博弈论课文翻译
博弈论(哈佛大学原版教程)
Note, that a pure Nash equilibrium is a (degenerate) mixed equilibrium, too.
1
the behavior of real-world people rather than come up with an explanation of how they should play a game, then the notion of NE and even even IDSDS can be too restricting. Behavioral game theory has tried to weaken the joint assumptions of rationality and common knowledge in order to come up with better theories of how real people play real games. Anyone interested should take David Laibson’s course next year. Despite these reservation about Nash equilibrium it is still a very useful benchmark and a starting point for any game analysis. In the following we will go through three proofs of the Existence Theorem using various levels of mathematical sophistication: • existence in 2 × 2 games using elementary techniques • existence in 2 × 2 games using a fixed point approach • general existence theorem in finite games You are only required to understand the simplest approach. The rest is for the intellectually curious.
博弈论
对 C 来 说 左 优 于 右
田忌赛马
• 比赛规则：马按奔跑的速度分为上中下三等，等次 不同装饰不同，各家的马依等次比赛，三赛二胜。 • 比赛前田忌按照孙膑的主意，第一场，用上等马鞍 将下等马装饰起来，冒充上等马， 与齐威王的上 等马比赛；第二场，用上等马与齐威王的中等马比 赛；第三场，田忌的中等马和齐威王的下等马比赛。 结果二比一，田忌赢了齐威王。 • 在双方条件相当时，对策得当可以战胜对方；在双 方条件相差很远时，对策得当也可将损失减低到最 低程度。
均衡策略
• 指的是只要另一方不变，双方进行 任何改变都不会使自己的状况更优。
C 左 右
上 R 下
（4，3） （10，6）
（12，8） （5，4）
囚徒困境模型
两人因盗窃被捕，警方怀疑其有抢劫行 为但未获得确凿证据可以判他们犯了抢劫罪， 除非有一人供认或两人都供认。即使两人都 不供认，也可以判他们犯盗窃物品的轻罪。 囚徒被分离审查，不允许他们之间或通 信息，并交代政策如下：如果两人都供认， 每个人都将因抢劫罪加盗窃罪被判3年监禁； 如果两人都拒供，则两人都将因盗窃罪被判 半年监禁；如果一人供认而另一个拒供，则 供认这被认为有功而免受处罚，拒供者将因 抢劫罪、盗窃罪以及拒供重判5年。
由于每个囚徒都发现供认是自己更好的选择，因此， 博弈的稳定结果是两个囚徒都会选择供认。这就是博 弈的纳什均衡。
攻守同盟？ 很难达成：隔离审查，每个人都担心对方背弃盟约。
囚徒困境的启示
“囚徒的两难选择”有着广泛而深刻的意义。
个人理性与集体理性的冲突，各人追求利己行为 而导致的最终结局是一个“纳什均衡”，也是对 所有人都不利的结局。他们两人都是在坦白与抵 赖策略上首先想到自己，这样他们必然要服长的
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.。
1博弈论第一章
n n
n
厂商i的收益：
i 1
qi P( qi ) cqi qi [ P( qi ) c]
i 1
厂商i的收益不仅与自己既定成本和产量有关，还与 其他厂商的产量决策有关。
1.3.1 博弈中的参与人 1.3.2 博弈中的策略 1.3.3 博弈中的收益 1.3.4 博弈的过程 1.3.5 博弈的信息 1.3.6 参与人的能力和理性 1.3.7 博弈的分类
哲理：相生相克，以柔克刚
猜硬币方 正 面 反 面 1， -1 -1， 1 参 与 人 1 石 头 剪 子 布 石 头 0， 0 -1， 1 1， -1 参与人2 剪 子 1， -1 0， 0 -1， 1 布 -1， 1 1， -1 0， 0
盖 硬 币 方
正 面 反 面
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1.2.3 产量决策Cournot模型
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1.3.1 博弈中的参与人
参与人：独立决策、独立承担博弈结果的个 人或组织。只要在一个博弈中统一决策、统一行动、
统一承担结果，不管一个组织有多大，甚至大到一个国家 或多国，都可以作为一个参与人。囚徒困境中的警察、田 忌赛马中的孙膑都不是参与人。
一、单人博弈——只有一个参与人的博弈
严格地讲，单人博弈由于不存在其他参与人的反应和反 作用，因此不属于博弈论的研究对象。但是讨论单人 博弈会使理论更完整，为多人博弈提供基础和启示。 例一：单人迷宫 扩展型
入口 右 A B 0 出口(奖金M) 右 M
A,1
左 B,1
博弈规则面前参与人之间平等，不因参与 人之间权利、地位的差异而改变 参与人数量对博弈结果和分析有影响。根
两个囚徒的收益矩阵
[讲解]博弈论课文翻译
![[讲解]博弈论课文翻译](https://img.taocdn.com/s3/m/85e5e03fbc64783e0912a21614791711cc79790f.png)
搏弈论阿维纳什•迪克斯特&巴里•内尔巴夫1搏弈是有关策略的科学。
它试图以数学和逻辑的方法来帮助搏弈者作出决策，在一系列纷繁复杂的搏弈中应采取何种策略来保证自己获得最大利益。
搏弈论研究的搏弈的范围包括了从下棋到抚育儿童，从网球竞技到公司转手。
但是所有的博弈都具有一个共同的特征：相互作用。
也就是说,每一个博弈者的博弈结果取决于所有博弈参与者的策略选择。
在零和搏弈中，搏弈者的利益之间是完全冲突的，因此一方的得利必然导致另一方的损失。
更多具有代表性的例子还有会导致共同得利(正和)搏弈和共同损失(负和)搏弈，同样的情况还会发生在另外一些冲突中。
2搏弈论研究的先驱者是普林斯顿数学家约翰•冯•诺依曼。
在早先的一段时间里，研究的重点被放在了完全冲突(零和)搏弈(非合作搏弈)上，其他的搏弈当时被认为是以合作形式出现。
也就是说，搏弈要求参与者共同地选择和实施他们的行为.最近的研究则把重点放在了那些既不属于零和搏弈也不属于绝对合作搏弈的情况上，在这些搏弈中，搏弈者自主地选择搏弈行为，但他们之间的相互关系中充满了合作与竞争。
3搏弈行为与我们在中性环境中所作的各种决定有着根本性的不同。
要说明这一点，我们可以思考一下伐木工人和军队将军所作决定之间的不同。
当伐木工人决定要如何砍树时，他不会考虑树木本身会有什么反抗，他所处的环境为中性。
而当将军决定要消灭敌军时，他必须提前预料到并消除敌军的反抗。
与这一例子中的将军相类似，一个搏弈者必须认识到他与其他机智且怀有争胜之心的竞争者之间的相互作用，他自己所作的决定也必须能够同时应对可能出现的合作或冲突。
‖4搏弈的实质是搏弈者采取策略之间的相互依赖性。
这种策略性的相互依赖表现为两个不同的类别：连续策略之间的相互作用以及联立策略之间的相互作用。
就前者而言，搏弈者依次采取行动，每个人都会注意其他搏弈者先前的行为。
就后者而言，搏弈者同时采取行动，每个人都会忽略其他搏弈者当前的行为。
5对连续策略博弈中的某一博弈这来说，一个普遍的原则就是放眼前方，及时反思和总结。
博弈论 Game theory (全)
博弈论 Game Theory博弈论亦名“对策论”、“赛局理论”，属应用数学的一个分支, 目前在生物学，经济学，国际关系，计算机科学, 政治学，军事战略和其他很多学科都有广泛的应用。
在《博弈圣经》中写到：博弈论是二人在平等的对局中各自利用对方的策略变换自己的对抗策略，达到取胜的意义。
主要研究公式化了的激励结构间的相互作用。
是研究具有斗争或竞争性质现象的数学理论和方法。
也是运筹学的一个重要学科。
博弈论考虑游戏中的个体的预测行为和实际行为，并研究它们的优化策略。
表面上不同的相互作用可能表现出相似的激励结构（incentive structure），所以他们是同一个游戏的特例。
其中一个有名有趣的应用例子是囚徒困境（Prisoner's dilemma）。
具有竞争或对抗性质的行为称为博弈行为。
在这类行为中，参加斗争或竞争的各方各自具有不同的目标或利益。
为了达到各自的目标和利益，各方必须考虑对手的各种可能的行动方案，并力图选取对自己最为有利或最为合理的方案。
比如日常生活中的下棋，打牌等。
博弈论就是研究博弈行为中斗争各方是否存在着最合理的行为方案，以及如何找到这个合理的行为方案的数学理论和方法。
生物学家使用博弈理论来理解和预测演化（论）的某些结果。
例如，约翰·史密斯（John Maynard Smith）和乔治·普莱斯（George R. Price）在1973年发表于《自然》杂志上的论文中提出的“evolutionarily stable strategy”的这个概念就是使用了博弈理论。
其余可参见演化博弈理论（evolutionary game theory）和行为生态学（behavioral ecology）。
博弈论也应用于数学的其他分支，如概率，统计和线性规划等。
历史博弈论思想古已有之，我国古代的《孙子兵法》就不仅是一部军事著作，而且算是最早的一部博弈论专著。
博弈论最初主要研究象棋、桥牌、赌博中的胜负问题，人们对博弈局势的把握只停留在经验上，没有向理论化发展。
博弈论(部分英文版翻译)
博弈论(部分英文版翻译)博弈论托马斯·S.Ferguson/translator:·xly第一部分:公平组合游戏1.外卖游戏1.1简单的外卖游戏1.2什么是组合游戏？1.3 P状态和N状态1.4游戏1.5相关练习2.尼姆游戏初步分析尼姆和多堆尼姆游戏布顿理论证明守财奴版尼姆游戏相关练习3.图形游戏有向图形游戏SG函数相关例子的一般图的SG函数4.组合游戏和N图游戏及SG定理的相关应用与休息游戏相关的练习5.硬币游戏的例子二维空间中的硬币旋转游戏尼姆复杂的网格游戏练习6.绿色哈肯布什竹竿树木上的绿色哈肯布什普通根图练习的绿色引导参考材料第一部分:公平组合游戏1。
外卖游戏组合游戏是两人游戏。
如果有足够的条件，当一方不能继续经营时，游戏的结果就会出来。
这个游戏的结果取决于一系列的状态，包括初始状态和准备操作的玩家。
游戏双方轮流操作，直到达到最终状态。
最终状态意味着该状态不能再运行。
此时，结果已经出现分歧。
这里有两个关于组合游戏的主要材料。
一部是康威的《论数字与游戏》，学术出版社1976年出版。
这本书介绍了这一领域的许多基本思想，加速了这一领域今天的发展。
另一本更适合这门课的参考书是学术出版社于1982年出版的两卷本平装本，书名是《柏林坎普、康威和盖伊的数学游戏制胜之道》。
这本书介绍了许多有趣的游戏，学习数学的本科生可以理解。
这些理论可以分为两类。
公平游戏指的是任何给定的状态，游戏双方要采取的行动是相同的。
另一方面，游击队游戏意味着给定一个状态，游戏双方将采取不同的行动。
例如，国际象棋是一种游击队游戏。
在第一部分，我们只研究“公平竞争”。
公平组合游戏的介绍可以在理查德·盖伊写的公平游戏中找到(发表在1989年的COMAP数学探索系列中)。
让我们从一个简单的例子开始。
1.1一个简单的外卖游戏。
这是这个公平组合游戏的一些规则(从一堆筹码中取一些):(1)有两个玩家，我们分别将他们标记为1号和2号；(2)桌上有一堆筹码，总共21个筹码；(3)一次操作可以取1、2、3个筹码，至少要取一个筹码，最多要取3个筹码。
英语第一章阅读 game theory 原文及翻译
The essence of a game is the interdependence of player strategies. There are two distinct types of strategic interdependence: sequential and simultaneous. In the former the players move in sequence, each aware of the others’ previous actions. In the latter the players act at the same time, each ignorant of the others’ actions.
Game theory was pioneered by Princeton mathematician john von Neumann. In the early years the emphasis was on games of pure conflict (zero-sum games). Other games were considered in a cooperative form. That is, the participants were supposed to choose and implement their actions jointly. Recent research has focused on games that are neither zero sum nor purely cooperative. In these games the players choose their actions separately, but their links to others involve elements of both competition and cooperation.
博弈论-耶鲁大学公开课第一次课电子版
