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
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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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2015/12/5
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.
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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。
‖
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。
‖
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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