(完整版)耶鲁公开课--博弈论笔记
耶鲁大学公开课英文讲义—博弈论第一节
Game Theory: Lecture 1 TranscriptProfessor Ben Polak: So this is Game Theory Economics 159. If you're here for art history, you're either in the wrong room or stay anyway, maybe this is the right room; but this is Game Theory, okay. You should have four handouts; everyone should have four handouts. There is a legal release form--we'll talk about it in a minute--about the videoing. There is a syllabus, which is a preliminary syllabus: it's also online. And there are two games labeled Game 1 and Game 2. Can I get you all to look at Game 1 and start thinking about it. And while you're thinking about it, I am hoping you can multitask a bit. I'll describe a bit about the class and we'll get a bit of admin under our belts. But please try and look at--somebody's not looking at it, because they're using it as a fan here--so look at Game 1 and fill out that form for me, okay?So while you're filling that out, let me tell you a little bit about what we're going to be doing here. So what is Game Theory? Game Theory is a method of studying strategic situations. So what's a strategic situation? Well let's start off with what's not a strategic situation. In your Economics - in your Intro Economics class in 115 or 110, you saw some pretty good examples of situations that were not strategic. You saw firms working in perfect competition. Firms in perfect competition are price takers: they don't particularly have to worry about the actions of their competitors. You also saw firms that were monopolists and monopolists don't have any competitors to worry about, so that's not a particularly strategic situation. They're not price takers but they take the demand curve. Is this looking familiar for some of you who can remember doing 115 last year or maybe two years ago for some of you? Everything in between is strategic. So everything that constitutes imperfect competition is a strategic setting. Think about the motor industry, the motor car industry. Ford has to worry about what GM is doing and what Toyota is doing, and for the moment at least what Chrysler is doing but perhaps not for long. So there's a small number of firms and their actions affect each other.So for a literal definition of what strategic means: it's a setting where the outcomes that affect you depend on actions, not just on your own actions, but on actions of others. All right, that's as much as I'm going to say for preview right now, we're going to come back and see plenty of this over the course of the next semester.So what I want to do is get on to where this applies. It obviously applies in Economics, but it also applies in politics, and in fact, this class will count as a Political Science class if you're a Political Science major. You should go check with the DUS in Political Science. It count - Game Theory is very important in law these days. So for those of you--for the half of you--that aregoing to end up in law school, this is pretty good training. Game Theory is also used in biology and towards the middle of the semester we're actually going to see some examples of Game Theory as applied to evolution. And not surprisingly, Game Theory applies to sport.So let's talk about a bit of admin. How are you doing on filling out those games? Everyone managing to multitask: filling in Game 1? Keep writing. I want to get some admin out of the way and I want to start by getting out of the way what is obviously the elephant in the room. Some of you will have noticed that there's a camera crew here, okay. So as some of you probably know, Yale is undergoing an open education project and they're videoing several classes, and the idea of this, is to make educational materials available beyond the walls of Yale. In fact, on the web, internationally, so people in places, maybe places in the U.S. or places miles away, maybe in Timbuktu or whatever, who find it difficult to get educational materials from the local university or whatever, can watch certain lectures from Yale on the web.Some of you would have been in classes that do that before. What's going to different about this class is that you're going to be participating in it. The way we teach this class is we're going to play games, we're going to have discussions, we're going to talk among the class, and you're going to be learning from each other, and I want you to help people watching at home to be able to learn too. And that means you're going to be on film, at the very least on mike.So how's that going to work? Around the room are three T.A.s holding mikes. Let me show you where they are: one here, one here, and one here. When I ask for classroom discussions, I'm going to have one of the T.A.s go to you with a microphone much like in "Donahue" or something, okay. At certain times, you're going to be seen on film, so the camera is actually going to come around and point in your direction.Now I really want this to happen. I had to argue for this to happen, cause I really feel that this class isn't about me. I'm part of the class obviously, but it's about you teaching each other and participating. But there's a catch, the catch is, that that means you have to sign that legal release form.So you'll see that you have in front of you a legal release form, you have to be able