Professor Ben Polak:Okay, so last time we looked at and played this game.

You had to choose grades, so you had to choose Alpha andBeta, and this table told us what outcome would arise.

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In particular, what grade you would get andwhat grade your pair would get.

So, for example, if you had chosen Beta and your pair had chosen Alpha, then you would get a C and your pair would get an A.

One of the first things we pointed out, is that this is notquite a game yet.

It's missing something.

This has outcomes in it, it's an outcome matrix, but it isn't a game, because for a game we need toknow payoffs.

Then we looked at some possiblepayoffs, and now it is a game.

So this is a game, just to give you some more jargon, this is a normal-formgame.

And here we've assumed thepayoffs are those that arise if players only care about theirown grades, which I think was true for a lot of you.

It wasn't true for the gentleman who's sitting therenow, but it was true for a lot of people.

We pointed out, that in this game, Alpha strictly dominates Beta.

What do we mean by that?We mean that if these are your payoffs, no matter what yourpair does, you attain a higher payoff from choosing Alpha, than you do from choosing Beta.

Let's focus on a couple oflessons of the class before I come back to this.

One lesson was, do not play a strictlydominated strategy.

Everybody remember that lesson?Then much later on, when we looked at some morecomplicated payoffs and a more complicated game, we looked at a different lesson which was this:put yourself in others' shoes to try and figure out whatthey're going to do.

So in fact, what we learnedfrom that is, it doesn't just matter whatyour payoffs are — that's obviously important — it's alsoimportant what other people's payoffs are, because you want to try and figure out what they're going todo and then respond appropriately.

So we're going to return to both of these lessons today.

Both of these lessons will reoccur today.

Now, a lot of today is going to be fairly abstract, so I just want to remind you that Game Theory has some realworld relevance.

Again, still in the interest ofrecapping, this particular game is called the Prisoners'Dilemma.

It's written there, the Prisoners' Dilemma.

Notice, it's Prisoners, plural.

And we mentioned some examples last time.

Let me just reiterate and mention some more examples whichare actually written here, so they'll find their way intoyour notes.

So, for example, if you have a joint project that you're working on, perhaps it's a homework assignment, or perhaps it's a video project like these guys, that can turn into a Prisoners' Dilemma.

Why? Because each individual mighthave an incentive to shirk.

Price competition — two firmscompeting with one another in prices — can have a Prisoners'Dilemma aspect about it.

Why?Because no matter how the other firm, your competitor, prices you might have an incentive to undercut them.

If both firms behave that way, prices will get driven downtowards marginal cost and industry profits will suffer.

In the first case, if everyone shirks you end upwith a bad product.

In the second case, if both firms undercut each other, you end up with lowprices, that's actually good for consumers but bad for firms.

Let me mention a third example.

Suppose there's a commonresource out there, maybe it's a fish stock ormaybe it's the atmosphere.

There's a Prisoners' Dilemmaaspect to this too.

You might have an incentive toover fish.

Why?Because if the other countries with this fish stock–let's saythe fish stock is the Atlantic–if the other countriesare going to fish as normal, you may as well fish as normaltoo.

And if the other countriesaren't going to cut down on their fishing, then you want to catch the fish now, because there aren't going to be any there tomorrow.

Another example of this would be global warming and carbonemissions.

Again, leaving aside thescience, about which I'm sure some of you know more than mehere, the issue of carbon emissions is a Prisoners'Dilemma.

Each of us individually has anincentive to emit carbons as usual.

If everyone else is cutting down I don't have too, and if everyone else does cut down I don't have to, I end up using hot water and driving a big car and so on.

In each of these cases we end up with a bad outcome, so this is socially important.

This is not just some abstractthing going on in a class in Yale.

We need to think about solutions to this, right from the start of the class, and we already talkedabout something.

We pointed out, that this is not just a failure of communication.

Communication per se will not get you out of a Prisoners'Dilemma.

You can talk about it as muchas you like, but as long as you're going to go home andstill drive your Hummer and have sixteen hot showers a day, we're still going to have high carbon emissions.

You can talk about working hard on your joint problem sets, but as long as you go home and you don't work hard, it doesn't help.

In fact, if the other person isworking hard, or is cutting back on theircarbon emissions, you have every bit moreincentive to not work hard or to keep high carbon emissionsyourself.

So we need something more andthe kind of things we can see more: we can think aboutcontracts; we can think about treatiesbetween countries; we can think about regulation.

All of these things work by changing the payoffs.

Not just talking about it, but actually changing theoutcomes actually and changing the payoffs, changing theincentives.

Another thing we can do, a very important thing, is we can think about changingthe game into a game of repeated interaction and seeing how muchthat helps, and we'll come back and revisitthat later in the class.

One last thing we can think ofdoing but we have to be a bit careful here, is we can think about changing the payoffs by education.

I think of that as the “Maoist” strategy.

Lock people up in classrooms and tell them they should bebetter people.

That may or may not work — I'mnot optimistic — but at least it's the same idea.

We're changing payoffs.

So that's enough for recap andI want to move on now.

And in particular, we left you hanging at the end last time.

We played a game at the very end last time, where each of you chose a number — all of you chose anumber — and we said the winner was going to be the person whogets closest to two-thirds of the average in the class.

Now we've figured that out, we figured out who the winneris, and I know that all of you have been trying to see if youwon, is that right? I'm going to leave you insuspense.

I am going to tell you todaywho won.

We did figure it out, and we'll get there, but I want to do a little bitof work first.

So we're just going to leave itin suspense.

That'll stop you walking outearly if you want to win the prize.

So there's going to be lots of times in this class when we getto play games, we get to have classroomdiscussions and so on, but there's going to be sometimes when we have to slow down and do some work, and the next twenty minutes are going to be that.

So with apologies for being a bit more boring for twentyminutes, let's do something we'll call formal stuff.

In particular, I want to develop and make surewe all understand, what are the ingredients of agame? So in particular, we need to figure out what formally makes something into agame.

The formal parts of a game arethis.

We need players — and whilewe're here let's develop some notation.

So the standard notation for players, I'm going to use thingslike little i and little j.

So in that numbers game, the game when all of you wrote down a number and handed it inat the end of last time, the players were who?The players were you.

