MIT14_12F12_chapter2Gamethory

8/13/2019 MIT14_12F12_chapter2Gamethory

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Chapter2DecisionTheory2.1

The

basic

theory

of

choice

We consider a set of alternatives. Alternatives are mutually exclusive in the sensethatonecannotchoosetwodistinctalternativesatthesametime. Wealsotakethesetoffeasiblealternativesexhaustivesothataplayerschoicesisalwayswell-defined.1

We are interested in a players preferences on . Such preferences are modeledthrougharelationon,whichissimplyasubsetof. A relation issaidtobecomplete ifandonly if, givenany, either or. A relation issaidtobetransitive ifandonlyif,givenany

,

[and].A relation is a preferencerelation ifandonlyifitiscompleteandtransitive. Givenanypreferencerelation, we can definestrictpreferenceby

6[and]andtheindifferenceby

[and]1 Thisisamatterofmodeling. Forinstance,ifwehaveoptionsCoffeeandTea,wedefinealternatives

as=CoffeebutnoTea,=Tea but no Coffee,=CoffeeandTea,and=noCoffeeandnoTea.

9


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10 CHAPTER2. DECISIONTHEORYA preference relation can be represented by a utility function : R in the

followingsense: ()()

Thisstatement

can

be

spelled

out

as

follows.

First,

if

()

(),

then

the

player

finds

alternativeasgoodasalternative . Second, andconversely, if theplayerfindsatleast as good as , then () must be at least as high as (). In other words, theplayeractsas if heistryingtomaximizethevalueof().

Thefollowingtheoremstatesfurtherthatarelationneedstobeapreferencerelationinordertoberepresentedbyautilityfunction.Theorem2.1 Let befinite. Arelationcanbe presentedbyautilityfunction ifandonly if it is complete and transitive. Moreover, if :

R represents , and if

:RR isastrictly increasingfunction,thenalsorepresents.By the last statement, such utility functions are calledordinal, i.e., only the order

informationisrelevant.Inordertousethisordinaltheoryofchoice,weshouldknowtheplayerspreferences

onthe alternatives. As we have seen intheprevious lecture, in game theory, aplayerchoosesbetweenhisstrategies,andhispreferencesonhisstrategiesdependonthestrate-giesplayedbytheotherplayers. Typically,aplayerdoesnotknowwhichstrategiestheotherplayersplay. Therefore,weneedatheoryofdecision-makingunderuncertainty.2.2 Decision-makingunderuncertaintyConsider afinite set of prizes, and let be the set of all probability distributionsP

: [01] on , where () = 1. We call these probability distributionslotteries. A lottery can be depicted by a tree. For example, in Figure 2.1, Lottery 1depictsasituationinwhichtheplayergets$10withprobability1/2(e.g. ifacointossresultsinHead)and$0withprobability1/2(e.g. ifthecointossresultsinTail).

Intheabovesituation,theprobabilitiesaregiven,asinacasino,wheretheprobabili-tiesaregeneratedbyamachine. Inmostreal-worldsituations,however,theprobabilitiesare not given to decision makers, who may have an understanding of whether a givenevent ismore likelythananothergivenevent. Forexample, inagame, aplayer isnot


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112.2. DECISION-MAKINGUNDERUNCERTAINTY

Lottery 1

Figure2.1:givenaprobability distribution regardingthe other players strategies. Fortunately, ithasbeenshownbySavage(1954)undercertainconditionsthataplayersbeliefscanberepresented by a (unique) probability distribution. Using these probabilities, one canrepresentthedecisionmakersactsbylotteries.

Wewouldliketohaveatheorythatconstructsaplayerspreferencesonthelotteriesfromhispreferencesontheprizes. Therearemanyofthem. Themostwell-knownandthemostcanonicalandthemostusefuloneisthetheoryofexpectedutilitymaximiza-tion by Von Neumann and Morgenstern. A preference relation on is said to berepresentedbyavonNeumann-Morgensternutilityfunction:Rifandonlyif

X X () ()() ()()() (2.1)

foreach. Thisstatementhastwocrucialparts:

1. : Rrepresents intheordinalsense. That is, if()(),thentheplayerfindslotteryasgoodaslottery. Andconversely,iftheplayerfindsatleastasgoodas, then ()mustbeatleastashighas().