So this is Game Theory Economics 159.欢迎来到经济学159 博弈论If you're here for art history,如果你是来上艺术史的you're either in the wrong room or stay anyway,那你走错教室了不过你不妨留下来maybe this is the right room; but this is Game Theory, okay.没准你会喜欢这门课这里是博弈论You should have four handouts; everyone should have four handouts.你们每人都应该有四页讲义人手一份There is a legal release form--里面有一授权书we'll talk about it in a minute--about the videoing.是关于视频采集的我们一会再说这个There is a syllabus, which is a preliminary, it's also online.有一初步教学大纲上也有And there are two games labeled Game 1 and Game 2.有两页分别印有游戏1和游戏2的传单Can I get you all to look at Game 1 and start thinking about it.大家先浏览下游戏1 然后思考一下And while you're thinking about it, i am hoping you can multitask a bit. 你们一边思考一边听我说I'll describe a bit about the class我来简单介绍下这门课程and we'll get a bit of admin under our belts.还有这门课程的评分细则But please try and look at--somebody's not looking at it,请大家看下游戏1 我知道有人没看because they're using it as a fan here--他们用那纸扇风呢So look at Game 1 and fill out that form for me, okay?快浏览下游戏1 然后填完好吗So while you're filling that out, let me tell you a little你们一边写一边听我说bit about what we're going to be doing here.我来简单地介绍一下这门课So what is Game Theory?什么是博弈论Game Theory is a method of studying strategic situations.博弈论研究策略形势So what's a strategic situation?那什么是策略形势呢Well let's start off with what's not a strategic situation.我们先看看什么不属于策略形势In your Economics - in your Intro Economics class in 115 or 110,在你们经济学基础课115或者110中you saw some pretty good examples of situations that were not strategic. 应该讲过一些不是策略形势的案例You saw firms working in perfect competition.比如说自由竞争企业Firms in perfect competition are price takers:这些企业是价格承受者they don't particularly have to worry about他们不必担心the actions of their competitors.他们的竞争对手的策略You also saw firms that were monopolists又比如说垄断企业and monopolists don't have any competitors to worry about,垄断企业没有竞争对手so that's not a particularly strategic situation.所以这也不是策略形势They're not price takers but they take the demand curve.他们不是价格承受者但面对需求曲线Is this looking familiar for some of you who can remember对于你们这些学过经济学115的来说doing 115 last year or maybe two years ago for some of you?这些听起来都不陌生吧Everything in between is strategic.介于这两种情况之间的就是策略形势So everything that constitutes imperfect competition也就是说不完全竞争的情况is a strategic setting.就是策略形势Think about the motor industry, the motor car industry.比如汽车产业在汽车产业里Ford has to worry about what GM is doing and what Toyota is doing,福特关注通用和丰田的决策and for the moment at least现在可能暂时what Chrysler is doing but perhaps not for long.还得关注克莱斯勒的决策So there's a small number of firms and their actions affect each other. 少数几家公司的决策会互相影响So for a literal definition of what strategic means:策略形势书面定义是it's a setting where the outcomes that affect you depend on actions, 行为影响结果然而结果不仅not just on your own actions, but on actions of others.取决于你的行为还取决于其他人的行为All right, that's as much as I'm going to say for preview right now. 好了暂时就先介绍这么多We're going to come back and see plenty of this通过学习本学期的课程over the course of the next semester.你们会接触更多类似案例So what I want to do is get on to where this applies.下面我们讲讲博弈论适用于什么领域It obviously applies in Economics, but it also applies in politics, 显然适用于经济学但也适用于政治学and in fact, this class will count as a Political Science class实际上对政治专业的同学来说if you're a Political Science major.它也算作政治课You should go check with the DUS in Political Science. It counts.回去咨询下本科教学主管确实算的Game Theory is very important in law these days.近年来博弈论在法学上也举足轻重So for those of you--for the half of you我估计你们多数人可能会去读法学院--that are going to end up in law school, this is pretty good training. 博弈论对你们来说是门不错的基础课Game Theory is also used in biology博弈论同样适用于生物学and towards the middle of the semester在本学期期中的时候we're actually going to see some examples of我们会探讨一下Game Theory as applied to evolution.博弈论应用于生物进化论的案例And not surprisingly, Game Theory applies to sport.而且毫不奇怪博弈论适用于体育So let's talk about a bit of admin.我们接下来谈谈评分标准How are you doing on filling out those games?这两个游戏你们填好了没有Everyone managing to multitask.每个人都要学会一心多用啊Filling in Game 1? Keep writing.还在填游戏1 好吧继续填I want to get some admin out of the way我们继续讲评分标准and I want to start by getting out of the way不过我得先介绍一下what is obviously the elephant in the room.教室里的大家伙Some of you will have noticed that there's a camera crew here, okay. 有些人注意到了这几位摄像人员So as some of you probably know,可能你们已经知道了Yale is undergoing an open education project耶鲁大学正在开展一个教育项目and they're videoing several classes,我们在录制一些课程的录像and the idea of this,这样做的目的是让is to make educational materials available beyond the walls of Yale. 耶鲁外的人也能享受到这些教学资源In fact, on the web, internationally,课程将被发布到网络上so people in places, maybe places in the U.S.全球各地的人都能看到比如美国人or places miles away, maybe in Timbuktu or whatever,或者更遥远的国度像廷巴克图who find it difficult to get educational materials他们可能在当地大学from the local university or whatever,不容易得到这样的教育资源can watch certain lectures from Yale on the web.他们就可以到耶鲁观看这些视频Some of you would have been in classes that do that before.有些同学可能之前参加过这种课程What's going to different about this class is that这门课的不同之处在于you're going to be participating in it.大家都要参与到课程中来The way we teach this class is we're going to play games,课堂上我们通过做游戏来博弈we're going to have discussions, we're going to talk among the class, 课上会有随堂讨论课堂交流and you're going to be learning from each other,我们相互学习这样看视频的学生and I want you to help people watching at home to be able to learn too. 也能收到良好的教学效果And that means you're going to be on film, at the very least on mike. 但同时这意味着你要被拍到视频中来So how's that going to work?至少也会被录音课程怎么拍摄呢Around the room are three T.A.S holding mikes.教室里有三位助教持有麦克风Let me show you where they are: one here, one here, and one here.我来告诉大家助教坐在哪里When I ask for classroom discussions,当我们进行随堂讨论时I'm going to have one of the T.A.S go to you with a microphone我会请一位助教拿着麦克风到你那much like in "Donahue" Or something. Okay.就想多纳休那样At certain times, you're going to be seen on film,有时候你会被拍摄到so the camera is actually going to come around and point in your direction. 此时摄像机会转到你所在的方向Now I really want this to happen.我希望大家多多配合I had to argue for this to happen, cause I really feel that我来解释一下我的理由我觉得this class isn't about me.这门课不应该仅仅由我来讲I'm part of the class obviously,没错我是课程的一部分but it's about you teaching each other and participating.但你们也要参与到课堂中来But there's a catch, the catch is,但是这会有个条件that that means you have to sign that legal release form.