博弈论(部分英文版翻译)
博弈论英文课件 (3)
Solving matching pennies
Player 2
Head
Tail
Expected payoffs
Head Player 1
Static (or SimultaneousMove) Games of Complete Information
Matching pennies
Player 2
Head
Tail
Player 1
Head Tail
-1 , 1 1 , -1 1 , -1 -1 , 1
n Head is Player 1’s best response to Player 2’s strategy Tail n Tail is Player 2’s best response to Player 1’s strategy Tail
( 1(H)=0.5, 1(T)=0.5 ) is a Mixed strategy. That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively.
( 1(H)=0.3, 1(T)=0.7 ) is another Mixed strategy. That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively.
Ø Player 2’s expected payoff of playing s22:
EU2(s22, (r, 1-r))= r×u2(s11, s22)+(1-r)×u2(s12, s22)
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.。
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.
“the game theory＂的汉译法
馆２ ０ ０ ３年 ３月 出版 的《 综 合 英 汉 科技 大词 典 》 中 的 翻译 是 “ （ 数） 对策 （ 博弈 ） 论” ２ ， 张 柏 然 主编 、 商 务
个体间利益与冲突、 竞争与合作等 内容。后来该术 语 从 最初 的数 学用语 转 用于政 治学 、 经 济学 、 社会 学 和国际关系等领域 ， 被我 国译介者套用《 论语》 中的
２ ０ １ ３年 １ ０月
第２ ７卷 第 ５期
新 乡学 院学报（ 社会科学版 ）
Ｊ ｏ ｕ ｎａ ｒ ｌ ｏ ｆ Ｘ ｉ ｎ ｘ ｉ ａ ｎ ｇ Ｕ ｎ ｉ ｖ ｅ ｒ ｓ ｉ ｔ ｙ （ Ｓ ｏ ｃ ｉ ａ ｌ Ｓ ｃ ｉ ｅ ｎ ｃ ｅ Ｅ ｄ ｉ ｔ ｉ ｏ ｎ ）
翻译是 “ 博弈 （ 对策） 论” ＿ １ Ｊ ， 顾 仁敖 主编 、 商务 印 书
１ ９ ４ ４年合 作完成 的著作 Ｔ ｈ ｅ ｏ ｒ ｙ ｏ ｆ Ｇ ａ ｍ ｅ ｓ ａ ｎ ｄ Ｅ ｃ ｏ —
ｎ ｏ ｍｉ ｃ Ｂ ｅ ｈ ａ ｖ ｉ ｏ ｒ 。它归属 于数 学 的 一 个 分 支 学科 ， 其 研究 重点 是不 同个 体 行 为 的 相互 影 响或 作 用 ， 不 同
近 年来 ， “ 博 弈论 ” 这 个 词 汇经 常会 出 现 在一 些 媒体 上 ， 人们 看 到 “ 博弈 ” 二字 ， 总 会 觉 得 这 门学 问 既雅 致 又深 不可 测 ， 致 使许 多读 者 望而 却步 ， 不敢 问 津 。其 实 ， “ 博 弈论” 是英文 “ ｔ ｈ ｅ ｇ ａ ｍｅ ｔ ｈ ｅ ｏ ｒ ｙ ” 的 汉 译 。英 文看上 去 简 单 明 了 的 “ ｔ ｈ ｅ ｇ ａ ｍｅ ｔ ｈ ｅ ｏ ｙ” ｒ 怎 么 会被 译成 了令 人 望而生 畏 的 “ 博 弈论 ” 呢？ 查 阅最近 几 年 出版 发 行 的众 多 科 技 英 汉词 典 ，
博弈论用英文怎么说英语是什么
博弈论用英文怎么说英语是什么博弈论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。
那么你知道博弈论的英文怎么说吗?下面店铺为大家带来博弈论的英文说法，供大家参考学习。
博弈论的英文说法：game theory英 [ɡeim ˈθiəri]美 [ɡem ˈθiəri]博弈论相关英文表达：博弈论算法 Algorithmic Game Theory合作博弈论 Cooperative Games Theory经济博弈论 Economic Game Theory重复博弈论 repeated game approach博弈论英文说法例句：1. Game theory is a powerful weapon to decision - making of multi - person.博弈论是解决多人竞争决策问题的有利武器.2. Game theory is an equilibrium problem in decision influence.博弈论是在选择中的决策影响和均衡问题.3. Conflict and cooperation are the two fundamental issues in game theory.冲突与合作是博弈论研究的两大基本问题.4. The main research way is the analytical method of Game Theory.研究方法主要是博弈论中的博弈分析方法.5. Game theory; High - tech SMEs; principal - agent; performance management; incentive mechanism.博弈论; 高新技术中小企业; 委托—代理; 绩效管理; 激励机制.6. Textbook farce and textbook game theory – how delightful!经典的闹剧,经典的博弈论——多有趣啊 !7. Contract Theory, Information Economics, Applied Game Theory, Corporate Finance.契约理论, 信息经济学, 应用博弈论, 公司财务,政治经济学.8. This story illustrates an important distinction between ordinary decision theory and theory.这个故事说明了普通决策理论和博弈论之间的一个重要的区别.9. The backward induction is an important reasoning method in game theory.逆向归纳法(倒推法)是博弈论中的一种重要的推理方法.10. What's the definition of the game theory?[灌水]博弈论的定义是什么 ?11. Game theory studies mutual roles among rational agents.博弈论是研究理性的行动者相互作用的理论.12. So the penalty kick, for instance, is like this laboratory mile of game theory.打个比方, 罚点球, 就像博弈论里面的实验室.13. Based on principal - agency, the thesis analyzes budget management system from the angle of game theory.本文以委托代理理论为基础, 从博弈论视角对企业预算管理制度进行了分析.14. Evolutionary game theory provides a uniform frame to study the evolution of cooperation.进化博弈论为理解合作行为的演化提供了一个统一的框架.15. Proving the possibility and inevitability of the tax evasion by the basic theory of GAME.用博弈论的基本理论证明企业偷税的可能性和必然性.。
博弈论英文参考
Journal of Mathematical Psychology 42,215 226(1998)Strategy and Equity:An ERC-Analysis ofthe Gu th van Damme GameGary E.BoltonSmeal College of Business ,Pennsylvania State UniversityandAxel OckenfelsUniversity of MagdeburgGu th and van Damme's three-person bargaining experiment challengesconventional thinking about how self-interest,as well as fairness,influencesbehavior.Among other things,the experiment demonstrates that people careabout receiving their own fair share,but care far less about how the remainderis divided among the other bargainers.The ERC model posits that,along withpecuniary gain,people are motivated by their own relative payoff standing.Beyond this,ERC employs standard game theoretic concepts.We describethe general ERC model,and show that it predicts many of the keyphenomena observed in the experiment. 