to sign it, and what that says is that we can use you being shown in class. Think of this as a bad hair day release form. All right, you can't sue Yale later if you had a bad hair day. For those of you who are on the run from the FBI, your Visa has run out, or you're sitting next to your ex-girlfriend, now would be a good time to put a paper bag over your head.All right, now just to get you used to the idea, in every class we're going to have I think the same two people, so Jude is the cameraman; why don't you all wave to Jude: this is Jude okay. And Wes is our audio guy: this is Wes. And I will try and remember not to include Jude and Wes in the classroom discussions, but you should be aware that they're there. Now, if this is making you nervous, if it's any consolation, it's making me very nervous. So, all right, we'll try and make this class work as smoothly as we can, allowing for this extra thing. Let me just say, no one's making any money off this--at least I'm hoping these guys are being paid--but me and the T.A.s are not being paid. The aim of this, that I think is a good aim, it's an educational project, and I'm hoping you'll help us with it. The one difference it is going to mean, is that at times I might hold some of the discussions for the class, coming down into this part of the room, here, to make it a little easier for Jude.All right, how are we doing now on filling out those forms? Has everyone filled in their strategy for the first game? Not yet. Okay, let's go on doing a bit more admin. The thing you mostly care about I'm guessing, is the grades. All right, so how is the grade going to work for this class? 30% of the class will be on problem sets, 30% of the grade; 30% on the mid-term, and 40% on the final; so 30/30/40.The mid-term will be held in class on October 17th; that is also in your syllabus. Please don't anybody tell me late - any time after today you didn't know when the mid-term was and therefore it clashes with 17 different things. The mid-term is on October 17th, which is a Wednesday, in class. All right, the problem sets: there will be roughly ten problem sets and I'll talk about them more later on when I hand them out. The first one will go out on Monday but it will be due ten days later. Roughly speaking they'll be every week.The grade distribution: all right, so this is the rough grade distribution. Roughly speaking, a sixth of the class are going to end up with A's, a sixth are going to end up with A-, a sixth are going to end up with B+, a sixth are going to end up with B, a sixth are going to end up with B-, and the remaining sixth, if I added that up right, are going to end up with what I guess we're now calling the presidential grade, is that right?That's not literally true. I'm going to squeeze it a bit, I'm going to curve it a bit, so actually slightly fewer than a sixth will get straight A's, and fewer than a sixth will get C's and below. We'll squeeze the middle to make them be more B's. One thing I can guarantee from past experience in this class, is that the median grade will be a B+. The median will fall somewhere in the B+'s. Just as forewarning for people who have forgotten what a median is,that means half of you--not approximately half, it means exactly half of you--will be getting something like B+ and below and half will get something like B+ and above.Now, how are you doing in filling in the forms? Everyone filled them in yet? Surely must be pretty close to getting everyone filled in. All right, so last things to talk about before I actually collect them in - textbooks. There are textbooks for this class. The main textbook is this one, Dutta'sbook Strategy and Games. If you want a slightly tougher book, more rigorous book, try Joel Watson's book, Strategies. Both of those books are available at the bookstore.But I want to warn everybody ahead of time, I will not be following the textbook. I regard these books as safety nets. If you don't understand something that happened in class, you want to reinforce an idea that came up in class, then you should read the relevant chapters in the book and the syllabus will tell you which chapters to read for each class, or for each week of class, all right. But I will not be following these books religiously at all. In fact, they're just there as back up.In addition, I strongly recommend people read, Thinking Strategically. This is good bedtime reading. Do any of you suffer from insomnia? It's very good bedtime reading if you suffer from insomnia. It's a good book and what's more there's going to be a new edition of this book this year and Norton have allowed us to get advance copies of it. So if you don't buy this book this week, I may be able to make the advance copy of the new edition available for some of you next week. I'm not taking a cut on that either, all right, there's no money changing hands.All right, sections are on the syllabus sign up - sorry on the website, sign up as usual. Put yourself down on the wait list if you don't get into the section you want. You probably will get into the section you want once we're done. All right, now we must be done with the forms. Are we done with the forms? All right, so why don't we send the T.A.s, with or without mikes, up and down the aisles and collect in your