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  • Professor Ben Polak: Okay, so last time we came

  • across a new idea, although it wasn't very new for

  • a lot of you, and that was the idea of Nash

  • Equilibrium. What I want to do today is

  • discuss Nash Equilibrium, see how we find that

  • equilibrium into rather simple examples.

  • And then in the second half of the day I want to look at an

  • application where we actually have some fun and play a game.

  • At least I hope it's fun. But let's start by putting down

  • a formal definition. We only used a rather informal

  • one last week, so here's a formal one.

  • A strategy profile--remember a profile is one strategy for each

  • player, so it's going to be S_1*,

  • S_2*, all the way up to

  • S_M* if there are M players playing the game--so

  • this profile is a Nash Equilibrium (and I'm just going

  • to write NE in this class for Nash Equilibrium from now on)

  • if, for each i –so for each

  • player i, her choice--so her choice here is S_i*,

  • i is part of that profile is a best response to the other

  • players' choices. Of course, the other players'

  • choices here are S_--i* so everyone is

  • playing a best response to everyone else.

  • Now, this is by far the most commonly used solution concept

  • in Game Theory. So those of you who are

  • interviewing for McKenzie or something, you're going to find

  • that they're going to expect you to know what this is.

  • So one reason for knowing what it is, is because it's in the

  • textbooks, it's going to be used in lots of applications,

  • it's going to be used in your McKenzie interview.

  • That's not a very good reason and I certainly don't want you

  • to jump to the conclusion that now we've got to Nash

  • Equilibrium everything we've done up to know is in some sense

  • irrelevant. That's not the case.

  • It's not always going to be the case that people always play a

  • Nash Equilibrium. For example,

  • when we played the numbers game, the game when you chose a

  • number, we've already discussed last

  • week or last time, that the equilibrium in that

  • game is for everyone to choose one,

  • but when we actually played the game, the average was much

  • higher than that: the average was about 13.

  • It is true that when we played it repeatedly,

  • it seemed to converge towards 1,

  • but the play of the game when we played it just one shot first

  • time, wasn't a Nash Equilibrium. So we shouldn't form the

  • mistake of thinking people always play Nash Equilibrium or

  • people, "if they're rational," play Nash Equilibrium.

  • Neither of those statements are true.

  • Nevertheless, there are some good reasons for

  • thinking about Nash Equilibrium other than the fact it's used by

  • other people, and let's talk about those a

  • bit. So I want to put down some

  • motivations here--the first motivation we already discussed

  • last time. In fact, somebody in the

  • audience mentioned it, and it's the idea of "no

  • regrets." So what is this idea?

  • It says, suppose we're looking at a Nash Equilibrium.

  • If we hold the strategies of everyone else fixed,

  • no individual i has an incentive to deviate,

  • to move away. Alright, I'll say it again.

  • Holding everyone else's actions fixed, no individual has any

  • incentive to move away. Let me be a little more careful

  • here; no individual has any

  • strict incentive to move away.

  • We'll see if that actually matters.

  • So no individual can do strictly better by moving away.

  • No individual can do strictly better by deviating,

  • holding everyone else's actions.

  • So why I call that "no regret"? It means, having played the

  • game, suppose you did in fact play a Nash Equilibrium and then

  • you looked back at what you had done,

  • and now you know what everyone else has done and you say,

  • "Do I regret my actions?" And the answer is,

  • "No, I don't regret my actions because I did the best I could

  • given what they did." So that seems like a fairly

  • important sort of central idea for why we should care about

  • Nash Equilibrium. Here's a second idea,

  • and we'll see others arise in the course of today.

  • A second idea is that a Nash Equilibrium can be thought of as

  • self-fulfilling beliefs. So, in the last week or so

  • we've talked a fair amount about beliefs.

  • If I believe the goal keeper's going to dive this way I should

  • shoot that way and so on. But of course we didn't talk

  • about any beliefs in particular. These beliefs,

  • if I believe that--if everyone in the game believes that

  • everyone else is going to play their part of a particular Nash

  • Equilibrium then everyone, will in fact,

  • play their part of that Nash Equilibrium.

  • Now, why? Why is it the case if everyone

  • believes that everyone else is playing their part of this

  • particular Nash Equilibrium that that's so fulfilling and people

  • actually will play that way? Why is that the case?

  • Anybody? Can we get this guy in red?

  • Student: Because your Nash Equilibrium matches the

  • best response against both. Professor Ben Polak:

  • Exactly, so it's really--it's almost a repeat of the first

  • thing. If I think everyone else is

  • going to play their particular--if I think players 2

  • through N are going to play S_2* through

  • S_N*--then by definition my best response is

  • to play S_1* so I will in fact play my part in the Nash

  • Equilibrium. Good, so as part of the

  • definition, we can see these are self-fulfilling beliefs.

  • Let's just remind ourselves how that arose in the example we

  • looked at the end last time. I'm not going to go back and

  • re-analyze it, but I just want to sort of make

  • sure that we followed it. So, we had this picture last

  • time in the partnership game in which people were choosing

  • effort levels. And this line was the best

  • response for Player 1 as a function of Player 2's choice.

  • And this line was the best response of Player 2 as a

  • function of Player 1's choice. This is the picture we saw last

  • time. And let's just look at how

  • those--it's no secret here what the Nash Equilibrium is:

  • the Nash Equilibrium is where the lines cross--but let's just

  • see how it maps out to those two motivations we just said.

  • So, how about self-fulfilling beliefs?

  • Well, if Player--sorry, I put 1, that should be 2--if

  • Player 1 believes that Player 2 is going to choose this

  • strategy, then Player 1 should choose

  • this strategy. If Player 1 thinks Player 2

  • should take this strategy, then Player 1 should choose

  • this strategy. If Player 1 thinks Player 2 is

  • choosing this strategy, then Player I should choose

  • this strategy and so on; that's what it means to be best

  • response. But if Player 1 thinks that

  • Player 2 is playing exactly her Nash strategy then Player 1's

  • best response is to respond by playing his Nash strategy.

  • And conversely, if Player 2 thinks Player 1 is

  • playing his Nash strategy, then Player 2's best response

  • indeed is to play her Nash strategy.

  • So, you can that's a self-fulfilling belief.

  • If both people think that's what's going to happen,

  • that is indeed what's going to happen.

  • How about the idea of no regrets?

  • So here's Player 1; she wakes up the next

  • morning--oh I'm sorry it was a he wasn't it?

  • He wakes up the next morning and he says, "I chose

  • S_1*, do I regret this?"

  • Well, now he knows what Player 2 chose;

  • Player 2 chose S_2* and he says, "no that's the best

  • I could have done. Given that Player 2 did in fact

  • choose S_2*, I have no regrets about

  • choosing S_1*; that in fact was my best

  • response." Notice that wouldn't be true at

  • the other outcome. So, for example,

  • if Player 1 had chosen S_1* but Player 2 had

  • chosen some other strategy, let's say S_2 prime,

  • then Player I would have regrets.

  • Player I would wake up the next morning and say,

  • "oh I thought Player 1 was going to play S_2*;

  • in fact, she chose S_2 prime.

  • I regret having chosen S_1*;

  • I would have rather chosen S_1 prime.

  • So, only at the Nash Equilibrium are there no

  • regrets. Everyone okay with that?

