Econ Problems

ECON: Game Theory A Nontechnical Introduction to the Analysis of Strategy

Homework Homework assignment due Oct. 30. Remember that you are not required to do all homework problems, but are required to do some from each set, and to accumulate 60 homework points during the class. 1. Road Rage. Consider the following simple game, which we may call the “road rage” game. There are two players, Al and Bob. Bob has two choices: to aggress (perhaps by cutting Al off in traffic) or not to aggress. If Bob chooses “do not aggress,” then there is no choice of strategies for Al, but if Bob aggresses, Al can choose between strategies “retaliate” (perhaps by dangerous driving or by taking a shot at Bob’s car with a firearm) or not retaliate. An example in tabular “normal form” is shown in Table 12.E1.

Table 12.E1. The Road Rage Game

Bob aggress don’t

Al

If Bob aggresses then retaliate; if not, do nothing (-50,-100) (5,4)

If Bob aggresses then don’t retaliate; if not, do nothing (4,5) (5,4)

As usual, the payoff to the left is the payoff to Al, while the payoff to the right is the payoff to Bob. Draw the tree diagram for the game.

a. What are the subgames of this game? b. Which subgames are basic? c. Determine the subgame perfect equilibrium of this game. d. Does it seem that the subgame perfect equilibrium is what occurs in the real world?

Explain your answer. Although many governments have tried to discourage “Road Rage” by penalizing retaliation, the Washington State Police adopted a policy to discourage road rage by increasing the penalty for aggressive driving. Does this make sense in terms of game theory?

 

 

 

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Figure 1

This game has two subgames: the game itself and Al’s decision whether to retaliate or not. The second one is basic. Since the solution to the basic subgame is “don’t retaliate” for 4,5, Bob’s decision is “aggress” for 5 rather than 4, and this is subgame perfect. The subgame perfect solution happens sometimes in the real world, but not always, for two reasons. First, sometimes people do drive courteously and not aggressively. Second, retaliation does happen sometimes. Since people who retaliate are ignoring the bad payoff, they may ignore the risk of legal penalties as well. However, those who aggress are acting in the way that maximizes the payoffs, and strong legal enforcement could change those payoffs and thus prevent both aggression and retaliation. Thus, it seems that the Washington State Police could be on the right track. 2. Make or Buy Alfa Corp. makes computer chips. Beta, Ltd retails computers and can either make their own chips or buy them from Alfa. Alfa has an option either to mount a costly campaign to convince the public that computers are better with “Alfa inside” or not; and, if Beta elects to buy from them, they can ask a high or a low price. Their game in extensive form is shown in Figure 12. E3. below. The first payoff is to Alfa. Determine the subgame perfect equilibrium for this game. Will Alfa advertise? Explain in detail.

 

Bob

Al

-100,-50 B , A

5,4 B,A

don’t

don’t agg

res s

reta liat

e

4,5 B,A

 

 

 

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Figure 12. E3. Make or Buy

Alfa will advertise, because it is subgame perfect. In the following figure, the two subgames that are highlighted are basic.

Solving them, we have the following reduced game:

 

 

 

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Beta’s choice in the basic subgames in this reduced game will be buy in the upper branch and make in the lower branch. Thus we have the further reduced game:

In this case, clearly, Alfa will choose to advertise for 15 rather than not for 12. 3. It’s Good to be The Dean? A few years ago, Pixel University was negotiating with a candidate for a position as Dean of the College of Neat Stuff. The negotiation went on for months, with the candidate insisting that a new building should be built for the college as a condition for him to take the position. Finally the university floated a bond issue for the building, and the candidate signed the contract. The diagram In Figure 13.E1 represents this negotiation as a game in extended form. P stands for Pixel and D stands for Dean. At the last stage, if it has not floated a bond issue, (strategy “don’t”) but the candidate accepts the offer anyway (strategy “sign”), then Pixel has the options of promising the Dean that the building will be built, hiring him, and then refusing to build the new building but renovating Old Main anyway (strategy “hire”), hiring him and building

15, 12

12, 8

 

 

 

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the college building (strategy “college”), and declining to hire him but proceeding with the renovation of Old Main (strategy “main”) The diagram is based on the assumption that once the University floated the loan they could no longer refuse to build the college building, since the bond issue could not be used for other purposes. Therefore, if the university chooses “float’ and the candidate chooses “sign,” the university has only the choices of building the college building or refusing to hire the candidate.

a. What is the subgame perfect equilibrium of this game, and why did the Dean sign the contract?

b. Who changed the rules in this game? c. How were they able to do that?