Game #1; not Game #2, just Game #1.Just while we're doing that, I think the reputation of this class--I think--if you look at the course evaluations online or whatever, is that this class is reasonably hard but reasonably fun. So I'm hoping that's what the reputation of the class is. If you think this class is going to be easy, I think it isn't actually an easy class. It's actually quite a hard class, but I think I can guarantee it's going to be a fun class. Now one reason it's a fun class, is the nice thing about teaching Game Theory - quieten down folks--one thing about teaching Game Theory is, you get to play games, and that's exactlywhat we've just been doing now. This is our first game and we're going to play games throughout the course, sometimes several times a week, sometimes just once a week.We got all these things in? Everyone handed them in? So I need to get those counted. Has anyone taken the Yale Accounting class? No one wants to - has aspirations to be - one person has. I'll have a T.A. do it, it's all right,we'll have a T.A. do it. So Kaj, can you count those for me? Is that right? Let me read out the game you've just played."Game 1, a simple grade scheme for the class. Read the following carefully. Without showing your neighbor what you are doing, put it in the box below either the letter Alpha or the letter Beta. Think of this as a grade bid. I will randomly pair your form with another form and neither you nor your pair will ever know with whom you were paired. Here's how the grades may be assigned for the class. [Well they won't be, but we can pretend.] If you put Alpha and you're paired with Beta, then you will get an A and your pair a C. If you and your pair both put Alpha, you'll both get B-. If you put Beta and you're paired with Alpha, you'll get a C and your pair an A. If you and your pair both put Beta, then you'll both get B+."So that's the thing you just filled in.Now before we talk about this, let's just collect this information in a more useful way. So I'm going to remove this for now. We'll discuss this in a second, but why don't we actually record what the game is, that we're playing, first. So this is our grade game, and what I'm going to do, since it's kind of hard to absorb all the information just by reading a paragraph of text, I'm going to make a table to record the information. So what I'm going to do is I'm going to put me here, and my pair, the person I'm randomly paired with here, and Alpha and Beta, which are the choices I'm going to make here and on the columns Alpha and Beta, the choices my pair is making.In this table, I'm going to put my grades. So my grade if we both put Alpha is B-, if we both put Beta, was B+. If I put Alpha and she put a Beta, I got an A, and if I put Beta and she put an Alpha, I got a C. Is that correct? That's more or less right? Yeah, okay while we're here, why don't we do the same for my pair? So this is my grades on the left hand table, but now let's look at what my pair will do, what my pair will get.So I should warn the people sitting at the back that my handwriting is pretty bad, that's one reason for moving forward. The other thing I should apologize at this stage of the class is my accent. I will try and improve the handwriting, there's not much I can do about the accent at this stage.So once again if you both put Alpha then my pair gets a B-. If we both put Beta, then we both get a B+; in particular, my pair gets a B+. If I put Alpha and my pair puts Beta, then she gets a C. And if I put Beta and she puts Alpha, then she gets an A. So I now have all the information that was on the sheet of paper that you just handed in.Now there's another way of organizing this that's standard in Game Theory, so we may as well get used to it now on the first day. Rather then drawing two different tables like this, what I'm going to do is I'm going to take the second table and super-impose it on top of the first table. Okay, so let me do that and you'll see what I mean. What I'm going to do is draw a larger table, the same basic structure: I'm choosing Alpha and Beta on the rows, my pair is choosing Alpha and Beta on the columns, but now I'm going to put both grades in. So the easy ones are on the diagonal: you both get B- if we both choose Alpha; we both get B+ if we both choose Beta. But if I choose Alpha and my pair chooses Beta, I get an A and she gets a C. And if I choose Beta and she chooses Alpha, then it's me who gets the C and it's her who gets the A.So notice what I did here. The first grade corresponds to the row player, me in this case, and the second grade in each box corresponds to the column player, my pair in this case. So this is a nice succinct way of recording what was in the previous two tables. This is an outcome matrix; this tells us everything that was in the game.Okay, so now seems a good time to start talking about what people did. So let's just have a show of hands. How many of you chose Alpha? Leave your hands up so that Jude can catch that, so people can see at home, okay. All right and how many of you chose Beta? There's far more Alphas - wave your hands the Beta's okay. All right, there's a Beta here, okay. So it looks like a lot of - well we're going to find out, we're going to count--but a lot more Alpha's than