  • This is just revisiting really what we did last time and

  • underlining these points. So, I want to spend quite a lot

  • of time today just getting used to the idea of Nash Equilibrium

  • and trying to find Nash Equilibrium.

  • (I got to turn off that projector that's in the way

  • there. Is that going to upset the

  • lights a lot?) So okay, so what I want to do is I want

  • to look at some very simple games with a small number of

  • players to start with, and a small number of

  • strategies, and I want us to get used to how we would find the

  • Nash Equilibria in those simple games.

  • We'll start slowly and then we'll get a little faster.

  • So, let's start with this game, very simple game with two

  • players. Each player has three

  • strategies and I'm not going to motivate this game.

  • It's just some random game. Player 1 can choose up,

  • middle, or down, and Player 2 can choose left,

  • center, and right and the payoffs are

  • as follows: (0,4) (4,0) (5,3) (4,0) (0,4) (5,3) again (3,5)

  • (3,5) and (6,6). So, we could discuss--if we had

  • more time, we could actually play this game--but isn't a very

  • exciting game, so let's leave our playing for

  • later, and instead, let's try and figure out what

  • are the Nash Equilibria in this game and how we're going to go

  • about finding it. The way we're going to go about

  • finding them is going to mimic what we did last time.

  • Last time, we had a more complicated game which was two

  • players with a continuum of strategies,

  • and what we did was we figured out Player 1's best responses

  • and Player 2's best responses: Player 1's best response to

  • what Player 2's doing and Player 2's best response to what Player

  • 1 is doing; and then we looked where they

  • coincided and that was the Nash Equilibrium.

  • We're going to do exactly the same in this simple game here,

  • so we're going to start off by figuring out what Player 1's

  • best response looks like. So, in particular,

  • what would be Player 1's best response if Player 2 chooses

  • left? Let's get the mikes up so I can

  • sort of challenge people. Anybody?

  • Not such a hard question. Do you want to try the woman in

  • the middle? Who's closest to the woman in

  • the middle? Yeah.

  • Student: The middle. Professor Ben Polak:

  • Okay, so Player 1's best response in this case is middle,

  • because 4 is bigger than 0, and it's bigger than 3.

  • So, to mark that fact what we can do is--let's put a

  • circle--let's make it green--let's put a circle around

  • that 3. Not hard, so let's do it again.

  • What is Player 1's best response if Player 2 chooses

  • center? Let's get somebody on the other

  • side. (Somebody should feel free to

  • cold call somebody here.) How about cold calling the guy with

  • the Yale football shirt on? There we go.

  • Student: Up. Professor Ben Polak: All

  • right, so up is the correct answer.

  • (So, one triumph for Yale football).

  • So 4 is bigger than 3, bigger than 0,

  • so good, thank you. Ale why don't you do the same?

  • Why don't you cold call somebody who's going to tell me

  • what's the best response for Player 1 to right.

  • Shout it out. Student: Down.

  • Professor Ben Polak: Down, okay so best response is

  • down because 6 is bigger than 5, which is bigger than 5;

  • I mean 6 is bigger than 5. So what we've done here is

  • we've found Player 1's best response to left,

  • best response to center, and best response to right and

  • in passing notice that each of Player 1 strategies is the best

  • response to something, so nothing would be knocked out

  • by our domination arguments here, and nothing would be

  • knocked out by our never a best response arguments here for

  • Player 1. Let's do the same for Player 2;

  • so why don't I keep the mikes doing some cold calling,

  • so why don't we--you want to cold call somebody at the back

  • to tell me what is Player 2's best response against up?

  • Student: Left. Professor Ben Polak: So,

  • the gentleman in blue says left because 4 is bigger 3,

  • and 3 is bigger than 0. Let's switch colors and switch

  • polygons and put squares around these.

  • I don't insist on the circles and the squares.

  • If you have a thing for hexagons you can do that

  • to0--whatever you want. What's Player 2's--let's write

  • it in here--the answer, the answer was that the best

  • response was left, and Player 2's best response to

  • middle--so do you want to grab somebody over there?

  • Shout it out! Student: Center.

  • Professor Ben Polak: Because 4 is bigger than 3,

  • and 3 is bigger than 0, so that's center.

  • So, in my color coding that puts me at--that gives me a

  • square here. And finally,

  • Player 2's best response to down?

  • And your turn Ale, let's grab somebody.

  • Student: Left--right. Professor Ben Polak:

  • Right, okay good. Because 6 is bigger than 5

  • again, so here we go. What we've done now is found

  • Player 1's best response function that was just the

  • analog of finding the best response line we did here.

  • Here we drew it as a straight line in white,

  • but we could have drawn it as a sequence of little circles in

  • green; it would have looked like this.

  • The same idea, I'm not going to do the whole

  • thing but there would be a whole bunch of circles looking like

  • this. And then we found Player 2's

  • best response for each choice of Player 1 and we used pink

  • rectangles and that's the same idea as we did over here.

  • Here we did it in continuous strategies in calculus,

  • but it's the same idea. So, imagine I'm drawing lots of

  • pink rectangles. And again, I'm not going to do

  • it, but you can do it in your notes;

  • same basic idea. Just as the Nash Equilibrium

  • must be where these best response lines coincided here

  • because that's the point at which each player is playing a

  • best response to each other, so the Nash Equilibrium over

  • here is--do you want to cold call somebody?

  • Do you want to just grab somebody in the row behind you

  • or something, wherever?

  • Anybody? Yeah, I think you in the Italia

  • soccer shirt; I think you're going to get

  • picked on, so okay what's the Nash Equilibrium here?

  • Student: The down right. Professor Ben Polak: So

  • the down-right square. So, the Nash Equilibrium here

  • is--no prizes--no World Cup for that one--is down right.

  • So, why is that the Nash Equilibrium?

  • Because at that point Player 1 is playing a best response to

  • Player 2 and Player 2 is playing a best response to Player I.

  • Now, I want to try and convince you, particularly those of you

  • who are worried by the homework assignments, that that was not

  • rocket science. It's not hard to find Nash

  • Equilibria in games. It didn't take us very long,

  • and we went pretty slow actually.

  • So let's look at another example.

  • Oh, before I do, notice this--before I leave

  • this game I'm just going to make one other remark.

  • Notice that in this game that each strategy of Player 1 is a

  • best response to something, and each strategy of Player 2

  • is a best response to something. So, had we used the methods of

  • earlier on in the class, that's to say,

  • deleting dominated strategies or deleting strategies that are

  • never a best response, we'd have gotten nowhere in

  • this game. So, Nash Equilibrium goes a

  • little further in narrowing down our predictions.

  • But we also learned something from that.

  • We argued last time, in the last few weeks,

  • that rationality, or even mutual knowledge of

  • rationality or even common knowledge of rationality

  • couldn't really get us much further than deleting dominated

  • strategies, or if you like,

  • deleting strategies that are never best responses.

  • So, here we've concluded that the Nash Equilibrium of this

  • game is down right, a very particular prediction,

  • but notice a perfectly rational player could here,

  • could choose middle. The reason they could choose

  • middle, Player 1 could choose middle is--could be that-- they

  • say (rationally) they should choose middle because they think

  • Player 2 is choosing left. Then you say,

  • "well, why would you think Player 2 is going to choose

  • left?" Player 1 could say,

  • "I think Player 2 is going to choose left because Player 2

  • thinks I'm going to play up." And then you say,

  • "but how could Player 2 possibly think that you're going

  • to play up?" And then Player 1 would say,

  • "I think Player 2 thinks I'm going to play up because Player

  • 2 thinks that I think that he's going to play center."

