Game Theory An Introduction Answers

Preview text © Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.Solution ManualGame Theory: An IntroductionSteve TadelisJanuary 31, 2013© Copyright, Princeton University Press.

No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.ABSTRACT This Solution Manual includes only the even numbered questions andis available for public access. It is still incomplete. It will be updated every 2-3 weeksto add the solutions to problems as they become available. A complete version isexpected by March 15, 2013.© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.ContentsIIIDynamic Games of Complete Information1627 Preliminaries638 Credibility and Sequential Rationality73© Copyright, Princeton University Press.

An Introduction to Applicable Game Theory Robert Gibbons ame theory is rampant in economics. Having long ago invaded industrial organization, game-theoretic modeling is now commonplace in interna-tional, labor, macro and public finance, and it is gathering steam in de-velopment and economic history. Nor is economics alone: accounting, finance, law. View Introduction to Game Theory Homework 1 Answers from ECON 1200 at University of Pittsburgh. Economics 1200 Spring 2014 Homework # 1 Answer Key (Answers to questions are shown in boldface.

No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.Part IRational Decision Making2© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.41. The Single-Person Decision Problem¥(b) Imagine that you don’t care about distance and that your preferencesfor movies is alphabetic (i.e., you like Aliens the most and The Matrixthe least.) Using payoff values 1 through 6 complete the decision treeyou drew in part (a). What option would you choose?Answer:¥(c) Now imagine that your car is in the shop, and the cost of walking eachmile is equal to one unit of payoff. Logitech quickcam 8.2.0 driver for mac. Update the payoffs in the decisiontree.

Would your choice change?Answer:¥3.4. Alcohol Consumption: Recall the example in which you needed to choosehow much to drink. Imagine that your payoff function is given by  − 42© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.1.

The Single-Person Decision Problem5where  is a parameter that depends on your physique. Every person mayhave a different value of , and it is known that in the population () thesmallest  is 02; () the largest  is 6; and () larger people have higher ’sthan smaller people.(a) Can you find an amount of drinking that no person should drink?Answer: The utility from drinking 0 is equal to 0. If a decision makerdrinks  = 2 then, if he has the largest  = 6, his payoff is  = 6 × 2 −4 × (2)2 = −4 and it is easy to see that decision makers with smallervalues of  will obtain an even more negative payoff from consuming = 2. Hence, no person should choose  = 2. ¥(b) How much should you drink if your  = 1? If  = 4?Answer: The optimal solution is obtained by maximizing the payofffunction () =  − 42. The first-order maximization condition is − 8 = 0 implying that  = 8 is the optimal solution.

For  = 1 thesolution is  = 18 and for  = 4 it is  = 12. ¥(c) Show that in general, smaller people should drink less than larger people.Answer: This follows from the solution in part (b) above. For everytype of person , the solution is () = 8 which is increasing in , andlarger people have higher values of  ¥(d) Should any person drink more than one bottle of wine?Answer: No.

Even the largest type of person with  = 6 should onlyconsume  = 34 of a bottle of wine. Fruit Trees: You have room for up to two fruit bearing trees in your garden.The fruit trees that can grow in your garden are either apple, orange or pear.The cost of maintenance is $100 for an apple tree, $70 for an orange tree and$120 for a pear tree. Your food bill will be reduced by $130 for each apple© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.1. The Single-Person Decision Problem7(e) Now imagine that the food bill reduction is half for the second tree ofthe same kind (you like variety). That is, the first apple still reducesyour food bill by $130, but if you plant two apple trees your food billwill be reduced by $130 + $65 = $195.

(Similarly for pear and orangetrees.) What will a rational player choose now?Answer: An apple tree is still the best choice for the first tree, but nowthe second tree should be a pear tree. ¥© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.81. The Single-Person Decision Problem© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.102. Introducing Uncertainty and Time(b) What is the optimal decision, or best response, as a function of .Answer: The expected payoffs from each of the remaining two choicesare given by,(Football) =  × 1 + (1 − ) × 2 = 2 −  (Boxing) =  × 3 + (1 − ) × 0 = 3 which implies that football is a better choice if and only if2 −  ≥ 3 or,  ≤ 12, and boxing is better otherwise. Drilling for Oil: An oil drilling company must decide whether or not toengage in a new drilling activity before regulators pass a law that bans drillingat that site.