2. The function takesaparticular form: foreach lottery,() istheexpectedPvalueofunder. That is, () ()(). Inotherwords,theplayeractsas if he wants to maximize the expected value of . For instance, the expected

1/2

1/2

10

0

utilityofLottery1fortheplayeris((Lottery1))= 12

(10)+12

(0).2

Inthe sequel, Iwill describe thenecessaryand sufficientconditions forarepresen-

2 If wereacontinuum, likeR,wewouldcomputetheexpectedutilityofby

tation as in (2.1). The first condition states that the relation is indeed a preferencerelation: R

()().


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12 CHAPTER2. DECISIONTHEORY

d d PPPPPPPPP

PPPPPPPPP

Figure2.2: TwolotteriesAxiom2.1 iscompleteandtransitive.

This isnecessarybyTheorem2.1, for represents inordinalsense. Thesecondcondition iscalled independence axiom,statingthataplayerspreferencebetweentwolotteries

and

does

not

change

if

we

toss

acoin

and

give

him

afi

xedlottery

if

tail

comesup.Axiom2.2 For any, and any (01],+ (1 )+ (1 )

.Let and bethelotteriesdepictedinFigure2.2. Then,thelotteries+ (1)

and + (1 ) can be depicted as in Figure 2.3, where we toss a coin between afixedlotteryandourlotteriesand. Axiom2.2stipulatesthattheplayerwouldnotchangehismindafterthecointoss. Therefore,theindependenceaxiomcanbetakenasanaxiomofdynamicconsistencyinthissense.

Thethirdcondition ispurelytechnical, andcalledcontinuity axiom. Itstatesthattherearenoinfinitelygoodorinfinitelybadprizes.Axiom2.3 For any with, there exist (01)suchthat+ (1 ) and+ (1 ).

Axioms2.1and2.2implythat,givenany andany[01],if, then + (1 ) + (1 ) (2.2)

Thishastwoimplications:1. Theindifferencecurvesonthelotteriesarestraightlines.


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132.2. DECISION-MAKINGUNDERUNCERTAINTY

1

dPPPP

PPPPPd@

@1 @

@@

+ (1 )

(2)6

@@

@@

@@

@H0

HHHHHHHHH@

@0 @

dPPPP

PPPPPd@

@1 @

@@

+ (1 )Figure2.3: Twocompoundlotteries

@

@

HHHHHHHHHHHHHHHH0 @

@ @

@ @

@ @ @ -(1)

0 1Figure2.4: Indifferencecurvesonthespaceoflotteries


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14 CHAPTER2. DECISIONTHEORY2. Theindifferencecurves,whicharestraightlines,areparalleltoeachother.To illustrate these facts, consider three prizes 0 1, and 2, where 2 1 0.

A lottery canbedepictedonaplanebytaking(1)asthefirstcoordinate (onthehorizontal

axis),

and

(2)as

the

second

coordinate

(on

the

vertical

axis).

The

remaining

probability(0) is1 (1) (2). [SeeFigure2.4forthe illustration.] Givenanytwolotteriesand,theconvexcombinations+ (1 ) with[01]formthelinesegmentconnectingto. Now, taking =,wecandeducefrom(2.2)that, if,then + (1 ) + (1 ) = for each [01]. That is, the line segmentconnectingtoisanindifferencecurve. Moreover,ifthelinesand0 areparallel,then=|0|||, where ||and|0|arethedistancesofand0 totheorigin,respectively.Hence,taking=,wecomputethat0 =+ (1 ) 0 and0 =+ (1 ) 0,where0 isthelotteryattheoriginandgives0 withprobability1. Therefore,by(2.2),ifisanindifferencecurve,0 isalsoanindifferencecurve,showingthattheindifferencecurvesareparallel.

Linecanbedefinedbyequation1(1) + 2(2) =forsome1 2 R. Since 0 isparallelto, then 0 canalsobedefinedbyequation1(1) + 2(2) =0 forsome0. Sincetheindifferencecurvesaredefinedbyequality1(1) + 2(2) =forvariousvalues of ,thepreferencesarerepresentedby

() = 0 + 1(1) + 2(2) (0)(0) + (1)(1) + (2)(2)

where(0) = 0(1) = 1(2) = 2

givingthedesiredrepresentation.Thisistrueingeneral,asstatedinthenexttheorem:

Theorem2.2 A relation on can be represented by a von Neumann-Morgensternutilityfunction:asin(2.1)ifandonlyifsatisfiesAxioms2.1-2.3. Moreover,and representthesamepreferencerelationifandonly if =+ forsome 0andR.