  • And then you could say, "how could Player 2 play

  • …" etc., etc., etc.

  • And you could see you could work around a little cycle

  • there. Nobody would be irrational

  • about anything: Everything would be perfectly

  • well justified in terms of beliefs.

  • It's just we wouldn't have a nice fixed point.

  • We wouldn't have a nice point where it's a simple argument.

  • In the case of down right, Player 1 thinks it's okay to

  • play down because he thinks 2 is going to play right,

  • and Player 2 thinks they're going to play right because he

  • thinks Player 1's going to play down.

  • So, just to underline what I'm saying rather messily there,

  • rationality, and those kinds of arguments

  • should not lead us to the conclusion that people

  • necessarily play Nash Equilibrium.

  • We need a little bit more; we need a little bit more than

  • that. Nevertheless,

  • we are going to focus on Nash Equilibrium in the course.

  • Let's have a look at another example.

  • Again, we'll keep it simple for now and we'll keep to a

  • two-player, three-strategy game; up, middle, down,

  • left, center, right and this time the payoffs

  • are as follows: (0,2) (2,3) (4,3) (11,1) (3,2)

  • (0,0) (0,3) (1,0) and (8,0). So, this is a slightly messier

  • game. The numbers are really just

  • whatever came into my head when I was writing it down.

  • Once again, we want to find out what the Nash Equilibrium is in

  • this game and our method is exactly the same.

  • We're going to, for each player,

  • figure out their best responses for each possible choice of the

  • other player. So, rather than write it out on

  • the--let me just go to my green circles and red squares and let

  • me get my mikes up again, so we can cold call people a

  • bit. Can we get the--where's the

  • mike? Thank you.

  • So, Ale do you want to just grab somebody and ask them

  • what's the best response against left?

  • Grab anybody. What's the best one against

  • left? Student: Middle.

  • Professor Ben Polak: Middle, okay.

  • Good, so middle is the best response against left,

  • because 11 is bigger than 0. Myto , do you want to grab

  • somebody? What's the best response

  • against center? Student: Middle.

  • Professor Ben Polak: Middle again because 3 is bigger

  • than 2 is bigger than 1. And grab somebody at the back

  • and tell me what the best response is against right?

  • Anybody just dive in, yep. Student: Down.

  • Professor Ben Polak: Okay down, good.

  • Let's do the same for the column player,

  • so watch the column player's--Ale,

  • find somebody; take the guy who is two rows

  • behind you who is about to fall asleep and ask him what's the

  • best response to up? Student: Best response

  • is--for which player? Professor Ben Polak: For

  • Player 2. What's Player 2's best response

  • for up? Sorry that's unfair,

  • pick on somebody else. That's all right;

  • get the guy right behind you. What's the best response for up?

  • Student: So it's [inaudible]

  • Professor Ben Polak: So, this is a slightly more tricky

  • one. The best response to up is

  • either c or r. This is why we got you to sign

  • those legal forms. Best ones to up is either c or

  • r. The best response to middle?

  • Student: Center. Professor Ben Polak: Is

  • center, thank you. And the best response to down?

  • Student: Left. Professor Ben Polak: So,

  • the best response to down is left.

  • Here again, it didn't take us very long.

  • We're getting quicker at this. The Nash Equilibrium in this

  • case is--let's go back to the guy who was asleep and give him

  • an easier question this time. So the Nash Equilibrium in this

  • case is? Student: Nash

  • Equilibrium would be the two [inaudible]

  • Professor Ben Polak: Exactly, so the middle of

  • center. Notice that this is a Nash

  • Equilibrium. Why is it the Nash Equilibrium?

  • Because each player is playing a best response to each other.

  • If I'm Player 1 and I think that Player 2 is playing center,

  • I should play middle. If I'm Player 2,

  • I think Player 1 is playing middle, I should play center.

  • If, in fact, we play this way neither person

  • has any regrets. Notice they didn't do

  • particularly well in this game. In particular,

  • Player 1 would have liked to get this 11 and this 8,

  • but they don't regret their action because,

  • given that Player 2 played center, the best they could have

  • got was 3, which they got by playing middle.

  • Notice in passing in this game, that best responses needn't be

  • unique. Sometimes they can be a tie in

  • your best response. It could happen.

  • So, what have we seen? We've seen how to find Nash

  • Equilibrium. And clearly it's very closely

  • related to the idea of best response, which is an idea we've

  • been developing over the last week or so.

  • But let's go back earlier in the course, and ask how does it

  • relate to the idea of dominance? This will be our third concept

  • in this course. We had a concept about

  • dominance, we had a concept about best response,

  • and now we're at Nash Equilibrium.

  • It's, I think, obvious how it relates to best

  • response: it's when best responses coincide.

  • How about how it relates to dominance?

  • Well, to do that let's go back. What we're going to do is we're

  • going to relate the Nash Equilibrium to our idea of

  • dominance, or of domination,

  • and to do that an obvious place to start is to go back to a game

  • that we've seen before. So here's a game where Player 1

  • and Player 2 are choosing ά and β and the payoffs are

  • (0,0) (3,-1) (-1,3) and (1,1). So, this is the game we saw the

  • first time. This is the Prisoner's Dilemma.

  • We know in this game--I'm not going to bother to cold call

  • people for it--we know in this game that β

  • is dominatedis strictly dominated by ά.

  • It's something that we learned the very first time.

  • Just to check: so against ά,

  • choosing ά gets you 0, β

  • gets you -1. Against β

  • choosing ά gets you 3, β

  • gets you 1, and in either case ά is strictly better,

  • so ά strictly dominates β, and of course it's the

  • same for the column player since the game's completely symmetric.

  • So now, let's find the Nash Equilibrium in this game.

  • I think we know what it's going to be, but let's just do it in

  • this sort of slow way. So, the best response to ά

  • must be ά. The best response to β

  • must be ά, and for the column player the

  • best response to ά must be ά,

  • and the best response to β must be ά.

  • Everyone okay with that? I'm just kind of--I'm rushing

  • it a bit because it's kind of obvious, is that right?

  • So, the Nash Equilibrium in this game is (ά,

  • ά). In other words,

  • it's what we would have arrived at had we just deleted strictly

  • dominated strategies. So, here's a case where ά

  • strictly dominates β, and sure enough Nash

  • Equilibrium coincides, it gives us the same answer,

  • both people choose ά in this game.

  • So, there's nothing--there's no news here, I'm just checking

  • that we're not getting anything weird going on.

  • So, let's just be a bit more careful.

  • How do we know it's the case--I'm going to claim it's

  • the case, that no strictly dominated strategy--in this case

  • β--no strictly dominated strategy could ever be played in

  • a Nash Equilibrium. I claim--and that's a good

  • thing because we want these ideas to coincide--I claim that

  • no strictly dominated strategy could ever be played in the Nash

  • Equilibrium. Why is that the case?

  • There's a guy up here, yeah. Student: The strictly

  • dominated strategy isn't the best response of anything.

  • Professor Ben Polak: Good.