1998Academic Press1.INTRODUCTION:MOTIVES AND THE GU TH VAN DAMME EXPERIMENT Motives drive decision making.While most economic and business models posit self-interested material gain as the sole driver,this is of course a modeling abstrac-tion.People are motivated by many things.Some the drive to procreate,for example are without a doubt as fundamental as material gain.The question then is whether material gain alone is sufficient to explain the variety of economic activities in which people participate.When confined to casual empiricism,the right answer is hard to judge:People do struggle for profits in highly competitive markets.But they also demand fair treatment in the workplace.People strike mutually beneficial bargains;other times,negotiations collapse in bitter disagree-ment.People ``free ride''on the public domain and contribute substantially to charity.The control afforded by the laboratory permits a precision of analysis rarely open to the casual empiricist.And as illustrated by papers in this issue,the variety of Article No.MP9812112150022-2496Â98 25.00Copyright 1998by Academic PressAll rights of reproduction in any form reserved.Correspondence should be addressed to Gary Bolton,310Beam,Penn State University,University Park,PA 16802,USA;(814)865-0611;fax (814)863-2381;geb3Ä,or Axel Ockenfels,University of Magdeburg,FWW,Postfach 4120,D-39016Magdeburg,Germany;(+391)67-12197;fax (+391)67-12971;axel.ockenfels Äww.uni-magdeburg.de.216BOLTON AND OCKENFELSbehavior suggested by casual empiricism is mirrored in the lab:Experiments featur-ing market institutions often produce the type of competitive behavior we associate with the struggle for material gain(Hoffman,Liebcap,6Shachat,1998). Experiments featuring simple negotiations yield results suggesting a role for fairness (Gu th6van Damme,1998).Some,but not all,subjects in public good games choose to cooperate more than self-interest would dictate(Croson6Marks,1998, and Nagel6Tang,1998).Even when simply given the option of keeping a sum of money or sharing with an anonymous other,many choose to share(Cason6Mui, 1998).While different investigators give these observations different interpretations, we would say the pattern of evidence compels an investigation of whether economic behavior is motivated by more than just material gain.1Gu th and van Damme's bargaining experiment clarifies some central issues although in doing so,it deepens the puzzle.The experiment concerns a three-person bargaining game,in which one bargainer,the proposer X,proposes a division of 120points among the three(10points worth1Dutch guilder,and in some cases worth2).A minimal amount,5points,must be allocated to each player,but otherwise the proposer is free to allocate as he chooses.A second bargainer,the responder Y,either accepts or rejects the proposal.If accepted,the money is distributed accordingly.If rejected,all receive nothing.The third bargainer,the dummy Z,has no say in the negotiation,and no choice but to accept any agreement set by the other two.The game was played in three conditions,each distinguished by the information the responder is given about the proposal.In the xyz-condition,the responder knows the full proposal at the time of accepting or rejecting.In the y-condition,the responder knows only his own allocation.In the z-condition,the responder knows only the dummy's allocation.In some treatments,all games played had the same information condition(the constant mode).In other treatments,games were rotated through all three conditions(the cycle mode).The prediction of subgame perfection,a standard game theoretic solution based on the self-interested material gain assumption,is invariant to both the information condition and treatment mode:Every feasible proposal gives each bargainer a positive amount,so the responder always makes more money accepting than reject-ing.The proposer should therefore ask for the maximum allowable.As an alter-native prediction,the experimenters consider a hypothesis they call``strong intrinsic motivation for