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Hi, folks. So, it's Matt again. And, we're now going to talk a little bit

Â about solving extensive form games with incomplete and moving a little bit beyond

Â subgame perfection. And this is just to give you some

Â impression of, of stuff that's out there in game theory.

Â We're not going to spend too much time on this, but it'll give you a flavor of it.

Â And yu know, the, the idea of, of solving these kinds of games is, that, that makes

Â things difficult is, you know, subgame perfection and backward induction had a

Â lot of bite in games with complete information because we could analyze

Â parts of the tree there were lots of subgames, figure out what's going on in

Â that, and that would tell us then we could simplify that and, and that gives

Â us an idea of, of what's going to happen in other parts of the game.

Â With incomplete information, there's no proper subgames. So, players don't really

Â know exactly which node they're at in the game and that can be difficult.

Â So, they may not be many proper subgames. So, the, the basic reasoning doesn't

Â apply, subgame perfection does not apply directly in a lot of games, doesn't have

Â much bite. But, there are ways of extending the

Â reasoning. So, there are ways to take the same kind

Â of credibility ideas that are behind subgame perfection and apply them in

Â these kinds of games. So, we'll just take a peek at that and

Â just give you a taste of it but we're not going to go into it in too much depth.

Â Okay, so let's look at a simple game. And this game is one where it's an entry

Â decision by one say one firm of Player 1. so they have a decision of either E or N.

Â So, think of E as Enter, N as Not. And Player 2 is another firm say, in a

Â market. So, they are already in a market place selling a particular good, and Firm

Â 1 is deciding, should I enter into this market and compete with the other firm,

Â okay? So, I've offered you know, there's a

Â coffee shop open on a particular corner. There's somebody else thinking, okay,

Â should I enter right across the street and have a competing coffee shop? So,

Â Firm 1 is now thinking about entering from 2's already there.

Â And the question is, what happens once the, the Firm 1 answered? So if, if the

Â terms of payoffs here, if Firm 1 does not enter, if this player does not enter,

Â Player 1 gets zero, and Player 2 ends up getting two.

Â so, the, the payoff for Player 2 here is 2, if Firm 1 does not enter.

Â And that's, that's true either way it happens, if Firm 1 doesn't enter.

Â And then if Firm 1 enters, then the payoffs depend on whether the, the

Â incumbent coffee shop say, is one that's going to fight.

Â so F stands for fight or A for acquiesce. So basically, they can either say, okay,

Â look, live and let live. We'll have two coffee shops, we'll lose some of our

Â business. Or we can go toe-to-toe by offering

Â special coupons, discounts we, we're going to make this miserable for the

Â other, for the other company. And so, the payoffs actually depend on

Â whether Firm 2 fights or not. and moreover, the, the incomplete

Â information here is about the strength. How good Player 1 is.

Â So, they could be a strong player, probability of half, or they could be a

Â weak player. So, the node up here is a move by nature.

Â So, nature moves first, randomly picks whether Player 1 is strong or not, strong

Â or weak. So, with probability of half, they pick a

Â strong player, with probability of half, they pick a weak player.

Â And Player 1 gets to see the outcome of that.

Â So, Player 1, this new coffee shop, I know whether I've got really good coffee

Â or not. Player 2 doesn't know what the quality

Â of, of Firm 1 is when Firm 1 enters. So, Firm 1 is the strong one, or Firm 1

Â is the weak one. Player 2 cannot distinguish between those two different

Â situations and that's why we have have this information set connected here.

Â Okay? So, that's the structure of the game.

Â And basically, where is the strong and weak manifest itself in terms of payoffs?

Â it manifests itself in terms of, for instance, what happens if Firm 2 fights?

Â So, Firm 2 fights, a, a strong Firm 1, they both get -1, so they both lose.

Â If, if Firm 1 is strong Firm 2 fights, that's going to be costly for both of

Â them. If Firm 1 Firm 2 fights a weak entrant,

Â then Firm 2 gets zero and Firm 1 gets -2. So, weakness means that they'd do less

Â well in, in, in fighting. we can also, in this particular game,

Â have a situation here where, where the you know, Firm 1 the weak version of Firm

Â 1, even if Firm 2 is accommodating, is eventually going to go out of business.

Â They're just, they, you know, they've got really lousy coffee.

Â they're not going to make it. Okay, so let's try to analyze this game

Â using subgame perfection. well, with subgame perfection, there's

Â actually many equilibrium of this game. and part of the problem is that when

Â we're trying to look at subgames, we can't Just chop off this part and say

Â it's a subgame because it's not. this node is connected to this node for

Â Player 2, they're not sure whether they're over here or over here.

Â So, we can't chop off this small pieces, and essentially the only game is the

Â whole game. So, the only subgame in this games is

Â the, the whole game. And so a subgame perfection is just the

Â same as Nash equilibrium in this game. So, if we're looking at, at Nash

Â equilibrium, let's look for a couple of them.

Â let's take a peak at one where Firm 1 does not enter,

Â right? So, no matter what, Firm 1 does not enter whether they're strong or weak.

Â And Firm 2 plans on fighting, okay? So, Firm 2 says, I'm going to fight

Â you if you enter. And Firm 1 says, oh, that's bad. I'm

Â going to get negative payoff, therefore they don't enter, okay? So, that's one

Â Nash equilibrium. A Nash equilibrium is

Â one if there strong, they don't enter. If they're weak, they don't enter. And from

Â two only has one information setting, they, they fight,

Â right? So, that's a Nash Equilibrium. Okay.