  • Student: It should be the best response to find the

  • Nash Equilibrium. Professor Ben Polak:

  • Good, very good. So a strictly dominated

  • strategy is never a best response to anything.

  • In particular, the thing that dominates it

  • always does better. So, in particular,

  • it can't be a best response to the thing being played in the

  • Nash Equilibrium. So that's actually a very good

  • proof that proves that no strictly dominated strategy

  • could ever be played in a Nash Equilibrium.

  • But now we have, unfortunately,

  • a little wrinkle, and the wrinkle,

  • unfortunately, is weakly dominated--is weak

  • domination. So we've argued,

  • it was fairly easy to argue, that no strictly dominated

  • strategy is ever going to reappear annoyingly in a Nash

  • Equilibrium. But unfortunately,

  • the same is not true about weakly dominated strategies.

  • And in some sense, you had a foreshadowing of this

  • problem a little bit on the homework assignment,

  • where you saw on one of the problems (I think it was the

  • second problem on the homework assignment) that deleting weakly

  • dominated strategies can lead you to do things that might make

  • you feel a little bit uneasy. So weak domination is really

  • not such a secure notion as strict domination,

  • and we'll see here as a trivial example of that.

  • And again, so here's the trivial example,

  • not an interesting example, but it just makes the point.

  • So here's a 2 x 2 game. Player 1 can choose up or down

  • and Player 2 can choose left or right, and the payoffs are

  • really trivial: (1,1) (0,0) (0,0) (0,0).

  • So, let's figure out what the Nash Equilibrium is in this game

  • and I'm not going to bother cold calling because it's too easy.

  • So the best response for Player 1, if Player 2 plays left is

  • clear to choose up. And the best response of Player

  • 1 if Player 2 chooses right is--well, either up or down will

  • do, because either way he gets 0.

  • So these are both best responses.

  • Is that correct? They're both best responses;

  • they both did equally well. Conversely, Player 2's best

  • response if Player 1 chooses up is, sure enough,

  • to choose left, and that's kind of the answer

  • we'd like to have, but unfortunately,

  • when Player 1 plays down, Player 2's best response is

  • either left or right. It makes no difference.

  • They get 0 either way. So, what are the Nash

  • Equilibria in this game? So, notice the first

  • observation is that there's more than one Nash Equilibrium in

  • this game. We haven't seen that before.

  • There's a Nash Equilibrium with everybody in the room I think,

  • I hope, thinks it's kind of the sensible prediction in this

  • game. In this game,

  • I think all of you, I hope, all of you if you're

  • Player 1 would choose up and all of you for Player 2 would choose

  • left. Is that right?

  • Is that correct? It's hard to imagine coming up

  • with an argument that wouldn't lead you to choose up and left

  • in this game. However, unfortunately,

  • that is--this isn't unfortunate, up left is a Nash

  • Equilibrium but so is down right.

  • If Player 2 is choosing right, your best response weakly is to

  • choose down. If Player 1 is choosing down,

  • Player 2's best response weakly is to choose right.

  • Here, this is arriving because of the definition of Nash;

  • it's a very definite definition. I deleted it now.

  • When we looked at the definition we said,

  • something is a Nash Equilibrium was each person is playing a

  • best response to each other; another way of saying that is

  • no player has a strict incentive to deviate.

  • No player can do strictly better by moving away.

  • So here at down right Player 1 doesn't do strictly better;

  • it's just a tie if she moves away.

  • And Player 2 doesn't do strictly better if he moves

  • away. It's a tie.

  • He gets 0 either way. So, here we have something

  • which is going to worry us going on in the course.

  • Sometimes we're getting--not only are we getting many Nash

  • Equilibria, that's something which--that shouldn't worry us,

  • it's a fact of life. But in this case one of the

  • Nash Equilibria seems silly. If you went and tried to

  • explain to your roommates and said, "I predict both of these

  • outcomes in this game," they'd laugh at you.

  • It's obvious in some sense that this has to be the sensible

  • prediction. So, just a sort of worrying

  • note before we move on. So, this was pretty formal and

  • kind of not very exciting so far, so let's try and move on to

  • something a little bit more fun. So, what I want to do now is I

  • want to look at a different game.

  • Again, we're going to try and find Nash Equilibrium in this

  • game but we're going to do more than that,

  • we're going to talk about the game a bit, and a feature of

  • this game which is--to distinguish it from what we've

  • done so far--is the game we're about to look at involves many

  • players, although each player only has a

  • few strategies. So, what I want to do is I want

  • to convince you how to find--to discuss how to find Nash

  • Equilibria in the game which, unlike these games,

  • doesn't just have two players--it has many

  • players--but fortunately, not many strategies per player.

  • So let me use this board. So this is going to be called

  • The Investment Game and we're going to play this game,

  • although not for real. So, the players in this game,

  • as I've just suggested, the players are going to be

  • you. So, everyone who is getting

  • sleepy just looking at this kind of analysis should wake up now,

  • you have to play. The strategies in this game are

  • very simple, the strategy sets, or the strategy alternatives.

  • Each of you can choose between investing nothing in a class

  • project, $0, or invest $10. So, I'm sometimes going to

  • refer to investing nothing as not investing,

  • is that okay? That seems like a natural to do.

  • You're either going to invest $10 or nothing,

  • you're not going to invest. So that's the players and those

  • are the strategies, so as usual we're missing

  • something. What we're missing are the

  • payoffs. So here are the payoffs;

  • so the payoffs are as follows: if you don't invest,

  • if you do not invest, you invest nothing,

  • then your payoff is $0. So nothing ventured nothing

  • gained: natural thing. But if you do invest $10,

  • remember each of you is going to invest $10 then your

  • individual payoffs are as follows.

  • Here's the good news. You're going to get a profit of

  • $5. The way this is going to work

  • is you're going to invest $10 so you'll make a gross profit of

  • $15 minus the $10 you originally invested for a net of $5.

  • So a net profit--so $5 net profit, that's the good news.

  • But that requires more than 90% of the class to invest,

  • greater than or equal to 90% of the class to invest.

  • If more than 90% of the class invests, you're going to make

  • essentially 50% profit. Unfortunately,

  • the bad news is you're going to lose your $10,

  • get nothing back so this is a net loss, if fewer than 90% of

  • the class invest. I mean a key rule here;

  • you're not allowed to talk to each other: no communication in

  • the class. No hand signals,

  • no secret winks, nothing else.

  • So, everyone understand the game?

  • Including up in the balcony, everyone understand the game?

  • So, what I want you to do--I should say first of all,

  • we can't play this game for real because there's something

  • like 250 of you and I don't have that kind of cash lying around.

  • So we're not--pretend we're playing this for real.

  • So, without communicating I want each of you to write on the

  • corner of your notepad whether you're going to invest or not.

  • You can write Y if you're going to and N if you're not going to

  • invest. Don't discuss it with each

  • other; just write down on the corner

  • of your notepad Y if you're going to invest and N if you're

  • not going to invest. Don't discuss it guys.

  • Now, show your neighbor what you did, just so you can--your

  • neighbor can make you honest. Now, let's have a show of

  • hands, so what I want to do is I want to have a show of hands,

  • everybody who invested. Don't look around;

  • just raise your hands, everyone who invested?

  • Everyone who didn't invest! Oh that's more than 10%.