fairness.''Again,the predictions are invariant to the information condition and treatment mode:Each bargainer gets a one-third share.Hence the experimenters pit a hypothesis predicated on the material gain motive against one predicated on fairness.In the introduction to their paper,Gu th and van Damme cite five important regularities that emerge from their experiment.We discuss them later in detail;here is a brief summary:First,proposals depend on the information condition,with the responder sometimes getting a large share.Second,the amount the dummy receives1Of course,one of the main advantages of the laboratory is that we can test competing explanations against one another.This has,and continues to be,done.See Roth(1995)for an overview of hypotheses and experiments concerning bargaining games.is in all conditions very small.Third,some proposals are rejected,although a smaller proportion than usually observed in two-person versions of the game,where there is just a proposer and a responder.Fourth,there is a learning trend.And fifth,there are some differences across constant and cycle treatment modes.Most of these observations are inconsistent with one or both hypotheses.We might speculate that the data represent some convex combination of the two.But Gu th and van Damme point out that the way proposers and responders treat the dummy is inconsistent with even a moderated concern for fairness,at least if we understand the concept of fairness to be connected in some way to that of altruism (Section 6):The experimental data clearly refute the idea that proposers are intrinsically motivated by considerations of fairness:they only allocate marginal amounts to the dummy and they give little to the responder in information condition m =z .(Also responders don't show concern for the dummy.)In sum,conventional understandings of self-interest and fairness,whether taken separately or in combination,appear inadequate to explain the data.In this paper,we show that the ERC model predicts four of the five regularities cited by Gu th and van Damme,not only as the general form stated above,but also in detail;for example,the ERC model accurately predicts the direction proposals move across information conditions.Another paper,Bolton 6Ockenfels (1997),demonstrates that the ERC model is consistent with the behavior observed in a wide variety of other laboratory games,including those thought to exhibit behavior reflecting ``equity,''``reciprocity,''and ``competitiveness;''hence the moniker ERC .The ERC model is constructed from standard game theory,save for the motiva-tional premise:ERC players are motivated by both the monetary payoff from the experiment,as well as by their own ``relative payoff,''a measure of how the individual's monetary payoff compares to that of the rest of the group.Put another way,the model asserts that individuals are motivated by the interaction of two things:own absolute (monetary)payoff,and own relative payoff.The distribution of payoffs among other players does not enter in the player's calculation.Hence we see immediately that ERC is consistent with Gu th and van Damme's observation that other players show very little concern for the dummy.2We can say more,and in greater detail.2.THE ERC MODELWe concern ourselves with n-player lab games,n 1,where players are randomly drawn from the population,and anonymously matched.All game payoffs are monetary and non-negative y i ,i =1,2,...,n .ERC posits that each player i217STRATEGY AND EQUITY 2Fehr and Schmidt's (1997)model of ``biased inequality aversion''has some features in common with ERC.One major difference is that the biased inequality model implies that people care about the dif-ference in payoff between self and each of the other individuals.maximizes the expected value of the motivation function,v i(y i,*i).We refer to y i as i's absolute payoff and*i as i's relative payoff,where*i(y i,c,n)={y iÂc1Ân=n c y i,if c>01,if c=0if i's proportion of the social reference share,1Ân;and c= nj=1y j is the size of the pie that is distributed among all players.The``social reference share''is the proportion of the total payoff that i would receive if all players received the same payoff.The motivation function is characterized as follows:A0.v i is continuous and twice differentiable on R+_R+.A1.