博弈论 战略分析入门第三章课后题答案

Analysis of Strategy

Chapter 3. Dominant Strategies and Social Dilemmas

0. Problem Alert

In exercise 2 in this chapter, as printed, there is an inconsistency between the verbalassumptions and the payoff table. The payoff table shows greater total payoffs when onlyone makes an effort than when both do. (This was corrected when the effort dilemmarecurs in a later chapter but slipped by in this chapter). If you assign this problem Isuggest you use the following payoff table instead.

Table E1. An Effort Dilemma

Mr. Joneswork

shirk2,155,5

Mr.Smith

workshirk

10,1015,2

Answers will be given for both versions. The corrected problem will be given ona separate page in the appendix to this chapter to make it easy to print out as ahandout.

1. Objectives and Concepts

The major objective of this chapter is to introduce the ideas of dominant strategiesand dominant strategy equilibria, together with the range of issues to which they can beapplied. Since dominant strategy equilibrium is our first concept of equilibrium in thebook, the concept of equilibrium in game analyses (in itself) is also a key concept for thechapter. Among important subsidiary concepts are Social Dilemmas, that is, Prisoner’sDilemmas in wider applications, and Cooperative Dominant Games. Since both of theserest on a contrast between the dominant strategy equilibrium and the cooperativesolution, a rudimentary concept of cooperative solution must also be brought into thechapter at this point. This is a long list of “big” concepts, but they are interdependentconcepts and are useful and applicable in themselves, a point the chapter tries to stress.One of my business students reported back that she found the concept of a dominantstrategy itself simplified some business decision problems enough that her boss asked herto explain how she did it. She managed to keep him in the dark – and perhaps that washer dominant strategy in the circumstances!

The concepts of dominant strategies and dominant strategy equilibria rest on thebest response, and I have adopted a minor novelty, here – best response tables, that is,tables that show one player’s best response to each of the strategies the other playerchooses. This seems to help to stress the major difference between dominant strategiesand other strategies that may enter into Nash equilibria – that the best response is alwaysthe same, regardless of which strategy the other player chooses. With one unavoidableexception, all of the examples for this chapter are examples of strongly dominantstrategies. I will not introduce weak dominance until Chapter 9. (For nonmathematical

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students, this is a fairly hard distinction and a bit of a distraction from the more importantconcepts here, in my opinion).

Cooperative Dominant games are also a minor novelty. The point is to show that adominant strategy equilibrium needs not be bad – an important logical point! – and to setup the Acceptance Game, which underscores a point often made by free-marketeconomists: one way to solve a Social Dilemma is to sign an enforceable contract.

This chapter also gives the first examples of games with more than two strategiesand gives one example to illustrate the fact that a game needs not have a dominantstrategy equilibrium. This is an important logical point as well as a practical one, sincewe will take up many games in the balance of the book that do not have dominantstrategy equilibria, and, as always, it seems good pedagogy to make the point explicitrather than relying on the student’s mathematical intuition, which varies widely.

The voting example is an adaptation of Hotelling’s location model, which, whenapplied to political platforms, implies that all parties locate in the center.

In summary, the key concepts are(Best Response Strategies)Dominant Strategies

Dominant Strategy Equilibrium

Social Dilemmas

Cooperative Dominant GamesCooperative Solutions

Contracts and Acceptance GamesApplication to Voting and ElectionsGames with Three or More Strategies

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2. Common Study Problems

Students who do not have the habit of thinking mathematically tend (in myexperience) to construct their mental models around ideas like “The idea of the game is towin, and how can the player win if he does not get the best possible payoff?” or “Theidea of the game is to win, and how can the player win if he does not get a biggerpayoff?” This creates some problems in assimilating especially the Social Dilemmaanalysis. It will probably help to continue to stress that 1) best response means bestresponse to a strategy, not to the situation as a whole, and 2) that we ASSUME that theplayers are interested in the absolute, not the relative payoff, and that if they are pursuingthe biggest relative payoff, this will lead to a different kind of analysis and also todifferent results. There are some examples of that in later chapters, including principallyChapter 20.

3. For Business Students

Collaborative Product Development is the main business example for this chapter.The ideas of human capital and spin-off technologies are widely recognized as importantstrategic factors for technologically-oriented businesses. More generally, businessstudents should profit from the idea of contracts as a solution to dilemmas whendilemmas do happen. The catchphrase (and book title) Coopetition expresses somethingof the importance of interdependence in the modern business world, as illustrated byCollaborative Product Development.