You'all were the players.

Useful text and expression meaning you plural.

In the numbers game, you'all, were the players.

Second ingredient of the game are strategies.

(There's a good clue here.

If I'm writing you should bewriting.

) Notation: so I'm going to use little”s_i” to be a particular strategy of Player i.

So an example in that game might have been choosing thenumber 13.

Everyone understand that?Now I need to distinguish this from the set of possiblestrategies of Player I, so I'm going to use capital”S_i” to be what? To be the set of alternatives.

The set of possible strategies of Player i.

So in that game we played at the end last time, what were the set of strategies?They were the sets 1, 2, 3, all the way up to 100.

When distinguishing a particular strategy from the setof possible strategies.

While we're here, our third notation for strategy, I'm going to uselittle “s” without an “i, ” (no subscripts):little “s” without an “i, ” to mean a particular play of thegame.

So what do I mean by that?All of you, at the end last time, wrote down this number andhanded them in so we had one number, one strategy choice foreach person in the class.

So here they are, here's my collected in, sort of strategy choices.

Here's the bundle of bits of paper you handed in last time.

This is a particular play of the game.

I've got each person's name and I've got a number from eachperson: a strategy from each person.

We actually have it on a spreadsheet as well:so here it is written out on a spreadsheet.

Each of your names is on this spreadsheet and the number youchose.

So that's a particular play ofthe game and that has a different name.

We sometimes call this “a strategy profile.

“So in the textbook, you'll sometimes see the term astrategy profile or a strategy vector, or a strategy list.

It doesn't really matter.

What it's saying is onestrategy for each player in the game.

So in the numbers game this is the spreadsheet — or an exampleof this is the spreadsheet.

(I need to make it so you canstill see that, so I'm going to pull down theseboards.

And let me clean something.

) Soyou might think we're done right?We've got players.

We've got the choices theycould make: that's their strategy sets.

We've got those individual strategies.

And we've got the choices they actually did make:that's the strategy profile.

Seems like we've got everythingyou could possibly want to describe in a game.

What are we missing here? Shout it out.

“Payoffs.

” We're missing payoffs.

So, to complete the game, we need payoffs.

Again, I need notation for payoffs.

So in this course, I'll try and use “U” for utile, to be Player i's payoff.

So “U_i” will dependon Player 1's choice … all the way to Player i's ownchoice … all the way up to Player N'schoices.

So Player i's payoff”U_i, ” depends on all the choices in the class, in this case, including her own choice.

Of course, a shorter way of writing that would be”U_i(s), ” it depends on the profile.

So in the numbers game what is this?In the numbers game “U_i(s)” can be twothings.

It can be 5 dollars minus yourerror in pennies, if you won.

I guess it could be something if there was a tie, I won't bother writing that now.

And it's going to be 0 otherwise.

So we've now got all of the ingredients of the game:players, strategies, payoffs.

Now we're going to make an assumption today and for thenext ten weeks or so; so for almost all the class.

We're going to assume that these are known.

We're going to assume that everybody knows the possiblestrategies everyone else could choose and everyone knowseveryone else's payoffs.

Now that's not a very realisticassumption and we are going to come back and challenge it atthe end of semester, but this will be complicatedenough to give us a lot of material in the next ten weeks.

I need one more piece of notation and then we can getback to having some fun.

So one more piece of notation, I'm going to write “s_-i” to mean what?It's going to mean a strategy choice for everybody exceptperson “i.

” It's going to be useful to havethat notation around.

So this is a choice for allexcept person “i” or Player i.

So, in particular, if you're person 1 and then “s_-i” would be”s_2, s_3, s_4″ up to “s_n” but it wouldn'tinclude “s_1.

” It's useful why?Because sometimes it's useful to think about the payoffs, as coming from “i's” own choice and everyone else's choices.

It's just a useful way of thinking about things.

Now this is when I want to stop for a second and I know thatsome of you, from past experience, are somewhat mathphobic.

You do not have to wave yourhands in the air if you're math phobic, but since some of youare, let me just get you all to take a deep breath.

This goes for people who are math phobic at home too.

So everyone's in a slight panic now.

You came here today.

You thought everything wasgoing to fine.

And now I'm putting math on theboard.

Take a deep breath.

It's not that hard, and in particular, notice that all I'm doing here is writing down notation.

There's actually no math going on here at all.

I'm just developing notation.

I don't want anybody to quitthis class because they're worried about math or mathnotation.

So if you are in that categoryof somebody who might quit it because of that, come and talk to me, come and talk to the TAs.

We will get you through it.

It's fine to be math phobic.

I'm phobic of all sorts of things.

Not necessarily math, but all sorts of things.

So a serious thing, a lot of people get put off bynotation, it looks scarier than it is, there's nothing going on here except for notation at thispoint.

So let's have an example tohelp us fix some ideas.

(And again, I'll have to cleanthe board, so give me a second.

) I think an example might helpthose people who are disturbed by the notation.

So here's a game which we're going to discuss briefly.

It involves two players and we'll call the Players I and IIand Player I has two choices, top and bottom, and Player II has three choices left, center, and right.

It's just a very simpleabstract example for now.

And let's suppose the payoffsare like this.

They're not particularlyinteresting.

We're just going to do it forthe purpose of illustration.

So here are the payoffs:(5, -1), (11, 3), (0, 0), (6, 4), (0, 2), (2, 0).

Let's just map the notation wejust developed into this game.

So first of all, who are the players here? Well there's no secret there, the players are — let's just write it down why don't we.

The players here in this game are Player I and Player II.

What about the strategy sets or the strategy alternatives?So here Player I's strategy set, she has two choices top orbottom, represented by the rows, which are hopefully the top rowand the bottom row.

Player II has three choices, this game is not symmetric, so they have different numberof choices, that's fine.

Player II has three choicesleft, center, and right, represented by theleft, center, and right column in the matrix.

Just to point out in passing, up to now, we've been lookingmostly at symmetric games.

Notice this game is notsymmetric in the payoffs or in the strategies.

There's no particular reason why games have to be symmetric.

Payoffs: again, this is not rocket science, but let's do it anyway.

So just an example of payoffs.