Beta's. Let me try and find out some reasons why people chose.So let me have the Alpha's up again. So, the woman who's in red here, can we get a mike to the - yeah, is it okay if we ask you? You're not on the run from the FBI? We can ask you why? Okay, so you chose Alpha right? So why did you choose Alpha?Student: [inaudible] realized that my partner chose Alpha, therefore I chose [inaudible].Professor Ben Polak: All right, so you wrote out these squares, you realized what your partner was going to do, and responded to that. Any otherreasons for choosing Alpha around the room? Can we get the woman here? Try not to be intimidated by these microphones, they're just mikes. It's okay.Student: The reason I chose Alpha, regardless of what my partner chose, I think there would be better outcomes than choosing Beta.Professor Ben Polak: All right, so let me ask your names for a second-so your name was?Student: Courtney.Professor Ben Polak: Courtney and your name was?Student: Clara Elise.Professor Ben Polak: Clara Elise. So slightly different reasons, same choice Alpha. Clara Elise's reason - what did Clara Elise say? She said, no matter what the other person does, she reckons she'd get a better grade if she chose Alpha. So hold that thought a second, we'll come back to - is it Clara Elise, is that right? We'll come back to Clara Elise in a second. Let's talk to the Beta's a second; let me just emphasize at this stage there are no wrong answers. Later on in the class there'll be some questions that have wrong answers. Right now there's no wrong answers. There may be bad reasons but there's no wrong answers. So let's have the Beta's up again. Let's see the Beta's. Oh come on! There was a Beta right here. You were a Beta right? You backed off the Beta, okay. So how can I get a mike into a Beta? Let' s stick in this aisle a bit. Is that a Beta right there? Are you a Beta right there? Can I get the Beta in here? Who was the Beta in here? Can we get the mike in there? Is that possible? In here - you can leave your hand so that - there we go. Just point towards - that's fine, just speak into it, that's fine. Student: So the reason right?Professor Ben Polak: Yeah, go ahead.Student: I personally don't like swings that much and it's the B-/B+ range, so I'd much rather prefer that to a swing from A to C, and that's my reason. Professor Ben Polak: All right, so you're saying it compresses the range.I'm not sure it does compress the range. I mean if you chose Alpha, you're swinging from A to B-; and from Beta, swinging from B+ to C. I mean those are similar kind of ranges but it certainly is a reason. Other reasons for choosing? Yeah, the guy in blue here, yep, good. That's all right. Don't hold the mike; just let it point at you, that's fine.Student: Well I guess I thought we could be more collusive and kind of work together, but I guess not. So I chose Beta.Professor Ben Polak: There's a siren in the background so I missed the answer. Stand up a second, so we can just hear you.Student: Sure.Professor Ben Polak: Sorry, say again.Student: Sure. My name is Travis. I thought we could work together, but I guess not.Professor Ben Polak: All right good. That's a pretty good reason. Student: If you had chosen Beta we would have all gotten B+'s but I guess not.Professor Ben Polak: Good, so Travis is giving us a different reason, right? He's saying that maybe, some of you in the room might actually care about each other's grades, right? I mean you all know each other in class. You all go to the same college. For example, if we played this game up in the business school--are there any MBA students here today? One or two. If we play this game up in the business school, I think it's quite likely we're going to get a lot of Alpha's chosen, right? But if we played this game up in let's say the Divinity School, all right and I'm guessing that Travis' answer is reflecting what you guys are reasoning here. If you played in the Divinity School, you might think that people in the Divinity School might care about other people's grades, right? There might be ethical reasons--perfectly good, sensible, ethical reasons--for choosing Beta in this game. There might be other reasons as well, but that's perhaps the reason to focus on. And perhaps, the lesson I want to draw out of this is that right now this is not a game. Right now we have actions, strategies for people to take, and we know what the outcomes are, but we're missing something that will make this a game. What are we missing here?Student: Objectives.Professor Ben Polak: We're missing objectives. We're missing payoffs. We're missing what people care about, all right. So we can't really start analyzing a game until we know what people care about, and until we know what the payoffs are. Now let's just say something now, which I'll probably forget to say in any other moment of the class, but today it's relevant.Game Theory, me, professors at Yale, cannot tell you what your payoff should be. I can't tell you in a useful way what it is that your goals in life should be or whatever. That's not what Game Theory is about. However, once we know what your payoffs are, once we know what your goals are, perhaps Game Theory can you help you get there.So we've had two different kinds of payoffs mentioned