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4. Class Agenda

First hour:

1) Quiz on earlier material

2) Introductory presentation: Dominant Strategies and Social Dilemmas

• Assignments

3) Play a social dilemma in class. For this hour, match the students at random andplay it without allowing the students to discuss their responses before they makethem. A handout for the game follows on page 3.6.4) Discussion: The Refi Game

A handout sheet is on the page 3.7. One might pose the question in this class and revealthe answer with the handout in the following class, after the students have had a chanceto think about it.

Second Hour:

1) Discussion of quiz and assignments

2) Play the same social dilemma in class, but for this hour, match the students withtheir neighbors in the class, allowing the students to discuss their responses beforethey make them.

3) Discussion: What are the differences and similarities of the two times playing thesocial dilemma game? How can you account for them?

Discussion: Continue discussion of the Refi Game.

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A Random-Matching Two-Person Game

This game is based on a general Social Dilemma. Each person chooses between thestrategies of collusion or defecting from the collusive 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 (2,2)(0,3)

(3,0)

(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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The Refi Game

In an article1 from the Philadelphia Metro newspaper we read “With mortgageinterest rates remaining at near-record lows, … Some owners are refinancing as

frequently as three or four times a year, even though the new loan’s interest rate will godown just 0.25 percent. … One result of frequent refinances by individual homeowners isthat it tends to drive up [processing] fees. Yet, … it is more of an ethical considerationthan a legal one. … with the onset of so many (early loan payoffs), the industry hasanswered by increasing interest rates to offset these losses. In the long run, everyone ispaying higher fees resulting from this churning trend.” The implication seems to be that ifonly a few borrowers refinance their loans, they can do so cheaply and benefit, but ifmany do so their benefits are less. Can we express this in terms of a two-person game innormal form?

In a two-person game there can be either 0, 1, or 2 refinancing, so the increase incost will have to come with the step from one to two.

Table 1. The Refi Game, Part 1.

BrefirefiA

no

This would be a clear Social Dilemma. In the article we read “it is more of anethical consideration” which may refer to the idea that, as long as both players act in theirisolated rational self-interest, they will refinance. Social dilemmas are often made thebasis for ethical arguments.

2,7

5,53,3

no7,2

1

James Woodard/CNS, Frequent Refinancing Leads to Higher Rates, PhiladelphiaMetro, April 3, 2003, p. 7.

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5. Answers to Exercises and Discussion Questions

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

The dominant strategy equilibrium provides a very powerful and persuasiveanswer to the question “what would a rational self-interested person do” in the game, andit provides an explanation for “dilemmas” in which people pursuing their self-interestinteract in ways that make them all worse off. The other side of that coin is that it focusesstrongly on the noncooperative side of these “dilemmas,” overlooking the tendency forpeople to find ways to act in their common self-interest to put the cooperative solution tothe game into action. Also, not all games have dominant strategy equilibria, so there aremany games for which this concept provides no answer at all.

2. Effort Dilemmas. One family of \"social dilemmas\" arises where a group of people areinvolved in some task that depends on the efforts of each of them. The strategy choicesare \"work\" and \"shirk.\" In an effort dilemma, one person's shirking places the burden ofincreased effort on the other(s). Then the payoff table could be something like Table E1.

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Table E1. An Effort Dilemma

Mr. Joneswork

Mr.Smith

workshirk

10,1015,2

shirk2,155,5

Are there any dominant strategies in this game? What? Has this game a dominantstrategy equilibrium? What would be the cooperative equilibrium in this game?

In the Pacific Northwest of the United States, in the middle of the twentiethcentury, between two and three dozen plywood companies were worker owned. Theworker-owners controlled the companies by majority rule and hired and fired themanagers. However, the managers typically had discretion to assign and discipline work,and were, if anything, more powerful than managers in similar profit-seeking companies.Does the example suggest an explanation of this?

(a) Are there any dominated strategies in this game? What?

The dominant strategy for this game is the \"shirk\" strategy. It is the one strategy thatyields a higher payoff than the other strategy, regardless of which strategy the otherplayer chooses.

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If Mr. Smith's strategy isWorkShirk

The best response for Mr. Jones IsShirkShirk

Since this game is symmetrical, Mr. Smith and Mr. Jones both have the same dominantstrategy.

(b) Has this game a dominant strategy equilibrium?