So Player I's payoff, if she chooses top and PlayerII chooses center, we read by looking at the toprow and the center column, and Player I's payoff is thefirst of these payoffs, so it's 11.

Player II's payoff, from the same choices, top for Player I, center for Player II, again we go along the top row and the center column, but this time we choose Player II's payoff, which is the second payoff, so it's 3.

So again, I'm hoping this is calming down the math phobics inthe room.

Now how do we think this gameis going to be played? It's not a particularlyinteresting game, but while we're here, why don't we just discuss it for a second.

If our mike guys get a little bit ready here.

So how do we think this game should be played?Well let's ask somebody at random perhaps.

Ale, do you want to ask this guy in the blue shirt here, does Player I have a dominated strategy?Student: No, Player I doesn't have adominated strategy.

For instance, if Player II picks left then Player I wants to pick bottom, but if Player II picks center, Player I wants to pick center.

Professor Ben Polak: Good.

Excellent.

Very good.

I should have had you stand up.

I forgot that.

Never mind.

But that was very clear, thank you.

Was that loud enough so peoplecould hear it? Did people hear that?People in the back, did you hear it?So even that wasn't loud enough okay, so we we really need toget people–That was very clear, very nice, but we need peopleto stand up and shout, or these people at the backcan't hear.

So your name is?Student: Patrick.

Professor Ben Polak:What Patrick said was: no, Player I does not have adominated strategy.

Top is better than bottomagainst left — sorry, bottom is better than topagainst left because 6 is bigger than 5, but top is better than bottom against center because 11 isbigger than 0.

Everyone see that?So it's not the case that top always beats–it's not the casethat top always does better than bottom, or that bottom alwaysdoes better than top.

What about, raise hands thistime, what about Player II? Does Player II have a dominatedstrategy? Everyone's keeping their handsfirmly down so as not to get spotted here.

Ale, can we try this guy in white?Do you want to stand up and wait until Ale gets there, and really yell it out now.

Student: I believe rightis a dominated strategy because if Player I chooses top, then Player II will choose center, and if– I'm gettingconfused now, it looks better on my paper.

But yeah, right is never the best choice.

Professor Ben Polak: Okay, good.

Let's be a little bit careful here.

So your name is? Student: Thomas.

Professor Ben Polak: So Thomas said something which wastrue, but it doesn't quite match with the definition of adominated strategy.

What Thomas said was, right is never a best choice, that's true.

But to be a dominated strategy we need something else.

We need that there's another strategy of Player II thatalways does better.

That turns out also to be truein this case, but let's just see.

So in this particular game, I claim that center dominatesright.

So let's just see that.

If Player I chose top, center yields 3, right yields 0: 3 is bigger than 0.

And if Player I chooses bottom, then center yields 2, right yields 0: 2 is bigger than 0 again.

So in this game, center strictly dominatesright.

What you said was true, but I wanted something specifically about dominationhere.

So what we know here, we know that Player II should not choose right.

Now, in fact, that's as far as we can getwith dominance arguments in this particular game, but nevertheless, let's just stick with it asecond.

I gave you the definition ofstrict dominance last time and it's also in the handout.

By the way, the handout on the web.

But let me write that definition again, using or making use of the notation from the class.

So definition so: Player i's strategy”s'_i” is strictly dominated by Player i's strategy”s_i, ” and now we can use ournotation, if “U_I” from choosing “s_i, “when other people choose “s_-i, ” is strictlybigger than U_I(s'_i)when other people choose “s_-i, “and the key part of the definition is, for all “s_-i.

” So to say it in words, Player i's strategy “s'_i” is strictlydominated by her strategy “s_i, “if “s_i” always does strictly better — always yieldsa higher payoff for Player i — no matter what the other peopledo.

So this is the same definitionwe saw last time, just being a little bit morenerdy and putting in some notation.

People panicking about that, people look like deer in theheadlamps yet? No, you look all right:all rightish.

Let's have a look at anotherexample.

People okay, I can move this?All right, so it's a slightly more exciting example now.

So imagine the following example, an invader is thinkingabout invading a country, and there are two ways — thereare two passes if you like — through which he can lead hisarmy.

You are the defender of thiscountry and you have to decide which of these passes or whichof these routes into the country, you're going to choose to defend.

And the catch is, you can only defend one ofthese two routes.

If you want a real worldexample of this, think about the third CenturyB.

C.

, someone can correct me afterwards.

I think it's the third Century B.

C.

when Hannibal is thinking of crossing the Alps.

Not Hannibal Lecter: Hannibal the general in thethird Century B.

C.

.

The one with the elephants.

Okay, so the key here is going to be that there are two passes.

One of these passes is a hard pass.

It goes over the Alps.

And the other one is an easypass.

It goes along the coast.

If the invader chooses the hard pass he will lose one battalionof his army simply in getting over the mountains, simply in going through the hard pass.

If he meets your army, whichever pass he chooses, if he meets your army defending a pass, then he'll lose anotherbattalion.

I haven't given you–I've givenyou roughly the choices, the choice they're going to befor the attacker which pass to choose, and for the defender which pass to defend.

But let's put down some payoffs so we can start talking aboutthis.

So in this game, the payoffs for this game are going to be as follows.

It's a simple two by two game.

This is going to be theattacker, this is Hannibal, and this is going to be thedefender, (and I've forgotten whichgeneral was defending and someone's about to tell methat).

And there are two passes youcould defend: the easy pass or the hard pass.

And there's two you could use to attack through, easy or hard.

(Again, easy pass here justmeans no mountains, we're not talking aboutsomething on the New Jersey Turnpike.

) So the payoffs hereare as follows, and I'll explain them in asecond.

So his payoff, the attacker's payoff, is how many battalions does heget to bring into your country? He only has two to start withand for you, it's how many battalions of his get destroyed?So just to give an example, if he goes through the hardpass and you defend the hard pass, he loses one of those battalions going over themountains and the other one because he meets you.

So he has none left and you've managed to destroy two of them.

Conversely, if he goes on the hard pass and you defend theeasy pass, he's going to lose one of those battalions.

He'll have one left.

He lost it in the mountains.

But that's the only one he's going to lose because you weredefending the wrong pass.

Everyone understand the payoffsof this game? So now imagine yourself as aRoman general.