  • Let's do that again. So everyone who invested raised

  • their handsand everyone who didn't invest

  • raise their hands. So I don't know what that is,

  • maybe that's about half. So now I'm thinking we should

  • have played this game for real. I want to get some discussion

  • going about this. I'm going to discuss this for a

  • while; there's a lot to be said about

  • this game. Let me borrow that,

  • can I borrow this? So this guy;

  • so what did you do? Student: I invested.

  • Professor Ben Polak: Why did you invest?

  • Student: Because I think I can lose $10.

  • Professor Ben Polak: All right, so he's rich,

  • that's okay. You can pay for lunch.

  • Who didn't invest, non-investors?

  • Yeah, here's one, so why didn't you invest?

  • Shout it out so everyone can hear.

  • Student: I didn't invest because to make a profit there

  • needs to be at least a 2:1 chance that everyone else--that

  • 90% invest. Professor Ben Polak: You

  • mean to make an expected profit? Student: Yeah,

  • you only get half back but you would lose the whole thing.

  • Professor Ben Polak: I see, okay so you're doing some

  • kind of expected calculation. Other reasons out there?

  • What did you do? Student: I invested.

  • Professor Ben Polak: I'm finding the suckers in the room.

  • Okay, so you invested and why? Student: You usually

  • stand to gain something. I don't see why anybody would

  • just not invest because they would never gain anything.

  • Professor Ben Polak: Your name is?

  • Student: Clayton. Professor Ben Polak: So

  • Clayton's saying only by investing can I gain something

  • here, not investing seems kind of--it

  • doesn't seem brave in some sense.

  • Anyone else? Student: It's basically

  • the same game as the (1,1) (0,0) game in terms they're both Nash

  • Equilibrium, but the payoffs aren't the same

  • scale and you have to really be risk adverse not to invest,

  • so I thought it would be-- Professor Ben Polak: So

  • you invested? Student: I invested,

  • yeah. Professor Ben Polak: All

  • right, so let me give this to Ale before I screw up the sound

  • system here. Okay, so we got different

  • answers out there. Could people hear each other's

  • answer a bit? Yeah.

  • So, we have lots of different views out here.

  • We have half the people investing and half the people

  • not investing roughly, and I think you can make

  • arguments either way. So we'll come back to whether

  • it's exactly the same as the game we just saw.

  • So the argument that-- I'm sorry your name--that Patrick

  • made is it looks a lot like the game we saw before.

  • We'll see it is related, clearly.

  • It's related in one sense which we'll discuss now.

  • So, what are the Nash Equilibria in this game?

  • Let me get the woman up here? Student: No one invests

  • and everyone's happy they didn't lose anything;

  • or everyone invests and nobody's happy.

  • Professor Ben Polak: Good, I've forgotten your name?

  • Student: Kate. Professor Ben Polak: So

  • Kate's saying that there are two Nash Equilibria and (with

  • apologies to Jude) I'm going to put this on the board.

  • There are two Nash Equilibria here, one is all invest and

  • another one is no one invest. These are both Nash Equilibria

  • in this game. And let's just check that they

  • are exactly the argument that Kate said.

  • That if everyone invests then no one would have any regrets,

  • everyone's best response would be to invest.

  • If nobody invests, then everyone would be happy

  • not to have invested, that would be a best response.

  • It's just--in terms of what Patrick said,

  • it is a game with two equilibria like the previous

  • game and there were other similarities to the previous

  • game, but it's a little bit--the

  • equilibria in this case are not quite the same as the other

  • game. In the other game that other

  • equilibrium, the (0,0) equilibrium seemed like a silly

  • equilibrium. It isn't like we'd ever predict

  • it happening.Whereas here, the equilibrium with nobody

  • investing actually is quite a plausible equilibrium.

  • If I think no one else is investing then I strictly prefer

  • not to invest, is that right?

  • So, two remarks about this. Do you want to do more before I

  • do two remarks? Well, let's have one remark at

  • least. How did we find those Nash

  • Equilibria? What was our sophisticated

  • mathematical technique for finding the Nash Equilibria on

  • this game? We should - can you get the

  • mike on - is it Kate, is that right?

  • So how do you find the Nash Equilibria?

  • Student: [Inaudible] Professor Ben Polak: All

  • right, so in principle you could--I mean that works but in

  • principle, you could have looked at every

  • possible combination and there's lots of possible combinations.

  • It could have been a combination where 1% invested

  • and 99% didn't, and 2% invested and 98% didn't

  • and so on and so forth. We could have checked

  • rigorously what everyone's best response was in each case,

  • but what did we actually end up doing in this game?

  • What's the method? The guy in--yeah.

  • Student: [Inaudible] Professor Ben Polak:

  • That's true, so that certainly makes it easier.

  • I claim, however--I mean you're being--both of you are being

  • more rigorous and more mathematical than I'm tempted to

  • be here. I think the easy method

  • here--the easy sophisticated math here is "to guess."

  • My guess is the easy thing to do is to guess what flight would

  • be the equilibrium here and then what?

  • Then check; so a good method here in these

  • games is to guess and check. And guess and check is really

  • not a bad way to try and find Nash Equilibria.

  • The reason it's not a bad way is because checking is fairly

  • easy. Guessing is hard;

  • you might miss something. There might be a Nash

  • Equilibrium hidden under a stone somewhere.

  • In America it's a "rock." It may hidden under a rock

  • somewhere. But checking with something

  • that you--some putative equilibrium, some candid

  • equilibrium, checking whether it is an

  • equilibrium is easy because all you have to do is check that

  • nobody wants to move, nobody wants to deviate.

  • So again, in practice that's what you're going to end up

  • doing in this game, and it turns out to be very

  • easy to guess and check, which is why you're able to

  • find it. So guess and check is a very

  • useful method in these games where there are lots and lots of

  • players, but not many strategies per

  • player, and it works pretty well.

  • Okay, now we've got this game up on the board,

  • I want to spend a while discussing it because it's kind

  • of important. So, what I want to do now is I

  • want to remind us what happened just now.

  • So, what happened just now? Can we raise the yeses again,

  • the invest again. Raise the not invested,

  • not invest. And I want to remind you guys

  • you all owe me $10. What I want to do is I want to

  • play it again. No communication,

  • write it down again on the corner of your notepad what

  • you're going to do. Don't communicate you guys;

  • show your neighbor. And now we're going to poll

  • again, so ready. Without cheating,

  • without looking around you, if you invested--let Jude get a

  • good view of you--if you invested raise your hand now.

  • If you didn't invest--okay. All right, can I look at the

  • investors again? Raise your hands honestly;

  • we've got a few investors still, so these guys really owe

  • me money now, that's good.

  • Let's do it again, third time, hang on a second.

  • So third time, write it down,

  • and pretend this is real cash. Now, if you invested the third

  • time raise your hand. There's a few suckers born

  • everyday but basically. So, where are we heading here?

  • Where are we heading pretty rapidly?

  • We're heading towards an equilibrium;

  • let's just make sure we confirm that.

  • So everyone who didn't invest that third time raise your

  • hands. That's pretty close;

  • that show of hands is pretty close to a Nash Equilibrium

  • strategy, is that right? So, here's an example of a

  • third reason from what we already mentioned last time,

  • but a third reason why we might be interested in Nash

  • Equilibria. There are certain circumstances

  • in which play converges in the natural sense--not in a formal

  • sense but in a natural sense--to an equilibrium.