Hint: think of the negotiation game as being imbedded in a larger game.

 

Figure 13.E1. The Dean’s Negotiation The basic proper subgames are shown as Figures 1 and 2 and are decisions made by Pixel University. The first payoff is to Pixel, and they will choose “hire” for a payoff of 15, leaving the dean at –5 without his building. In Figure 2, Pixel chooses “college” for 10, rather than “main” for 8, leaving the dean with his building, represented as a payoff of 10. This leaves us with the reduced game shown in Figure 3.

Figure 1

P

D 0,0

D

P

0,0 8,-5

15,-5

PFloat

Do n’t

Don’t

sign

sig n

hir e

college

Main

10,10

Don’t

Main 8,-5

col leg

e 10,10

P 8,-5

15,-5 college 10,10

 

 

 

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Figure 2 This game has basic proper subgames as shown in Figures 4 and 5. In Figure 4, the Dean chooses Don’t Sign, to avoid a negative payoff, leaving payoffs 0,0. In Figure 5, the Dean chooses sign, for a payoff of 10 rather than 0, and the payoffs are 10,10. This leaves us with the reduced game shown in Figure 6, a no-brainer for Pixel to float the bond issue. Thus the subgame perfect sequence is float, sign, college. The dean signed because he was confident that, once the University had floated its bond issue, the building would be built. Whoever proposed the bond issue changed the rules of the negotiation by imbedding the negotiation game as a subgame in a larger game.

Figure 3

P 8,-5

10,10

P

D

0,0

D

15,-5

0,0

10,10

 

 

 

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Figure 4

Figure 5

Figure 6 In the real-world incident that suggested this example, the dean did sign, a few years have passed, and he does not yet have his building. So, this game analysis failed. Why did it fail? There are three possibilities. One is that the building is on the way, and GTTT — good things take time. In that case the game analysis didn’t fail after all, and it is just too soon to tell. The second possibility is that the assumption that the bond issue committed the university to the building was mistaken. The third possibility is that the dean overestimated the rationality of the university administration that hired him, and the university has made a losing, irrational decision not to build the building. I do not have evidence to judge which of these might be right. 4. Agency. “Agency models” are used in a large literature of applied game theory in economics. For example, a corporate executive is the agent of the shareholders. Critics of corporations, such as Adam Smith, focus on this point. Adam Smith wrote “The directors of such companies, however, being the managers rather of other people’s money than of their own, it cannot well be expected that they should watch over it with the same anxious vigilance with which the partners in a private copartnery frequently watch over their own. … Negligence and profusion, therefore, must always prevail, more or less, in the management of the affairs of such a company. … To

D

15,-5

0,0

D

0,0

10,10

P 0,0

10,10

 

 

 

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establish a joint stock company [corporation] … would certainly not be reasonable. …” However, many modern corporations encourage the executives to “watch over” the assets of the company by offering them stock options in lieu of part of their salary. For a very simple example, suppose a corporate chief executive can choose either high or low effort (H or L) and the share value of the company will be 20 (million? billion?) if he chooses H and 10 if he chooses L; except that his compensation is deducted from the share value. Thus, for example, if he is paid a salary of 3 and chooses L the payoff to the shareholders is 10-3=7. The executive’s game payoff is his pay if he chooses L and his pay minus 2 if he chooses H. (The deduction of 2 is the effort cost of making the greater effort H). Thus, for example, if the executive is paid a salary of 3 and chooses H his payoff is 3-2=1. The shareholders can choose between paying him a salary of 3 or stock options worth a fraction q of the gross stock value. Thus, for example, if the executive is paid with stock options and chooses H, his game payoff is 20q, and the shareholders get 20(1-q). The shareholders choose their strategy first and commit to it by signing an employment contract.

a. Draw the tree diagram for this problem. b. What is the minimum value of q that will give the executive an incentive to choose H? c. What is the maximum q that will make it subgame perfect for the stockholders to offer a

stock option? What is the upper limit on the payoff to the stockholders? Can they get as much as they would if they paid the salary and the executive chose H?

d. Since the corporate accounting scandals of 2001-2002 and the economic downturn of 2008-2009, there has been much discussion of the large amounts paid to corporate CEOs in the form of stock options. What determines the maximum compensation that a CEO could get in the form of stock options in this model?

