博弈论 战略分析入门第四章课后题答案

Analysis of Strategy

Chapter 4. Nash Equilibrium

1. Objectives and Concepts

The principle objective of this chapter is to introduce the Nash equilibrium and toconvey some notion of the range of possibilities and applications, including the

possibilities that there may be no Nash equilibria in pure strategies and the possibility thatthere may be plural Nash equilibria. (Since mixed strategy equilibria are not introduceduntil Chapter 8, it is not possible to give a meaningful definition of pure strategies at thispoint, and is necessary to talk around it a bit.) Important subsidiary concepts arecoordination games and Schelling points (or focal point equilibria), heuristic methods offinding the Nash equilibria, such as underlining, and refinement of Nash equilibrium.

The chapter begins with an example that is based on Warren Nutter’s game-theoretic version of Bertrand competition, except that in this instance a kind of qualitycompetition is considered. The solution to this game can be found by iterated eliminationof dominated strategies (which will not be covered until Chapter 11) and reflects theintuition that it is best to be just one step ahead of the competition. Thus, while it does nothave a dominant strategy equilibrium, it has some dominated strategies and a uniqueNash equilibrium, and hopefully forms a natural bridge from the study of dominantstrategy equilibrium.

Games with plural equilibria are introduced with the game of Choosing RadioFormats. The idea that history (or other clues) can establish a Schelling point also comesin with this example. The Market Day game reinforces the idea that plural Nash

equilibria can have explanatory value – explaining the persistence of what seem to bearbitrary conventions. Games without Nash equilibria (in pure strategies) are introducedwith an escape-evasion game. This is an important category in itself, though the mostimportant applications are in differential games and thus beyond the scope of the book.

Accordingly, the concepts areNash Equilibrium

Unique Nash EquilibriaFinding Nash EquilibriaPlural Nash Equilibria

The difficulty of choosing among plural Nash equilibriaSchelling Points

Custom, convention and history as Schelling pointsSchelling points from the logic of the gameRefinement

Games without Nash equilibria in pure strategies

2. Common Study Problems

Students who have not yet grasped the best-response idea will find Nash

equilibria even more difficult than dominant strategy equilibria. This is the crisis point forstudents who have not “got” best response. The best response tables (such as table 2 inthe chapter) are designed to make this a little easier, so urge the student to rely on themand on underlining as intermediate steps in their analysis. I sometimes suggest to my

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students that they physically move their fingers along the column or row to pick out thebiggest payoff. Making the solution as mechanical as possible will help students over thathump. Another (less troubling) problem is the relationship between Nash and dominantstrategy equilibria. Taking dominant strategy equilibria first is a pedagogical

convenience, since it is a little easier and will be familiar to students who have seen thePrisoner’s Dilemma in another class, but it can produce the impression that dominantstrategy equilibria are not Nash equilibria. The Venn diagram (Figure 1) is meant tospeak to that problem, and may need some stress in class.

3. For Business Students

The key business concepts for this chapter are strategies of location and marketniche, in the Location, Location, Location example, but also in the Radio Formatsexample and in the Hairstyle example in the exercises and discussion questions.

4. Class Agenda

First hour:

1) Quiz on earlier material

2) Introductory presentation: Nash Equilibria

• Assignments

3) Discussion: The Blonde Problem AgainSecond Hour:

1) Discussion of quiz and assignments

2) Play a coordination game in class, with random matching and without discussion.A handout description of the game is given on the next page.

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Another Random-Matching Two-Person Game

Once again, each person chooses between the strategies of collusion or defecting from thecollusive arrangement.

Put in your name and circle one of the two statements: either \"my strategy is collude\" or\"my strategy is defect.\" Your instructor will tell you whether to follow directions A) orB) below.

A) After you turn it in, your strategy choice will be matched with that of anotherclass member AT RANDOM, and your bonus points will be based on the payofftable above. There is to be no discussion of your strategy choices.

B) You will be matched with your neighbor and may discuss your strategy choice ifyou wish.

Payoffs are in GameBucks.

Table

Art's StrategyColludeDefect (3,3)(0,2)

(2,0)

(1,1)

Collude

Bob's Strategy

Defect

What will you do? Go for the big reward with a \"collude\" strategy or protect yourself

with an \"defect\" strategy?

Student name ____________________________

My strategy is (circle one)

Collude

Defect

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3) Discussion:

a. Results of the in-class game.

b. Give other examples of Schelling points in coordination games. Ideally,these should come from the students, but the following instances maystimulate the discussion if it comes slowly:

i. Driving on the right or left-hand side of the road.ii. Speaking the same language.

iii. Choosing a profession. Assumption: if both choose the same

profession, it does not pay well because it is too crowded. Howmany business majors in the class? Engineering? Communications,etc?