(a)Narrow self-interest:v i1 0,v i11 0.(b)Monotonicity:Fixing a*i,given two choices where v i(y1i,*i)=v i(y2i ,*i)and y1i>y2iplayer i chooses(y1i,*i).parative effect:v i2=0for*i=1,and v i22<0.A0is posited for mathematical convenience.A1implies that,fixing the relative payoff,i has preferences over the absolute payoff like those assumed in traditional economics models.A2is the main innovation of the ERC model.It implies that,fix-ing the absolute payoff,v i takes it maximum where i receives the social reference share.Let k=cÂn be the average absolute payoff.Fixing k,i's motivation function can be written as v ki(*i):#v i(k*i,*i).A3insures risk aversion with respect to*i:A3.Risk aversion:v ki"(*i) 0.Define{i(k):=arg max*i v ki(*i)and_i(k):v i(k_i,_i)=v i(0,1).The value{i is the proportion of the social reference share that i would ideally assign to self given the average absolute payoff k.A0 A3insure that{i#[1, )and the value is unique up to i and k>0.By definition,player i is indifferent between a distribution in which i receives the proportion of the social reference share_i and a distribution in which all players receive nothing.With the addition of A4, _i#(0,1]and the value is unique up to i and k>0:A4.Strong equity effect:_i 1.In essence,A4guarantees that i prefers a distribution in which i receives more than the social reference share to a distribution in which all players receive nothing. A5provides an explicit characterization of the heterogeneity that exists among players,stated in terms of{i and_i:A5.Heterogeneity:Let f and g be density functions and k>0.Then f({i|k)>0on[1, )and g(_i|k)>0on(0,1].218BOLTON AND OCKENFELSThe ERC model presented here is basically equivalent to the ERC model proposed in Bolton 6Ockenfels (1997).That paper provides an extensive discussion of the assumptions and their implications.The present model posits three slight modifications that make it easier to apply ERC to the game of Gu th and van Damme (GvD game).First,we define the relative payoff of player i as i 's proportion of the social reference share rather than as i 's proportion of the monetary pie c .These formulations are equivalent when we confine our attention to a fixed number of players n .The present analysis allows us to do comparative statics across games that have differing numbers of players.Second,in A3we assume risk aversion rather than a weaker quasiconcavity assumption.These two modifications are used exclusively to derive proposition ERC7below.Third,we state A4as a basic assumption,rather than a special one necessary for specific propositions.We emphasize that none of these modifications are inconsistent with any of the results in Bolton 6Ockenfels (1997).2.1.Solving the ModelWe solve the model by applying Bayesian perfect equilibrium to the class of motivation functions characterized above.Specifically,we derive predictions under the assumption that players choose the strategy that maximizes the expected value of their motivation function given the information they have about their playing partners'motivation functions.Playing partners in the GvD experiment were anonymous to one another,meaning a player could not know the exact charac-teristics of his partners'motivation functions.We assume that players are sufficiently experienced with one another to know the distribution of motivation functions from which the partners are randomly,and independently selected.In particular,we suppose that proposers know the distribution of _i ,defined above in A5.3.ERC PREDICTIONS AND THE GVD DATAIn this section,we derive a series of seven ERC predictions and compare them to the GvD game data.We organize the analysis (roughly)around the major obser-vations cited by Gu th and van Damme.Following Gu th and van Damme,let x (resp.y ,z )be the points or ``payoffs''received by the proposer X (resp.responder Y ,dummy Z ).3.1.Proposer and Responder Behavior :Fairness and Selfishness in the xyz-and y-conditionsERC asserts that individuals are motivated by their own absolute and their own