This is going to be a littlebit of a stretch for imagination, but imaginationyourself as a Roman general, and let's figure out whatyou're going to do.

You're the defender.

What are you going to do? So let's have a show of hands.

How many of you think you should defend the easy pass?Raise your hands, let's raise your hands so Judecan see them.

Keep them up.

Wave them in the air with a bit of motion.

Wave them in the air.

We should get you flags okay, because these are the Romans defending the easy pass.

And how many of you think you're going to defend the hardpass? We have a huge number of peoplewho don't want to be Roman generals here.

Let's try it again, no abstentions, right? I'm not going to penalize youfor giving the wrong answer.

So how many of you think you'regoing to defend the easy pass? Raise your hands again.

And how many think you're going to defend the hard pass?So we have a majority choosing easy pass — had a largemajority.

So what's going on here?Is it the case that defending the easy pass dominatesdefending the hard pass? Is that the case?Is it the case that defending the easy pass dominatesdefending the hard pass? You can shout out.

No, it's not.

In fact, we could check that ifthe attacker attacks through the easy pass, not surprisingly, you do better if you defend the easy pass than the hard pass:1 versus 0.

But if the attacker was toattack through the hard pass, again not surprisingly, you do better if you defend the hard pass than the easy pass.

So that's not an unintuitive finding.

It isn't the case that defending easy dominatesdefending hard.

You just want to match with theattacker.

Nevertheless, almost all of you chose easy.

What's going on?Can someone tell me what's going on?Let's get the mikes going a second.

So can we catch the guy with the–can we catch this guy withthe beard? Just wait for the mike to getthere.

If you could stand up:stand up and shout.

There you go.

Student: Because you want to minimize the amount ofenemy soldiers that reach Rome or whatever location it is.

Professor Ben Polak: You want to minimize the number ofsoldiers that reach Rome, that's true.

On the other hand, we've just argued that youdon't have a dominant strategy here;it's not the case that easy dominates hard.

What else could be going on? While we've got you up, why don't we get the other guy who's got his hand up there inthe middle.

Again, stand up and shout inthat mike.

Point your face towards themike.

Good.

Student: It seems as though while you don't have adominating strategy, it seems like Hannibal isbetter off attacking through–It seems like he would attackthrough the easy pass.

Professor Ben Polak:Good, why does it seem like that?That's right, we're on the right lines now.

Why does it seem like he's going to attack through the easypass? Student: Well if you'renot defending the easy pass, he doesn't lose anyone, and if he attacks through the hard pass he's going to lose atleast one battalion.

Professor Ben Polak: Solet's look at it from–Let's do the exercise–Let's do thesecond lesson I emphasized at the beginning.

Let's put ourselves in Hannibal's shoes, they're probably boots or something.

Whatever you do when you're riding an elephant, whatever you wear.

Let's put ourselves inHannibal's shoes and try and figure out what Hannibal's goingto do here.

So it could be–From Hannibal'spoint of view he doesn't know which pass you're going todefend, but let's have a look at his payoffs.

If you were to defend the easy pass and he goes through theeasy pass, he will get into your country with one battalion andthat's the same as he would have got if he went through the hardpass.

So if you defend the easy pass, from his point of view, it doesn't matter whether hechooses the easy pass and gets one in there or the hard pass, he gets one in there.

But if you were to defend thehard pass, if you were to defend the mountains, then if he chooses the easy pass, he gets both battalions inand if he chooses the hard pass, he gets no battalions in.

So in this case, easy is better.

We have to be a little bitcareful.

It's not the case that forHannibal, choosing the easy pass to attack through, strictly dominates choosing the hard pass, but it is the casethat there's a weak notion of domination here.

It is the case — to introduce some jargon — it is the casethat the easy pass for the attacker, weakly dominates thehard pass for the attacker.

What do I mean by weaklydominate? It means by choosing the easypass, he does at least as well, and sometimes better, than he would have done had he chosen the hard pass.

So here we have a second definition, a new definition fortoday, and again we can use our jargon.

Definition- Player i's strategy, “s'_i” isweakly dominated by her strategy “s_i” if–now we'regoing to take advantage of our notation–if Player i's payofffrom choosing “s_i” against “s_-i” isalways as big as or equal, to her payoff from choosing”s'_i” against “s_-i” and this has tobe true for all things that anyone else could do.

And in addition, Player i's payoff from choosing”s_i” against “s_-i” is strictlybetter than her payoff from choosing “s'_i”against “s_-i, ” for at least one thing thateveryone else could do.

Just check, that exactlycorresponds to the easy and hard thing we just had before.

I'll say it again, Player i's strategy”s'_i” is weakly dominated by her strategy”s_i” if she always does at least as well bychoosing “s_i” than choosing “s'_i”regardless of what everyone else does, and sometimes she does strictly better.

It seems a pretty powerful lesson.

Just as we said you should never choose a strictlydominated strategy, you're probably never going tochoose a weakly dominated strategy either, but it's a little more subtle.

Now that definition, if you're worried about what I've written down here and youwant to see it in words, on the handout I've already puton the web that has the summary of the first class, I included this definition in words as well.

So compare the definition of words with what's written herein the nerdy notation on the board.

Now since we think that Hannibal, the attacker, is not going to play a weakly dominated strategy, we think Hannibal is not going to choose the hard pass.

He's going to attack on the easy pass.

And given that, what should we defend?We should defend easy which is what most of you chose.

So be honest now: was that why most of you choseeasy? Yeah, probably was.

We're able to read this.

So, by putting ourselves inHannibal's shoes, we could figure out that hishard attack strategy was weakly dominated.

He's going to choose easy, so we should defend easy.

Having said that of course, Hannibal went through themountains which kind of screws up the lesson, but too late now.

Now then, I promised you we'dget back to the game from last time.

So where have we got to so far in this class.

We know from last time that you should not choose a dominatedstrategy, and we also know we probably aren't going to choosea weakly dominated strategy, and we also know that youshould put yourself in other people's shoes and figure outthat they're not going to play strongly or strictly or weaklydominated strategies.

That seems a pretty good way topredict how other people are going to play.

So let's take those ideas and go back to the numbers game fromlast time.

Now before I do that, I don't need the people at home to see this, but how many of youwere here last time? How many of you were not.