Yes, this game does have a dominated strategy equilibrium. It is “shirk, shirk.”

(c) What would be the cooperative equilibrium in this game?

The cooperative equilibrium in this game would be the work, work. This would be thestrategy that the players would choose if they could commit themselves to a coordinatedchoice of strategies.

(d) Does the example explain about the work-owners controlled plywood companies’managerial discretion to assign and discipline work and how these managers were morepowerful than others in similar profit seeking companies?

By having the \"cooperative\" solution to this game as a real life solution, themanagers of the plywood companies were expected to get the work done. The

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owner/operators knew that the payoff would be the greatest if everyone did the right thingand all \"worked\" and none \"shirked\". Those that did were probably disciplined and therest of the workers approved.

These answers apply to the MODIFIED version given in the appendix. With theoriginal payoffs, the cooperative solution is either “work, shirk” or “shirk, work,” for atotal payoff of 22, and the application to the plywood cooperatives is less clear. By theway, the source for the story of the plywood cooperatives is Katrina Berman, Worker-Owned Plywood Companies, Washington State University Press, 1967. In my classes,several students wanted the managers to be part of the game -- smart, but to make thatwork we would need a 3-person game, at least.

3. The Training Game: Two firms hire labor from the same unskilled pool. Each firmcan either train its labor force or not. Training increases productivity but there arespillovers in that the rival can hire the trained workers away. Thus, both firms alwaysface the same productivity net of pay, although the net productivity is higher if moreworkers are trained. The relation between the proportion trained and net productivity isshown in Table E2.

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Table E2. Training

Proportion trained050%100%

net productivity57.510

(Each firm either trains all of its workers or none, so 0, 50%, and 100% trained are theonly possibilities). Each firm's profit is the net productivity shown in the table minustraining cost. Training cost is 3 if the firm chooses to train its employees and zerootherwise.

Who are the \"players\" in this game? What are their strategies? Express this as agame in normal form. Are there dominant strategies in this game? What? Is there adominant strategy equilibrium? What?

(a) Who are the players in this game? What are their strategies?

In this game there are two players, Firm #1 and Firm #2. There are two strategies foreach firm, to Train or not to Train.

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(b) Express this game in normal form.

FIRM #1Train

FIRM #2

TrainNot Train

7,77.5, 4.5

Not Train4.5,7.55,5

(c) Are there dominant strategies in this game? What?

There are dominant strategies in this game. They are the same for both firms. That is toNOT Train.

If Firm #29s strategy isTrainNot Train

The best response for Firm #1 IsNot TrainNot Train

(d) Is there a dominant strategy equilibrium in this game? What?

The dominant strategy equilibrium in this game in the \"NOT Train, NOT Train\" Strategy.

4. Poison Gas. Allemand and Angleterre are rival nations, often at war, and both canproduce and deploy poison gas on the battlefield. In any battle, the payoffs to using gasare as in Table E3.

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Table E3. Gas

Angleterre

gasno

Allemandgas-8,-83,-10

no-10,30,0

Are there any dominant strategies in this game? What? Has this game a dominantstrategy equilibrium? What would be the cooperative equilibrium in this game? Is thisgame a social dilemma? Why?

Historically, poison gas was used in World War I, but not in World War II,although Germany opposed France and Britain in the second war as it had in the first.The consensus explanation was that if one side were to use gas in a battle, the other sidewould retaliate with gas in subsequent battles, with the result that the first user wouldnevertheless be worse off as a result. Discuss the limitations of the Dominant StrategyEquilibrium in the analysis of social dilemmas in the light of this contrast.

The dominant strategy is given by:

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If AllemandUses gasDoesn’t usegas

Then

AngleterreUses gasUses gas

The table indicates the dominant equilibrium is (Gas,Gas) with a payoff of (-8,-8). Thereis a social dilemma present because use of cooperative strategy yields a better result forboth sides. In this specific example the payoffs give no edge to either side to use gas. Ifboth Allemand and Angleterre use gas then they can be assured their opponent will fire itback (assuming both sides have possession). Fear of retaliation probably explains thenonuse of gas in World War II. This example shows a deterrent to using dominatestrategies in the event a mutually assured destruction is present.