你们都要签署授权书So you'll see that you have in front of you a legal release form,你们的面前应该有授权书you have to be able to sign it, and what that says is that授权书的大概容是we can use you being shown in class.允许我们把你拍摄进来Think of this as a bad hair day release form.大家考虑好日后你要是在视频里All right, you can't sue Yale later if you had a bad hair day.发现你糟糕的发型可别起诉耶鲁大学For those of you who are on the run from the FBI,被FBI通缉的签证失效的your Visa has run out, or you're sitting next to your ex-girlfriend, 还有前女友就坐在旁边的now would be a good time to put a paper bag over your head.赶快找个纸袋子把脸给蒙上All right, now just to get you used to the idea,还要说一下每节课我们都会in every class we're going to have I think the same two people,看到这两名工作人员so Jude is the cameraman; why don't you all wave to Jude.裘德是摄影师大家跟裘德打个招呼This is Jude okay. And Wes is our audio guy: this is Wes.这位就是裘德韦斯是录音师And I will try and remember not to我会尽量注意include Jude and Wes in the classroom discussions,不要让他们两位加入课堂讨论的but you should be aware that they're there.但是你们也要注意到他们的存在Now, if this is making you nervous,要是这令你很紧if it's any consolation, it's making me very nervous.它会让我更紧这权当安慰了吧So, all right, we'll try and make this class我会尽量让work as smoothly as we can, allowing for this extra thing.课堂顺利地进行下去还有一件事Let me just say, no one's making any money off this--让我说完没有人会从中得到好处at least I'm hoping these guys are being paid--虽然我也希望他们会得到酬劳but me and the T.A.S are not being paid.但至少我和助教们是没有酬劳的The aim of this, that I think is a good aim, it's an educational project, 这完全出于好意这是个教育项目and I'm hoping you'll help us with it.我也希望你们也能尽己所能把它做好The one difference it is going to mean, is that还有希望大家都能相互理解一下at times I might hold some of the discussions for the class,当我们举行课堂讨论的时候coming down into this part of the room, here,请你们靠前站出来一点to make it a little easier for Jude.这样便于裘德拍摄All right, how are we doing now on filling out those forms?言归正传你们填好了没Has everyone filled in their strategy for the first game?各位是否都填好第一个游戏了Not yet. Okay, let's go on doing a bit more admin.还没有那我们多介绍下评分标准The thing you mostly care about i'm guessing, is the grades.我猜你们最关心的应该是成绩了All right, so how is the grade going to work for this class?那么这门课是怎么评分的呢30% of the class will be on problem sets,课堂作业占总分数的30%30% of the grade; 30% on the mid-term,期中考试占30%and 40% on the final; so 30/30/40.期末考试占40% 即30/30/40The mid-term will be held in class on October 17;期中考试在十月十七日that is also in your syllabus.教学大纲上也写了Please don't anybody tell me late -所以以后千万别跟我说any time after today you didn't know when the mid-term你不知道什么时候期中考试was and therefore it clashes with 17 different things.然后还在考试那天安排大一堆事儿去做The mid-term is on October 17, which is a Wednesday, in class.期中考试在十月十七号周三随堂举行All right, the problem sets:关于课堂作业there will be roughly ten problem sets我应该会安排十次左右的课堂作业and I'll talk about them more later on when I hand them out.在留作业的时候我再细说The first one will go out on Monday but it will be due ten days later. 周一留第一份作业十天之交上来Roughly speaking they'll be every week.差不多每周都有课堂作业The grade distribution:有关分数的分布all right, so this is the rough grade distribution.我介绍下大体的分数的分布Roughly speaking, a sixth of the class are going to end up with A's, 总体来说六分之一的人会得到Aa sixth are going to end up with A-,六分之一的人会得到A-a sixth are going to end up with B+,六分之一的人会得到B+a sixth are going to end up with B,六分之一的人会得到Ba sixth are going to end up with B-,六分之一的人会得到B-and the remaining sixth, if I added that up right,剩下的六分之一如果我没算错are going to end up with what I guess we're now calling这些人将会得到the presidential grade, is that right?传说中的"超优"成绩That's not literally true. I'm going to squeeze it a bit,但我可能稍微压缩下分数段I'm going to curve it a bit,我会让分数段更有弹性so actually slightly fewer than a sixth will get straight A's,实际上略少于六分之一的人会得到Aand fewer than a sixth will get C's and below.略少于六分之一会得到CWe'll squeeze the middle to make them be more B's.我们这样压缩会使更多的人得到BOne thing I can guarantee from past experience in this class,不过从教学经验看is that the median grade will be a B+.中间成绩应该是B+The median will fall somewhere in the B+'s.分数中位数会在B+的附近Just as forewarning for people who have forgotten what a median is, 稍微提示下忘记中位数含义的同学that means half of you--not approximately half,这也就是说不是一半左右it means exactly half of you--是绝对一半的同学will be getting something like B+ and below会得到B+的成绩或更低的成绩and half will get something like B+ and above.另一半会到B+的成绩或更高的成绩Now, how are you doing in filling in the forms?那两表你们填得怎么样了Everyone filled them in yet?每个人都填好了吗Surely must be pretty close to getting everyone filled in.应该差不多每个人都填好了吧All right, so last things to talk about在我把表收回之前before I actually collect them in - textbooks.我再讲最后一件事课本There are textbooks for this class.这门课的课本The main textbook is this one, Dutta's book Strategy and Games.主要是这本杜塔的《策略与博弈》If you want a slightly tougher book, more rigorous book,要是你想要一本更难更有挑战的try Joel Watson's book, Strategies.试试乔尔?沃森的《策略》Both of those books are available at the bookstore.这两本书在书店都有卖的But I want to warn everybody ahead of time,不过我先提个醒I will not be following the textbook.我不会完全参照课本I regard these books as safety nets.课本只不过是救命稻草If you don't understand something that happened in class,要是你课堂上没听懂you want to reinforce an idea that came up in class,或者你课下想深入了解下课上的容then you should read the relevant chapters in the book你应该去读书上对应的章节and the syllabus will tell you which chapters to read for each class, 教学大纲会告诉你每堂课讲哪章or for each week of class, all right.或者每周讲哪几章But I will not be following these books religiously at all.但是我不会严格按照课本来讲课In fact, they're just there as back up.重申一下课本就是救命稻草In addition, I strongly recommend people read Thinking Strategically. 对了我强烈建议读一读《战略思想》This is good bedtime reading. Do any of you suffer from insomnia?这是本不错的睡前读物你们失眠吗It's very good bedtime reading if you suffer from insomnia.你要是失眠它绝对是最好的睡前读物It's a good book and what's more这也的确是本好书there's going to be a new edition of this book this year而且今年会有新版上市and Norton have allowed us to get advance copies of it.出版商会让我们提前得到新版的So if you don't buy this book this week,如果你这周不买I may be able to make the advance copy of the new可能下周你们edition available for some of you next week.就能买到新版的了I'm not taking a cut on that either, all right,我不会从中得到提成there's no money changing hands.这不是推销All right, sections are on the syllabus sign up小组讨论的安排在教学大纲上sorry on the website, sign up as usual.哦我说错了像平常一样在网上注册Put yourself down on the wait list如果你想换个小组if you don't get into the section you want.请在等候名录里预约You probably will get into the section you want once we're done.