博弈论战略分析入门第一章课后题答案.pdf

Instructors Guide to Game Theory: A Nontechnical Introduction to the Analysis of Strategy Chapter 1. Conflict, Strategy, and Games 1. Objectives and Concepts The major objective of this chapter is to introduce the student to the idea that “serious” interactions can be usefully treated as games what I have called the “scientific metaphor” at the root of game theory. Secondary objectives are to introduce the concepts of best-response strategies and the representation of games in normal form. Thus, the chapter starts with an example from war, which most people without preparation in game theory would think of as a most natural field for thinking of strategy, and the chapter begins with an example presented in extensive form, because it seems to be a more intuitive and natural way of thinking about strategy. Interweaved with this are some discussions of the origins of game theory. The chapter also takes up an episode from the movie version of “A Beautiful Mind,” since it seems very likely that many students will have seen the movie and it may be a major source of whatever ideas they have about game theory. The Prisoners Dilemma is the one example they are most likely to have seen in one or more other classes, so it belongs here, too. Using the Karplus Learning Cycle as a major organizing principle, I open with an example the Spanish Rebellion and only then introduce the general ideas it illustrates, and then follow with another example, NIM. Again, the discussion of the game in normal form begins with an example, the familiar Prisoners Dilemma, then proceeds to the general principles and follows with two more examples, the one from the movie and an advertising dilemma. This procedure is “psycho-logical” rather than logical, and some 1.2 instructors may not be familiar with it. However, I think it works well with most students, who can understand the general principles better if they have an example already in mind. Accordingly, the key concepts are Definition of Game Theory History and emergence of Game Theory Game Theory as applicable to more than what we ordinarily think of as games. Representation in extensive form (tree diagrams) Best Response Representation in Normal Form 2. Common Study Problems The most important study problem probably will not actually emerge for a few class periods, but the roots are here in the first chapter: the concept of best response is difficult for some students, including some very good ones. Confusion may show up later in the form of a real difficulty in answering questions about social dilemmas: “How can this be a best response if it makes everybody worse off?” At this point, it may be helpful to emphasize that “best response” means the best response to other strategies that other players might choose, NOT necessarily a best response to the situation as a whole. Some (often very good) students may want to dispute whether the analysis of the Spanish Rebellion is really right. They have a point. It could be more completely represented as follows: 1.3 Hirtuleius Pius Pius New Carthage Laminium Pius wins Good Chance for Pius Good Chance for Hirtuleius Hirtuleius wins big New Carthage New Carthage Laminium River Baetis Hirtuleius Stay at Laminium Sure win for Pius delay decision But a) it doesnt make any difference, since Hirtuleius will never choose to stay at Laminium, and give Pius a sure win. (That would not be subgame perfect, a concept we will get into in Chapter 14). b) Therefore, at the first step Hirtuleius commits himself to meeting Pius at the River Baetis, and it is that commitment that is shown by the first decision node. c) All game theory examples are simplified and abstracted in some ways, and we always need to take care that we have a simplification that focuses on the important points, rather than missing them. So it really is a good point to make, and this is a good example of the ways we need to be careful about our simplifying assumption. 3. For Business Students The major bait for business students in this chapter is purposely given a high profile as the last example, the advertising game. 4. Class Agenda First hour 1) Get organized 1.4 a) Class Details b) Assignments 2) Introductory presentation: What is Game Theory? Second hour 1) Discussion of assignments, homework, etc. 2) Discussion on Game Theory as a Scientific Metaphor Discussion question: One issue in environmental policy is the passage of resources on to the next and following generation. For example, forests and underground aquifers can be of use to each generation, if they are preserved. However, if one generation uses them so intensively that they are destroyed, then future generations are deprived of that benefit. How might we capture this as a “game?” Who are the players? What are the rules? Payoffs? Is the play sequential or simultaneous? 3) Play “The Environment Game” in class. Handout follows on the next page for convenience in printing and copying. 