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152.3. MODELINGSTRATEGICSITUATIONSBy the last statement in our theorem, this representation is unique up to affine

transformations. Thatis,adecisionmakerspreferencesdonotchangewhenwechangehisvonNeumann-Morgenstern(VNM)utilityfunctionbymultiplyingitwithapositivenumber,oraddingaconstanttoit;buttheydochangewhenwetransformitthroughanon-lineartransformation. Inthissense,thisrepresentation iscardinal. Recallthat,in ordinal representation, the preferences wouldnt change even if the transformation

were non-linear, so long as it was increasing. For instance, under certainty, = and would representthe samepreference relation, while (when there is uncertainty)

the VNM utility function = represents a very differentset of preferences on thelotteriesthanthosearerepresentedby.

2.3 ModelingStrategicSituationsInagame,whenaplayerchooseshisstrategy, inprinciple,hedoesnotknowwhattheother players play. That is, he faces uncertainty about the other players strategies.Hence, in order to define the players preferences, one needs to define his preferenceunder such uncertainty. In general, this makes modeling a difficult task. Fortunately,using the utility representation above, one can easily describe these preferences in acompactway.

Consider two players Alice and Bob with strategy sets and . If Alice plays andBobplays, then the outcome is ( ). Hence, it sufficestotakethesetofoutcomes = = {( ) | } as the set of prizes. ConsiderAlice. When she chooses her strategy, she has a belief about the strategies of Bob,represented by a probability distribution on , where () is the probabilitythatBobplays,foranystrategy. Givensuchabelief,eachstrategy inducesalottery, whichyieldstheoutcome( ) withprobability(). Therefore, wecanconsidereachofherstrategiesasalottery.

Example2.1 Let = { } and = { }. Then, the outcome set is ={}. Suppose that Alice assigns probability () = 13 to and() = 23 to . Then, under this belief, her strategies and yield thefollowing


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16 CHAPTER2. DECISIONTHEORYlotteries:

TLTLTLTL00

1/31/3TRTRTRTR 00

2/32/3BB1/1/ 1/333 1/3TT1/1/33

BLBLBL00 BL2/32/3

00BLBLBLBL

Ontheotherhand, ifsheassignsprobability() = 12 toand() = 12 to,thenherstrategies and yieldthefollowing lotteries:

TLTLTLTL001/21/2

TRTRTRTR 001/1/22

BB 1/1/21/2 1/22TT1/1/22BLBLBL00 BL

1/21/200

BLBLBLBL

The objective of a game theoretical analysis is to understand what players believeabouttheotherplayersstrategiesandwhattheywouldplay. Inotherwords,theplayersbeliefs, and,aredeterminedattheendoftheanalysis,andwedonotknowthemwhenwemodelthesituation. Hence,inordertodescribeaplayerspreferences,weneedtodescribehispreferencesamongall the lotteriesasabove foreverypossiblebelief hemayhold. Intheexampleabove,weneedtodescribehowAlicecomparesthelotteries

TLTLTLTL00

ppTRTRTRTR 00

1-1-ppBB ppp pTTpp

BLBLBL0 BL01-1-pp

00BLBL BLBL (2.3)


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172.3. MODELINGSTRATEGICSITUATIONSforevery[0 1]. Thatisclearlyachallengingtask.

Fortunately,underAxioms2.1-2.3,whichwewillassumethroughoutthecourse,wecandescribethepreferencesofAlicebyafunction

: RSimilarly,wecandescribethepreferencesofBobbyafunction

: RIn the example above, all we need to do is tofind four numbers for each player. ThepreferencesofAliceisdescribedby( ),( ),( ), and ( ).Example2.2 Inthepreviousexample,assumethatregardingthelotteriesin(2.3),thepreferencerelationofAlice issuchthat

if 14 (2.4)

if = 14

if 14

and she is indifferent between the sure outcomes ( ) and ( ). Under Axioms 2.1-2.3,wecanrepresentherpreferencesby