5. Running for Office. Richard Nixon, a highly successful Republican politician, saidthat the way for a Republican to be elected to public office was to \"run to the right in theprimary election, but run to the center in the general election.\" The political example inthis chapter gives an explanation for the advice to \"run to the center in the generalelection.\" How would you explain \"run to the right in the primary?\" Note: in the politicalsystems of many American states, including California (Nixon's home state), primaryelections are held to determine the nominees of the major parties, and are limited tovoters registered as members of those respective parties. The nominees selected thencompete in a general election. Hint: it may be that registered members of the RepublicanParty are not distributed over the political spectrum as the general population is.

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It seems fairly certain that registered Republicans are more predominantly on the rightthan are the general population of voters, who include also Democrats and Independents(who are not registered as members of any party) and a very few who register as minorparty members. Thus, the payoffs could be more nearly as follows:

Table. Vote Payoffs for Two Candidates in a Republican Primary

Candidate Rleft.

left

Candidate S

middleright

50,5075,2590,10

middle25,7550,5070,30

right.10,9030,7050,50

This game has a dominant strategy equilibrium: right, right.

What is remarkable (to me) is the success that Republicans, including Mr. Nixon,have had in doing this. By contrast Democrats have alternated between compromising inthe primaries and nominating centrists (such as Clinton) or nominating mild leftistssuitable to their active registered wing (such as McGovern) and “self-destructing.” I haveno explanation that would not sound partisan.

6. Happy Hour. Refer to problem 4 in Chapter 1.

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Are there any dominant strategies in this game? What? Has this game a dominant strategyequilibrium? What would be the cooperative equilibrium in this game? Is this game asocial dilemma? Why?

JIM'S GIN MILL

Offer Snack

Tom's

The dominated strategies in this game are to offer snacks. This is true for both thetavern and the gin mill. As such it has a dominated strategy equilibrium, the \"offersnacks, offer snacks\" strategy.

The cooperative equilibrium would be the \"don’t offer snacks, don't offer snacks\"strategy. If both bars agreed not to offer snacks, both would be better off, with profits of30 rather than 20.

Yes, this game is a social dilemma as it has a dominant strategy equilibrium andthe dominant strategy solution is different than the cooperative solution to the game.

6. Quiz question

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

Don't Offer Snack

-20,50

30,20

Offer Snack

20,20

Don't Offer Snack50,-20

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Student name ____________________________

Quiz – Game Theory

Amanda and Buffy are playing a game in which they choose strategies by pushingbuttons. They are isolated, in separate rooms, choose simultaneously, and do not knowwhom they are playing. Amanda’s strategies are to punch the red or the green button,while Buffy’s strategies are to punch the blue or the yellow button. The payoff table is:

Buffyblue

red

Amanda

green

1,2

3,1

0,0

yellow2,1000

Are there any dominant strategies? If so, what? Is there a dominant strategy equilibrium?If so, what?

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Answer:

Yes, there is one dominant strategy. Amanda’s dominant strategy is green. However,Buffy’s best response to red is yellow, while her best response to green is blue, she doesnot have a dominant strategy and there is no dominant strategy equilibrium.

Some (very good) students may spontaneously discover the method of eliminatingdominated strategies, observing that once the red strategy is eliminated for Amanda, theblue strategy is dominant for Buffy. That is quite right, but when the student goes on toclaim that this should be thought of as a dominant strategy equilibrium, she is pushing theinsight too far. I respond by taking off the very minimum penalty, giving great praise forinsight, but stressing the importance of using the definitions carefully and not bendingthem – since bending definitions will lead to confusion sooner or later.

7. Appendix: Revised Problem

Given on next page for convenience in printing and copying.

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Effort Dilemmas. One family of \"social dilemmas\" arises where a group of people areinvolved in some task that depends on the efforts of each of them. The strategy choicesare \"work\" and \"shirk.\" In an effort dilemma, one person's shirking places the burden ofincreased effort on the other(s). Then the payoff table could be something like Table E1.

Table E1. An Effort Dilemma

Mr. Joneswork

shirk2,155,5

Mr.Smith

workshirk

10,1015,2

Are there any dominant strategies in this game? What? Has this game a dominantstrategy equilibrium? What would be the cooperative equilibrium in this game?

In the Pacific Northwest of the United States, in the middle of the twentiethcentury, between two and three dozen plywood companies were worker owned. Theworker-owners controlled the companies by majority rule and hired and fired themanagers. However, the managers typically had discretion to assign and discipline work,and were, if anything, more powerful than managers in similar profit-seeking companies.Does the example suggest an explanation of this?

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