relative payoff.The distribution of payoffs among other players does not enter the motivation function.The following propositions show that,according to the model,and consistent with the data,neither the proposer X nor the responder Y behave altruistically towards the dummy Z if the information condition is either xyz or y :219STRATEGY AND EQUITY220BOLTON AND OCKENFELSERC1.In information conditions xyz and y,an offer of the social reference share or more to the responder(y 40)is never rejected,regardless of the dummy-payoff z.Proof.Since the pie size is c=120,the social reference share cÂn=120Â3is40. By A1we have that each player i prefers(40,1)to(0,1)so that y=40is never rejected.Moreover,A3and A4imply that y>40(*i>1)is never rejected. Evidence.For information conditions xyz and y(constant and cycle modes combined),Gu th and van Damme(1994)report a total of252offers of y that are greater than or equal to40.None of these offers is rejected.Moreover,when the dummy Z is offered the minimum payoff(5),responders reject in only about70 of the88total cases in conditions xyz and y combined.Gu th and van Damme con-clude from their analysis that,``there is not a single rejection that can be clearly attributed to a low share for the dummy''(Section1).The next three predictions of ERC capture some empirical properties of the proposed distributions(x,y,z)and show that the proposer treats the dummy Z with substantially less regard than the responder Y.All of these results make use of the following lemma:Lemma.The probability that an offer in which y<40is rejected,increases as y decreases.Proof.Follows directly from the heterogeneity assumption,A5.ERC2.In the information conditions xyz and y,the proposer allocates himself at least the social reference share(x 40).Proof.By A1and A2we have that a proposer X always strictly prefers x=y=z=40to any allocation with x<40.The proof of ERC1shows that x=y=z=40carries no risk of rejection.Evidence.True in all but one out of360cases.ERC3.In information conditions xyz and y,the dummy never receives more than the social reference share(z 40).Proof.Suppose that z>40.Then either x<40or y<40.If x<40,then X can improve his situation by redistributing some money from Z to X.This increases the absolute payoff x(and increases the value of the proposer's motivation function) without altering the probability of rejection.If y<40,then X can improve his situation by redistributing money from Z to Y.This decreases the probability of rejection while holding the absolute payoff x constant.Evidence.True in all but two out of360cases.Note that the upper bound for the dummy's payoff,as derived in ERC3,is valid neither empirically nor theoretically for the payoff of the responder Y.The responder's theoretical upper payoff bound is75rather than40,because x may be only40(ERC2)so that y can be as large as75(recall that the minimum value for z is5).In91of the180cases of the information condition xyz(constant and cyclemodes combined),the responder receives a payoff that is greater than the social reference share.The mechanism underlying the asymmetric treatment of the responder Y and the dummy Z becomes even clearer in the next proposition,which states that as long as the probability of rejection is positive,the dummy receives only his minimum payoff.In essence,the responder is served first .However,once the probability of rejection is zero,and X has taken all he wants,any additional amount is,by the theory,allocated indeterminantly:Z might get more than the minimum payoff,or Y might get more than the social reference share,or both might happen.ERC 4.In information conditions xyz and y :If the proposer offers y <40,then z =5,the minimum value allowed .Proof .As long as y <40and z >5,X can redistribute money from Z to Y .This redistribution does not change X 's relative and absolute outcome but increases the probability of acceptance.Evidence .In the constant mode,ERC4is true in all but one out of 75cases with y <40.Evidence in the cycle is less conclusive:In 44out of 108cases with y <40we have z >5.In the constant mode data,the responder Y is clearly served first.While almost none of the dummies receive