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# ECON 159: Game Theory

## Lecture 2

## - Putting Yourselves into Other People's Shoes

### Overview

At the start of the lecture, we introduce the “formal ingredients” of a game: the players, their strategies and their payoffs. Then we return to the main lessons from last time: not playing a dominated strategy; and putting ourselves into others’ shoes. We apply these first to defending the Roman Empire against Hannibal; and then to picking a number in the game from last time. We learn that, when you put yourself in someone else’s shoes, you should consider not only their goals, but also how sophisticated are they (are they rational?), and how much do they know about you (do they know that you are rational?). We introduce a new idea: the iterative deletion of dominated strategies. Finally, we discuss the difference between something being known and it being commonly known.

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html## Game Theory## ECON 159 - Lecture 2 - Putting Yourselves into Other People's Shoes## Chapter 1. Recap of Previous Lecture: Prisoners’ Dilemma and Payoffs [00:00:00]
One of the first things we pointed out, is that this is not quite 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 to know payoffs. Then we looked at some possible payoffs, and now it is a game. So this is a game, just to give you some more jargon, this is a normal-form game. And here we’ve assumed the payoffs are those that arise if players only care about their own grades, which I think was true for a lot of you. It wasn’t true for the gentleman who’s sitting there now, 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 your pair does, you attain a higher payoff from choosing Alpha, than you do from choosing Beta. Let’s focus on a couple of lessons of the class before I come back to this. One lesson was, do not play a strictly dominated strategy. Everybody remember that lesson? Then much later on, when we looked at some more complicated 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 what they’re going to do. So in fact, what we learned from that is, it doesn’t just matter what your payoffs are – that’s obviously important – it’s also important what other people’s payoffs are, because you want to try and figure out what they’re going to do 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 real world relevance. Again, still in the interest of recapping, 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 which are actually written here, so they’ll find their way into your 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 might have an incentive to shirk. Price competition – two firms competing 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 down towards marginal cost and industry profits will suffer. In the first case, if everyone shirks you end up with a bad product. In the second case, if both firms undercut each other, you end up with low prices, that’s actually good for consumers but bad for firms. Let me mention a third example. Suppose there’s a common resource out there, maybe it’s a fish stock or maybe it’s the atmosphere. There’s a Prisoners’ Dilemma aspect to this too. You might have an incentive to over fish. Why? Because if the other countries with this fish stock–let’s say the fish stock is the Atlantic–if the other countries are going to fish as normal, you may as well fish as normal too. And if the other countries aren’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 carbon emissions. Again, leaving aside the science, about which I’m sure some of you know more than me here, the issue of carbon emissions is a Prisoners’ Dilemma. Each of us individually has an incentive 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 abstract thing 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 talked about 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 much as you like, but as long as you’re going to go home and still 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 is working hard, or is cutting back on their carbon emissions, you have every bit more incentive to not work hard or to keep high carbon emissions yourself. So we need something more and the kind of things we can see more: we can think about contracts; we can think about treaties between countries; we can think about regulation. All of these things work by changing the payoffs. Not just talking about it, but actually changing the outcomes actually and changing the payoffs, changing the incentives. Another thing we can do, a very important thing, is we can think about changing the game into a game of repeated interaction and seeing how much that helps, and we’ll come back and revisit that later in the class. One last thing we can think of doing 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 be better people. That may or may not work – I’m not optimistic – but at least it’s the same idea. We’re changing payoffs. So that’s enough for recap and I 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 