 

Figure 16

Figure 16 is the tree diagram. S stands for stockholders and E stands for the CEO. In the upper basic proper subgame, E will never choose H. In the lower basic subgame, he will choose H only if 20q-2>10q, i.e. q>20%. This implies that the upper limit for the shareholders is 0.8*20=16, less than they would get if, somehow, they could pay a salary of three and get an effort of H, but still more than the 7 they get if they pay a salary. The shareholders are better off with options as long as 20(1-q)>7, i.e. q<13/20 = 0.65. The maximum the CEO could get is 0.65*20=13.

17,1

7,3

20(1-q),20q-2 S

sal ary

options E

E

H

H

L

L 10(1-q),10q

 

 

 

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5. Free Samples Acme Cartoon Equipment (ACE) Corp. relies on other companies to supply the semifinished parts it assembles into cartoon equipment, under contract. There are, however, two types of suppliers: Type R (reliable) and Type U (unreliable). When ACE interviews a potential supplier, ACE has two strategies: buy or not buy. Some suppliers give free samples and others do not. With a type R supplier, the game in extensive form is as shown in Figure 13.E2, in which the first payoff is to ACE and the second is to the supplier. Node S is the decision node for the supplier, to give free samples or not, while node B is the node for the buyer, to buy or not. When the supplier is Type U, however, the payoffs are as shown in Figure 13.E3. ACE does not know which suppliers are R and U. Suppose that the probability that the supplier is type R is 0.5.

 

Figure E2. Acquisition Game with Type R Supplier

 

 

 

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Figure 13.E3. Acquisition Game with Type U Supplier

a. What are the subgame perfect equilibria in the two games taken separately? b. Given the probabilities of 0.5 that the supplier is R or U, what is the optimum strategy for

the buyer and what is the payoff from it? c. Suppose that ACE were to adopt the rule of buying only from suppliers who give free

samples, and suppliers know this. What would the results be? Explain this in terms of imbedded games.

In both cases, “don’t give” is the subgame perfect strategy for the seller, since the buyer’s decision will be the same regardless whether samples are given or not. Thus it is “don’t give,” “buy” for the first game and “don’t give,” “don’t buy” for the second game. Given the probabilities of 0.5 that the supplier is R or U, what is the optimum strategy for the buyer and what is the payoff from it? Acme is playing a game against nature, with nature choosing R or U as its strategies. The game looks like Figure 13. Nature’s play is labeled N. If a sample is not given, we are in the lower branch of the diagram. Acme does not know what kind of seller they are dealing with, as shown by the information set. The expected value payoff of “buy” for Acme is –0.5, while the expected value payoff of “don’t buy” is 0; so “don’t buy” is the buyer’s best response and “don’t” pays 0 to a seller of either type. On the other hand, if a sample is given, then Acme can determine from the sample whether the seller is of type R or U. Acme can then adopt a contingent strategy, “if R, buy; otherwise, don”t.” Thus, for a type R seller, the payoff of giving a sample is 2, while for a type U supplier, it is –2. Type R suppliers will offer samples in a subgame perfect equilibrium and type U suppliers will not. Thus, Acme can tell whether they are type R or type U just by observing whether samples are offered or not. In the jargon of microeconomics, the sample is a “market signal,” since it is correlated with the supplier type when the suppliers act rationally in their own interest. The analysis of the games separately is misleading because they are nested (not imbedded) in a bigger game in which nature plays a part.

 

 

 

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Figure 13

Suppose that ACE were to adopt the strategy of buying only from suppliers who give free samples, and suppliers know this. What would the results be? Explain this in terms of imbedded games. If we consider a decision to require samples, this would mean that the game above is imbedded in a larger game, in which the decision to require samples would rule out the entire lower branch of Figure 13. This would assure Acme of getting the information it needs. Type U suppliers, knowing they could not make a sale, presumably would drop out of the market – another strategy choice we would need to include in the game. As we have just seen, the result will not be changed, since Acme will get the information they need anyway, so long as all sellers act rationally. Perhaps Acme might adopt the requirement to guard against mistakes resulting from “trembling hand” behavior, i.e. against the possibility that a type R seller might mistakenly fail to offer samples.

S

BN

R

U

don’t

don’t

bu y

bu y

gi ve

don’t

2,2

1,-1

-1,-1

0,-2

1,3

0,0

-2,1

0,0

N

R

U B

B don’t

don’t

bu y

buy