here. We had the kind of payoff where we care about our own grade, and Travis has mentioned the kind of payoff where you might care about other people's grades. And what we're going to do today is analyze this game under both those possible payoffs. To start that off, let's put up some possible payoffs for the game. And I promise we'll come back and look at some other payoffs later. We'll revisit the Divinity School later.All right, so here once again is our same matrix with me and my pair, choosing actions Alpha and Beta, but this time I'm going to put numbers in here. And some of you will perhaps recognize these numbers, but that's not really relevant for now. All right, so what's the idea here? Well the first idea is that these numbers represent utiles or utilities. They represent what these people are trying to maximize, what they're to achieve, their goals.The idea is - just to compare this to the outcome matrix - for the person who's me here, (A,C) yields a payoff of--(A,C) is this box--so (A,C) yields a payoff of three, whereas (B-,B-) yields a payoff of 0, and so on. So what's the interpretation? It's the first interpretation: the natural interpretation that a lot of you jumped to straight away. These are people--people with these payoffs are people--who only care about their own grades. They prefer an A to a B+, they prefer a B+ to a B-, and they prefer a B- to a C. Right, I'm hoping I the grades in order, otherwise it's going to ruin my curve at the end of the year. So these people only care about their own grades. They only care about their own grades.What do we call people who only care about their own grades? What's a good technical term for them? In England, I think we refer to these guys - whether it's technical or not - as "evil gits." These are not perhaps the most moral people in the universe. So now we can ask a different question. Suppose, whether these are actually your payoffs or not, pretend they are for now. Suppose these are all payoffs. Now we can ask, not what did you do, but what should you do? Now we have payoffs that can really switch the question to a normative question: what should you do? Let's come back to - was it Clara Elise--where was Clara Elise before? Let's get the mike on you again. So just explain what you did and why again.Student: Why I chose Alpha?Professor Ben Polak: Yeah, stand up a second, if that's okay.Student: Okay.Professor Ben Polak: You chose Alpha; I'm assuming these were roughly your payoffs, more or less, you were caring about your grades.Student: Yeah, I was thinking -Professor Ben Polak: Why did you choose Alpha?Student: I'm sorry?Professor Ben Polak: Why did you choose Alpha? Just repeat what you said before.Student: Because I thought the payoffs - the two different payoffs that I could have gotten--were highest if I chose Alpha.Professor Ben Polak: Good; so what Clara Elise is saying--it's an important idea--is this (and tell me if I'm paraphrasing you incorrectly but I think this is more or less what you're saying): is no matter what the other person does, no matter what the pair does, she obtains a higher payoff by choosing Alpha. Let's just see that. If the pair chooses Alpha and she chooses Alpha, then she gets 0. If the pair chooses Alpha and she chose Beta, she gets -1. 0 is bigger than -1. If the pair chooses Beta, then if she chooses Alpha she gets 3, Beta she gets 1, and 3 is bigger than 1. So in both cases, no matter what the other person does, she receives a higher payoff from choosing Alpha, so she should choose Alpha. Does everyone follow that line of reasoning? That's a stronger line of reasoning then the reasoning we had earlier. So the woman, I have immediately forgotten the name of, in the red shirt, whose name was -Student: Courtney.Professor Ben Polak: Courtney, so Courtney also gave a reason for choosing Alpha, and it was a perfectly good reason for choosing Alpha, nothing wrong with it, but notice that this reason's a stronger reason. It kind of implies your reason.So let's get some definitions down here. I think I can fit it in here. Let's try and fit it in here.Definition: We say that my strategy Alpha strictly dominates my strategy Beta, if my payoff from Alpha is strictly greater than that from Beta, [and this is the key part of the definition], regardless of what others do.Shall we just read that back? "We say that my strategy Alpha strictly dominates my strategy Beta, if my payoff from Alpha is strictly greater than that from Beta, regardless of what others do." Now it's by no means my main aim in this class to teach you jargon. But a few bits of jargon are going to be helpful in allowing the conversation to move forward and this is certainly one. "Evil gits" is maybe one too, but this is certainly one.Let's draw out some lessons from this. Actually, so you can still read that, let me bring down and clean this board. So the first lesson of the class, and there are going to be lots of lessons, is a lesson that emerges immediately from the definition of a dominated strategy and it's this. So Lesson One of the course is:do not play a strictly dominated strategy. So with apologies to Strunk and White, this is in the passive form, that's dominated, passive voice. Do not play a strictly dominated strategy. Why? Somebody