The cost of drilling is $1,000,000. After drilling is completed andthe drilling costs are incurred, then the company will learn if there is oil ornot. If there is oil, operating profits generated are estimated at $4,000,000.If there is no oil, there will be no future profits.(a) Using  to denote the likelihood that drilling results in oil, draw thedecision tree of this problem.Answer: Two decision branches: drill or not drill. Following drilling,Nature chooses oil with probability , with the payoff of $3 million (4minus the initial investment). With probability 1 −  Nature choosesno-oil with a payoff $ − 1 million.

¥(b) The company estimates that  = 06. What is the expected value ofdrilling? Should the company go ahead and drill?Answer: The expected payoff (in millions) from drilling is  × 3 − (1 −) × 1 = 4 − 1 = 06, which means that the company should drill. ¥© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.2.

Introducing Uncertainty and Time11(c) To be on the safe side, the company hires a specialist to come up with amore accurate estimate of . What is the minimum vale of  for whichit would be the company’s best response to go ahead and drill?Answer: The minimum value of  for which drilling causes no expectedloss is calculated by solving  × 3 − (1 − ) × 1 ≥ 0, or  ≥ 14  ¥5.6.

Real Estate Development: A real estate developer wishes to build a newdevelopment. Regulations impose an environmental impact study that willyield an “impact score,” which is an index number based on the impact thedevelopment will likely have on traffic, air quality, sewage and water usage,etc. The developer, who has lots of experience, knows that the score willbe no less than 40, and no more than 70. Furthermore, he knows that anyscore between 40 and 70 is as likely as any other score between 40 and 70(use continuous values). The local government’s past behavior implies thatthere is a 35% chance that it will approve the development if the impactscore is less than 50, a 5% chance that it will approve the development ifthe score is between 50 and 55, and if the score is greater than 55 then theproject will surely be halted. The value of the development to the developer is $20,000,000. Assuming that the developer is risk neutral, what is themaximum cost of the impact study such that it is still worthwhile for thedeveloper to have it conducted?Answer: Observe that there is a 13 probability of getting a score between40 and 50 given that 40 to 50 is one-third of the range 40 to 70.

There isa 16 probability of getting a score between 50 and 55 given that 50 to 55 isone-sixth of the range 40 to 70. Hence, the expected value of doing a studyis111× 35 × $20 000 000 + × 05 × $20 000 000 + × 0 × $20 000 000362= $2 500 000Hence, the most the developer should pay for the study is $2,500,000. ¥© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.2. Introducing Uncertainty and Time13to be done? What answers do you give him?Answer: Bozoni should do the steps sequentially in this order: first testthe cold chain, then the ripening process, then do the test-marketing.The expected value of profits is 1.84 million. Observe that it would notbe profitable to launch the product if Bozoni had to do all the stepssimultaneously.

This is an example of real options–by sequencing thesteps, Bozoni creates options to switch out of a doomed project beforetoo much money gets spent. Ziffenboeffel calls you back. Since Table 1 was produced (see below),Bozoni has found a small research firm that can perform the necessarytests for the ripening process at a lower cost than Bozoni’s in-houseresearch department.Table 1: Data on launching the Cherimoya juiceStepProbability of success CostRipening process0.71,000,000Test marketing0.35,000,000Cold chain0.6500,000At the same time, the EU has raised the criteria for getting approvalfor new food producing facilities, which raises the costs of these tests.Mr.

Ziffenboeffel would, therefore, like to know how your answer to (a)changes as a function of the cost of the ripening test. What do you tellhim?Answer: This is sensitivity analysis for the cost of testing the ripeningprocess. This can be done by varying the cost for ripening, and seeingwhich expected payoff (highlighted yellow) is highest for which values ofthe cost. For example, whenever we set the cost below 375,000 it turnsout that the payoff from the sequence  →  →  gives the highestpayoff among the six possible sequences. (Excel’s GoalSeek is a particularly handy way for finding the threshold values quickly).Specifically, the optimal sequence is© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.142. Introducing Uncertainty and Timei)  →  →  if the cost of  ≤ 375 000ii)  →  →  if the cost of 375 000 ≤  ≤ 2 142 857iii)  →  →  if the cost of 2 142 857 ≤  ≤ 8 640 000iv) don’t launch if  costs more than 8 640 000where “” stands for the ripening process, “” stands for the coldchain, and “ ” stands for test marketing.