I asked the wrong question.

How many of you were not herelast time? So we handed out again thatgame.

We handed out again the gamewith the numbers, but just in case, let me just read out the game you played.

This was the game you played.

“Without showing your neighborwhat you are doing, put it in the box below a wholenumber between 1 and a 100.

We will (and in fact have)calculated the average number chosen in the class and thewinner of this game is the person who gets closest totwo-thirds times the average number.

They will win five dollars minus the difference inpennies.

” So everybody filled that inlast time and I have their choices here.

So before we reveal who won, let's discuss this a littlebit.

Let me come down hazardouslyoff this stage, and figure out–Let's get themics up a bit for a second, we can get some mics ready.

So let me find out from people here and see what people did asecond.

You can be honest here sinceI've got everything in front of me.

So how many of you chose some number like 32, 33, 34? One hand.

Actually I can tell you, nine of you did.

So should I read out the names? Should I embarrass people?We've got Lynette, Lukucin, we've Kristin, Bargeon; there's nine of you here.

Let's try it again.

How many of you chose numbers between 32 and34? Okay, a good number of you.

Now we're seeing some hands up.

So keep your hands up a second, those people.

So let me ask people why?Can you get your hand into the guy?What's your name? If we can get him to stand up.

Stand up a second and shout out to the class.

What's your name? Student: Chris.

Professor Ben Polak: Chris, you're on this listsomewhere.

Maybe you're not on this listsomewhere.

Never mind, what did you choose?Student: I think I chose 30.

Professor Ben Polak: Okay 30, so that's pretty close.

So why did you choose 30? Student: Because Ithought everyone was going to be around like the 45 range because66 is two-thirds, or right around of 100, and they were going to go two-thirds less than that and Idid one less than that one.

Professor Ben Polak:Okay, thank you.

Let's get one of the others.

There was another one in here.

Can you just raise your handsagain, the people who were around 33, 34.

There's somebody in here.

Can we get you to stand up (andyou're between mikes).

So that would be–Yep, go ahead.

Shout it out.

What's your name first of all?Student: Ryan.

Professor Ben Polak:Ryan, I must have you here as well, never mind.

What did you choose? Student: 33, I think.

Professor Ben Polak: 33.

Oh you did.

You are Ryan Lowe? Student: Yeah.

Professor Ben Polak: You are Ryan Lowe, okay.

Good, go ahead.

Student: I thought similar to Chris actually and Ialso thought that if we got two-thirds and everyone waschoosing numbers in between 1 and 100 ends up with 33, would be around the number (indiscernible).

Professor Ben Polak: So just to repeat the argument thatwe just heard.

Again, you have to shout it outmore because I'm guessing people didn't hear that in room.

So I'll just repeat it to make sure everyone hears it.

A reason for choosing a number like 33 might go as follows.

If people in the room choose randomly between 1 and 100, then the average is going to be around 50 say and two-thirds of50 is around 33, 33 1/3 actually.

So that's a pretty good piece of reasoning.

What's wrong with that reasoning?What's wrong with that? Can we get the guy, the woman in the striped shirt here, sorry.

We haven't had a woman for a while, so let's have a woman.

Thank you.

Student: That even ifeveryone else had the same reasoning as you, it's still going to be way too high.

Professor Ben Polak: So in particular, if everyone else had the same reasoning as you, it's going to be way too high.

So if everyone else reasonsthat way then everyone in the room would choose a number like33 or 34, and in that case, the average would be what?Sorry, that two-thirds of the average would be what?Something like 22.

So the flaw in the argumentthat Chris and Ryan had — it isn't a bad argument, it's a good starting point — but the flaw in the argument, the mistake in the argument was the first sentence in theargument.

The first sentence in theargument was, if the people in the roomchoose random, then they will choose around50.

That's true.

The problem is that people in the room aren't going to chooseat random.

Look around the room a second.

Look around yourselves.

Do any of you look like arandom number generator? Actually, from here I can seesome of the people, but I'm not going to put.

Actually looking at some of your answers maybe some of youare.

On the whole, Yale students are not random number generators.

They're trying to win the game.

So they're unlikely to choosenumbers at random.

As a further argument, if in fact everyone thought that way, and if you figured outeveryone was going to think that way, then you would expect everyone to choose a number like 33 andin that case you should choose a number like 22.

How many of you, raise your hands a second.

How many of you chose numbers in the range 21 through 23?There's way more of you than that.

I'll start reading you out as well.

Actually about twelve of you, raise your hands.

There should be twelve hands going up somewhere.

There's two, three hands going up, four, five hands going up.

There's actually 12 people whochose exactly 22, so considerably more if include23 and 21.

So those people, I'm guessing, were thinking this way, is that right? Let me get one of my 22's upagain.

Here's a 22.

You want to get this guy? What's your name sir?Stand up and shout.

Student: RyanProfessor Ben Polak: You chose 22?Student: I chose 22 because I thought that mostpeople would play the game dividing by two-thirds a coupleof times, and give numbers averagingaround the low 30's.

Professor Ben Polak: Soif you think people are going to play a particular way, in particular if you think people are going to choose thestrategy of Ryan and Chris, and choose around 33, then 22 seems a great answer.

But you underestimate your Yalecolleagues.

In fact, 22 was way too high.

Now, again, let's just iterate the point here.

Let me just repeat the point here.

The point here is when you're playing a game, you want to think about what other people are trying to do, to try and predict what they're trying to do, and it's not necessarily a great starting point to assumethat the people around you are random number generators.

They have aims- trying to win, and they have strategies too.

Let me take this back to the board a second.

So, in particular, are there any strategies herewe can really rule out? We said already people are notrandom.

Are there any choices we canjust rule out? We know people are not going tochoose those choices.

Let's have someone here.

Can we have the guy in green? Wait for Ale, there we go.

Good.

Stand up.

Give me your name.

Student: My name's Nick.

Professor Ben Polak: Shout it out so people can hear.

Student: No one is going to choose a number over 50.

Professor Ben Polak: No one is going to choose a numberover 50.

Okay, I was going–okay that'sfair enough.

Some people did.

That's fair enough.

I was thinking of something alittle bit less, that's fine.