want to tell me why? Do you want to get this guy? Stand up - yeah.Student: Because everyone's going to pick the dominant outcome and then everyone's going to get the worst result - the collectively worst result.Professor Ben Polak: Yeah, that's a possible answer. I'm looking for something more direct here. So we look at the definition of a strictly dominated strategy. I'm saying never play one. What's a possible reason for that? Let's - can we get the woman there?Student: [inaudible]Professor Ben Polak: "You'll always lose." Well, I don't know: it's not about winning and losing. What else could we have? Could we get this guy in the pink down here?Student: Well, the payoffs are lower.Professor Ben Polak: The payoffs are lower, okay. So here's an abbreviated version of that, I mean it's perhaps a little bit longer. The reason I don't want to play a strictly dominated strategy is, if instead, I play the strategy that dominates it, I do better in every case. The reason I never want to play a strictly dominated strategy is, if instead I play the strategy that dominates it, whatever anyone else does I'm doing better than I would have done. Now that's a pretty convincing argument. That sounds like a convincing argument. It sounds like too obvious even to be worth stating in class, so let me now try and shake your faith a little bit in this answer.。
耶鲁大学开放课程博弈论笔记
耶鲁大学开放课程博弈论笔记博弈论，是一门研究决策者之间互动行为的学科，它在经济学、政治学、社会学等多个领域发挥着重要作用。
耶鲁大学开放课程中的博弈论课程为我们提供了深入理解和掌握博弈论的机会。
在本篇文章中，我将分享我在学习耶鲁大学开放课程博弈论时所做的笔记和心得体会。
一、博弈论的基本概念和原理1.1 构成博弈论的基本要素博弈论研究的基本要素包括玩家、策略和支付。
玩家是博弈中的决策者，策略是玩家可选择的行动方案，支付是博弈的结果对玩家所产生的效用。
1.2 纳什均衡纳什均衡是博弈论中最重要的概念之一。
在一个博弈中，若每个参与者选择了一个策略，并且没有一个参与者愿意改变自己的策略，那么这种策略组合就被称为纳什均衡。
纳什均衡是一个非合作博弈中的稳定状态。
1.3 合作博弈与非合作博弈博弈论可分为合作博弈和非合作博弈两大类。
合作博弈强调玩家之间的合作与协调，而非合作博弈中玩家之间是相互独立的，没有直接的合作关系。
二、博弈论的应用领域2.1 经济学中的博弈论应用在经济学中，博弈论被广泛应用于市场竞争、拍卖、企业策略等方面。
通过博弈论的模型和方法，我们能够更好地理解各种经济行为和市场现象，并提供决策方案。
2.2 政治学中的博弈论应用政治学中，博弈论主要应用于研究选举、政策制定等政治行为。
博弈论揭示了政治参与者之间的互动关系和利益博弈，为我们分析政治决策提供了一种新的视角。
2.3 社会学中的博弈论应用博弈论在社会学中的应用主要涉及合作与互助、社会规范等方面。
通过博弈论的分析，我们能够更好地理解人类社会中的合作关系、道德行为和社会规范的形成。
三、耶鲁大学开放课程博弈论学习心得在学习耶鲁大学开放课程博弈论的过程中，我深刻体会到博弈论的重要性和应用广泛性。
通过学习博弈论，我不仅了解了博弈论的基本概念和原理，还学会了运用博弈论的方法分析和解决实际问题。
耶鲁大学开放课程博弈论课程的教学内容十分丰富，通过生动的案例分析和实践操作，课程帮助我更好地理解了博弈论的核心思想和应用方法。
(完整版)耶鲁大学公开课博弈论原版资料
Syllabusby (course_default) — last modified 10-14-2008 04:00 PMDocument Actions•This course is an introduction to game theory and strategic thinking. Ideas such as dominance, backward induction, Nash equilibrium, evolutionary stability, commitment, credibility, asymmetric information, adverse selection, and signaling are discussed and applied to games played in class and to examples drawn from economics, politics, the movies, and elsewhere.ECON 159: Game Theory (Fall, 2007)SyllabusProfessor:Ben Polak, Professor of Economics and Management, Yale University Description:This course is an introduction to game theory and strategic thinking. Ideas such as dominance, backward induction, Nash equilibrium, evolutionary stability, commitment, credibility, asymmetric information, adverse selection, and signaling are discussed and applied to games played in class and to examples drawn from economics, politics, the movies, and elsewhere.Texts:A. Dixit andB. Nalebuff. Thinking Strategically, Norton 1991J. Watson. Strategy: An Introduction to Game Theory, Norton 2002P.K. Dutta. Strategies and Games: Theory And Practice, MIT 1999 Requirements:Who should take this course?This course is an introduction to game theory. Introductory microeconomics (115 or equivalent) is required. Intermediate micro (150/2)is not required, but it is recommended. We will use calculus (mostly one variable) in this course. We will also refer to ideas like probability and expectation. Some may prefer to take the course next academic year once they have more background. Students who have already taken Econ 156b should not enroll in this class.Course Aims and Methods.Game theory is a way of thinking about strategic situations. One aim of the course is to teach you some strategic considerations to take into account making your choices. A second aim is to predict how other people or organizations behave when they are in strategic settings. We will see that these aims are closely related. We will learn new concepts, methods and terminology. A third aim is to apply these tools to settings from economics and from elsewhere. The course will emphasize examples. We will also play several games in class.Outline and Reading.Most of the reading for this course comes from the first ten chapters of Dutta or from the first two parts of Watson. There will be a reading packet for weeks 6-7. The readings are not compulsory, but they will help back up the class material.Grading:Problem sets: 30%Midterm examination: 30%Final examination: 40%Transcript 1 - Introduction: five first lessonsby mvd4 — last modified 09-15-2011 09:34 AMDocument Actions•We introduce Game Theory by playing a game. We organize the game into players, their strategies, and their goals or payoffs; and we learn that we should decide what our goals are before we make choices. With some plausible payoffs, our game is a prisoners' dilemma. We learn that we should never choose a dominated strategy; but that rational play by rational players can lead to bad outcomes. We discuss some