( ) = 3 ( ) = 1( ) = 0 ( ) = 0

Thederivationisasfollows. Byusingthefactthatsheisindifferentbetween( )and( ), we reckon that ( ) = ( ). By the second part of Theorem 2.2, we can set ( ) = 0(oranyothernumberyoulike)! Moreover,in(2.3),thelotteryyields

() =( ) + (1 ) ( ) andthe lottery yields

() =( ) + (1 ) ( ) = 0


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18 CHAPTER2. DECISIONTHEORYHence,thecondition(2.4)canberewrittenas

( ) + (1 ) ( ) 0 if 14( ) + (1 ) ( ) = 0 if= 14( ) + (1 ) ( ) 0 if 14

That is,1 3

( ) + ( ) = 04 4

and( ) ( )

In other words, all we need to do is tofind numbers ( ) 0 and ( ) 0with ( ) = 3( ), as in our solution. (Why would any such two numbersyield

the

same

preference

relation?)

2.4 AttitudesTowardsRiskHere, wewillrelatetheattitudesofan individualtowardsrisktothepropertiesofhisvon-Neumann-Morgenstern utility function. Towards this end, consider the lotterieswithmonetaryprizesandconsideradecisionmakerwithutilityfunction: RR.

Alotteryissaidtobeafair gambleifitsexpectedvalueis0. Forinstance,consideralottery

that

gives

with

probability

and

with

probability

1

;

denote this lottery

by ( ;). Suchalotteryisafairgambleifandonlyif + (1 )= 0

A decision maker is said to be risk-neutral if and only if he is indifferent betweenacceptingandrejectingall fairgambles. Hence, adecisionmakerwithutility functionisrisk-neutralifandonlyifX X

() () = (0) whenever () = 0This is true if andonly if the utility function is linear, i.e., () = + for somerealnumbersand. Therefore,anagent isrisk-neutral ifandonly ifhehasa linearVon-Neumann-Morgensternutilityfunction.

A decision maker is strictly risk-averse if and only if he rejects all fair gambles,exceptforthegamblethatgives0withprobability1. Thatis,X X

() () (0) = ()


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192.4. ATTITUDESTOWARDSRISKHere,the inequalitystatesthatherejectsthe lottery, and the equality is by the fact thatthelotteryisafairgamble. Asinthecaseofriskneutrality,itsufficestoconsiderthebinarylotteries(;),inwhichcasetheaboveinequalityreducesto

() + (1 )() (+ (1 ))This isafamiliar inequalityfromcalculus: afunction issaidtobestrictlyconcave ifandonlyif

(+ (1 )) () + (1 )()forall(01). Therefore,strictrisk-aversionisequivalenttohavingastrictlyconcaveutility function. Adecisionmaker issaidtoberisk-averse iffhe has a concave utilityfunction, i.e., (+ (1 ))() + (1 )() foreach,, and . Similarly,adecisionmakerissaidtobe(strictly)riskseeking iffhehasa(strictly)convex utilityfunction.

ConsiderFigure2.5. Ariskaversedecisionmakersexpectedutilityis() = (1) + (1 ) (2) if he has a gamble that gives 1 with probability and 2withprobability1. Ontheotherhand,ifhehadtheexpectedvalue1+(1 ) 2forsure,hisexpectedutilitywouldbe(1+ (1 ) 2). Hence,thecordABistheutilitydifferencethatthisrisk-averseagentwouldlosebytakingthegamble insteadofits expected value. Likewise, the cord BC is the maximum amount that he is willingtopayinordertoavoidtakingthegamble insteadofitsexpectedvalue. Forexample,suppose that 2 is his wealth level; 2 1 is the value of his house, and is theprobability that the house burns down. In the absence offire insurance, the expectedutilityofthis individual is(gamble),whichislowerthantheutilityoftheexpectedvalue of the gamble.2.4.1 Risksharing

Consideranagentwithutilityfunction:7 . Hehasa(risky)assetthatgives$100withprobability1/2andgives$0withprobability1/2. Theexpectedutilityoftheasset fortheagentis0 = 12 0 + 12 100 = 5. Consideralsoanotheragentwhoisidenticalto this one, in the sense that he has the same utility function and an asset that pays$100withprobability1/2andgives$0withprobability1/2. Assumethroughoutthatwhatanassetpaysisstatisticallyindependentfromwhattheotherassetpays. Imagine