Ziffenboeffel calls you back yet again. The good news is the EUregulations and the outsourcing of the ripening tests “balance” eachother out, so the cost of the test remains 1,000,000. Now the problemis that his marketing department is suggesting that the probability thatthe market research will result in good news about the demand couldbe different in light of some recent data on the sales of other subtropicalfruit products. He would, therefore, like to know how your answer to(a) changes as a function of the probability of a positive result from themarket research. What do you tell him?Answer: This can be found by varying the probability of success fortest marketing (highlighted by blue in the excel sheet) between 0 and1.

The optimal sequence turns out to bei) don’t launch if  ≤ 01905ii)  →  →  if   01905where  is the probability that the test marketing will be successful.¥9.10. Surgery: A patient is very sick, and will die in 6 months if he goes untreated.The only available treatment is risky surgery. The patient is expected to livefor 12 months if the surgery is successful, but the probability that the surgeryfails and the patient dies immediately is 0.3.© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.162. Introducing Uncertainty and Timepopulation, call these the “treatable” patients, the surgery is successful, while for the other 30% (“untreatable”) it is not, and previouslythe patient did not know which population he belongs to. The test canbe thought of as detecting which population the patient belongs to.The above description means that if the patient is treatable then thetest will claim he is treatable with probability 0.9, while if the patientis untreatable then the test will claim he is treatable with probability0.1.

Hence, 63% of the population are treatable and detected as such(0.7×09), while 3% of the population are untreatable but are detectedas treatable (0.3×01). Hence, of the population of people for whom the63= 0955 ¥test is positive, the probability of successful surgery is 63+3(d) Assuming that the patient has the test done, at no cost, and the resultis positive, should surgery be performed?Answer: The value from not having surgery is (3) = 08, and a positivetest updates the probability of success to 0955 with the expected payoffbeing 0955 × 1 so the patient should have surgery done. ¥(e) It turns out that the test may have some fatal complications, i.e., thepatient may die during the test.

Draw a decision tree for this revisedproblem.Answer: Given the data above, we know that without taking the testthe patient will not have surgery because the expected value of surgeryis 0.7 while the value of living 3 months is 0.8. Also, we showed abovethat after a positive test the patient will choose to have surgery, and itis easy to show that after a negative test he won’t (the probability of7= 0206) Hence, the decision tree can bea successful outcome is 7+27collapsed as follows 9the decision to have surgery have been collapsedto the relevant payoffs):© Copyright, Princeton University Press. No part of this book may bedistributed, posted, or reproduced in any form by digital or mechanicalmeans without prior written permission of the publisher.2. Introducing Uncertainty and Time17¥(f) If the probability of death during the test is 0.005, should the patientopt to have the test prior to deciding on the operation?Answer: From the decision tree in part (e), the expected value conditional on surviving the test is equal to07(09 × 1 + 01 × 08) + 03(01 × 0 + 09 × 08) = 0902which implies that if the test succeeds with probability 0.995 then theexpected payoff from taking the test is0995 × 0902 + 0005 × 0 = 0897which implies that the test should be taken because 0897  08. More Oil: Chevron, the No. 2 US oil company, is facing a tough decision.The new oil project dubbed “Tahiti” is scheduled to produce its first commercial oil in mid-2008, yet it is still unclear how productive it will be.

“Tahitiis one of Chevron’s five big projects,” told Peter Robertson, vice chairmanof the company’s board to the Wall Street Journal.1 Still, it was unclear1 “Chevron’sTahitiFacilityBetsBigonGulfOilBoom.”Jun27,2007.pg.B5C.http://proquest.umi.com/pqdweb?did=&sid=1&Fmt=3&clientId=1566&RQT=309&VName=PQD.