一般你只需要预约一次就可以了All right, now we must be done with the forms.你们现在应该填完了吧Are we done with the forms?都填完了吗All right, so why don't we send the T.A.S, with or without mikes,我们把表交给助教吧up and down the aisles and collect in your Game #1;让他们来帮我收一下游戏1not Game #2, just Game #1.不要游戏2 只要游戏1Just while we're doing that, I think the reputation of this class-- 收表的时候我来说说这门课的名声I think--if you look at the course evaluations online or whatever,你们要是在上看了教学评估is that this class is reasonably hard but reasonably fun.就会发现这门课有点难但也很有趣So I'm hoping that's what the reputation of the class is.我也希望这门课是这样的If you think this class is going to be easy,你要是认为这门课挺简单I think it isn't actually an easy class. It's actually quite a hard class, 这不是轻松的课事实上它有点难but I think I can guarantee it's going to be a fun class.我保证这门课一定很有趣Now one reason it's a fun class,为什么这门课很有趣is the nice thing about teaching Game Theory因为学习博弈论- quieten down folks--大家安静一下one thing about teaching Game Theory is, you get to play games,在课堂上我们通过一些游戏来学习博弈and that's exactly what we've just been doing now.我们不刚做完一个游戏吗This is our first game and we're going to play games throughout the course, 这是第一游戏整个课程充满了游戏sometimes several times a week, sometimes just once a week.有时候一周做几次有时一周一次We got all these things in?表收齐了吗Everyone handed them in?每个人都交了吧So I need to get those counted.下面要开始统计了Has anyone taken the Yale Accounting class?有人上过会计课吗No one wants to - has aspirations to be - one person has.有人想帮忙统计吗这有一个人想I'll have a T.A. do it, it's all right, we'll have a T.A. do it.我还是安排助教做吧So Kaj, can you count those for me? Is that right?卡茄你帮我统计一下好吗Let me read out the game you've just played.我们回顾一下刚才进行的游戏"Game 1, a simple grade scheme for the class."游戏1是一个简单的成绩博弈Read the following carefully.请仔细阅读以下条款Without showing your neighbor what you are doing,在不被你同桌看到的情况下put it in the box below either the letter Alpha or the letter Beta. 在方框中填写字母α或者字母βThink of this as a grade bid.把这看成成绩的赌注I will randomly pair your form with another form我会随机把你们分成两两一组and neither you nor your pair will ever know with whom you were paired. 你们不知道会跟谁分到一组Here's how the grades may be assigned for the class.按如下方法给出你们的成绩Well they won't be,纯属娱乐大家别当真If you put Alpha and you're paired with Beta,如果你选α而你对手选βthen you will get an A and your pair a C.那么你得A你对手得CIf you and your pair both put Alpha, you'll both get B-.如果你们都选α 那么你们都得B-If you put Beta and you're paired with Alpha,如果你选β你对手选αyou'll get a C and your pair an A.你得C你对手得AIf you and your pair both put Beta, then you'll both get B+."如果你们都选β 你们都得B+"So that's the thing you just filled in.这就是你们之前填的东西Now before we talk about this,在我们讨论之前let's just collect this information in a more useful way.先有效地整理一下这些信息So I'm going to remove this for now.我先把它们擦掉We'll discuss this in a second,一会再开始分析but why don't we actually record what the game is, that we're playing first.我们先记录一下这个游戏的容So this is our grade game, and what I'm going to do,就是我们的成绩博弈since it's kind of hard to absorb all the information因为这么长一段文字just by reading a paragraph of text,从中摘取信息有点困难I'm going to make a table to record the information.不如我们列表整理信息So what I'm going to do is I'm going to put me here,那么我把我写在这里and my pair, the person I'm randomly paired with here,我对手就是被随机分到的人写在这and Alpha and Beta, which are the choices I'm going to make here还有是α和β 就是我要做的选择and on the columns Alpha and Beta, the choices my pair is making.还有我对手要做的选择In this table, I'm going to put my grades.我把我的成绩填到表格里So my grade if we both put Alpha is B-, if we both put Beta, was B+. 如果我们都选α我得B- 都选β我得B+If I put Alpha and she put a Beta, I got an A,如果我选α她选β 我得Aand if I put Beta and she put an Alpha, I got a C.如果我选β她选α 那么我得CIs that correct? That's more or less right?这些都对吧Yeah, okay while we're here, why don't we do the same for my pair?好了我们再把对手的情况也写下来So this is my grades on the left hand table,我的成绩写在左边的表格里but now let's look at what my pair will do, what my pair will get.我们再看看我对手她会得到什么成绩So I should warn the people sitting at the back提醒一下坐在后面的人that my handwriting is pretty bad, that's one reason for moving forward. 我的板书写得很糟所以最好往前坐The other thing I should apologize at this stage of the class is my accent. 还有我得检讨的是我的口音I will try and improve the handwriting,我会努力写好板书there's not much I can do about the accent at this stage.但是口音暂时不太好改So once again if you both put Alpha then my pair gets a B-.如果我们都选α 我对手得B-If we both put Beta, then we both get a B+;如果我们都选β 我们都会得到B+in particular, my pair gets a B+.此情况下我对手得B+If I put Alpha and my pair puts Beta, then she gets a C.如果我选α我对手选β 那么她得CAnd if I put Beta and she puts Alpha, then she gets an A.如果我选β她选α 那么她得ASo I now have all the information由此我就把这纸上的that was on the sheet of paper that you just handed in.所有信息都包含进来了Now there's another way of organizing this that's standard in Game Theory,下面再用博弈论的标准方式来整理信息so we may as well get used to it now on the first day.从今天以后大家要习惯用这种标准Rather then drawing two different tables like this,与其画两个不同的表格what I'm going to do is I'm going to take the second table不如把第二个表格的容插入到and super-impose it on top of the first table.第一个表中Okay, so let me do that and you'll see what I mean.我来做个示你就明白我的意思了What I'm gonna do is draw a larger table, the same basic structure: 我要画个结构相同的大表格I'm choosing Alpha and Beta on the rows,用列来表示我选α还是βmy pair is choosing Alpha and Beta on the columns,用行来表示我对手选α还是βbut now I'm going to put both grades in.但我要把两人成绩写在一个单元格里So the easy ones are on the diagonal:这样就更直观了you both get B- if we both choose Alpha;如果我们都选α的话得B-we both get B+ if we both choose Beta.如果我们都选β的话得B+But if I choose Alpha and my pair chooses Beta,要是我选α我对手选βI get an A and she gets a C.我得A 她得CAnd if I choose Beta and she chooses Alpha,如果我选β她选αthen it's me who gets the C and it's her who gets the A.我得C而她得ASo notice what I did here.注意下我的表示方法The first grade corresponds to the row player, me in this case,单元格第一个成绩是我的成绩and the second grade in each box corresponds to the column player, 每个单元格第二个成绩是my pair in this case.我对手的成绩So this is a nice succinct way of这样就把前两表格的信息recording what was in the previous two tables.更简明地表述出来了This is an outcome matrix; this tells us everything that was in the game. 它是一个列出了博弈所有容的矩阵Okay, so now seems a good time to start talking about what people did. 现在我们来考虑一下我们的选择So let's just have a show of hands.我们举手表决吧How many of you chose Alpha?多少人选了α?Leave your hands up so that Jude can catch that,先不要放下手让裘德拍一下so people can see at home, okay.