more than their minimum payoff in the case of y <40,a majority of dummies receive a payoff z >5in the 69cases with y 40.In essence,ERC4says that proposers allocate money to where it has the greatest marginal effect.So a proposer who allocates self x >75,allocates the remainder to the responder Y (except the minimum payoff for the dummy)because giving to the dummy only improves the relative standing,while giving to the responder has an additional positive effect:It reduces the risk of rejection.On the other hand,once the proposer is satiated,and the risk of rejection is zero (y 40),ERC leaves the distribution of the remaining money indeterminant.In fact,there is evidence in the constant mode that proposers do not much care how the money they distribute to the others is allocated:For proposals with y 40,the distribution of the adjusted payoffs y :=y &40and z :=z &5do not differ significantly (Mann Whitney U-test,N =88,two-sided p-value=0.579;the corre-sponding test for the cycle mode yields significance).Gu th and van Damme observe that a strong intrinsic motivation for fairness would imply that each player receives40.But this kind of mitigation of payoffs would imply that dummy Z should receive what is not needed to insure acceptance.The distributions of y and z show that proposers do not have a strong tendency to mitigate payoffs.Rather,proposers in the constant mode appear to give arbitrarily once acceptance is insured.(Bolton et al .,in press,make a similar observation in the context of the dictator game.)3.2.Proposals Are Sensitive to the Information ConditionWe now bring the z-condition into the discussion.Gu th and van Damme emphasize that ``proposers react systematically and strategically to the information that responders receive about the proposal''(Section 1).We might speculate that 221STRATEGY AND EQUITY222BOLTON AND OCKENFELSproposers behave strategically by trying to signal a generous offer y in the z-condi-tion,where the responder Y receives information solely about the offer z.But what kind of offer to z signals that y is large?There are two possible hypotheses.First, one might speculate that a generous offer to Z signals that the proposer X is an altruist,and therefore increases the probability of a generous offer to Y.We will call this the altruism-signaling hypothesis.It implies a negative correlation between z and the probability of rejection.In contrast,the ERC-signaling hypothesis suggests that z is negatively correlated with the responder's expectation of y:Suppose that all proposers want to realize their optimal proportion of the social reference share {X in the z-condition.Then the distribution of{X can be associated with a distribu-tion of total offers y+z.Hence,there is a negative correlation between observed z and expected y.And a proposer who wants to signal that y is large should choose a small z regardless of her{X.As it happens,the constant mode data exhibits no evidence for signaling of any sort.Specifically,there is no correlation between z and y(Spearman rank correla-tion coefficient of0.018,p=0.88).There is,however,a significant correlation between z and y in the cycle mode(correlation coefficient of0.37,p=0.00).Because the correlation is positive,we can rule out ERC-signaling.On the other hand,we expect altruism-signaling to be accompanied by a negative correlation between z and the rejection rate.There is no evidence for this;as Gu th and van Damme put it(Section3)``...responders view high z-values with suspicion,the percentage of rejected proposals does not decrease with z.''In sum,there is no clear evidence for any form of signaling.Therefore,the follow-ing propositions are derived under the assumption that signaling does not take place.That is,we assume that the proposal z does not offer any information that influences the rejection probability.Of course,ERC predicts that very large offers to the dummy,for example z=120,are rejected.However,z is greater than40in only three cases and is always smaller or equal to55.Therefore,we can safely ignore these sorts of offers.As in Gu th and van Damme,let p(x)(resp.p(y),p(z))be the amount the proposer allocates to