5. Answers to Exercises and Discussion Questions

1. Solving the Game. Explain the advantages and disadvantages of NashEquilibrium as a solution concept for noncooperative games.

Nash equilibrium is based on the idea that each player chooses the best response tothe strategy chosen by the other player. This is a clear concept of rationality wheneach person chooses in isolation from the other. Among the shortcomings are 1)Nash equilibrium may not be unique, posing the problem of determining which oftwo or more Nash equilibria may actually be chosen by rational agents, and 2)considering only the list of strategies for the game in normal form, that is, the“pure” strategies, there may not be a Nash equilibrium.

2. Location, Location, Location (Again) Not all location problems have similarsolutions. Here is another one: Gacey's and Mimbel's are deciding where to put

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their stores in Metropolis, the town across the river from Gotham City. Thethree strategies for Metropolis are to locate downtown, in Old Town, or in theGarden District. The payoffs are shown in Table E1.

Table E1 Payoffs in a New Location Game

Gacey's

Downtown

Downtown

Mimbel's

Old TownGardenDistrict

70,60110,70120,80

Old Town60,12040,40110,120

GardenDistrict80,100120,11050,50

Does this game have Nash equilibria? What strategies, if so? Which strategieswould you predict that Gacey's and Mimbel's would choose? Compare and contrast thisgame with the location game in the chapter. What would you say about the relativeimportance of congestion in the location decisions of the firms in the two cases?

A table modified to show the highest payouts for each player for each decision is asfollows:

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Mimbel'sDowntownOld TownGarden District

Gacey's

DowntownOld Town70, 6060, 120110, 70120, 8040, 40110, 120Garden District

80, 100120, 11050, 50

There are two Nash Equilibria. When Gacey’s locates in Old Town, Mimbels willlocate in the Garden District, and vice versa. Which solution will actually be chosen isnot definite.

This problem is different from the one in the chapter since there are 2 NashEquilibriums instead of one, which requires a little guesswork as to which one will be thefinal solution. It is similar in that there is not a dominant strategy equilibrium.

Congestion must be more of a problem in this scenario than in the chapterproblem. There is never a Nash equilibrium when both pick the same site. This could beexplained by the congestion problem

3. Drive on. Two cars meet, crossing, at the intersection of Pigtown Pike and

Hiccup Lane. Each has two strategies: wait or go. The payoffs are shown in Table E2.

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Table E2. The Drive On Game

Mercedeswait

Buick

waitgo

0,05,1

go1,5-100,-100

Discuss this game, from the point of view of noncooperative solutions. Does ithave a dominant strategy equilibrium? Does it have Nash equilibria? What strategies, ifso? Would you predict which strategies rational drivers would choose in this game?Which? Why? Pigtown Borough has decided to put a stoplight at this intersection. Howcould that make a difference in the game?

Here is a table modified to show the maximum payout for each driver:

WaitGo

MercedesWaitGo0, 05, 15, 1-100,-100

Once again, there are 2 Nash Equilibria. They are for the Buick to wait and theMercedes go, or vice versa.

To determine which will happen requires guesswork. The personality of thedrivers might determine what happens. If I were in the Mercedes, I would probably notwant to risk an expensive car getting damaged. Someone else, say in a CL600, might

Buick4.8

figure that his car is faster and that he can beat the other driver. Also, one of the driversmight just wave the other on rather than have both wait or both go.

It is possible that both drivers might wait rather than run the risk of an accident,i.e. choose a risk dominant strategy.

The stoplight would provide a Schelling Point to select for the equilibrium atwhich the driver with the green light chooses go.

4. Rock, Paper, Scissors. Here is another common school-yard game called Rock, Paper,Scissors. Two children (we will call them Susan and Tess) simultaneously choose asymbol for rock, paper or scissors. The rules for winning and losing are:Paper covers rock (paper wins over rock)Rock breaks scissors (rock wins over scissors)Scissors cut paper (scissors win over paper)

The payoff table is shown as Table E3.

Table E3. Rock, Paper, Scissors

Susan

paper

paper

Tess

stonescissors

0,0-1,11,-1

stone1,-10,0-1,1

scissors.-1,11,-10,0

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Discuss this game, from the point of view of noncooperative solutions. Does ithave a dominant strategy equilibrium? Does it have Nash equilibria? What strategies, ifso? How do you think the little girls will try to play the game?

Here is a table modified to show the best responses.

Susan

paper

paper

Tess

stonescissors

0,0-1,11,-1stone1,-10,0-1,1scissors.-1,11,-10,0

We see that there are no dominant strategies, nor are there Nash equilibriain terms of the strategies shown here. We have no basis (so far) to decide how thegirls will play the game.