这样看视频的人也能看到All right and how many of you chose Beta?多少人选了β?There's far more Alphas - wave your hands the Beta's okay.远远少于选α的先不要放下手All right, there's a Beta here, okay.这里有一个选β的So it looks like a lot of - well we're going to find out,我们来数一下确认一下we're going to count--but a lot more Alpha's than Beta's.选α的人要远远多于选βLet me try and find out some reasons why people chose.我们来探讨下你们为什么这么选So let me have the Alpha's up again.来看看都谁选α了So, the woman who's in red here,穿红色衣服的女士can we get a mike to the -yeah, is it okay if we ask you?递给她一个麦克风You're not on the run from the FBI? We can ask you why?请问没被FBI通缉吧?Okay, so you chose Alpha right? So why did you choose Alpha?你选了α对吧你为什么要选α呢Realized that my partner chose alpha, therefore I chose .我感觉我对手会选α因此我也选αAll right, so you wrote out these squares,这么说你也写下了这些矩阵you realized what your partner was going to do, and responded to that. 你分析出了对手的选择并依此做出的选择Any other reasons for choosing Alpha around the room?还有别的原因使你们选α吗Can we get the woman here?把麦克风递给那位女士好吗Try not to be intimidated by these microphones,它们只不过是麦克风而已they're just mikes. It's okay.别紧The reason I chose Alpha, regardless of what my partner chose,我选α是因为不管我对手选什么I think there would be better outcomes than choosing Beta.我选α得到的成绩总会比选β的要好All right, so let me ask your names for a second-so your name was?请问你叫什么名字?Courtney.考特尼Courtney and your name was?考特尼另一位女士的名字是Clara Elise.克莱拉?埃莉斯So slightly different reasons, same choice Alpha.虽然原因不尽相同但她们都选了αClara Elise's reason - what did Clara Elise say?克莱拉?埃莉斯的理由是她怎么说She said, no matter what the other person does,她说无论别人怎么选she reckons she'd get a better grade if she chose Alpha.她觉得选α会使自己得到最优的结果So hold that thought a second, we'll come back to先保留这个说法我们稍后再做讨论is it Clara Elise, is that right?这是克莱拉?埃莉斯的理由We'll come back to Clara Elise in a second.我们一会再讨论这个理由Let's talk to the Beta's a second;我们先来讨论一下选β的理由let me just emphasize at this stage there are no wrong answers.我要强调一下这里没有错误的答案Later on in the class there'll be some questions that have wrong answers. 在以后在课上可能会有错误的答案Right now there's no wrong answers.但是现在没有错误的答案There may be bad reasons but there's no wrong answers.有不充分的理由但没有错误的理由So let's have the Beta's up again. Let's see the Beta's.请选β的再举下手看看有谁选了βOh come on! There was a Beta right here.之前这里有选β的啊You were a Beta right? You backed off the Beta, okay.你选过β对吗? 哦你改变主意了So how can I get a mike into a Beta?能不能把麦克风给一位选了β的同学Let' s stick in this aisle a bit.我们往过道这边来一点Is that a Beta right there? Are you a Beta right there?那有人选β对吗你是不是选了βCan I get the Beta in here? Who was the Beta in here?这边有人选β的吗Can we get the mike in there? Is that possible?能把麦克拿到那里吗In here - you can leave your hand so that - there we go.这有一个选β的先别放下手对了Just point towards - that's fine, just speak into it, that's fine. 把麦克对着他请讲So the reason right?为什么选β对吗Yeah, go ahead.没错继续说I personally don't like swings that much我不喜欢成绩波动很大的and it's the B-/B+ range,比如B-/B+这个围so I'd much rather prefer that to a swing from A to C,所以我还是喜欢像A到C这样小点的and that's my reason.这就是我的原因All right, so you're saying it compresses the range.你说的意思是压缩围I'm not sure it does compress the range.但是我不太确定这是否压缩了围I mean if you chose Alpha, you're swinging from A to B-;要是你选α 你的分数会在A到B-and from Beta, swinging from B+ to C.选β 分数围会在B+到CI mean those are similar kind of ranges这两个围大小差不多but it certainly is a reason.但这也算是个原因Other reasons for choosing?还有别的原因吗Yeah, the guy in blue here, yep, good.。
博弈论1
Prisoner Dilemma
B confess confess deny
A
deny
-8,-8 -10,0
0,-10 -1,-1
Question: What will A and B do?
Applications of prisoner dilemma
Professor vs. Cheaters
How to learn game theory?
When is 0>2, 2>5, and 0>5? Rock, Paper, Scissors
当你用瓦片做赌注时 ， 技艺可以发挥的淋漓 尽致，但倘若换成黄 就会大失水准。 金，就会大失水准。
Types of games
Zero-sum games Non-zero-sum games
美 丽 心 灵
金发女郎
约翰 追求 金发女郎 追求 金发女郎 追求 其他女士 0，0 ， 1，2 ， 追求 其他女士 2，1 ， 1，1 ，
雷哈德
Prisoner Dilemma
Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal. If one testifies (defects from the other) for the prosecution against the other and the other remains silent (cooperates with the other), the betrayer goes free and the silent accomplice receives the full 8-year sentence. If both remain silent, both prisoners are sentenced to only one year in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. If we assume that each player cares only about minimizing his or her own time in jail, how should the prisoners act?
博弈论(哈佛大学原版教程)
1.2
Example II
L M R
U
2,2
1,1
4,0
D
1,2
4,1
3,5
2
In this game the unique Nash equilibrium is (U,L). It is easy to see that (U,L) is a NE because both players would lose from deviating to any other strategy. To show hat there are no other Nash equilibria we could check each other ∞ ∞ strategy profile, or note that S1 = {U } and S2 = {L} and use: Proposition 1 If s∗ is a pure strategy Nash equilibrium of G then s∗ ∈ S ∞ .
α−c 2β
−
qj 2
−c if qj ≤ αβ otherwise
The NE is the solution to q1 = BR1 (q2 ) and q2 = BR2 (q1 ). This system has exactly one solution. This can be shown algebraically or simply by looking at the intersections of the BR graphs in figure 1. Because of symmetry we know that q1 = q2 = q ∗ . Hence we obtain: q∗ = α − c q∗ − β 2
博弈文言文翻译
原文：“兵者，诡道也。
故能而示之不能，用而示之不用，近而示之远，远而示之近。
利而诱之，信而诱之，亲而诱之，动而诱之。
利而舍之，信而舍之，亲而舍之，动而舍之。
此四者，兵家之诡道也。
”翻译：兵法之道，变幻莫测。
故能者，使人不知其能；用者，使人不知其用；近者，使人不知其近；远者，使人不知其远。
以利诱之，以信诱之，以亲诱之，以动诱之。
得利而不取，得信而不守，得亲而不交，得动而不动。
此四者，兵家之诡道也。
原文：“兵者，诡道也。
故善战者，致人而不致于人。
能使敌人自至者，利也；能使敌人自至者，信也；能使敌人自至者，亲也；能使敌人自至者，动也。
故善战者，能为之下，而不能为之上。
”翻译：兵法之道，变幻莫测。
故善于作战者，能引导敌人，而不被敌人所引导。
能使敌人自行前来，是因为利诱；能使敌人自行前来，是因为信任；能使敌人自行前来，是因为亲近；能使敌人自行前来，是因为行动。
故善于作战者，能屈居人下，而不能高居人上。
原文：“兵者，诡道也。
故能而示之不能，用而示之不用，近而示之远，远而示之近。
利而诱之，信而诱之，亲而诱之，动而诱之。
此四者，兵家之诡道也。
”翻译：兵法之道，变幻莫测。
故能者，使人不知其能；用者，使人不知其用；近者，使人不知其近；远者，使人不知其远。
以利诱之，以信诱之，以亲诱之，以动诱之。
此四者，兵家之诡道也。
原文：“兵者，诡道也。
故善战者，能以形固。
形固而攻者削，攻者削而守者坚。
故善战者，能为之下，而不能为之上。
”翻译：兵法之道，变幻莫测。
故善于作战者，能以形态稳固。
形态稳固，攻者削弱；攻者削弱，守者坚固。
故善于作战者，能屈居人下，而不能高居人上。
原文：“兵者，诡道也。
故善战者，能以少击众，以弱击强。
故善战者，能为之下，而不能为之上。
”翻译：兵法之道，变幻莫测。
故善于作战者，能以少胜多，以弱胜强。
故善于作战者，能屈居人下，而不能高居人上。
博弈之术，非一日之功，需勤学苦练，方能领悟其真谛。
今日所译，仅为博弈之术之冰山一角，愿与诸君共勉。
博弈论(哈佛大学原版教程)
P rob(fj(vj) ≤ b) = P rob(vj ≤ fj−1(b)) = F (fj−1(b)) = fj−1(b)
(1)
The last equation follows because F is the uniform distribution. The expected utility from bidding b is therefore:
In all our auctions there are n participants and each participant has a valuation vi and submits a bid bi (his action).