a number – and we said the winner was going to be the person who gets closest to two-thirds of the average in the class. Now we’ve figured that out, we figured out who the winner is, and I know that all of you have been trying to see if you won, is that right? I’m going to leave you in suspense. I am going to tell you today who won. We did figure it out, and we’ll get there, but I want to do a little bit of work first. So we’re just going to leave it in suspense. That’ll stop you walking out early if you want to win the prize. ## Chapter 2. The Formal Ingredients of a Game [00:06:47]So there’s going to be lots of times in this class when we get to play games, we get to have classroom discussions and so on, but there’s going to be some times 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 twenty minutes, let’s do something we’ll call formal stuff. In particular, I want to develop and make sure we all understand, what are the ingredients of a game? So in particular, we need to figure out what formally makes something into a game. The formal parts of a game are this. We need players – and while we’re here let’s develop some notation. So the standard notation for players, I’m going to use things like little i and little j. So in that numbers game, the game when all of you wrote down a number and handed it in at 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 be writing.) Notation: so I’m going to use little “s When distinguishing a particular strategy from the set of possible strategies. While we’re here, our third notation for strategy, I’m going to use little “s” without an “i,” (no subscripts): little “s” without an “i,” to mean a particular play of the game. So what do I mean by that? All of you, at the end last time, wrote down this number and handed them in so we had one number, one strategy choice for each 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 each person: 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 you chose. So that’s a particular play of the game and that has a different name. We sometimes call this “a strategy profile.” So in the textbook, you’ll sometimes see the term a strategy profile or a strategy vector, or a strategy list. It doesn’t really matter. What it’s saying is one strategy for each player in the game. So in the numbers game this is the spreadsheet – or an example of this is the spreadsheet. (I need to make it so you can still see that, so I’m going to pull down these boards. And let me clean something.) So you might think we’re done right? We’ve got players. We’ve got the choices they could 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 everything you 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 So in the numbers game what is this? In the numbers game “U I need one more piece of notation and then we can get back to having some fun. So one more piece of notation, I’m going to write “s Now this is when I want to stop for a second and I know that some of you, from past experience, are somewhat math phobic. You do not have to wave your hands in the air if you’re math phobic, but since some of you are, 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 was going to fine. And now I’m putting math on the board. 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 quit this class because they’re worried about math or math notation. So if you are in that category of 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 by notation, it looks scarier than it is, there’s nothing going on here except for notation at this point. ## Chapter 3. Weakly Dominant Strategies [00:16:01]So let’s have an example to help us fix some ideas. (And again, I’ll have to clean the board, so give me a second.) I think an example might help those 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 II and Player I has two choices, top and bottom, and Player II has three choices left, center, and right. It’s just a very simple abstract example for now. And let’s suppose the payoffs are like this. They’re not particularly interesting. We’re just going to do it for the 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 we just 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 or bottom, represented by the rows, which are hopefully the top row and the bottom row. Player II has three choices, this game is not symmetric, so they have different number of choices, that’s fine. Player II has three choices left, center, and right, represented by the left, center, and right column in the matrix. Just to point out in passing, up to now, we’ve been looking mostly at symmetric games. Notice this game is not symmetric 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 Player II chooses center, we read by looking at the top row and the center column, and Player I’s payoff is the first 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 in the room. Now how do we think this game is going to be played? It’s not a particularly interesting 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?