prisoners' dilemmas in the real world and some possible real-world remedies. With other plausible payoffs, our game is a coordination problem and has very different outcomes: so different payoffs matter. We often need to think, not only about our own payoffs, but also others' payoffs. We should put ourselves in others' shoes and try to predict what they will do. This is the essence of strategic thinking.Game Theory: Lecture 1 TranscriptSeptember 5, 2007 << backChapter 1. What Is Strategy? [00:00:00]Professor Ben Polak:So this is Game Theory Economics 159. If you're here for art history, you're either in the wrong room or stay anyway, maybe this is the right room; but this is Game Theory, okay. You should have four handouts; everyone should have four handouts. There is a legal release form--we'll talk about it in a minute--about the videoing. There is a syllabus, which is a preliminary syllabus: it's also online. And there are two games labeled Game 1 and Game 2. Can I get you all to look at Game 1 and start thinking about it. And while you're thinking about it, I am hoping you can multitask a bit. I'll describe a bit about the class and we'll get a bit of admin under our belts. But please try and lookat--somebody's not looking at it, because they're using it as a fan here--so look at Game 1 and fill out that form for me, okay?So while you're filling that out, let me tell you a little bit about what we're going to be doing here. So what is Game Theory? Game Theory is a。
博弈论读书笔记
1、 人面对自然物时如何行为？寻找人类如何最大化的使用自然物的途径，主要借鉴自然科学所积累的知识及其提升的工具理性。
2、 人面对着他人或社会时如何行为？探究如何充分运用人的理性以及实现社会需求的最大化，需啊哟分析具体环境下人的行为方式和偏好。
3、 博弈思维的联合理性就具有双重特性：一是相互依存，即博弈中的任何博弈方都受到其他博弈方行为的影响；二是理性行为，即博弈方的决策必定建立在预测其他博弈方的行动之上。
4、 非合作的纳什均衡存在以下问题：纳什均衡的非唯一性；不考虑博弈方的策略选择如何影响对手的策略；允许不可信威胁的存在。
5、 完美信息是指一个博弈方对其他博弈方的行动都有准确的了解，即么个信息集只包含一个值。
完全信息则是指自然不首先行动和自然地初始行动被所有博弈方准确观察到，即没有事前的不确定性。
不完全信息意味着不完美信息，但不完美信息并不意味着不完全信息。
6、 在不完全信息博弈中，首先行动的是“自然”，“自然”决定了博弈方以多大的可能性采取某种行动，由“自然”决定的每个博弈方以多大的可能性采取某种行动的情况只有每个博弈方个人知道，其他博弈方都不知道。
确定博弈是指不存在由“自然”作出行动的博弈，否则就是不确定博弈。
7、 严格占优均衡是指无论对手选择何种策略，均衡状态时的策略都是博弈方的最好选择；纳什均衡则是指在对手不改变当前策略的条件下，均衡状态时的策略是博弈方的最好选择。
8、 在对策G 中，如果策略组合1(,,)n s s **是一个纳什均衡，那么它的严格占优策略在重复剔除过程中就不会被剔除掉。
如果策略组合是剔除的严格占优策略均衡，那么他一定是一个纳什均衡。
9、 一般地，要使得任何有限博弈都存在纳什均衡这一命题，就必须有个前提条件：允许博弈方选择混合策略，即博弈方以一定的概率选择某种策略。
设想在多次反复博弈中，博弈方的最终收益状况可以从平均得益上表现出来。
一般地，如果一个策略规定博弈方在每个给定的信息情况下只选择一种特定的行动，就称该策略为纯策略；相反，如果一个策略规定博弈方在每一个给定的信息情况下以某种概率分布随机地选择不同的行动，就称为混合策略。
博弈论知识点总结完整版
博弈论（一）：基本知识1.1定义:博弈论，又称对策论，是使用严谨的数学模型研究冲突对抗条件下最优决策问题的理论，是研究竞争的逻辑和规律的数学分支。
即，博弈论是研究决策主体在给定信息结构下如何决策以最大化自己的效用，以及不同决策主体之间的均衡。
1.2基本要素：参与人、各参与人的策略集、各参与人的收益函数，是博弈最重要的基本要素。
1.3博弈的分类：博弈论根据其所采用的假设不同而分为合作博弈理论和非合作博弈理论。
两者的区别在于参与人在博弈过程中是否能够达成一个具有约束力的协议（binding agreement）。
倘若不能，则称非合作博弈（Non-cooperative game）。
合作博弈强调的是集体主义，团体理性，是效率、公平、公正；而非合作博弈则主要研究人们在利益相互影响的局势中如何选择策略使得自己的收益最大，强调个人理性、个人最优决策，其结果有时有效率，有时则不然。
目前经济学家谈到博弈论主要指的是非合作博弈，也就是各方在给定的约束条件下如何追求各自利益的最大化，最后达到力量均衡。
博弈的划分可以从参与人行动的次序和参与人对其他参与人的特征、战略空间和支付的知识、信息，是否了解两个角度进行。
把两个角度结合就得到了4种博弈：a、完全信息静态博弈，纳什均衡，Nash(1950)b、完全信息动态博弈，子博弈精炼纳什均衡，泽尔腾（1965）c、不完全信息静态博弈，贝叶斯纳什均衡，海萨尼（1967-1968）d、不完全信息动态博弈，精炼贝叶斯纳什均衡，泽尔腾（1975）Kreps, Wilson(1982) Fudenberg, Tirole(1991)1.4课程主要内容：完全信息静态博弈完全信息动态博弈不完全信息静态博弈机制设计合作博弈1.5博弈模型的两种表示形式：策略式表述(Strategic form), 扩展式表述（Extensive form）1.6占优均衡：a、占优策略：在博弈中如果不管其他参与人选择什么策略，一个参与人的某个策略给他带来的支付值始终高于其他策略，或至少不劣于其他策略，则称该策略为该参与人的严格占优策略或占优策略。
完整版)博弈论知识点总结
完整版)博弈论知识点总结博弈论是研究决策主体在相互作用中做出的决策以及均衡问题的学科。
该学科的研究假设包括：1）决策主体是理性的，会尽可能地最大化自己的收益；2）完全理性是共同知识；3）每个参与者都能对环境和其他参与者的行为形成正确的信念和预期。
博弈中涉及到的变量包括：参与人、行动、战略和信息。
完全信息指每个参与人都了解其他参与人的支付函数，而完美信息则指在博弈过程中，每个参与人都能观察和记忆之前的行动选择。
不完全信息则表示参与人没有完全掌握其他参与人的信息，存在不确定性因素。
博弈与传统决策的区别在于，博弈是决策主体之间的相互作用，需要考虑其他决策者的选择和效用函数。
博弈的表示形式包括战略式博弈和扩展式博弈，其中战略式博弈适用于描述不需要考虑博弈进程的完全信息静态博弈问题，而扩展式博弈则更适用于描述动态博弈问题。
与战略式博弈不同，扩展式博弈更注重参与者在博弈过程中面临的决策问题的序列结构分析，而不是仅关注博弈结果的描述。
扩展式博弈包括参与人集合、参与人的行动顺序、序列结构和参与人的支付函数等要素。
战略式博弈是一种静态模型，而扩展式博弈是一种动态模型。
博弈论可以分为合作博弈和非合作博弈，其中合作博强调团体理性、团体最优决策和效率，而非合作博弈强调个人理性和个人最优决策。
根据参与人行动先后顺序的不同，博弈可以分为静态博弈和动态博弈，后者包括先行动者获得先行动者行动信息的情况。
根据参与人对信息的掌握程度，博弈可以分为完全信息和不完全信息博弈。
根据决策主体对信息的掌握程度和行动的先后顺序，博弈可以分为完全信息静态博弈、完全信息动态博弈、不完全信息静态博弈和不完全信息动态博弈。
不同类型的博弈有不同的均衡类型和求解方法，顺序的不同也会影响均衡结果。
Hotelling价格竞争模型是一种重要的扩展式博弈，用于描述两个企业在同一市场上的价格竞争。
相对应。
占有均衡是指在博弈中存在一组参与人的战略选择，使得每个参与人都无法通过改变自己的战略来提高自己的支付。
(完整word)耶鲁大学博弈论_精简版
第一讲导论—五个入门结论1。
通过成绩博弈模型可以知道，不选择严格劣势策略，因为每次博弈会得到更好的收益.2。
通过囚徒的困境博弈模型可以知道，理性选择导致次优的结果（协商难以达成目的的原因不是因为缺少沟通，而是没有强制力）。
3。
通过愤怒天使博弈模型可以知道，汝欲得之,必先知之；永远选择优势策略，选择非劣势策略，损失小，如果对手有优势策略则应以此作为选择策略的指导.4.如果想要赢，就应该站在别人的立场去分析他们会怎么做.第二讲学会换位思考1.构成博弈要素包括，参与人,参与人的策略以及收益.2。
所谓严格优势策略，就是指不论对方采取什么策略，采取的这个策略总比采取其他任何策略都好的策略。
3。
在博弈中剔出某些选择时需要站在别人的角度去思考结果,因为对手不会选择劣势策略；同时要考虑到对手也是一个理性的参与人。
4.在博弈中剔除某些选择是一种直接思考，同时也是作为一个理性参与人的选择。
第三讲迭代剔除和中位选民定理1。
在选民投票博弈模型中,通过不断地迭代以及剔除来决定策略，由此,我们得到了一种新的选择策略的方法：迭代剔除法。
2.选民投票博弈模型的结果与现实存在偏差，主要是因为：现实中选民并不是均匀分布的;选民通常根据候选人的性格而非政治立场来进行投票，而政治立场只是单一维度；只适用于只有两个候选人的情况;④同时存在弃权票；⑤选民未必相信候选人所声明的立场。
3.建立模型，是为了更好的描述事实以激发灵感，模型是有重要的事是抽象而来，逐步增加约束条件完善模型观察结果，比较分析结果的变化。
第四节足球比赛与商业合作之最佳对策1。
点球博弈模型告诉我们，不要选择一个在任何情况或信念下都不是最佳对策的策略。
2.最佳对策：参与人针对对手策略的定义：参与人i的策略s^i（简写成BR）是对手策略S—i的最佳对策，如果参与人i在对手的策略S-i下选S^i的收益弱优于其它对策Si`，这对参与人i的所有Si`都适用，则策略S^i是其它参与人策略S—i的最佳对策。
博弈论
教授就此提到美国总统选举（这个问题本身就是通过政治家选举提出的嘛），尼克松，肯尼迪以及克林顿成功当选总统时演讲的政治立场，都是那种比较中立的，既不保守也不激进（可以把极端保守看作1，把极端激进看作10），从而拉拢中间选民。
经济学领域，这称之为“产品植入”。 /*手有点疼，今天暂时到这，这是注释*/。表现为商家分布普遍集中以争取附近的客源。
好了，来点正经的。
博弈的要素：参与者（player，i，j），策略（strategy，si），策略集（strategy set），策略组合（strategy profile），获益（payoff）