player P(P#[X,Y,Z])in the cycle mode and let p(cx)(resp. p(cy),p(cz))be the amount the proposer allocates to player P in the constant mode when the information condition is xyz(resp.y,z).Then,the following propositions state the predicted strategic adjustments of the proposals(x,y,z)to the change in the information conditions.ERC5.The proposer X demands more in the z-condition than in the xyz and y-conditions(x(cx),x(cy)<x(cz)and x(x),x(y)<x(z)).Likewise,the responder Y receives less in the z-condition than in the xyz-and y-condition(y(cx),y(cy)>y(cz) and y(x),y(y)>y(z)).Proof.In the z-condition,the rejection behavior is independent of z(no signal-ing)and therefore independent of the decision of X.Therefore,X should behave as if he or she is in a role of a dictator faced with two recipients.On the other hand, in the xyz-and y-conditions,the proposer is in an ultimatum situation.As shown in Bolton6Ockenfels(1997),the ultimatum situation creates an additionalstrategic incentive to give for all proposers who run the risk of rejection.Hence, ERC predicts lower offers in the z-condition if proposers are sufficiently selfish: {X>2.Proposers with{X 2offer the same total amount in both conditions.Evidence.ERC5is strongly supported by the data(Gu th6van Damme, Sections3and4).ERC6.Offers y and demands x do not differ across the xyz-and y-condition (x(cx)=x(cy)and x(x)=x(y);y(cx)=y(cy)and y(x)=y(y)).Proof.The responder Y is only interested in y and yÂc(ERC1).Since c is common knowledge,the full information condition does not give any additional decision-relevant information to the responder as compared to the y-information condition. Since the rejection behavior is equivalent in both information conditions,the proposer behavior is equivalent as well.Evidence.The data clearly supports ERC with respect to the responder's payoff y(Gu th6van Damme,Sections3and4).Gu th and van Damme find that x(cx)<x(cy)and x(x)<x(y).While these effects are statistically significant,in absolute terms they are``slight''(Gu th6van Damme,1994,Section4).3.3.Rejection RatesThe overall rejection rate for GvD games in information conditions xyz and y is about4percent.This is surprisingly low if one compares it with corresponding rates in the2-person ultimatum game,which typically run in the neighborhood of 15to20percent(see Roth's1995survey).ERC7offers an explanation.We suppose that the average size of the pie is fixed across games.The underlying idea of the proof is that a3-person GvD game creates more room to agree on a distribution of relative payoffs between the proposer and the responder than a2-person game.A proposer with{X 2will propose an offer with a zero probability of rejection in the GvD game,but not generally so in the ultimatum game.Risk aversion(A3)implies that the rest of the proposers,those with{X>2,will choose to use some resources to lower the probability of rejection relative to what it would be in the two-person game.ERC7.Holding the average pie size fixed across games,rejection rates are lower in the3-person GvD game in the information conditions xyz and y than in the2-person ultimatum game.Proof.Given that the average pie size,k,is fixed across games,i's motivation function v ki(*i)can be written as v i(*i)for both n=2and3.Note that if proposerX offers*2Y to the responder Y in the2-person game,then X receives*2X=2&*2Y.If X offers*3Y in the3-person game,he or she receives*3X=3&*3Y.We also knowthat the proposer X always prefers*nX =1to any*nX<1(ERC2).On the otherhand,*nY 1is never rejected(ERC1).Therefore,the optimal offer*nYis smaller orequal to n&1.Likewise,since*nY=0is always rejected(A1and A2),the optimaloffer*nY must be strictly positive.223STRATEGY AND EQUITYFixed average pie across games implies{i(k)#{i#[1, ).Suppose{X=1.Then,X chooses*nX =1independent of n,and by choosing*nY1,proposer X's offer isnever rejected,neither in the2-person game nor in the3-person game.Now sup-pose1<{X 2.Then,in the3-person game,the proposer can realize his or her optimal proportion of the social reference share{X with no risk of rejection by choosing*3Y1.However,on average the proposers with{X<2face a positive probability of rejection in the2-person game.