So in this particular game, I claim that center dominates right. 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 dominates right. What you said was true, but I wanted something specifically about domination here. So what we know here, we know that Player II should not choose right. Now, in fact, that’s as far as we can get with dominance arguments in this particular game, but nevertheless, let’s just stick with it a second. I gave you the definition of strict 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’ So to say it in words, Player i’s strategy “s’ Let’s have a look at another example. 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 thinking about invading a country, and there are two ways – there are two passes if you like – through which he can lead his army. You are the defender of this country and you have to decide which of these passes or which of these routes into the country, you’re going to choose to defend. And the catch is, you can only defend one of these two routes. If you want a real world example of this, think about the third Century B.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 the third 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 easy pass. It goes along the coast. If the invader chooses the hard pass he will lose one battalion of 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 another battalion. I haven’t given you–I’ve given you roughly the choices, the choice they’re going to be for 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 about this. 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 the attacker, this is Hannibal, and this is going to be the defender, (and I’ve forgotten which general was defending and someone’s about to tell me that). And there are two passes you could 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 just means no mountains, we’re not talking about something on the New Jersey Turnpike.) So the payoffs here are as follows, and I’ll explain them in a second. So his payoff, the attacker’s payoff, is how many battalions does he get to bring into your country? He only has two to start with and for you, it’s how many battalions of his get destroyed? So just to give an example, if he goes through the hard pass and you defend the hard pass, he loses one of those battalions going over the mountains 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 the easy 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 were defending the wrong pass. Everyone understand the payoffs of this game? So now imagine yourself as a Roman general. This is going to be a little bit of a stretch for imagination, but imagination yourself as a Roman general, and let’s figure out what you’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 Jude can 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 hard pass? We have a huge number of people who don’t want to be Roman generals here. Let’s try it again, no abstentions, right? I’m not going to penalize you for giving the wrong answer. So how many of you think you’re going 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 large majority. So what’s going on here? Is it the case that defending the easy pass dominates defending the hard pass? Is that the case? Is it the case that defending the easy pass dominates defending the hard pass? You can shout out. No, it’s not. In fact, we could check that if the 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 to attack 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 dominates defending hard. You just want to match with the attacker. 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 with the beard? Just wait for the mike to get there. If you could stand up: stand up and shout. There you go.
If you were to defend the easy pass and he goes through the easy pass, he will get into your country with one battalion and that’s the same as he would have got if he went through the hard pass. So if you defend the easy pass, from his point of view, it doesn’t matter whether he chooses the easy pass and gets one in there or the hard pass, he gets one in there. But if you were to defend the hard pass, if you were to defend the mountains, then if he chooses the easy pass, he gets both battalions in and if he chooses the hard pass, he gets no battalions in. So in this case, easy is better. We have to be a little bit careful. It’s not the case that for Hannibal, choosing the easy pass to attack through, strictly dominates choosing the hard pass, but it is the case that there’s a weak notion of domination here. It is the case – to introduce some jargon – it is the case that the easy pass for the attacker, weakly dominates the hard pass for the attacker. What do I mean by weakly dominate? It means by choosing the easy pass, 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 for today, and again we can use our jargon. Definition- Player i’s strategy, “s’ Just check, that exactly corresponds to the easy and hard thing we just had before. I’ll say it again, Player i’s strategy “s’ Now that definition, if you’re worried about what I’ve written down here and you want to see it in words, on the handout I’ve already put on 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 here in 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 chose easy? Yeah, probably was. We’re able to read this. So, by putting ourselves in Hannibal’s shoes, we could figure out that his hard attack strategy was weakly dominated. He’s going to choose easy, so we should defend easy. Having said that of course, Hannibal went through the mountains which kind of screws up the lesson, but too late now. ## Chapter 4. Rationality and Common Knowledge [00:35:29]Now then, I promised you we’d get 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 dominated strategy, and we also know we probably aren’t going to choose a weakly dominated strategy, and we also know that you should put yourself in other people’s shoes and figure out that they’re not going to play strongly or strictly or weakly dominated strategies. That seems a pretty good way to predict how other people are going to play. So let’s take those ideas and go back to the numbers game from last time. Now before I do that, I don’t need the people at home to see this, but how many of you were here last time? How many of you were not. I asked the wrong question. How many of you were not here last time? So we handed out again that game. We handed out again the game with the numbers, but just in case, let me just read out the game you played. This was the game you played. “Without showing your neighbor what you are doing, put it in the box below a whole number between 1 and a 100. We will (and in fact have) calculated the average number chosen in the class and the winner of this game is the person who gets closest to two-thirds times the average number. They will win five dollars minus the difference in pennies.” So everybody filled that in last time and I have their choices here. So before we reveal who won, let’s discuss this a little bit. Let me come down hazardously off this stage, and figure out–Let’s get the mics 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 a second. You can be honest here since I’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 and 34? 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?