提到了侵略者与防守者之间的关系，这个《博弈游戏》上也有讲过，比如汉尼拔将军翻过阿尔卑斯山，两条路走哪一条。引入部分优势策略：某一策略的获益至少大于等于其他策略，并且至少在一种策略中的获益大于其他策略。
以政治家选举的形式给出了“中位数选民定理”，课堂上的描述很繁琐，我用自己的语言描述一下，其实就是一个数字问题：博弈的两个人各自选择从1到10之间的一个数A和B，用数字相差的单位来表示距离。距离离A近的属于A范围，离B近的属于B范围，与A和B距离相等的数字属于中立，将被平分给A和B。最后谁的范围更大谁获胜。比如A为1，B为2，那么A的范围就是从1到1.5，B的范围是从1.5到10.所以B获胜，也就是选择2的人为数不少的人选择了32，33，34.理由是大家都在1-100之间随机选的话，平均数应该是50，所以平均数的三分之二就是33.但是这种想法的错误之处大家不是随机选择，并且显然不会有人选择比100的三分之二，也就是比66还大的数。所以67到100之间的选择可以直接被剔除。那么既然67不会有人选，大于44，45的数也可以完全被剔除.（平均数不可能超过67，所以其三分之二也不可能超过45，45到67之间的数在原博弈中并不是劣势策略的数，但是去掉67以上这些数之后就是劣势策略了）。所以综上所述，如果大家都是理性的话，不会有人选择超过45的数。那45的三分之二是30，所以30到45之间的数，它们在原博弈里不是劣势策略，即使去掉一次劣势策略也不是劣势策略，但是去掉2次的话就是了。再同理，30的三分之二是20，所以20到30之间的数也。。就这样一直下去，最后会一直到1，这样可能么？
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耶鲁公开课一博弈论笔记第一节、名词解释优势策略（Dominant strategy ）:不论其他局中人采取什么策略，优势策略对一个局中人而言都是最好的策略。
即某些时候它胜于其他策略，且任何时候都不会比其他策略差。
注：1、"优势策略”的优势是指你的这个策略对你的其他策略占有优势，而不是无论对手采用什么策略，都占有优势的策略。
2、采用优势策略得到的最坏的结果不一定比采用另外一个策略得到的最佳的结果略胜一筹。
严格劣势策略（strictly dominated strategy）:被全面的严格优势策略压住的那个策略，也就是说不是严格优势策略以外的策略。
弱劣势策略：原来不是严格劣势策略，但是经过剔除严格劣势策略后，这个策略就成了严格劣势策略。
例：囚徒困境甲沉默｛合作）甲认罪（背叛乙沉默（合作）二人同服刑半年甲即时获释！乙眼刑F评乙认罪（背扳）甲腮刑10年；乙即时获释二炯服刑2年囚徒到底应该选择哪一项策略，才能将自己个人的刑期缩至最短？两名囚徒由于隔绝监禁，并不知道对方选择；而即使他们能交谈，还是未必能够尽信对方不会反口。
就个人的理性选择而言，检举背叛对方所得刑期，总比沉默要来得低。
试设想困境中两名理性囚徒会如何作出选择：若对方沉默、背叛会让我获释，所以会选择背叛。
若对方背叛指控我，我也要指控对方才能得到较低的刑期，所以也是会选择背叛。
二人面对的情况一样，所以二人的理性思考都会得出相同的结论一一选择背叛。
背叛是两种策略之中的支配性策略。
因此，这场博弈中唯一可能达到的纳什均衡，就是双方参与者都背叛对方，结果二人同样服刑2年。
例：协和谬误20 世纪60 年代，英法两国政府联合投资开发大型超音速客机，即协和飞机。
该种飞机机身大、装饰豪华并且速度快，其开发可以说是一场豪赌，单是设计一个新引擎的成本就可能高达数亿元。
难怪政府也会被牵涉进去，竭力要为本国企业提供更大的支持。
项目开展不久，英法两国政府发现：继续投资开发这样的机型，花费会急剧增加，但这样的设计定位能否适应市场还不知道；但是停止研制也是可怕的，因为以前的投资将付诸东流。
随着研制工作的深入，他们更是无法做出停止研制工作的决定。
协和飞机最终研制成功，但因飞机的缺陷（如耗油大、噪音大、污染严重等）以及运营成本太高，不适合市场竞争，英法政府为此蒙受很大的损失。
在研制过程中，如果英法政府能及早放弃，本来可以使损失减少，但他们没能做到。
最后，英国和法国航空公司宣布协和飞机退出民航市场，才算是从这个无底洞中脱身。
这也是“壮士断腕”的无奈之举。
人们往往会陷入类似的误区：一项工作的成本越大，对它的后续投入就越多。
其实不仅是在制造协和飞机这样的重大项目上，就是在日常的生活中，人们在决定是否继续做一件事情的时候，不仅是看它对自己有没有好处，而且也过于注意自己是不是已经在这件事情上面有过投入。
我们把那些已经发生、不可收回的支出，如时间、金钱、精力称为“沉没成本”。
沉没的意思是说，你在正式完成交易之前投入的成本，如果一旦交易不成，就会白白损失掉。
但如果对沉没成本过分眷恋，就会继续原来的错误，造成更大的亏损。
在第一节课中得出的五个结论：1、不要选择劣势策略2、理性选择导致次优结果3 、站在他人立场分析他们会怎么做4 、先弄清你想要的，才能得到你想要的5 、人人都是自私的第二节、囚徒困境的解决之道：1、多次博弈；2、设立规章制度，惩罚违规者；3、思想教育（效果待定）。
博弈的要素：参与者i ;策略S;收益U符号的定义：Si 表示参与者i 的策略。
S-i 表示除参与者i 以外其他人的策略。
Ui 表示i 的收益。
名词解释：共同知识：我知道这件事；你也知道这事；我知道你知道这事；你知道我知道你知道这事这事；此后循环。
案例：老师在课堂上让每位学生从1-100 中选择一个数字。
选择到最接近全班平均数的2/3 的学生为胜利者。
学生共有50 个左右。
胜利者平分奖金5 美元。
解决方案：stepl、假设每个人都选择100，平均数100*2/3 = 66.66。
所以不能选择67-100之间的数（严格劣势策略）。
现实中有两名学生选择了。
step2 、剔除了step1 中的严格劣势策略后，重复迭代，66*2/3 = 44。
所以不能选择44-67之间的数（弱劣势策略）。
现实中有四名学生选择了。
step3、44*2/3 = 29,所以不能选择29-44之间的数。
现实中有13个左右选择了30-34 区间，。
选择这个数区间的学生想法是1-100 平均数是50，50*2/3 = 33,所以选择33附近的数可能比较接近。
这些学生低估了其同班同学的智商。
step4、29*2/3 = 19,所以不能选择19-29之间的数。
现实中有12个选择了。
选择这个区间的学生就像螳螂捕蝉中的螳螂,却没有想到还有更多的黄雀在后。
这么一直迭代下去,理论上如果所有学生都是理性人。
平均数应当是1。
现实中有12个学生选择了1 。
应该说选择了1 的学生都看出了这个博弈的窍门。
但是他们的选择不是最接近平均数的。
因为在现实中不可能所有人都是理性人。
最终1 2是最接近平均数2/3 的数。
有9人选择了这个数。
结论：迭代剔除劣势策略是个好的方法, 但在现实中不能过度迭代。
因为不是所有人都是理性人,而且不是所有人都有共同知识（概念见前述）。
应用案例：中间选民定理两个政治候选人,为了选举须确定自己的政治立场。
共有10个立场：1、2、3、4、5、6、7、8、9、10。
第个立场都有10%选票。
两个候选人要在一系列的政治主张中选择一个。
规则：选民会投票给观点最相近的候选人。
距离相等,该立场平分选票。
候选者希望选票最大化。
step1 :试证明:S2优于S1。
比较1号候选人选择S1,S2其利益U1的大小。
当2 号候选人选择1 号策略S1 时U1(1、1)［表示2号候选人选择S1, 1号候选人选择S1］为50% 号候选人选择S1］为90%< U1(2、1 ) ［表示2号候选人选择S1，1当2 号候选人选择U1(1,2)=10% < 当2 号候选人选择U1(1,3)=15% < 当2 号候选人选择U1(1,4)=20% < 2 号策略S2 时U1(2,2)=50% 3 号策略S3 时U1(2,3)=20% 4 号策略S4 时U1(2,4)=25%F面选择S2得票率都比S1大5%所以S2严格优于S1。
同理S9优于S10b step2 ：试证明：S3 优于S2剔除劣势策略S1 ，当2 号候选人选择U1(2,2)=50% < 当2 号候选人选择U1(2,3)=20% < 当2 号候选人选择U1(2,4)=25% < 当2 号候选人选择U1(2,5)=30% < S102 号策略S2 时U1(3,2)=80%3 号策略S3 时U1(3,3)=50%4 号策略S4 时U1(3,4)=30%5 号策略S5 时U1(3,5)=35%F面选择S3得票率都比S2大5%所以S3严格优于S2。
同理S8优于S9。
所以S2是弱劣势策略，以下同理可证S4优于S3, S5优于S4。
迭代剔除后将剩下S5, S6。
结论：政治家为了赢得更多选票，尤其是大量关键的“中间选民”，会表现的趋中，各个政 治家之间的差别会变得很小。
如美国选举时议题是：0.3%的税收差别，给不给移民发驾照,同性恋能否结婚之类对生活不会有重大影响的事项。
缺陷：在现实中每个立场的选民数非均匀分布；非所有人都投票；选民不只考虑政治立场, 还有性格，甚至外貌；政治候选人的口号与实际行动未必一致；候选人不止两位。
第三节、 之前的几节课中，各个案例都是有严格劣势策略的。
接下来的几个案例中没有严格劣势策略， 通过对这些没有严格劣势策略案例，可以模拟更复杂的现实情况，同时对数学的要求会加深。
例:分，两个玩家都想得到更高的得分。
在这个博弈中没有严格劣势策略，因为当 play2选择不同策略时，playl 的策略中没有一个 是始终劣势于其他策略的。
Plav 1 ■1>Rp5^0-2疳4*1冷4卩 2』2」S1=u,m,dS2=L,R 表格中的数值为play1,2选择不同策略时的得我们可以用画图的方式来分析没有严格劣势策略时Playl 应该如何选择策略的案例。
的如图：X轴P (r)表示、play2选择R策略的概率；Y轴表示playl的预期得分。
当P (r) = 0时，就是说play2选择L策略。
playl选择u,m,d时的得分分别是5, 1, 3。
当P(r)= 100%寸，就是说play2选择R策略。
playl选择u,m,d时的得分分别是0,4, 2。
将这六个点分别在图中标出，然后连成直线。
就得出了三个函数：U1(u,p(r))=5-5p(r); ... ...(playl 选择u 策略时，得分随play2选择R策略的概率变化而变化的函数)U1(m,p(r))=3p(r)+1;U1(d,p(r))=-2p(r)+4.其中三条直线有三个交点，分别位于P (r )= 1/3 ; 1/2 ; 3/5 三处。
结论：从图中可以看出，Play1要得分最高，要根据Play2的P(r)不同分三段来选择策略。
当P (r)小于1/3时，应该选择u策略；当P (r)大于1/3小于3/5时；应该选择中间的线外外代表的策略d;当P (r)大于3/5时，应该选择m策略。
上面这个案例是一个纯理论阐述，下面介绍足球比赛中点球时，射手应该如何选择的问题。
这个案例的数据是基于实际比赛中的统计数据。
例：点球守门员+表格中前列数字表示射手射中球的概率， 如4表示40%中球率。
L 表示左，R 表示右，M 表示中。
用前例中的方法画图:射手从图中可得出：为得到最高的点球成功率，当P(r)<50%时，应该射手应该选择踢左边；当P(r)》50%寸，应该射手应该选择踢右边；表示踢中路成功率的那条线始终没有最高概率， 所以射手最好不要选择踢中路。
这个模型的缺陷：没有考虑射手踢球的习惯； 没有考虑守门员守中路的情况 (考虑三个要素很复杂，而且中路是可以排除的严格劣势策略)；没有考虑球速。
最佳对策定义：Ui(SiA,S-i)>=Ui(Si'.S-i)Si A 表示对手策略 S-i 的最佳对策。
Si'表示Play i 的其它对策。
第四节、 例 合伙人博弈:2个股东都持有公司 50%殳份；两者平分利润；每个股东要选择为公司投入多少时间，用工作小时数代表双方策略 Si=(0,4)[0~4是连续的数，而非只能选整数]，双方可以在 0至4个小时之间选择。
这家公司利润：4*[S1+S2+b*S1*S2] (0<b<1/4); {S1+S2可以表示两个股东工作时间的简单相加对利润的贡献，b*S1*S2可以表示由于两个股东相互协作对利润的贡献；考虑到了这两个部分，所以这个公式可以很好的反映现实的情况}所以，U1(s1,s2)=1/2[4*(S1+S2+b*S1*S2)]-S1 的平方。