I was thinking of something a little bit less ambitious.

Somebody said 66.

So let's start analyzing this.

So, in particular, there's something about these strategy choices that aregreater than 67 at any rate.

Certainly, I mean 66 let's goup a little bit, so these numbers bigger than67.

What's wrong with numbersbigger than 67? What's wrong with–Raise yourhands if you have answer.

What's wrong?Can we get the guy in red who's right close to the mike?Stand up, give me your name.

Stand up.

Shout it out to the crowd.

Student: Peter.

Professor Ben Polak: Yep.

Student: If everyonechooses a 100 it would be 67.

Professor Ben Polak:Good, so even if everyone in the number–everyone in the roomdidn't choose randomly but they all chose a 100, a very unlikely circumstance, but even if everyone had chosen100, the highest, the average, sorry, the highest two-thirds of the average could possibly beis 66 2/3, hence 67 would be a pretty goodchoice in that case.

So numbers bigger than 67 seempretty crazy choices, but crazy isn't the word I'mlooking for here.

What can we say about thosechoices, those strategies 67 and above, bigger than 67, 68 andabove? What can we say about thosechoices? Somebody right behind you, the woman right behind you, shout it out.

Student: They have no payoffs for…Professor Ben Polak: They have no payoffs.

What's the jargon here? Let's use our jargon.

Somebody shout it out, what's the jargon about that?They're dominated.

So these strategies aredominated.

Actually, they're only weaklydominated but that's okay, they're certainly dominated.

In particular, a strategy like 80 is dominatedby choosing 67.

You will always get a higherpayoff from choosing 67, at least as high and sometimeshigher, than the payoff you would havegot, had you chosen 80, no matter what else happened inthe room.

So these strategies aredominated.

We know, from the very firstlesson of the class last time, that no one should choose thesestrategies.

They're dominated strategies.

So did anyone choose strategies bigger than 67?Okay, I'm not going to read out names here, but, turns out four of you did.

I'm not going to make you waveyour–okay.

So okay, for the four of youwho did, never mind, but …well mind actually, yeah.

So once we've eliminated the possibility that anyone in theroom is going to choose a strategy bigger than 67, it's as if those numbers 68 through 100 are irrelevant.

It's really as if the game is being played where the onlychoices available on the table are 1 through 67.

Is that right? We know no one's going tochoose 68 and above, so we can just forget them.

We can delete those strategies and once we delete thosestrategies, all that's left are choices 1 through 67.

So can somebody help me out now? What can I conclude, now I've concluded that the strategies 68 through 100essentially don't exist or have been deleted.

What can I conclude? Let me see if I can get a mikein here.

Stand up and wait for the mike.

And here comes the mike.

Good.

Shout out.

Student: That allstrategies 45 and above are hence also ruled out.

Professor Ben Polak: Good, so your name is?Student: Henry Professor Ben Polak: SoHenry is saying once we've figured out that no one shouldchoose a strategy bigger than 67, then we can go another step and say, if those strategies neverexisted, then the same argument rules out — or a similarargument rules out — strategies bigger than 45.

Let's be careful here.

The strategies that are lessthan 67 but bigger than 45, I think these strategies arenot, they're not dominated strategies in the original game.

In particular, we just argued that if everyonein the room chose a 100, then 67 would be a winningstrategy.

So it's not the case that thestrategies between 45 and 67 are dominated strategies.

But it is the case that they're dominated once we delete thedominated strategies: once we delete 67 and above.

So these strategies — let's be careful here with the wordweakly here — these strategies are not weakly dominated in theoriginal game.

But they are dominated –they're weakly dominated — once we delete 68 through 100.

So all of the strategies 45 through 67, are gone now.

So okay, let's have a look.

Did anyone choose — raise yourhands, Be brave here.

Did anyone choose a strategybetween 45 and 67? Or between 46 and 67?No one's raising their hand, but I know some of you didbecause I got it in front of me, at least four of you did and Iwon't read out those names yet, but I might read them out nexttime.

So four more people chose thosestrategies.

Now notice, there's a differentpart of this, this argument.

The argument that eliminates strategies 67 and above, or 68 upwards, that strategy just involves thefirst lesson of last time: do not choose a dominatedstrategy, admittedly weak here, but still.

But the second slice, strategies 45 through 67, getting rid of those strategies involves a little bit more.

You've got to put yourself in the shoes of your fellowclassmen and figure out, that they're not going tochoose 67 and above.

So the first argument, that's a straight forward argument, the second argumentsays, I put myself in other peoplesshoes, I realize they're not going to play a dominatedstrategy, and therefore, having realized they're not going to play a dominatedstrategy, I shouldn't play a strategybetween 45 and 67.

So this argument is an 'inshoes' argument.

Now what?Where can we go now? Yeah, so let's have the guy inthe beard, but let the mike get to him.

Yell out your name.

Student: You just repeatthe same reasoning again and again, and you eventually getdown to 1.

Professor Ben Polak:We'll do that but let's go one step at a time.

So now we've ruled out the possibility that anyone's goingto choose a strategy 68 and above because they're weaklydominated, and we've ruled out thepossibility that anyone's going to choose a strategy between 46and 67, because those strategies aredominated, once we've ruled out the dominated strategies.

So we know no one's choosing any strategies above 45.

, It's as if the numbers 46 and above don't exist.

So we know that the highest anyone could ever choose is 45, and two-thirds of 45 is roughly … someone help me outhere … 30 right: roughly 30.

So we know that all the numbers between 45 and 30, these strategies were not dominated.

And they weren't dominated even after deleting the dominatedstrategies.

But they are dominated once wedeleted not just the dominated strategies, but also thestrategies that were dominated once we deleted the dominatedstrategies.

I'm not going to try and writethat, but you should try and write it in your notes.

So without writing that argument down in detail, notice that we can rule out the strategies 30 through 45, not by just examining our own payoffs;not just by putting ourselves in other people's shoes andrealizing they're not going to choose a dominated strategy;but by putting our self in other people's shoes whilethey're putting themselves in someone else's shoes andfiguring out what they're going to do.

So this is an 'in shoes', be careful where we are here, this is an 'in shoes in shoes' argument, at which point youmight want to invent the sock.

Now, where's this going?We were told where it's going.