  • With the exception of a few dogged people who want to pay

  • for my lunch, almost everyone else was

  • converging to an equilibrium. So play converged fairly

  • rapidly to the Nash Equilibrium. But we discussed there were two

  • Nash Equilibria in this game.; Is one of these Nash

  • Equilibria, ignoring me for a second, is one of these Nash

  • Equilibria better than the other?

  • Yeah, clearly the "everyone investing" Nash Equilibrium is

  • the better one, is that right?

  • Everyone agree? Everyone investing is a better

  • Nash Equilibrium for everyone in the class, than everyone not

  • investing, is that correct? Nevertheless,

  • where we were converging in this case was what?

  • Where we're converging was the bad equilibrium.

  • We were converging rapidly to a very bad equilibrium,

  • an equilibrium which no one gets anything,

  • which all that money is left on the table.

  • So how can that be? How did we converge to this bad

  • equilibrium? To be a bit more formal,

  • the bad equilibrium and no invest equilibrium here,

  • is pareto dominated by the good equilibrium.

  • Everybody is strictly better off at the good equilibrium than

  • the bad equilibrium. It pareto dominates,

  • to use an expression you probably learned in 150 or 115.

  • Nevertheless, we're going to the bad one;

  • we're heading to the bad equilibrium.

  • Why did we end up going to the bad equilibrium rather than the

  • good equilibrium? Can we get the guy in the gray?

  • Student: Well, it was a bit of [inaudible]

  • Professor Ben Polak: Just now you mean?

  • Say what you mean a bit more, yeah that's good.

  • Just say a bit more. Student: So it seemed

  • like people didn't have a lot of confidence that other people

  • were going to invest. Professor Ben Polak: So

  • one way of saying that was when we started out we were roughly

  • even, roughly half/half but that was

  • already bad for the people who invested and then--so we started

  • out at half/half which was below that critical threshold of 90%,

  • is that right? From then on in,

  • we just tumbled down. So one way to say this--one way

  • to think about that is it may matter, in this case,

  • where we started. Suppose the first time we

  • played the game in this class this morning,

  • suppose that 93% of the class had invested.

  • In which case, those 93% would all have made

  • money. Now I'm--my guess is--I can't

  • prove this, my guess is, we might have converged the

  • other way and converged up to the good equilibrium.

  • Does that make sense? So people figured out that

  • they--people who didn't invest the first time--actually they

  • played the best response to the class, so they stayed put.

  • And those of you did invest, a lot of you started not

  • investing as you caught up in this spiral downward and we

  • ended up not investing. But had we started off above

  • the critical threshold, the threshold here is 90%,

  • and had you made money the first time around,

  • then it would have been the non-investors that would have

  • regretted that choice and they might have switched into

  • investing, and we might have--I'm not

  • saying necessarily, but we might have gone the

  • other way. Yeah, can we get a mike on?

  • Student: What if it had been like a 30/70 thing or

  • [inaudible] Professor Ben Polak:

  • Yes, that's a good question. Suppose we had been close to

  • the threshold but below, so I don't know is the answer.

  • We didn't do the experiment but it seems likely that the

  • higher--the closer we got to the threshold the better chance

  • there would have been in going up.

  • My guess is--and this is a guess--my guess is if we started

  • below the threshold we probably have come down,

  • and if was from above the threshold, we would probably

  • have gone up, but that's not a theorem.

  • I'm just speculating on what might have happened;

  • speculating on your speculations.

  • So, here we have a game with two equilibria,

  • one is good, one is bad;

  • one is really quite bad, its pareto dominated.

  • Notice that what happened here, the way we spiraled down

  • coincides with something we've talked about Nash Equilibrium

  • already, it coincides with this idea of

  • a self-fulfilling prediction. Provided you think other people

  • are not going to invest, you're not going to invest.

  • So, it's a self-fulfilling prediction to take you down to

  • not investing. Conversely, provided everyone

  • thinks everyone else is going to invest, then you're going to go

  • up to the good equilibrium. I think that corresponds to

  • what the gentleman said in the middle about a bare market

  • versus a bull market. If it was a bare market,

  • it looked like everyone else didn't have confidence in

  • everyone else investing, and then that was a

  • self-fulfilling prophesy and we ended up with no investment.

  • Now, we've seen bad outcomes in the class before.

  • For example, the very first day we saw a

  • Prisoner's Dilemma. But I claim that though we're

  • getting a bad outcome here in the class, this is not a

  • Prisoner's Dilemma. Why is this not a Prisoner's

  • Dilemma? What's different between--I

  • mean both games have an equilibrium which is bad.

  • Prisoner's Dilemma has the bad equilibrium when nobody tidies

  • their room or both guys go to jail, but I claim this is not a

  • Prisoner's Dilemma. Get the guy behind you.

  • Student: No one suffered but no one gets any payoffs.

  • Professor Ben Polak: Okay, so maybe the outcome isn't

  • quite so bad. That's fair enough,

  • I could have made it worse, I could have made it sort of--I

  • could have made that--I could have lowered the payoffs until

  • they were pretty bad. Why else is this not a

  • Prisoner's Dilemma? The woman who's behind you.

  • Student: Because there's no strictly dominated strategy.

  • Professor Ben Polak: Good;

  • so in Prisoner's Dilemma playing ά

  • was always a best response. What led us to the bad outcome

  • in Prisoner's Dilemma was that ά, the defense strategy,

  • the non-cooperative strategy, the not helping tidy your room

  • strategy, was always the best thing to do.

  • Here, in some sense the "good thing," the "moral thing" in

  • some sense, is to invest but it's not the case that not

  • investing dominates investing. In fact, if all the people

  • invest in the room you ought to invest, is that right?

  • So this is a social problem, but it's not a social problem

  • of the form of a Prisoner's Dilemma.

  • So what kind of problem is this? What kind of social problem is

  • this? The guy in front of you.

  • Student: Perhaps it's one of cooperation.

  • Professor Ben Polak: It's one of--okay.

  • So it's--it would help if people cooperated here but I'm

  • looking for a different term. The term that came up the first

  • day--coordination, this is a coordination game.

  • For those people that read the New Yorker,

  • I'll put an umlaut over the second "o."

  • Why is it a coordination game? Because you'd like everyone in

  • the class to coordinate their responses on invest.

  • In fact, if they did that, they all would be happy and no

  • one would have an incentive to defect and there would in fact

  • be an equilibrium. But unfortunately,

  • quite often, we fail to have coordination.

  • Either everyone plays non-invest or,

  • as happened the first time in class, we get some split with

  • people losing money. So, I claim that actually this

  • is not a rare thing in society at all.

  • There are lots of coordination problems in society.

  • There are lots of things that look like coordination games.

  • And often, in coordination games bad outcomes result and I

  • want to spend most of the rest of today talking about that,

  • because I think it's important, whether you're an economist or

  • whatever, so let's talk about it a bit.

  • What else has the structure of a coordination game,

  • and therefore can have the outcome that people can be

  • uncoordinated or can coordinate in the wrong place,

  • and you end up with a bad equilibrium?

  • What else looks like that? Let's collect some ideas.

