耶鲁大学公开课 博弈论 原文讲稿仔细整理注释 第2讲

ECON-159: GAME THEORYLecture 2 - Putting Yourselves into Other People's Shoes [September 10, 2007]Chapter 1.Recap of Previous Lecture: Prisoners' Dilemma and Payoffs [00:00:00]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 and Beta, and this table told us what outcome would arise. In particular, what grade you would get and what 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 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'allwere 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 i" to be a particular strategy of Player i. So an example in that game might have been choosing the number 13. Everyone understand that? Now I need to distinguish this from the set of possible strategies of Player I, so I'm going to use capital "S i"to be what? To be the set of alternatives.The set ofpossible 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 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 i" will depend on Player 1's choice … all the way to Player i's own choice … all the way up to Player N's choices. 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 two things. It can be 5 dollars minus your error 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 the next 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 possible strategies everyone else could choose and everyone knows everyone else's payoffs. Now that's not a very realistic assumption and we are going to come back and challenge it at the end of semester, but this will be complicated enough to give us a lot of material in the next ten weeks.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-i" to mean what? It's going to mean a strategy choice for everybody except person "i." It's going to be useful to have that notation around. So this is a choice for all except person "i" or Player i. So, in particular, if you're person 1 and then "s-i" would be "s2, s3, s4" up to "s n" but it wouldn't include "s1." 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 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, soit'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?Student: No, Player I doesn't have a dominated 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 people could 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 to get people--That was very clear, very nice, but we need people to stand up and shout, or these people at the back can't hear. So your name is?Student: Patrick.Professor Ben Polak: What Patrick said was: no, Player I does not have a dominated strategy. Top is better than bottom against left -- sorry, bottom is better than top against left because 6 is bigger than 5, but top is better than bottom against center because 11 is bigger than 0. Everyone see that? So it's not the case that top always beats--it's not the case that top always does better than bottom, or that bottom always does better than top. What about, raise hands this time, what about Player II? Does Player II have a dominated strategy? Everyone's keeping their hands firmly downso 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 right is a dominated strategy because if Player I chooses top, then Player II will choose center, and if-- I'm getting confused 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 was true, but it doesn't quite match with the definition of a dominated 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 that always does better. That turns out also to be true in this case, but let's just see.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'i" is strictly dominated by Player i's strategy "s i," and now we can use our notation, if "U I" from choosing "s i," when other people choose "s-i," is strictly bigger 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 strictly dominated by her strategy "s i," if "s i" always does strictly better -- always yields a higher payoff for Player i -- no matter what the other people do. So this is the same definition we saw last time, just being a little bit more nerdy and putting in some notation. People panicking about that, people look like deer in the headlamp s yet? No, you look all right: all rightish.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 andyou'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.Student: Because you want to minimize the amount of enemy soldiers that reach Rome or whatever location it is.Professor Ben Polak: You want to minimize the number of soldiers that reach Rome, that's true. On the other hand, we've just argued that you don'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 in the middle. Again, stand up and shout in that mike. Point your face towards the mike. Good.Student: It seems as though while you don't have a dominating strategy, it seems like Hannibal is better off attacking through--It seems like he would attack through 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 easy pass?Student: Well if you're not defending the easy pass, he doesn't lose anyone, and if he attacks through the hard pass he's going to lose at least one battalion.Professor Ben Polak: So let's look at it from--Let's do the exercise--Let's do the second 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 in Hannibal's shoes and try and figure out what Hannibal's going to do here. So it could be--From Hannibal's point of view he doesn't know which pass you're going to defend, but let's have a look at his payoffs.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'i" is weakly dominated by her strategy "s i" if--now we're going to take advantage of our notation--if Player i's payoff from choosing "s i" against "s-i" is always as big as or equal, to her payoff from choosing "s'i" against "s-i" and this has to be true for all things that anyone else could do. And in addition, Player i's payoff from choosing "s i" against "s-i" is strictly better than her payoff from choosing "s'i" against "s-i," for at least one thing that everyone else could do.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'i" is weakly dominated by her strategy "s i" if she always does at least as well by choosing "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 strictly dominated strategy, you're probably never going to choose 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 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.。