We're able to rule out 68 andabove.

Then we were able to rule out46 and above.

Now we're able to rule out 31and above.

By the next slice down we'll beable to eliminate — what is it — about 20 and above, so 30 down to above 20, and this will be an 'in shoes, in shoes, in shoes'.

These strategies aren'tdominated, nor are they dominated once you delete thedominated strategies, nor once we dominated thestrategies dominated once we've deleted the dominatedstrategies, but they are dominated once wedelete the strategies that have been dominated in the–you getwhat I'm doing here.

So where is this argument goingto go? Where's this argument going togo? It's going to go all the waydown to 1: all the way down to 1.

We could repeat this argument all the way down to 1.

Notice that once we've deleted the dominated strategies, you know I had said before about four people chose thisstrategy, and in here, about four people chose this strategy, but in this range 30through 45, I had lots of people.

How many of you chose a number between 30 and 45?Well more than that.

I can guarantee you more thanthat chose a number between 30 and 45.

In fact, the people who chose where we started off 33 chose inthat range.

A lot more of you chose numbersbetween 20 and 30, so we're really getting intothe meat of the distribution.

But we're seeing that these arechoices, that perhaps, are ruled out by this kind ofreasoning.

Now, I'm still not going toquite reveal yet who won.

I want to take this just onestep more abstract.

So I want to just discuss thisa little bit more.

I want to discuss theconsequence of rationality in playing games, slightly philosophical for a few minutes.

So I claim that if you are a rational player, by which I mean somebody who is trying to maximize their payoffsby their play of the game, that simply being rational, just being a rational player, rules out playing thesedominated strategies.

So the four of you who chosenumbers bigger than 67, whose names I'm not going toread out, maybe they were making a mistake.

However, the next slice down requires more than justrationality.

What else does it require?Yes, can I get this guy again, sorry?Shout out your name again, I've forgotten it.

Student: Nick.

Professor Ben Polak:Shout it out.

Student: Nick.

Professor Ben Polak: Yep.

Student: The assumptionthat your opponents are being rational as well.

Professor Ben Polak: Good.

To rule out the second slice, I need to be rational myself, and I need to know that others are rational.

That's illegible, but what it says is rationaland knowledge that other people are rational.

Now how about the next slice after that?Well now I need to be rational, I need to know that otherpeople are rational, and I need to know that otherpeople know that other people are rational.

So to get this slice, this next slice here, I need rationality; as some of you know that'swidely criticized in the social sciences these days.

Are we right to assume that people are rational?To get this slice I need rationality, I need knowledge ofrationality, let's call that KR and I need knowledge ofknowledge of rationality.

As I go down further, I'm going to need rationality, I need to know people arerational; I need to know that people knowthat people are rational, and I need to know that peopleknow that people know that people are rational.

Now let's just make this more concrete for you.

These people, the four people who chose this, they made a mistake.

What about the four people whochose numbers between 45 and 67? What can we conclude aboutthose people? The people who chose between 45and 67? Should I read out their names?No, I won't, perhaps I better not.

What can we conclude about these people?Yeah.

We're never going to get themike to this — try and get the mike in there.

Come forward as far as you can and then really shout, yep.

Student: They thinktheir classmates are pretty dumb.

Professor Ben Polak: Right, right.

It's not necessarily that the four people who chose between 46and 67 are themselves “thick, ” it's that they think the rest ofyou are “thick.

” Down here, this doesn't requirepeople to be thick, or to think the rest of you arethick, they're just people who thinkthat you think, sorry, they're just people whothink that you think that they're thick and so on.

But again, all the way to 1 we're going to need very, very many rounds of knowledge, of knowledge, of knowledge … of rationality.

Does anyone know what we call it if we assume an infinitesequence of “I know that you know that I know that you knowthat I know that you know that I know that you know” something?What's the expression for that? Believe it or not, technical expression.

The technical expression ofthat in philosophy is common knowledge, which I can neverspell, so I'm going to wing it.

Common knowledge is:”I know something, you know it, you know that I know it, I know that you know it, I know that you know that I know it, etc.

, etc.

etc.

: an infinite sequence.

But if we had common knowledge of rationality in this class, then the optimal choice would have been 1.

How many of you chose 1? Look around the room.

Let's just pan the room.

Keep your hands up a second.

How many of you chose 1? So actually a lot of you chose1.

1 was the modal answer in thisclass.

A lot of you chose 1.

So those people did pretty well.

They must have done–they mustbe thinking they're about to win… but they didn't win.

So it turns out that the average in this class, the average choice was about 13 1/3, which means two-thirds ofthe average was 9.

Two-thirds of the average was 9and some of you chose 9, so if you are here, stand up.

The following people chose 9, that's not right, where are the people who chose9? I've got them here somewhere?I'm sorry there's so many pages of people.

Here we go.

The following people chose 9.

So stand up if you're here and if you're that person's roommateif they're not here.

So Leesing Chang:is Leesing Chang here? Stand up if you're here.

A G.

Christopher Berrera:you can stand up, if you're here.

And William Fischel: are you here?I don't know if he is here.

Jed Glickstein: are you here?Jed Glickstein: stand up if you're here.

And Jeffrey Green: stand up if you're here.

And Allison Hoyt: stand up if you're here.

No Allison Hoyt, okay.

There's John Robinson.

All right so these people, stay up a second so the cameracan see you.

There you go, all the way around.

Wave.

Wave to mom at home.

Can we get a round of applausefor our winners? So Jude has trustworthilybrought back the five dollars.

I've got to focus for a secondjust to get it.

Here is the five dollars, we're going to tear this into nine pieces, except I'd getarrested and deported if I did that, so we're going to find a way to break this into change later.

Come and claim it afterwards, but you're all entitled towhatever a ninth, whatever that fraction of fivedollars is.

Okay, so why was it after allthat work — why was it that 1 wasn't the winning answer?Why wasn't 1 the winning answer? Let's have someone we haven'thad before.

Can we get the mike in way inthe back there? Can we get the mike in there onthe row you're on? See if you can point.

Actually good.

Stand up.

Shout.

Shout away.

Student: 1 would have been the winning answer[inaudible] Professor Ben Polak:Louder, louder, louder.