So the flaw in the argument that 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 the argument. The first sentence in the argument was, if the people in the room choose random, then they will choose around 50. That’s true. The problem is that people in the room aren’t going to choose at random. Look around the room a second. Look around yourselves. Do any of you look like a random number generator? Actually, from here I can see some of the people, but I’m not going to put. Actually looking at some of your answers maybe some of you are. On the whole, Yale students are not random number generators. They’re trying to win the game. So they’re unlikely to choose numbers at random. As a further argument, if in fact everyone thought that way, and if you figured out everyone was going to think that way, then you would expect everyone to choose a number like 33 and in 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 who chose exactly 22, so considerably more if include 23 and 21. So those people, I’m guessing, were thinking this way, is that right? Let me get one of my 22’s up again. Here’s a 22. You want to get this guy? What’s your name sir? Stand up and shout.
Let me take this back to the board a second. So, in particular, are there any strategies here we can really rule out? We said already people are not random. Are there any choices we can just rule out? We know people are not going to choose 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.
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 wave your–okay. So okay, for the four of you who did, never mind, but … well mind actually, yeah. So once we’ve eliminated the possibility that anyone in the room 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 only choices available on the table are 1 through 67. Is that right? We know no one’s going to choose 68 and above, so we can just forget them. We can delete those strategies and once we delete those strategies, 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 100 essentially don’t exist or have been deleted. What can I conclude? Let me see if I can get a mike in here. Stand up and wait for the mike. And here comes the mike. Good. Shout out.
So these strategies – let’s be careful here with the word weakly here – these strategies are not weakly dominated in the original 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 your hands, Be brave here. Did anyone choose a strategy between 45 and 67? Or between 46 and 67? No one’s raising their hand, but I know some of you did because I got it in front of me, at least four of you did and I won’t read out those names yet, but I might read them out next time. So four more people chose those strategies. Now notice, there’s a different part of this, this argument. The argument that eliminates strategies 67 and above, or 68 upwards, that strategy just involves the first lesson of last time: do not choose a dominated strategy, 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 fellow classmen and figure out, that they’re not going to choose 67 and above. So the first argument, that’s a straight forward argument, the second argument says, I put myself in other peoples shoes, I realize they’re not going to play a dominated strategy, and therefore, having realized they’re not going to play a dominated strategy, I shouldn’t play a strategy between 45 and 67. So this argument is an ‘in shoes’ argument. Now what? Where can we go now? Yeah, so let’s have the guy in the beard, but let the mike get to him. Yell out your name.
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 out here … 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 dominated strategies. But they are dominated once we deleted not just the dominated strategies, but also the strategies that were dominated once we deleted the dominated strategies. I’m not going to try and write that, 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 and realizing they’re not going to choose a dominated strategy; but by putting our self in other people’s shoes while they’re putting themselves in someone else’s shoes and figuring 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 you might want to invent the sock. Now, where’s this going? We were told where it’s going. We’re able to rule out 68 and above. Then we were able to rule out 46 and above. Now we’re able to rule out 31 and above. By the next slice down we’ll be able 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’t dominated, nor are they dominated once you delete the dominated strategies, nor once we dominated the strategies dominated once we’ve deleted the dominated strategies, but they are dominated once we delete the strategies that have been dominated in the–you get what I’m doing here. So where is this argument going to go? Where’s this argument going to go? It’s going to go all the way down 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 this strategy, and in here, about four people chose this strategy, but in this range 30 through 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 than that chose a number between 30 and 45. In fact, the people who chose where we started off 33 chose in that range. A lot more of you chose numbers between 20 and 30, so we’re really getting into the meat of the distribution. But we’re seeing that these are choices, that perhaps, are ruled out by this kind of reasoning. Now, I’m still not going to quite reveal yet who won. I want to take this just one step more abstract. So I want to just discuss this a little bit more. I want to discuss the consequence 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 payoffs by their play of the game, that simply being rational, just being a rational player, rules out playing these dominated strategies. So the four of you who chose numbers bigger than 67, whose names I’m not going to read out, maybe they were making a mistake. However, the next slice down requires more than just rationality. What else does it require? Yes, can I get this guy again, sorry? Shout out your name again, I’ve forgotten it.