Â it's also subgame perfection, given it's subgame, there's only one subgame in

Â this. what's strange about that equilibrium?

Â What's strange about that equilibrium is if you look at the fight decision of

Â Player 2, the fight decision is essentially a

Â dominated strategy in the sense that it gives -1 if the player's strong compared

Â to 1. If they were acquiescing.

Â and zero if it's against a week. Whereas, 1 if they acquiesce.

Â So, no matter what the type of the, the firm, 2 should really acquiesce, right?

Â They get a higher payoff from that. So, this is somehow not credible.

Â So, the we're losing credibility but it's, it's still consistent with Nash.

Â If Player 1 really believes Firm 2's going to fight, then that's fine.

Â And if Player 1 really never enters, well, Player 2 can say they're going to

Â fight and they never have to. So that following that strategy doesn't

Â hurt them in the sense that they're going to get to 2 no matter what.

Â And so, they don't need to deviate away from F if they're never called on to

Â move. Okay?

Â So that's, that's a Nash, but the, the what if here, the off-the-equilibrium

Â path behavior of Player 2 claiming they're going to fight is not really

Â credible in this game. So

Â what if Firm 2 is going to acquiesce? Right? So, there's another strategy

Â where, where for 2 for, for 2, we imagine them acquiesing.

Â So, what should 1 do? Well, if 1 then is strong, they should enter. They get a

Â path of 1 here, zero if they don't. If they're weak, what should they do? If

Â they're weak, well, they shouldn't enter, right?

Â Because they get a -1 here, a zero here. So weak should not enter,

Â okay? This is another Nash equilibrium. [SOUND] And, in some sense, it's a more

Â credible Nash equilibrium because in this situation from 2 is called on to, to

Â move, they're actually doing a best response.

Â So, they're following a best response of acquiescing.

Â And Firm 1 is doing the best it can. If it's strong it's entering, if it's weak

Â it's not. And this whole thing hangs together as

Â another Nash equilibrium. So here, there's a couple of Nash equilibrium.

Â There's actually more where you have Firm 2 doing some mixing and then Firm 1

Â staying out in some circumstances and, and not in others.

Â It depends on the particular mixtures you work on. So, there's actually a lot of

Â Nash equilibrium to this game. And so when we, when we want to analyze

Â this subgame perfection, of course, signs of Nash, it doesn't give us much bite in

Â terms of picking out one or the other. but one idea behind doing this in

Â analyzing these games is to try and build in the idea behind subgame confection in

Â terms of sequential rationality. And so, there are equilibrium concepts

Â that explicitly model player's beliefs about where they are in a tree for every

Â information set. And there's two, two solution concepts in

Â particular known as sequential equilibrium and perfect Bayesian

Â equilibrium that have key features where they have players, as part of the

Â equilibrium you specify what the beliefs of the players are.

Â And, it should be that the beliefs are not contradicted by the actual play of

Â the game, and players best respond to those beliefs.

Â So, you have best responded, and, and so forth.

Â But, you also make a requirement that the beliefs aren't contradicted by the actual

Â play of the game. And players have to best respond to their

Â beliefs even off the equilibrium path. And that's going to have bite in this

Â game. So, if we look at this game again and we

Â require that players have beliefs to different information sets.

Â So here, what we would have to have is now Player 2 has to say, what's the

Â probability that I'm here, what's the probability that I'm here.

Â So, they have some beliefs. But notice in this game, no matter what

Â those beliefs are, they should always acquiesce, right? So, once we give Player

Â 2 beliefs here and say they have to best respond to their beliefs in, in any any

Â node where they have beliefs, then that ties down and says, okay, if Player 2 has

Â to acquiesce, then for Player 1, if Player 2 is acquiescing, Player 1 is

Â strong. They should definitely enter.

Â If Player 2 is weak, they should definitely not enter.

Â So, we end up with a unique prediction in this game, whereas, when, with subgame

Â perfection, there were many. so the idea here is we, we have these

Â extra impositions that players have beliefs.

Â First of all, they're not contradicted so it has to be that what they're believing

Â is consistent with the way that other players are playing.

Â And players should best respond to their beliefs which is in imposing credibility

Â at every information set in, in the game. Okay.

Â So, this, this makes you, you know, ends up making a lot of, of predictions in

Â these kinds of games and they did, you know, the challenges here we see with

Â incomplete information, there may not be proper subgames.

Â the ideas of sequential rationality can be extended, but they require extra

Â layers of solution concepts. And, you know, once we do this, we're,

Â we're also layering on a lot more than we had before, and we've seen subgame

Â perfection already can be quite demanding of players.

Â Here now, they also have to be very good at inferring things based on where they

Â are. but when you begin to see things like

Â professional poker players playing, they're very much going through these

Â kinds of calculations. So, if another player raised a bet what

Â does that mean about what they're play, they're hand is likely to be?

Â should I be, you know, what, what should I do under different circumstances? If I

Â have a strong hand, should I call should I raise their and, and so forth.

Â So, so, what's going on in this kinds of solution concepts nonetheless are, are,

Â are, are very well suited to analyzing specific kinds of games.

Â So, there's a lot more to study even beyond the scope of this course.

Â but these are fascinating games to begin to wrap your head around.

Â