  • I got a hand way at the back. Can you get the guy who is just

  • way, way, way at the back, right up against the--yes,

  • there you go, great thank you.

  • Wait until the mike gets to you and then yell.

  • Student: A party on campus is a coordination game.

  • Professor Ben Polak: Yeah, good okay.

  • So a party on campus is a coordination game because

  • what--because you have to coordinate being at the same

  • place, is that right? That's the idea you had?

  • Go ahead. Student: [inaudible]

  • Professor Ben Polak: Good, okay that's good.

  • So that's another in which--there's two ways in which

  • a party can be a coordination problem.

  • One is the problem that if people don't show up it's a

  • lousy party, so you don't want to show up.

  • Conversely, if everyone is showing up, it's great,

  • it's a lot of fun, it's the place to be and

  • everyone wants to show up. Second--so there's two

  • equilibria there, showing up and--everyone not

  • showing up or everyone showing up.

  • Similar idea, the location of parties which

  • is what I thought you were driving at, but this similar

  • idea can occur. So it used to the case in New

  • Haven that there were different--actually there aren't

  • many anymore--but there used to be different bars around campus

  • (none of which you were allowed to go to,

  • so you don't know about) but anyway, lots of different bars

  • around campus. And there's a coordination game

  • where people coordinate on Friday night--or to be more

  • honest, the graduate students typically Thursday night.

  • So it used to be the case that one of those bars downtown was

  • where the drama school people coordinated,

  • and another one was where the economists coordinated,

  • and it was really good equilibrium that they didn't

  • coordinate at the same place. So one of the things you have

  • to learn when you go to a new town is where is the meeting

  • point for the kind of party I want to go to.

  • Again, you're going to have a failure of coordination,

  • everyone's wandering around the town getting mugged.

  • What other things look like coordination problems like that?

  • Again, way back in the corner there, right behind you,

  • there you go. Student: Maybe warring

  • parties in the Civil War signing a treaty.

  • Professor Ben Polak: Okay, that could be a--that

  • could be a coordination problem. It has a feeling of being a bit

  • Prisoner's Dilemmerish and in some sense, it's my disarming,

  • my putting down my arms before you putting down your arms,

  • that kind of thing. So it could be a coordination

  • problem, but it has a little bit of a flavor of both.

  • Go ahead, go on. Okay, other examples?

  • There's a guy there. While you're there why don't

  • you get the guy who is behind you right there by the door,

  • great. Student: Big sports

  • arena, people deciding whether or not they want to start a

  • chant or [inaudible] Professor Ben Polak:

  • Yeah, okay. I'm not quite sure which is the

  • good outcome or the bad outcome. So there's this thing called,

  • The Wave. There's this thing called The

  • Wave that Americans insist on doing, and I guess they think

  • that's the good outcome. Other things?

  • Student: Battle of the sexes.

  • Professor Ben Polak: Let's come back to that one.

  • Hold that thought and we'll come back to it next time.

  • You're right that's a good--but let's come back to it next time.

  • Anything else? Let me try and expand on some

  • of these meeting place ideas. We talked about pubs to meet

  • at, or parties to meet at, but think about online sites.

  • Online sites which are chat sites or dating sites,

  • or whatever. Those are--those online sites

  • clearly have the same feature of a party.

  • You want people to coordinate on the same site.

  • What about some other economic ideas?

  • What about some other ideas from economics?

  • What else has that kind of externality like a meeting

  • place? Student: I mean it's

  • excluding some people, but things like newspapers or

  • something, like we might want only one like global news site.

  • Professor Ben Polak: So, that's an interesting thought,

  • because it can go both ways. It could be a bad thing to have

  • one newspaper for obvious reasons, but in terms of having

  • a good conversation about what was in the newspaper yesterday

  • around the--over lunch, it helps that everyone's read

  • the same thing. Certainly, there's a bandwagon

  • effect to this with TV shows. If everybody in America watches

  • the same TV show, which apparently is American

  • Idol these days, then you can also talk about it

  • over lunch. Notice that's a really horrible

  • place to coordinate. There are similar examples to

  • the American Idol example. When I was growing up in

  • England, again I'm going to reveal my age;

  • everybody decided that to be an "in" person in England,

  • you had to wear flared trousers.

  • This was--and so to be in you had to wear flared trousers.

  • This is a horrible coordination problem, right?

  • So for people who don't believe that could ever have happened in

  • England that you could ever these sort of fashion goods that

  • you end up at a horrible equilibrium at the wrong place,

  • if you don't believe that could ever happen think about the

  • entirety of the Midwest. I didn't say that,

  • we're going to cut that out of the film later.

  • What else is like this? I'm going to get a few ideas

  • now, this gentleman here. Student: The

  • establishment of monopolies, because a lot of people use

  • Microsoft, say then everything is

  • compatible with Microsoft so more people would use it.

  • Professor Ben Polak: Good, I've forgotten your name.

  • Student: Steven. Professor Ben Polak: So,

  • Steven's pointing out that certain software can act this

  • way by being a network good. The more people who use

  • Microsoft and Microsoft programs, the bigger the

  • advantage to me using Microsoft, and therefore--because I can

  • exchange programs, I can exchange files if I'm

  • working with my co-authors and so on,

  • and so you can have different equilibria coordinating on

  • different software, and again, you could end up at

  • the wrong one. I think a lot of people would

  • argue--but I'm going to stay neutral on this--a lot of people

  • would argue that Microsoft wasn't necessarily the best

  • place for the world to end up. There are other technological

  • network goods like this. These are called network

  • externalities. An example here would be high

  • definition television. You want to have one

  • technological standard that everyone agrees on for things

  • like high definition televisions because then everyone can

  • produce TVs to that standard and goods that go along with that

  • standard, and of course it--each company

  • who's producing a TV and has a standard line would like theirs

  • to be chosen as the standard. Again, you could end up at the

  • wrong place. You could end up with a bad

  • equilibrium. How about political bandwagons?

  • In politics, particularly in primaries,

  • there may be advantage on the Democratic side or on the

  • Republican side, in having you all vote for the

  • same candidate in the primary, so they get this big boost and

  • it doesn't seem like your party's split and so on.

  • And that could end up--and again, I'm going to remain

  • neutral on this--it just could end up with the wrong candidate

  • winning. There's a political bandwagon

  • effect, the person who wins New Hampshire and Iowa tends then to

  • win everything, so that's another example.

  • Any other economic examples? Can we get this guy in here?

  • Student: Stock exchange. Professor Ben Polak:

  • Yeah, okay, so in particular which stock exchange to list on.

  • So there's a huge advantage having lots of stocks in the

  • same stock exchange. There are shared fixed costs;

  • there's also liquidity issues and lots of issues of that form

  • but mostly the form of fixed costs.

  • So there's a tendency to end up with one stock exchange.

  • We're not there yet but we do seem to be going that way quite

  • rapidly. The problem of course being,

  • that might not be the best stock exchange or it might give

  • that stock exchange monopoly power.

  • Let me give one--let me try one more example.

  • What about bank runs? What's a bank run?

  • Somebody--what's a bank run? Student: It's when the

  • public loses confidence in--because their security of

  • their money in banks, then they rush to withdraw

  • their deposits. Professor Ben Polak:

  • Good. So you can imagine a bank as a

  • natural case where there's two equilibria.