The rules of the auction determine the probability qi(b1, .., bn) that agent i wins the auction and the expected price pi(b1, .., bn) which he pays. His utility is simple ui = qivi − pi.a
(vi
−
f (vi))
−
vi
=
0
(4)
This is a differential equation and can be rewritten as follows:
vi = vif (vi) + f (vi)
(5)
We can integrate both sides and get:
1 2
vi2
博弈论最全完整-讲解
据“共同知识”的掌握分为完全信 息与不完全信息博弈。
完美信息博弈与不完美信息博弈
(games with perfect information and games with imperfect information)
Because We Had a Flat Tire”
“乘客侧前轮”看起来是一个合乎逻辑的选择。 但真正起作用的是你的朋友是否使用同样的
逻辑，或者认为这一选择同样显然。并且是 否你认为这一选择是否对他同样显然；反之， 是否她认为这一选择对你同样显然。……以 此类推。 也就是说，需要的是对这样的情况下该选什 么的预期的收敛。这一使得参与者能够成功 合作的共同预期的策略被称为焦点。心有灵 犀一点通。
例3：为什么教授如此苛刻？
问题是，一个好心肠的教授如何维持如 此铁石心肠的承诺？
他必须找到某种使拒绝变得强硬和可信 的方法。
拿行政程序或者学校政策来做挡箭牌 在课程开始时做出明确和严格的宣布 通过几次严打来获得“冷面杀手”的声
誉
导论
博弈均衡与一般均衡 博弈论与诺贝尔经济学奖获得者
博弈论的基本概念与类型 主要参考文献
2005年诺贝尔经济学奖获得者
以罗伯特·奥曼色列经济学家罗伯特 －奥曼（Robert J. Aumann）和美 国经济学家托马斯·谢林（Thomas C. Schelling）
获奖原因：“通过博弈论分析加强了 我们对冲突和合作的理解”所作出 的贡献而获奖。
罗伯特·奥曼
托马斯·谢林
导论
三、博弈论的基本类型
约翰·纳什 1928年生于美国
莱因哈 德·泽 尔腾， 1930 年生于 德国
博弈论1 (2)全篇
6
四、古典博弈论的三个基本假设：参与人是理性的；他们有 这些理性的共同知识；他们知道博弈规则。
理性的三个基本内涵：理性的局中人具有关于博弈的完全知 识；可以确切知道整个状态空间；具有相对无限的逻辑能力。
五、博弈的典型例子
1 ．囚徒困境 坦白
坦白
不坦白
-6，-6 -1，-8
3
2、博弈与一般决策的区别
二、博弈模型要素
1、参与人：博弈中的决策主体。他的目的是通过选择策 略以最大化自己的支付（效用）水平。
虚拟参与人（自然）：指决定外生的随机变量的概率分布 的机制。自然作为虚拟参与人没有自己的支付与目标函数， 即所有的结果对它都是无差异的。
2、策略：参与人在给定信息集（信息集包含了一个参与人有关 其他参与人之前行动的知识，可理解为参与人在特定时刻有关 变量值的知识。一个参与人无法准确知道的变量全体属于一个 信息集）的情况下的行动规则。它规定参与人在什么时候选择 什么行动。[策略是可供局中人选择对付其它局中人的完整行动 方案。]
21
例 某个地方的居民均匀地环绕一个圆形湖居 住。两小贩来此地推销商品。1）若居民都选 择离自己较近的小贩购买商品，问小贩选择推 销地点博弈的NE是什么？2）若有三个小贩同 时到此地推销商品，则推销地点博弈的NE又是 什么？3）若圆形湖的周长是1（千米），居民 的购买量是Q=1-D，D为居民与小贩推销点距 离，则两个和三个小贩博弈的NE各是什么？
14
例6 设 s1 [10,0] s2 [3,0]
u1 (s) 10s1 7s1s2 s12
u2 s 15s2 5s1s2 s22
求NE。
15
例7 设 s1 [10,0] s2 [3,0]
博弈论(部分英文版翻译)
博弈论By Thomas S.Ferguson /译者：xly 第一部分：公平的组合游戏1.Take-Away游戏1.1 一个简单的Take-Away游戏1.2 什么叫做组合游戏？1.3 P态与N态1.4 差集游戏1.5 相关练习2.Nim游戏初步分析Nim-sum多堆的Nim游戏Bounton理论的证明Misere版本的Nim游戏相关练习3.图表游戏有向图游戏SG函数相关例题一般图的SG函数4.组合游戏的和n个图表游戏的和SG定理有关应用Take-and-Break游戏相关练习5.Coin Turning游戏例子二维空间的Coin Turning游戏Nim复杂情况方格游戏练习6.Green Hackenbush竹竿Green Hackenbush on tressGreen hackenbush on general Rooted Graphs 练习参考资料第一部分：公平的组合游戏1.take-away游戏组合游戏是一种两人游戏，给定足够的条件时，当一方无法继续操作时，游戏的胜负就出来了。
这种游戏的胜负取决于一系列的状态，包括初始状态和正准备操作的玩家。
游戏双方轮流操作，直到达到最终状态。
最终状态的意思是，该状态已经不能被操作。
这时，胜负已分。
这里介绍两本关于组合游戏的主要资料。
一本是J.H.Conway写的On Numbers and Games，学术出版社1976年出版。
这本书介绍了很多关于这方面的基本思想，加快了今天这块领域的发展。
更适合这堂课的另一本参考书，Berlekamp，Conway和Guy写的winning ways for your mathematical plays，学术出版社1982年出版，是一套两册的平装本。
这本书介绍了很多有趣的游戏，学数学的本科生可以理解它。
这些理论可以分为两类，Impartial games是指任意给定一个状态，对游戏双方而言将要采取的操作是一样的；而partizan games是说，给定一个状态，游戏双方要采取的操作会有不同。
博弈翻译
Motivation and Decision TheoryMarkus M. Mobius February 7, 2004 1.Two Motivating ExperimentsExperiment 1Each of you (the students in this course) have to declare an integer between 0 and 100 to guess ”2/3 of the average of all the responses”.More precisely, each student who guesses the highest integer which is not higher than 2/3 of the average of all responses, will receive a prize of 10Dollars.How should you play this game? A naive guess would be that other players choose randomly a strategy. In that case the mean in the game would be around 50 and you should choose 33. But you realize that other players make the same calculation - so nobody should say a number higher than 33. That means that you should not name a number greater than 22 - and so on. The winning number was 13. That means that people did this iteration about 3times. But in fact, the stated numbers were all over the place - ranging from 0 to 40. That means that different students had different estimates of what their fellow students would do: ·Being aware of your fellow players’existence and trying to anticipate their moves is called strategic behavior. Game theory is mainly about designing models of strategic behavior.·In this game, the winner has to correctly guess how often hisfellow players iterate. Assuming infinite iterations would be consistent but those who bid 0 typically lose badly. Guessing higher numbers can mean two things: (a) the player does not understand strategic behavior or (b) the player understands strategic behavior but has low confidence1 in the ability of other players to understand that this is a strategic game. Interestingly, most people knew at least one other person in the class (hence there was at least some degree of what a game theorist would call common knowledge of rationality)Experiment 2I am going to auction a textbook (Osborn book). It costs about 60 Dollars on Amazon. Each of your can bid secretly on the book and the highest bidder wins the auction. However, all of you have to pay your bid regardless of whether you win or lose.In this game there is no optimal single bid for all players. You can check that for all cases where each player i bids some fixed bid bait least one of the players will regret her decision and try to reverse it - we say that there is no pure strategy Nash equilibrium in this game. Consider for example the case where all player bid 55 Dollars. Then some player should bid 55 and 5cents. No equilibrium!There is an equilibrium if we allow players to randomize. You can check that with two players who pick random numbers between 0 and 60 with equal probabilityno player would want