Solution Manual Game Theory: An Introduction

1 Solution ManualGame theory : An IntroductionSteve TadelisJanuary 31, 2013 Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means 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 be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. ContentsI Rational Decision Making21 The Single-Person Decision Problem32 Introducing Uncertainty and Time9II Static games of Complete Information213 Preliminaries234 Rationality and Common Knowledge275 Pinning Down Beliefs: Nash Equilibrium356 Mixed Strategies51 Copyright, Princeton University Press.

2 No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Contents 1 III Dynamic games of Complete Information627 Preliminaries638 Credibility and Sequential Rationality73 Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Part IRational Decision Making2 Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 1 The Single-Person Decision to the Movies:There are two movie theatres in your neighbor-hood: Cineclass, which is located one mile from your home, and Cineblast,located 3 miles from your home, each showing ,Gone with the WindandDr.

3 Strangelove, while Cineblastis showingThe Matrix,Blade RunnerandAliens. Your problem is to decidewhich movie to go to.(a) Draw a decision tree that represents this problem without : Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means 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 ( , you likeAliensthe most andThe Matrixthe least.) Using payoffvalues 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 in the decisiontree. Would your choice change?Answer: Consumption:Recall the example in which you needed to choosehow much to drink.

4 Imagine that your payofffunction is given by 4 2 Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 1. The Single-Person Decision Problem 5where is a parameter that depends on your physique. Every person mayhave a different value of , and it is known that in the population ( )thesmallest is0 2;( ) the largest is6;and( )largerpeoplehavehigher sthan smaller people.(a) Can youfind an amount of drinking that no person should drink?Answer:The utility from drinking 0 is equal to 0. If a decision makerdrinks =2then, if he has the largest =6,hispayoffis =6 2 4 (2)2= 4and it is easy to see that decision makers with smallervalues of will obtain an even more negative payofffrom consuming =2. Hence, no person should choose =2.

5 (b) How much should you drink if your =1?If =4?Answer:The optimal Solution is obtained by maximizing the payofffunction ( )= 4 maximization condition is 8 =0implying that = 8is the optimal Solution . For =1thesolution is =18and for =4it is =12. (c) Show that in general, smaller people should drink less than larger :This follows from the Solution in part (b) above. For everytype of person , the Solution is ( )= 8which 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 =6should onlyconsume =34of a bottle of wine. Trees:You have room for up to two fruit bearing trees in your fruit trees that can grow in your garden are either apple, orange or cost of maintenance is $100 for an apple tree, $70 for an orange tree and$120 for a pear tree.

6 Your food bill will be reduced by $130 for each apple Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 61. The Single-Person Decision Problemtree you plant, by $145 for each pear tree you plant and by $90 for eachorange tree you plant. You care only about your total expenditure in makingany planting decisions.(a) What is the set of possible actions and related outcomes?Answer:Youhavetwo slots thatcanbeleftempty,orhaveoneof3possible trees planted in each slot. Hence, you have 10 possible (b) What is the payoffof each action/outcome?Answer:To calculate the payoffsfromeachchoiceitisconvenienttouse atableasfollows:Choicecost food savings net payoffnothing00 0one apple tree10013030one orange tree709020one pear tree12014525two apple trees20026060two orange trees14018040two pear trees24029050apple and orange17022050apple and pear22027555pear and orange19023545(c) Which actions are dominated?

7 Answer:All but choosing two apple trees are dominated. (d) Draw the associated decision tree. What will a rational player choose?Answer:Thetreewillhavetenbranches withthepayoffs associatedwith the table above, and the optimal choice is two apple trees. 1 This is a probem of choosing 2 items out of 4 possibilities with replacement, which is equal to 4+2 12 =(4+2 1)!2!(4 1)!=5 42=10. Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 1. The Single-Person Decision Problem 7(e)Nowimaginethatthefoodbillreductionis halfforthesecondtreeofthesamekind(youlik evariety).Thatis,thefirst 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.)

8 What will a rational player choose now?Answer:An apple tree is still the best choice for thefirst tree, but nowthesecondtreeshouldbeapeartree. Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 81. The Single-Person Decision Problem Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 2 Introducing Uncertainty and Choices:A player has three possible venues to choose from:going to a football game, going to a boxing match, or going for a payofffrom each of these alternatives will depend on the weather. Thefollowing table gives the agent s payoffin each of the two relevant weatherevents:Alternativepayoffif Rain payoffif ShineFootball game12 Boxing Match30 Hike01 For Let denote the probability of rain.

9 (a) Is there an alternative that a rational player will never take regardlessof ? ( , it is dominated for any [0 1].)Answer:For this decision maker choosing the hike is always worse(dominated) by going to the football game, and he should never go onahike. Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means 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. 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.

10 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, 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 :Two decision branches: drill or not drill. Following drilling,Nature chooses oil with probability ,withthepayoffof $3 million (4minus the initial investment). With probability1 Nature choosesno-oil with a payoff$ 1million. (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.