Student: 1 would have been the winning answer hadeveryone assumed that the average would have beenconstantly compounded down to 1, but since a couple of peoplechose the, I mean not incorrect answers, but the higheraverages, then it was pushed up to 13.

Professor Ben Polak: Right, so to get all the way, — good — so to get all the way — thank you — So to getall the way to 1, we need a lot.

We need not just that you're all rational players, not just that you know each other's rational, but you know everyone else's rational.

I mean I know you all know each other because you've met atYale, but you also know each other well enough to know thatnot everyone in the room is rational, and you're pretty sure that not everyone knows that you'rerational and so on and so forth.

It's asking a lot to get to 1here, and in fact, we didn't get to 1.

In previous years we were even higher, so this was low thisyear.

In 2003, the average waseighteen and a half.

And in 2004, it was twenty-one and a half.

And in 2005, we had a class that didn't trust each other at all I guess, because the average was twenty-three.

And this year, it was thirteen and a third.

We're getting better there I think.

One nice thing, by the way — this is justchance I think — the median answer in the class was nine, which is spot on, so the median hit this bang on.

Now what I wanted you to do, is I want you all to playagain.

We haven't got time to do thisproperly, even though I've given you the sheets.

So write down — don't tell this to your neighbors — writedown a number.

Don't talk among yourselvesthat's cheating.

Write down a number.

If you haven't got a sheet in front of you, just write it on your notepad.

Write down a number.

Has everyone written down a number?I'm going to do a show of hands now.

How many — we'll get the camera on you — how many of youchose a number higher than 67? Oh there's some spoil makers inthe class.

How many of you chose a numberhigher than 20? How many of you chose a numberhigher than 10? How many chose a number between5 and 10? How many chose a number between0 — I'm sorry — between 1 and 5?How many of you, excluding the people who chose1 last time, how many of you chose a number that was lowerthan the number you chose last time?Now keep your hands up a second.

So almost all of you came down.

Why? Why are seeing this massivecontraction? I'm guessing the average numberin the class now is probably about 3 or 4, maybe even lower.

Why are we seeing this massivecontraction in the numbers being chosen?The woman in green, I've forgotten your name, I'm sorry? Student: Because we'vejust sat in lecture and you've told us we're not being rationalif we pick a high number.

Professor Ben Polak: Sopart of it is, you yourselves have figuredout, some of you, that you shouldn't choose ahigh number.

What else though?What else is going on here? Let's get somebody.

There's a guy waving an arm out there.

Do you want to stand up behind the hat?You.

Student: Because we'verepeated the game.

Professor Ben Polak:It's true we've repeated it.

It's true we repeated it butwhat is it about repeating it? What is it about talking aboutthis game that makes a difference?Let me hazard a guess here.

I think what makes a differenceis not only do you, yourselves, know better how toplay this game now, but you also know thateverybody around you knows better how to play the game.

Discussing this game raised not just each person'ssophistication, but it raised what you knowabout other people's sophistication, and you know that other people now know that you understand howto play the game.

So the main lesson I want youto get from this is that not only did it matter that you needto put yourself in other people's shoes and think aboutwhat their payoffs are.

You also need to put yourselfinto other people's shoes and think about how sophisticatedare they at playing games.

And you need to think about howsophisticated do they think you are at playing games.

And you need to think about how sophisticated do they think thatyou think that they are at playing games and so on.

This level of knowledge, these layers of knowledge, lead to very different play in the game.

And to make this more concrete, if a firm is competing againsta competitor it can be pretty sure, that competitor is a pretty sophisticated game player andknows that the firm is itself.

If a firm is competing againsta customer — let's say for a non-prime loan — perhaps thatassumption is not quite so safe.

It matters in how we take gamesthrough to the real world, and we're going to see more ofthis as the term progresses.

Now I've got five minutes, do I have five minutes left? So I've got five minutes totake a little small aside here.

We've been talking aboutknowledge and about common knowledge.

I just want to do a very quick experiment, so everyone stay intheir seat.

I'm going to get two T.

A.

's uphere, why don't I get Ale and Kaj up here.

And I wanted to show that common knowledge is not such anobvious a concept, as I've made it seem on theboard.

Come up on the stage a second.

You can leave the mike its okay.

Here we have two of our T.

A.

's, actually these are the two head T.

A.

's, and I want you to faceforward so you don't see what I'm doing.

I'm about to put on their heads a hat.

Here's a hat on Ale's head, and here's a hat on Kaj's head.

Let's move them this way so they're in focus.

Now you can all see these hats, and if they turn around to eachother, they can see each other's hat.

Now I want to ask you the question here.

Here is a fact, so is it common knowledge that– is it common knowledge that at least one of these people hasa pink hat on their head? Is it common knowledge?So I claim it's not common knowledge.

What is known here? Well I'll reveal the facts now:that in fact Ale knows that Kaj has a pink hat on his head.

So it's true that Ale knows that at least one person in theroom has a pink hat on their head.

And it's true that Kaj knows that Ale has a pink hat on hishead.

They both look absurd, but never mind.

But notice that Ale doesn'tknow the color of the hat on his own head.

So even though both people know, even though it is mutualknowledge that there's at least one pink hat in the room, Ale doesn't know what Kaj is seeing.

So Ale does not know that Kaj knows that there's a pink hat inthe room.

In fact, from Ale's point ofview, this could be a blue hat.

So again, they both know thatsomeone in the room has a pink hat on their head:it is mutual knowledge that there's a pink hat in the room.

But Ale does not know that Kaj knows that he is wearing a blue, a pink hat, and Kaj does not know that Ale knows that Kaj iswearing a pink hat.

Each of their hats — each oftheir own hats — might be blue.

So notice that common knowledge– thanks guys — common knowledge is a rather subtlething, thank you.

Common knowledge is a subtlething.

Mutual knowledge doesn't implycommon knowledge.

Common knowledge is a statementabout not just what I know.

It's about what do I know theother person knows that I know that the other person…and so on and so forth.

Even in this simple example, while you might think it's obviously common knowledge, it wasn't common knowledge that there was a pink hat in theroom.

Does anybody have smallersiblings or children of their own.

They can have a pink hat at the end of the class?We'll see you on Wednesday.

.