to change her pick - all picks will give her zero profit in expectation.The more players there are, the more the bid distribution is skewed to-wards 0 (check)!2What is game theory?Definition 1Game theory is a formal way to analyze interaction among a group of rational agents who behave strategically.This definition contains a number of important concepts which are dis-cussed in order:Group: In any game there is more than one decision maker who is referred to as player. If there is a single player the game becomes a decision problem.1There is common knowledge of rationality between players A and B if A knows that B is rational (and vice versa), if A knows that B knows that A is rational (and vice versa)etc.Interaction: What one individual player does directly affects at lea stone other player in the group. Otherwise the game is simple a series of independent decision problems.Strategic:Individual player account for this interdependence.Rational: While accounting for this interdependence each player chooses her best action. This condition can be weakened and we can assume that agents are boundless rational. Behavioral economicsanalyzes decision prob elms in which agents behave boundless rational. Evolutionary game theory is game theory with boundless rational agents.Example 1Assume that 10 people go into a restaurant. Every person pays for her own meal. This is a decision problem. Now assume that everyone agrees before the meal to split the bill evenly among st all 10 participants. Now we have a game.Game theory has found numerous applications in all fields of economics:1.Trade: Levels of imports, exports, prices depend not only on your own tariffs but also on tariffs of other countries.bor: Internal labor market promotions like tournaments: your chances depend not only on effort but also on efforts of others.3.IO: Price depends not only on your output but also on the output of your competitor (market structure ...).4.PF: My benefits from contributing to a public good depend on what everyone else contributes.5.Political Economy: Who/what I vote for depends on what everyone else is voting for.3Decision Theory under CertaintyIt makes sense to start by discussing trivial games - those we playagainst ourselves, e.g. decision problems. Agents face situations in which they have to make a choice. The actions of other agents do not influence my preference ordering over those choices - therefore there is no strategic interaction going on. Proper games will be discussed in the next lectures.A decision problem (A,۲) consists of a finite set of outcomes A = {a1,a2,..,an}and a preference relation ۲. The expression a ۲b should be interpreted as”b is at least as good as a”. We expect the preference relation to fulfill two simple axioms:Axiom 1Completeness. Any two outcomes can be ranked, e.g.a ۲b orb ۲ A.Axiom 2Transitivity implies that if a ≥b and b ≥c then a ≥c.Both axioms ensure that all choices can be ordered in a single chain without gaps (axiom 1) and without cycles (axiom 2).Although the preference relation is the basic primitive of any decision problem (and generally observable) it is much easier to work with a consistent utility function u : A →R because we only have to remember n real numbers{u1,u2,..,un}.Definition 2A utility function u : A →R is consist with the preference relationship of a decision problem (A,۲) if for all a,b ∈A:a ۲B if and only if u(a) ≤u(b)Theorem 1 Assume the set of outcomes isfinite. Then there exists a utility function u which is consistent.Proof:The proof is very simple. Simple collect all equivalent outcomes in equivalence classes. There are finitely many of those equivalence classes since there are only finitely many outcomes. Then we can order these equivalence classes in a strictly increasing chain due to completeness and transitivity.Note that the utility function is not unique.transformation of a consistent utility function gives another utility function which is also consistent.We can now define what a rational decision maker is.Definition 3A rational decision maker who faces a decision problem (A,۲)chooses an outcome a*∈A which maximizes his utility (or, equivalently, for each a ∈A we have a ۲ a*).Remark 1 When there are infinitely many choices we want to make sure that there is a continuous utility function. This requires one more axiom which makes sure that preferences are continuous. For that purpose, one has to define topology on the set of outcomes. We won’t deal with that since we won’t gain much insight from it.4.Decision Theory under UncertaintyLotteries are defined over the of outcomes A (which is again assumed to be finite to keep things simple).Definition 4A simple lottery is defined as the set {(a1,p1),(a2,p2),..(an,pn)}such that Ʃpi= 1 and 0 ≤pi≤1. In a simple lottery the outcome ai occurs with probability pi.When there are up to three outcomes we can conveniently describe the set of lotteries in a graphical way (see triangle).Under certainty the preference relationship can still be written down ex-plicitly for finite A (simply write down all of the uncertainty there are suddenly infinitely many lotteries.problems. First of all, it’s impractical to write a large number of lottery comparisons down. A second (and deeper) point is the observation that the preference relationship is in principle unobserved because of the infinite number of necessary comparisons.John von Neumann and Oscar Morgenstern showed that under some additional restrictions on preferences over lotteries there exists a utility function over outcomes such that the expected utility of a lottery provides a consistent ranking of all lotteries.n(n+1)2rankings).Definition 5Assume a utility function u over the outcomes A. The expected utility of the lottery L = {(a1,p1),(a2,p2),..(an,pn)} is defined asU(L)=Ʃ u (ai)piBefore we introduce the additional axioms we discuss the notion of com-pound (two stage) lotteries.u(L) =i=1u(ai)pi5全文翻译如下：一、两个激励实验实验一：你的每一个（学生在本课程）必须声明一个0到100之间的整数来猜“2 / 3的所有反应的平均水平。