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 who chose numbers between 45 and 67? What can we conclude about those people? The people who chose between 45 and 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 the mike to this – try and get the mike in there. Come forward as far as you can and then really shout, yep.
Does anyone know what we call it if we assume an infinite sequence of “I know that you know that I know that you know that 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 of that in philosophy is common knowledge, which I can never spell, 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 chose 1. 1 was the modal answer in this class. A lot of you chose 1. So those people did pretty well. They must have done–they must be 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 of the average was 9. Two-thirds of the average was 9 and 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 chose 9? 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 roommate if 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 camera can see you. There you go, all the way around. Wave. Wave to mom at home. Can we get a round of applause for our winners? So Jude has trustworthily brought back the five dollars. I’ve got to focus for a second just to get it. Here is the five dollars, we’re going to tear this into nine pieces, except I’d get arrested 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 to whatever a ninth, whatever that fraction of five dollars is. Okay, so why was it after all that work – why was it that 1 wasn’t the winning answer? Why wasn’t 1 the winning answer? Let’s have someone we haven’t had before. Can we get the mike in way in the back there? Can we get the mike in there on the row you’re on? See if you can point. Actually good. Stand up. Shout. Shout away.
In previous years we were even higher, so this was low this year. In 2003, the average was eighteen 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 just chance 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 play again. We haven’t got time to do this properly, even though I’ve given you the sheets. So write down – don’t tell this to your neighbors – write down a number. Don’t talk among yourselves that’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 you chose a number higher than 67? Oh there’s some spoil makers in the class. How many of you chose a number higher than 20? How many of you chose a number higher than 10? How many chose a number between 5 and 10? How many chose a number between 0 – I’m sorry – between 1 and 5? How many of you, excluding the people who chose 1 last time, how many of you chose a number that was lower than 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 massive contraction? I’m guessing the average number in the class now is probably about 3 or 4, maybe even lower. Why are we seeing this massive contraction in the numbers being chosen? The woman in green, I’ve forgotten your name, I’m sorry?
So the main lesson I want you to get from this is that not only did it matter that you need to put yourself in other people’s shoes and think about what their payoffs are. You also need to put yourself into other people’s shoes and think about how sophisticated are they at playing games. And you need to think about how sophisticated do they think you are at playing games. And you need to think about how sophisticated do they think that you 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 against a competitor it can be pretty sure, that competitor is a pretty sophisticated game player and knows that the firm is itself. If a firm is competing against a customer – let’s say for a non-prime loan – perhaps that assumption is not quite so safe. It matters in how we take games through to the real world, and we’re going to see more of this as the term progresses. ## Chapter 5. Common Knowledge vs. Mutual Knowledge [01:05:37]Now I’ve got five minutes, do I have five minutes left? So I’ve got five minutes to take a little small aside here. We’ve been talking about knowledge and about common knowledge. I just want to do a very quick experiment, so everyone stay in their seat. I’m going to get two T.A.’s up here, why don’t I get Ale and Kaj up here. And I wanted to show that common knowledge is not such an obvious a concept, as I’ve made it seem on the board. 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 face forward 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 each other, 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 has a 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 the room has a pink hat on their head. And it’s true that Kaj knows that Ale has a pink hat on his head. They both look absurd, but never mind. But notice that Ale doesn’t know the color of the hat on his own head. So even though both people know, even though it is mutual knowledge 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 in the room. In fact, from Ale’s point of view, this could be a blue hat. So again, they both know that someone 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 is wearing a pink hat. Each of their hats – each of their own hats – might be blue. So notice that common knowledge – thanks guys – common knowledge is a rather subtle thing, thank you. Common knowledge is a subtle thing. Mutual knowledge doesn’t imply common knowledge. Common knowledge is a statement about not just what I know. It’s about what do I know the other 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 the room. Does anybody have smaller siblings or children of their own. They can have a pink hat at the end of the class? We’ll see you on Wednesday. [end of transcript] Back to Top |
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