1.5 An In-Class Game From time to time in this class we will conduct some experiments with games, playing the games in class and discussing the results. Payoffs will be in GameBucks, and you will accumulate GameBucks throughout the class. Students GameBucks accumulations will be public knowledge. At the end of the class, students with above- average accumulations of GameBucks will get grade bonus points in proportion to the difference between the students accumulation and the class average. (Those below average will not be penalized). Your mastery of the principles of game theory should enable you to be more competitive in accumulating GameBucks. An Environment Game This chapter focuses on the idea that “real-world” problems and interactions can be thought of as games. Environmental problems are often studied in game theoretic terms. One issue in environmental problems is the passage of resources on to the next and following generation. For example, forests and underground aquifers can be of use to each generation, if they are preserved. However, if one generation uses them so intensively that they are destroyed, then future generations are deprived of that benefit. For this game, students play in order, for example, around a circle from left to right. The first student is given a certificate with “One GameBuck” written at each end. The student has the choices of passing the certificate on to the next student in order, or tearing it in half and returning it to the instructor in return for two GameBucks. Each student who receives the certificate has the same choices, except the last. Each student who passes the certificate gets one GameBuck on his record. The last student can only pass it back to the experimenter for one point. The succession of students represents the succession of generations, each of which has the potential to get one GameBuck of benefit from the resource if it is preserved. The maximum benefit is equal to the number of students. If a student early in the ordering takes the opportunity to get two GameBucks, the total number of GameBucks awarded may be considerably less than this. 1.6 5. Answers to Exercises and Discussion Questions 1. The Spanish Rebellion. In her story about the Spanish Rebellion, McCullough writes “There was only one thing Hirtuleius could do: march down onto the easy terrain . and stop Metellus Pius before he crossed the Baetis.“ Is McCullough right? Discuss. Yes, McCoullough is right. Hirtuleius must assume that Pius will respond to Hirtuleius choice, and anticipate that response. If Hirtuleius marches for New Carthage, Pius will respond by taking Laminium and breaking out, the worst outcome for Hirtuleius. If Hirtuleius waits and marches for the River Baetis, Pius will march for New Carthage, with a good chance of beating Hirtuleius Hirtuleius second worst outcome. But these are the only two possibilities, and second worst is better than very worse, so that is what Hirtuleius must choose. 2. Nim. Consider a game of Nim with three rows of coins, with one coin in the top row, two in the second row, and either one, two or three in the third row. A) Does it make any difference how many coins are in the last row? B) In each case, who wins? a) Suppose there are just 2 pennies in the last line. Then Anna can take the one from the top line. Barbara is left with one of two choices take 1 from either line, leaving the same game we had in the chapter, which we know Anna can win, or 1.7 take two from line, in which case Anna immediately takes the other two and wins. Thus first player wins in this case. b) Suppose there is just one in the last line. Then Anna can take the two from the middle, leaving Barbara to take one of the others so Anna takes the remaining one and wins. Here again the first player wins. c) However, try what you will, you will find there is no way that Anna can win if there are three coins in the last row. Here, second player wins, so it does make a difference. There is a mathematical trick to figure out more complex games, fortunately, since a tree diagram for a Nim game with 3 coins in the last row would start out with 6 options for Anna and have from 3 to 5 for Barbara at the next stage, it would get pretty unwieldy. Do a Google search on “Nim” if you are interested in the trick. 3. Matching Pennies. Matching pennies is a school-yard game. One player is identified as “even“ and the other as “odd.“ The two players each show a penny, with either the head or the tail showing upward. If both show the same side of the coin, then “even“ keeps both pennies. If the two show different sides of the coin, then “odd“ keeps both pennies. Draw a payoff table to represent the game of matching pennies in normal form. Odd HeadsTails Heads2,00,2Even Tails0,22,0 1.8 The standard of reading is assumed with the first payoff to even and the second to odd. (Even then odd.) 0- means wins no pennies; 2- means wins 2 pennies. Payoffs 1, -1 for wins one, loses one would be equally correct. 4. Happy Hour. Jims Gin Mill and Toms Turkey Tavern compete for pretty much the same crowd. Each can offer free snacks during happy hour, or not. The profits are 30 to each tavern if neither offers snacks, but 20 to each if they both offer snacks, since the taverns have to pay for the snacks they offer. However, if one offers snacks and the other does not, the one who offers snacks gets most of the business and a profit of 50, while the other loses 20. Discuss this example using concepts from this chapter. How is the competition between the two tavern owners like a game? What are the strategies? Represent this game in normal form. Jims Give SnacksNo Snacks Give Snacks20,2050,-20 TOMS No Snacks-20, 5030,30 This situation resembles a game because: There is more than one player Strategy is important There are outcomes that depend on each players choice of strategy Consider the strategies and payoffs involved here. The basic strategies are: offer 1.9 free snacks, do not offer free snacks. If both offer snacks, their payoff is lower than if both do not offer snac