博弈论 外文翻译2
第一章完全信息静态博弈在这一章中我们讨论像下面的简单形式的博弈：开始的时候由参与者同时选择行动，然后根据所有参与者刚刚选择的行动组合，每个参与者都会得到一个回报。
在此类静态（或各自同时行动）的博弈游戏中，我们仅限于关注完全信息博弈的情况，那就是说每一位参与者的收益函数（根据所有参与者选择行动的不同组合决定某一参与者收益的函数）在所有参与者之间是共同知识。
我们在本书的第二章和第四章讨论动态（或者序列行动）博弈，在本书的第三张和第四章分析不完全信息博弈（博弈中的一些参与者不知道其他参与者的收益函数，如拍卖中得每一个人都不清楚其他人到底原以为拍卖品出多高的价钱）。
在第1.1节首先介绍博弈论入门的两个最基本问题：如何描述一个博弈问题以及如何求得博弈问题的解。
我们扩展了将要用在分析完全信息静态博弈问题中用到的方法，并且这些依据我们将应用在分析后面章节中更丰富的博弈问题。
我们定义博弈的标准式表述和严格劣战略战的概念。
并说明有些博弈问题理性参与者不仅绝不会使用严格劣战略这一原则来解决问题，而且在另外的博弈问题里这种方法预测也是很准确（有时候预测像“想任何结果都有可能发生”之类）。
然后我们进一步引出并定义纳什均衡的概念—这个概念的用途很广泛，对很多种类的博弈都能作出比较严格的预测。
在第1.2节我们运用前面介绍的工具，分析其四个应用模型：(Cournot,1838)的不完全信息博弈竞争模型，（Bertrang,1883）的不完全竞争模型，（Farber，1980）的最后要价仲裁和公共财产问题（（Hume），1739年提出了这种问题，以后又不断被经济学家提出来讨论）。
在每个应用的例子中，我们首先把非标准的问题转化为博弈的标准形式，然后在求出这种博弈的纳什均衡。
（上面每个例子都存在唯一的纳什均衡，但是我们讨论的范围不仅仅是这点）。
在第1.3节回顾理论。
我们首先定义混合战略，它可以理解为一个参与者并不可以确定其他人将会如何行动。
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搏弈论
阿维纳什•迪克斯特&巴里•内尔巴夫
1搏弈是有关策略的科学。
它试图以数学和逻辑的方法来帮助搏弈者作出决策，在一系列纷繁复杂的搏弈中应采取何种策略来保证自己获得最大利益。
搏弈论研究的搏弈的范围包括了从下棋到抚育儿童，从网球竞技到公司转手。
但是所有的博弈都具有一个共同的特征：相互作用。
也就是说,每一个博弈者的博弈结果取决于所有博弈参与者的策略选择。
在零和搏弈中，搏弈者的利益之间是完全冲突的，因此一方的得利必然导致另一方的损失。
更多具有代表性的例子还有会导致共同得利(正和)搏弈和共同损失(负和)搏弈，同样的情况还会发生在另外一些冲突中。
2搏弈论研究的先驱者是普林斯顿数学家约翰•冯•诺依曼。
在早先的一段时间里，研究的重点被放在了完全冲突(零和)搏弈(非合作搏弈)上，其他的搏弈当时被认为是以合作形式出现。
也就是说，搏弈要求参与者共同地选择和实施他们的行为.最近的研究则把重点放在了那些既不属于零和搏弈也不属于绝对合作搏弈的情况上，在这些搏弈中，搏弈者自主地选择搏弈行为，但他们之间的相互关系中充满了合作与竞争。
3搏弈行为与我们在中性环境中所作的各种决定有着根本性的不同。
要说明这一点，我们可以思考一下伐木工人和军队将军所作决定之间的不同。
当伐木工人决定要如何砍树时，他不会考虑树木本身会有什么反抗，他所处的环境为中性。
而当将军决定要消灭敌军时，他必须提前预料到并消除敌军的反抗。
与这一例子中的将军相类似，一个搏弈者必须认识到他与其他机智且怀有争胜之心的竞争者之间的相互作用，他自己所作的决定也必须能够同时应对可能出现的合作或冲突。
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4搏弈的实质是搏弈者采取策略之间的相互依赖性。
这种策略性的相互依赖表现为两个不同的类别：连续策略之间的相互作用以及联立策略之间的相互作用。
就前者而言，搏弈者依次采取行动，每个人都会注意其他搏弈者先前的行为。
就后者而言，搏弈者同时采取行动，每个人都会忽略其他搏弈者当前的行为。
5对连续策略博弈中的某一博弈这来说，一个普遍的原则就是放眼前方，及时反思和总结。
每个博弈者应该弄清楚其他博弈者会对他当前的策略行为做出怎样的回应，他自己将如何应对等情况。
博弈者要预料到他的最初决定会最终导致何种结果，并且运用对形势的判断来计划好当前的最佳策略。
在考虑其他博弈者会如何应对时，博弈者必须能设身处地地换位思考，而不能把自己的主观判断强加与人。
6从理论上说，采取固定次序行动的任何连续博弈都可以圆满地完成。
我们可以通过预测每个可能的结果来决定各个博弈者的最佳策略。
例如象井字棋(tic-tac-toe)这样的简单游戏由于可以以这样的方式完成，因而并不具有挑战性。
但诸如象象棋等的其他博弈，即使是借助电脑的帮助，由于其本身的计算过程过于复杂而难以在实践中去实施。
因此，博弈者往往会会依据经验提前对形势作出判断并尽可能的评估最终的局面。
7与连续策略博弈的线形思维不同的是，采取共发性策略的博弈要求逻辑思维。
在忽略其他参与者当前策略的情况下，尽管博弈者们同时采取行动，每一个参与者必须清楚的意识同时还会有其他的参与者在依次关注整个博弈过程。
这时的思维模式可描述为：我想他认为我会这样考虑…。
因此，博弈者必须从全体博弈者的立场出发并努力判断出最终的博弈结果。
每个参与者的个人最佳行为都是全局谋划中不可或缺的一部分。
8运用普林斯顿数学家约翰•纳什提出的均衡概念，可以推导出这种逻辑思维的结论。
我们寻求一系列的策略组合，每个博弈者都会有自己的选择，当所有的对手们在实施他们决定的最佳策略时，我们所做的选择应该对自己是最有利的。
换句话说，每个博弈者都会对其他人的策略作出最优化的应对。
9有时, 无论其他博弈者如何行动，博弈的一方的最佳策略组合始终如一，这被称作这一博弈者的优策略。
在其他情况下，如果博弈者的策略始终于己不利，则被称作劣策略，其含义是指无论其他博弈者如何行动，对手的策略总是优于自己。
因此，谋求策略均衡应该从寻找优策略和消除劣策略开始。
10当我们把博弈的结果表述为一种均衡的时候，并不是基于以下的假设：即博弈的每个参与者的个人最佳策略将会带来共同的最优化结果。
确实也存在着一些糟糕的例子，比如囚徒困境(见下文)，由于囚徒们都追求个人私利的最大化而导致了全体参与者的困境。
11纳什的关于均衡的定义还不能完全解决联立策略博弈中逻辑推理思考的问题，有些博弈包含多种此类的均衡，而有些博弈却并不包含这样的均衡。
纳什均衡也还没有清楚地说明关于导致均衡的动态过程。
尽管有这样的一些缺陷，纳什均衡的定义已被证明在分析策略性互动时具有重要作用。
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12以下策略性互动的实例可以说明博弈论的一些基本理论框架:
13囚徒困境。
两个嫌疑犯分别被审问，每个人都可以招供或保持沉默。
如果嫌犯A保持沉默，嫌犯B可以通过招供而获得较轻的发落。
如果嫌犯A招供，嫌犯B最好选择招供以免被从重处理。
这时招供就是嫌犯B的优策略。
同样的情况也适用与B。
因此，在均衡的情况下双方都选择招供。
如果双方均缄口不言也都可相安无事。
由于合作分裂所导致的长远性损失远比招供而获得的暂时性得利严重，所以这样的合作行为会在博弈中反复进行下去。
在这种情况下，一般推荐采取针锋相对的策略。
14混合性策略。
在一些冲突中，任何规律性的行为都会被对手发觉并利用。
因而，通过采取混合性策略来迷惑对手就显的非常重要。
我们可以从体育运动中找到一些典型的例子，比如足球运动中的特定情况下选择跑位还是传球，网球比赛中选择击打近网球还是底线球，博弈论都为提高洞察力和掌握混合性策略恰当的火候提供了参考。
15战略性行动。
博弈者可以通过运用威吓和许诺来改变其他对手对其行动的预测，从而诱使他们采取对其有利的行为或改变他们对己不利的行为。
为了取得成功，威吓和许诺必须是可信的。
这往往会导致出现一些问题，原因就在于当需要采取以上策略时，通常实施威吓和许诺而获益往往会付出较大的代价。
博弈论研究几种提高可信度的方法，一个基本的原则就是威吓和许诺要在符合博弈者利益的前提下减少自身未来行动的自由度。
通过这样的方式，博弈者可以避免自己违背承诺，或者避免产生对对手冒犯的纵容。
16例如，当柯台斯到墨西哥后，烧掉了战船，故意没有给自己留下撤军的退路。
由于没有返城的船只，柯台斯面对的只有战胜并征服对手或被对手消灭掉两种可能。
尽管他的士兵在数量上处于绝对劣势，但这种血战至死的威胁使得对手的士气变的低落，印第安军队面对这样意志坚定的对手时，他们选择了退却。
宝丽来公司在拒绝共享即时成像市场时也采取了类似的策略，当时它决定与任何挤占该市场的对手拼个你死我活。
当柯达公司试图染指即时成像业时，宝丽来调动了所有的资源进行反击。
14年后, 宝丽来公司在与柯达公司的诉讼中获胜，重新赢得了在即时成像市场的垄断地位。
17使威吓变得可信的另一个方法是运用冒险性的边缘化策略，这一策略的风险在于: 如果其他博弈者未能按照威吓的要求去做的话，结果对大家将是一损俱损。
托马斯•谢林在他的《冲突策略》一书中介绍说，边缘化策略就是故意使局势变的无法控制的策略，正是由于局势的无法收拾可能令其他对手难以接受，从而迫使对手作出妥协。
18讨价还价。
两个博弈者一起来分蛋糕，每个人都想要一个大块，双方都倾向于尽快达成一致。
当两个人依次开始提议分配时，放眼前方和反思总结的博弈原则决定了均衡的份额，双方这时可以立即达成一致，但时间的延误决定了双方的份额，竞争中缺乏耐心的一方只能分得较小的蛋糕。
19隐瞒和公开信息。
当博弈的一方掌握其他各方不了解的信息时，他会急于隐瞒这一
信息(例如纸牌游戏中抽到的牌)，其他情况下他也可能想把信息令人信服的公布出来。
这两种情况下，一个基本的原则就是行胜于言。
要隐瞒信息，就要采取混合性策略，比如纸牌游戏中的作弊手法千变万化。
我们还可以回忆起温斯顿•丘吉尔关于把真相置于谎言保卫下的名言。
要公布信息，就要采用传递可信信息的策略，而且如果形势发生变化时策略也必须相应进行改变，例如，提供长期的质量保证是企业生产高质量产品信心的显示，对消费者来说也是一个可以信赖的信号。
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20最近博弈论研究的最新进展已经成功地解释和规范了在各种冲突和合作情况下应采取的恰当策略，然而博弈论研究仍有待进一步完善，从多角度设计成功策略也有待进一步去研究。
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