  • There's a good equilibrium, everyone has confidence in the

  • bank, everyone leaves their deposits in the bank.

  • The bank is then able to lend some of that money out on a

  • higher rate of interest on it. The bank doesn't want to keep

  • all that money locked up in the vault.

  • It wants to lend it out to lenders who can pay interest.

  • That's a good equilibrium for everybody.

  • However, if people lose confidence in the bank and start

  • drawing their deposits out then the bank hasn't got enough cash

  • in its vaults to cover those deposits and the bank goes

  • under. Now, I used to say at this

  • point in the class, none of you will have ever seen

  • a bank run because they stopped happening in America more or

  • less in the mid 30s. There were lots and lots of

  • bank runs in America before the 1930s, but since federal deposit

  • insurance came in, there's far fewer.

  • However, I really can't say that today because there's a

  • bank run going on right now. There's as bank run going on

  • actually in England with a company called Northern

  • Security--no, Northern Rock--it's called

  • Northern Rock, as we speak,

  • and it really is a bank run. I mean, if you looked at the

  • newspaper yesterday on The New York Times,

  • you'll see a line of depositors lined up, outside the offices in

  • London of this bank, trying to get their deposits

  • out. And you see the Bank of England

  • trying to intervene to restore confidence.

  • Just be clear, this isn't simple--this isn't a

  • simple case of being about the mortgage crisis.

  • This bank does do mortgages but it doesn't seem to be

  • particularly involved in the kind of mortgages that have been

  • attracting all the publicity. It really seems to be a shift

  • in confidence. A move from the good

  • equilibrium of everyone remaining invested,

  • to the bad equilibrium of everyone taking their money out.

  • Now, there are famous bank runs in American culture,

  • in the movies anyway. What is the famous bank run in

  • the movies, in American movies? Student: It's a

  • Wonderful Life. Professor Ben Polak:

  • It's a Wonderful Life; there's actually--there's one

  • in Mary Poppins as well, but we'll do It's a

  • Wonderful Life. How many of you have seen

  • It's A Wonderful Life? How many have you not seen

  • It's a Wonderful Life? Let me get a poll here.

  • How many people have not seen--keep your hands up a

  • second, keep your hands up. You need to know that if you're

  • on a green card, you can lose your green card if

  • you haven't sent It's a Wonderful Life.

  • So, in It's a Wonderful Life there's a run on the

  • bank--actually it's a savings and loan, but never mind.

  • We'll think of it as a bank. Luckily, it doesn't end up with

  • the savings or loan, or bank, going under.

  • Why doesn't the bank go under in It's a Wonderful Life?

  • Why doesn't it go under? Yeah, the guy in green?

  • Can we get the guy in green? Student: Everyone agrees

  • to only take as much as they need [inaudible]

  • Professor Ben Polak: For the people who didn't hear it,

  • everyone agrees only to take out a small amount,

  • perhaps even nothing and therefore the bank run ends.

  • Everyone realizes the bank isn't going to collapse,

  • and they're happy to leave their money in the bank.

  • Now, it's true everyone agrees but what do they agree?

  • What makes them agree? What makes them agree is Jimmy

  • Stewart, right? Everyone remember the movie?

  • So Jimmy Stewart gets up there and he says--he gives a speech,

  • and he says, look,

  • more or less,--I mean he doesn't say it in these words

  • but he would have done it if he had taken the class--he says,

  • look there are two equilibria in this game.

  • (I can't do the--whatever it is, the West Pennsylvania

  • accent, is that what it is, or whatever it is.) But anyway;

  • there are two equilibria in this game;

  • there's a bad one where we all draw our money out and we all

  • lose our homes eventually, and there's a good one where we

  • all leave our money in and that's better for everyone.,

  • So let's all leave our money in. But he gives this--he's a bit

  • more motivationally stimulating than me--but he leads people

  • leaving their money in. So, what I want to do is,

  • I want to pick on somebody in the class now--everyone

  • understands this game, everyone understands there's

  • two equilibria, everyone understands that one

  • equilibrium is better. Let's play the game again.

  • Let's choose the game again, but before I do I'm going to

  • give the mike to Patrick here and Patrick is going to have

  • exactly five seconds to persuade you.

  • Stand up. Patrick's going to have five

  • seconds to persuade you to tell you whatever he likes starting

  • now. Student: Okay,

  • so clearly if we all invest we're always better off,

  • so everybody should invest. Professor Ben Polak: All

  • right, now let's see if this is going work.

  • Okay, so let's see what happens now.

  • Everybody who is going to invest raise their hands and

  • everyone who's not investing raise their hands.

  • Oh we almost made it. We must have almost made it.

  • So notice what happened here. Give another round of applause

  • for Patrick. I think he did a good job there.

  • But there's a lesson here; the lesson here is the game

  • didn't change. It's the same game that we've

  • played three times already. This was the fourth time we've

  • played it. We had a totally different

  • response in playing this time. Almost all of you,

  • the vast majority of you, perhaps even 90% of you

  • invested this time and you did so in response to Patrick.

  • But Patrick wasn't putting any money down, he wasn't bribing

  • you, he wasn't writing a contract with you,

  • he wasn't threatening to break your legs, he just was pointing

  • out it's a good idea. Now, remember the Prisoner's

  • Dilemma, in the Prisoner's Dilemma, if Patrick--Patrick

  • could have got up in the Prisoner's Dilemma and given the

  • same speech and said look guys, we're all better off if we

  • choose β in the Prisoner's Dilemma than

  • if we choose ά; roughly the same speech.

  • What will you have done in the Prisoner's Dilemma?

  • You would have all chosen ά anyway.

  • So Patrick tried to persuade you, or Patrick communicating to

  • you that you do better by choosing β

  • in the Prisoner's Dilemma doesn't work but here--don't go

  • yet. Here it does work.

  • Why does Patrick--why is Patrick persuasive in this game

  • but he isn't persuasive in the Prisoner's Dilemma?

  • Can we get the mike on Katie again?

  • Why is Patrick persuasive in this game and not before?

  • Student: He's not trying to get you to play a strictly

  • dominated strategy. Professor Ben Polak:

  • He's not trying to get you play a strictly dominated strategy

  • and more than that, he's trying to persuade you to

  • play what? To play a Nash Equilibrium.

  • So, there's a lesson here, in coordination problems,

  • unlike Prisoner's Dilemma, communication--just

  • communication, no contracts--communication can

  • help. And in particular,

  • what we can persuade each other to do is to play the other Nash

  • Equilibrium. Now, this gives us one more

  • motivation for a Nash Equilibrium.

  • In a Prisoner's Dilemma, to get out of it we needed a

  • contract, we needed side payments, we needed to change

  • the payoffs of the game. But a Nash Equilibrium can be a

  • self-enforcing agreement. We can agree that we're going

  • to play invest in this game, and indeed we will play invest

  • without any side payments, without anybody threatening to

  • break your leg, without any contracts,

  • without any regulation or the law.

  • I'm assuming Patrick isn't that violent.

  • We're going to end up doing the right thing here because it's in

  • our own interest to do so. So coordination problems which

  • we've agreed are all over society, whether it comes to

  • bank runs or bubbles in the market,

  • or fashion in the Midwest, they're all over society.

  • Communication can make a difference and we'll pick that

  • theme up on Monday.

Professor Ben Polak: Okay, so last time we came

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