NOTE TO INSTRUCTOR For the purist, it is not correct to say here that “thereare no Nash equilibria,” since this game has a mixed-strategy equilibrium. But, ofcourse, we will not cover mixed strategy equilibria until a later chapter. Thecorrect statement is that there is no equilibrium in pure strategies.

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5. The Great Escape. Refer to Chapter 2, Question 2.

Discuss this game, from the point of view of noncooperative solutions. Does ithave a dominant strategy equilibrium? Does it have Nash equilibrium? What strategies, ifso? How can these two opponents each rationally choose a strategy?

Warden

Guardwalls

climb

Prisoner

dig

No escape, success inpreventing escapeEscape,failure

InspectcellsEscape,failure

No escape, success inpreventing escape

The numerical payoffs can be assigned in many different ways. Here is a simpleversion that interprets “no escape” as minus one for the prisoner, plus one for the warden,and “escape” as vice versa. As the underlines show, there is no Nash equilibrium. Thusfar, we have no basis to say how a rational person would choose strategies in this case.

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WardenGuardwalls

Prisoner

climbdig

-1,11,-1Inspectcells1,-1-1,16. Sibling Rivalry. Refer to Chapter 2, Question 1.

Discuss this game, from the point of view of noncooperative solutions. Does ithave a dominant strategy equilibrium? Determine all the Nash equilibria in this game. Dosome Nash Equilibria seem likelier to occur than others? Why?

Irismath

math

Julia

lit

3.8, 4.03.7, 4.03.7, 3.8

lit4.0, 4.0If the siblings act independently, rationally and with self- interest (non-cooperatively), we can find two Nash equilibrium's strategies: (literature, math), (math,literature).

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We note that there is a Schelling point in this game: (Math, Lit) yields a certain4.0 for both girls, which is a reason it might attract attention, and probably is more likelyto be observed.

7. Hairsyle.

Shaggmopp, Inc. and Shear Delight are hair-cutting salons in the same strip mall,each groping for a market niche. Each can choose one of three styles: punker,contemporary sophisticate, or traditional. Those are their strategies. They already havesomewhat different images, based on the personalities of the proprietors, as the namesmay suggest. The payoff table is shown as Table E4.

Table E4. Payoffs for Haircutters

Shear

punker

punker

Shaggmopp

sophisticatetraditional

35,2030,4020,40

sophisticate50,4025,20,45

traditional60,3035,5520,20

Are there any dominant strategies in this game? Is there a dominant strategyequilibrium? Are there any Nash equilibria? How many? Which? How do you know?

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Once again, here is the modified table:

Shear

Punker

ShaggmoppPunkerSophisticateTraditional

35, 2030, 4020, 40

Sophisticate50, 4025, 20, 45Traditional60, 3035, 5520, 20

Shaggmopp’s best strategy is to go punker regardless of what Shear does. This ishis dominant strategy. Since Shear has no such dominant strategy, there is no dominantstrategy equilibrium.

The only Nash equilibrium is when Shear decides to go with the sophisticate look.Since Shear knows that Shaggmopp will probably go punk rather than sophisticate, it willchoose sophisticate.

6. Quiz question

Placed on the next page for convenience in copying and printing.

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Student name ____________________________

Quiz – Game Theory

Felix and Oscarina share their home with two cats. Felix, who has a sharp senseof smell, would like for the cat boxes to be cleaned twice a week. Oscarina, whose senseof smell is less acute, would be satisfied if they were cleaned once a week. Each wouldprefer not to be the one to clean the cat boxes. Their payoffs are shown on the followingtable.

Oscarina

don't clean

don't clean

Felix

clean onceclean twice

-5,-3-2,40,5

clean once0,-15,21,3

clean twice7,-56,-42,-3

Find any and all Nash equilibria for the catbox game? Are there dominated strategies?Which? Is there a dominant strategy equilibrium? Explain.

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Answer:

A payoff table with best responses underlined follows:

Oscarina

don't clean

don't clean

Felix

clean onceclean twice

-5,-3-2,40,5clean once0,-15,21,3

clean twice7,-56,-42,-3

The Nash equilibrium is where Felix cleans the cat box twice and Oscarina nevercleans. “Clean twice” is a dominated strategy for Oscarina. Since the best response foreach person depends on the strategy chosen by the other, there is no dominant strategyequilibrium.

It seems that Felix, whose need is greater, will empty the catbox, if the twocompanions act noncooperatively. Now, it may seem odd that people who live togetherwould act noncooperatively , but life is strange, and odd things do happen. However, acouple of years ago, Oscarina gave Felix a Christmas present – a year of catbox cleaning– and has renewed the gift, so love triumphs after all.

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