't enter. And then if Firm 1 enters, then the payoffs depend on whether the, 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, okay, 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, we're going to make this miserable for the other, for the other company. And so, the payoffs actually depend on whether Firm 2 fights or not. and moreover, the, the incomplete information here is about the strength. How good Player 1 is. So, they could be a strong player, probability of half, or they could be a weak player. So, the node up here is a move by nature. So, nature moves first, randomly picks whether Player 1 is strong or not, strong or weak. So, with probability of half, they pick a strong player, with probability of half, they pick a weak player. And Player 1 gets to see the outcome of that. So, Player 1, this new coffee shop, I know whether I've got really good coffee or not. Player 2 doesn't know what the quality of, of Firm 1 is when Firm 1 enters. So, Firm 1 is the strong one, or Firm 1 is the weak one. Player 2 cannot distinguish between those two different situations and that's why we have have this information set connected here. Okay? So, that's the structure of the game. And basically, where is the strong and weak manifest itself in terms of payoffs? it manifests itself in terms of, for instance, what happens if Firm 2 fights? So, Firm 2 fights, a, a strong Firm 1, they both get -1, so they both lose. If, if Firm 1 is strong Firm 2 fights, that's going to be costly for both of them. If Firm 1 Firm 2 fights a weak entrant, then Firm 2 gets zero and Firm 1 gets -2. So, weakness means that they'd do less well in, in, in fighting. we can also, in this particular game, have a situation here where, where the you know, Firm 1 the weak version of Firm 1, even if Firm 2 is accommodating, is eventually going to go out of business. They're just, they, you know, they've got really lousy coffee. they're not going to make it. Okay, so let's try to analyze this game using subgame perfection. well, with subgame perfection, there's actually many equilibrium of this game. and part of the problem is that when we're trying to look at subgames, we can't Just chop off this part and say it's a subgame because it's not. this node is connected to this node for Player 2, they're not sure whether they're over here or over here. So, we can't chop off this small pieces, and essentially the only game is the whole game. So, the only subgame in this games is the, the whole game. And so a subgame perfection is just the same as Nash equilibrium in this game. So, if we're looking at, at Nash equilibrium, let's look for a couple of them. let's take a peak at one where Firm 1 does not enter, right? So, no matter what, Firm 1 does not enter whether they're strong or weak. And Firm 2 plans on fighting, okay? So, Firm 2 says, I'm going to fight you if you enter. And Firm 1 says, oh, that's bad. I'm going to get negative payoff, therefore they don't enter, okay? So, that's one Nash equilibrium. A Nash equilibrium is one if there strong, they don't enter. If they're weak, they don't enter. And from two only has one information setting, they, they fight, right? So, that's a Nash Equilibrium. Okay. it's also subgame perfection, given it's subgame, there's only one subgame in this. what's strange about that equilibrium? What's strange about that equilibrium is if you look at the fight decision of Player 2, the fight decision is essentially a dominated strategy in the sense that it gives -1 if the player's strong compared to 1. If they were acquiescing. and zero if it's against a week. Whereas, 1 if they acquiesce. So, no matter what the type of the, the firm, 2 should really acquiesce, right? They get a higher payoff from that. So, this is somehow not credible. So, the we're losing credibility but it's, it's still consistent with Nash. If Player 1 really believes Firm 2's going to fight, then that's fine. And if Player 1 really never enters, well, Player 2 can say they're going to fight and they never have to. So that following that strategy doesn't hurt them in the sense that they're going to get to 2 no matter what. And so, they don't need to deviate away from F if they're never called on to move. Okay? So that's, that's a Nash, but the, the what if here, the off-the-equilibrium path behavior of Player 2 claiming they're going to fight is not really credible in this game. So what if Firm 2 is going to acquiesce? Right? So, there's another strategy where, where for 2 for, for 2, we imagine them acquiesing. So, what should 1 do? Well, if 1 then is strong, they should enter. They get a path of 1 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 -1 here, a zero here. So weak should not enter, okay? This is another Nash equilibrium. [SOUND] And, in some sense, it's a more credible Nash equilibrium because in this situation from 2 is called on to, to move, they're actually doing a best response. So, they're following a best response of acquiescing. And Firm 1 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 equilibrium. There's actually more where you have Firm 2 doing some mixing and then Firm 1 staying out in some circumstances and, and not in others. It depends on the particular mixtures you work on. So, there's actually a lot of Nash equilibrium to this game. And so when we, when we want to analyze this subgame perfection, of course, signs of Nash, it doesn't give us much bite in terms of picking out one or the other. but one idea behind doing this in analyzing these games is to try and build in the idea behind subgame confection in terms of sequential rationality. And so, there are equilibrium concepts that explicitly model player's beliefs about where they are in a tree for every information set. And there's two, 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 responded, and, 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 to different information sets. So here, what we would have to have is now Player 2 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 2 beliefs here and say they have to best respond to their beliefs in, in any any node where they have beliefs, then that ties down and says, okay, if Player 2 has to acquiesce, then for Player 1, if Player 2 is acquiescing, Player 1 is strong. They should definitely enter. If Player 2 is weak, they should definitely not enter. So, we end up with a unique prediction in this game, whereas, when, with subgame perfection, there were many. so the idea here is we, 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 in imposing credibility at every information set in, in the game. Okay. So, this, this makes you, you know, ends up making a lot of, of predictions in these kinds of games and they did, you know, 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, you know, once we do this, we're, we're also layering on a lot more than we had before, and 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 they're play, they're hand is likely to be? should I be, you know, what, what should I do under different circumstances? If I have a strong hand, should I call should I raise their and, and so forth. So, so, what's going on in this kinds of solution concepts nonetheless are, are, are, 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.