(Here we implicitly assume that the population is not too risk averse in the sense that the probability of a proposer with 1<{X<2who demands more than half of the pie in the2-person game is positive.) Now,it remains to show that for proposers with{X>2,the rejection rate in the 2-person game is no smaller than in the3-person game.First,note that by ERC1 and ERC4the dummy always receives the minimum payoff from proposers with {X>2.Hence,without loss of generality,we can focus our analysis on the choice of the offer to the responder.We can write the problem of proposer X as(normalize v X(0,1)=0):max q(*nY )v X(n&*nY)with respect to*nY#(0,n&1],where q(*)=1& *g(_i|k)d_i is the probability that a randomly chosen responder accepts the offer of*.Suppose*2Y and*3Yare the solutions of the proposer's problem in the2-person and3-person game,respectively.We show that*2Y *3Ywhich implies q(*2Y) q(*3Y).Suppose to the contrary that*2Y >*3Y.Necessarily,in the3-person game,q(*3Y )v X(3&*3Y)>q(*2Y)v X(3&*2Y).(1)Define2q:=q(*3Y )&q(*2Y).By A5and because*2Y>*3Y,2q<0.Define2v:=v X(3&*3Y )&v X(3&*2Y).Then2v>0because*nX=n&*nY<{X and concavity(A3).Then(1)becomes[q(*2Y )+2q][v X(3&*2Y)+2v]>q(*2Y)v X(3&*2Y)or2qv X(3&*2Y )+[g(*2Y)+2q]2v>0.(2)In the2-person game,q(*3Y )v X(2&*3Y)<q(*2Y)v X(2&*2Y).(3)Define2v:=v X(2&*3Y )&v X(2&*2Y).Then2v>0.By the same series of substitutionsthat produce(2)we get2qv X(2&*2Y )+[q(*2Y)+2q]2v<0.(4)224BOLTON AND OCKENFELS。
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。
game theory4 博弈论 英文
If q>0: B is better than T If q=0: B is as good as S
{0} B1(q) =
if q > 0
{p: 0≤p≤1} if q = 0 {1} if p < 1/2 {q: 0≤q≤1} if p = 1/2 {0} if p > 1/2 MSNE: {(p,1-p); (0,1)} p≥1/2
• Game • Elimination of strategies that are strictly dominated by mixed strategies
• illustration • example
2 / 32
Review
MSNE
Elimination By Mixing
Summary
19 / 32
Review
MSNE
Elimination By Mixing
Summary
B1(q) =
{1} if q < 3/4 {p: 0≤p≤1} if q = 3/4 {0} if q > 3/4
B2(p) =
{0} if p < 1/3 {q: 0≤q≤1} if p = 1/3 {1} if p > 1/3
• P2 must be indifferent between L and R: p*1+(1-p)*2 = p*2+(1-p)*1 => p=1/2
9 / 32
Review
MSNE
Elimination By Mixing
Summary
2
1
T (p) B (1-p)
L (q) 0,1 2,2
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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。
(4)轮流进行，从玩家1开始；
(5)拿最后一个筹码的玩家赢(不能继续的玩家输)。
我们如何分析这个游戏？玩家有获胜的策略吗？你喜欢成为第一名还是第二名？这是个好决定吗？
我们将从最终状态到初始状态对游戏进行分析。
这种方法有时被称为逆向归纳法。
如果只剩下1、2或3个筹码，那么下一个玩家可以一次全部拿走并获胜。
假设还剩4份薯条。
那么下一个玩家肯定会留下1、2或3个筹码，然后他的对手会赢。
因此，四个筹码的状态对于下一个玩家是失败的，而对于前一个玩家是赢的。
如果还剩5、6或7个筹码，那么下一个玩家可以拿4个筹码赢。
如果还有8个筹码，下一个玩家肯定会留下5、6或7个筹码，这样前一个玩家就可以赢了。
这样，我们希望能够将状态改为0、4、8、12、16并获胜。
现在让我们来分析一下以下21份薯条的状况。
因为21不能除以4，所以我们先赢了。
唯一最好的办法就是先吃一份炸薯条，这样XXXX竞赛就成了一个正方形的棋盘。
表单，初始化为空。
玩家可以选择一个空方块，并在上面写下S或O。
谁先写连续的SOS，谁就赢了。
如果没人能写出来，那么这场比赛就是平局。
(a)假设n=4，首先在第一个正方形上写s。
证明最后一手会赢。
证明当n=7时，第一个玩家获胜。
证明n=XXXX提出了一个广义的尼姆对策和精致定理，称为
尼姆(k).尼姆(K)游戏基本上与尼姆游戏相同，只是必须在一次操作中从K堆筹码中取出筹码(给定K，并且必须从一堆筹码中取出至少一个筹码)。
当K=1时，这是尼姆游戏。
摩尔定理指出(x1，x2？当且仅当x1到xn是按位(二进制)时，xn)是p 状态
模(k+1)运算的结果是0。
(a)考虑练习3的灵活游戏，但稍有变化，即玩家每次操作时可以移动1。
或者左边两个硬币。
请注意，这实际上是尼姆(k)游戏，其中k=2。
用摩尔定理证明练习3的状态是n状态，并找到一种方法把它变成p状态。
证明摩尔定理。
考虑一下米莎游戏的最佳解决方案。
(我已经在这个问题上给出了证明，你可以问你需要什么。
)
3.图表游戏
我们现在将给出有向图游戏中组合游戏的等价描述。
这包括第1章和第2章中描述的游戏。
这可以通过用有向图的节点表示状态，用有向图的边表示操作来实现。
接下来，我们将定义一个名为SG(Sprague-Grundy)的函数，它包含比P和N状态更多的信息。
3.1有向图游戏。
让我们首先给出有向图的数学定义。
定义.有向图G可以表示为(x，f)，x是一组非空节点(即状态)；F是x(x∈X)的函数，X是X的子集。
对于给定的X，F(x)表示从状态X 开始可以到达的状态(称为X的追随者)。
如果F(x)为空，则x为最终状态。
两人游戏可以在地图上进行，从x0(初始状态)开始，并遵循以下规
则:(1)玩家1从x0开始；(2)双方轮流操作；
(3)当处于x状态时，执行操作的玩家将其改变为状态y∈f(x)；(4)如果在玩家操作之前游戏已经达到其最终状态，则判定玩家已经失败。
根据定义，图形游戏不能在无限制操作期间停止。
为了避免这种情况和其他问题，我们现在考虑这样一个图:无论从哪个x0状态，都有一个与x0相关的整数n，因此从x0开始的每条路的长度小于或等于n。
这样一个图被称为[渐进有界]。
如果x是一个有限集合，这意味着在这个图中没有环(环就是这样一条路径，x0，x1？在xm中，x0=xm，m>3)。
例如，我们在1.1节中分析的差集博弈，S={1，2，3}是一个图形博弈的代表。
让X={0，1，？，n}是状态集。
空堆是最终状态，所以F(0)是空集。
类似地，F(1)={0}，F(2)={0，1 }；2x}时。
事实上，这是一个非常愚蠢的游戏。
第一只手可以在第一次手术中拿走所有的薯条！计算SG函数有什么用？
这个问题将在下一章回答。
如果游戏考虑的是n堆而不是1堆的情况，那么游戏就不那么简单了。
下一章将要讨论的定理将告诉我们如何将nim-加法与SG函数相结合来解决多桩问题。
3.4更一般的SG功能。
让我们暂时看一下当图不再渐进有界或包含环时的问题。
首先，让我们假设渐进有界条件放宽到渐进有限(每条路都是有限长度)。
这个条件相当于我们所说的组合游戏的条件(6)(第1.2节)。
图中仍不允许出现环。
让我们举一个例子，它是进步的，而不是进步的。
这是一个渐进的有限图表，因为游戏将在有限的操作次数后结束。
然而，它不是一个渐进有界图，因为不是每条路的长度一开始就能知道它的上限。