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20 CHAPTER2. DECISIONTHEORY

EU

u(pW1+(1-p)W2)

EU(Gamble)

u

B

C

A

W1 W1+(1-p)W2 W2

Figure2.5:


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212.4. ATTITUDESTOWARDSRISKthatthetwoagentsformamutualfundbypoolingtheirassets,eachagentowninghalfofthemutualfund. Thismutualfundgives$200theprobability1/4(whenbothassetsyieldhighdividends),$100withprobability1/2(whenonlyoneontheassetsgiveshighdividend), and gives $0 with probability 1/4 (when both assets yield low dividends).Thus,eachagentsshareinthemutualfundyields$100withprobability1/4,$50withprobability 1/2, and $0 with probability 1/4. Therefore, his expected utility from the

1 1 1share in this mutual fund is = 0 = 60355. This is clearly 100+ 50+4 2 4largerthanhisexpectedutility fromhisownassetwhichyieldsonly5. Therefore,theaboveagentsgainfromsharingtheriskintheirassets.2.4.2 InsuranceImagine

a

world

where

in

addition

to

one

of

the

agents

above

(with

utility

function

: 7 and a risky asset that gives $100 with probability 1/2 and gives $0 withprobability1/2),wehavearisk-neutralagentwithlotsofmoney. Wecallthisnewagentthe insurancecompany. Theinsurancecompanycaninsuretheagentsasset,bygivinghim$100 ifhisassethappenstoyield$0. Howmuchpremium,,theagentwouldbewilling to pay to get this insurance? [A premium is an amount that is to be paid toinsurancecompanyregardlessoftheoutcome.]

Iftherisk-averseagentpayspremium andbuysthe insurance,hiswealthwillbe$100

forsure. Ifhedoesnot,thenhiswealthwillbe$100withprobability1/2and

$0withprobability1/2. Therefore,heiswillingtopay inordertogettheinsuranceiff

1 1 (100) (0)+ (100)

2 2i.e.,iff 1 1

100 0 + 1002 2

Theaboveinequalityisequivalentto10025=75

Thatis,heiswillingtopay75dollarspremiumforaninsurance. Ontheotherhand,ifthe insurancecompanysellsthe insurance forpremium, it gets forsureandpays$100withprobability1/2. Thereforeitiswillingtotakethedealiff

1 100=50

2


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22 CHAPTER2. DECISIONTHEORYTherefore, both parties would gain, if the insurance company insures the asset for apremium (5075),adealbothpartiesarewillingtoaccept.Exercise2.1 Now consider the case that we have two identical risk-averse agents asabove, and the insurance company. Insurance company is to charge the same premiumforeachagent, and therisk-averseagentshave anoptionofforming amutualfund.What istherangeofpremiumsthatareacceptabletoallparties?

2.5 ExerciseswithSolution1. [Homework1,2006]Inwhichofthefollowingpairsofgamestheplayerspreferences

overlotteriesarethesame?(a)

22 11 37

110 04 0421 17 15

121 50 3253 31 31

10 52 12(b)

12 70 4161 22 84

31 92 5015 71 4163 24 88

31 95 51Solution: RecallfromTheorem2.2thattwoutilityfunctionsrepresentthesamepreferencesoverlotteriesifandonlyifoneisanaffinetransformationoftheother.That is, we must have = + for some and where and are theutilityfunctionsontheleftandright,respectively,foreachplayer. In Part 1, the preferencesofplayer1aredifferentintwogames. Toseethis,notethat1() = 0 and 1() = 3. Hence, we must have = 3. Moreover, 1() = 1 and1() = 5. Hence, we must have = 2. But then, 1() + = 7 6=12 = 1(), showing that it is impossible to have an affine transformation.Similarly, one can check that the preferences of Player 2 are different in Part 2.


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232.5. EXERCISESWITHSOLUTIONNow,comparisonsofpayoffs for () and()yieldthat= 2 and= 1, but thenthepayoffs for () donotmatchundertheresultingtransformation.

2. [Homework 1, 2011] Alice and Bob want to meet in one of three places, namelyAquarium

(denoted

by

),

Boston

Commons

(denoted

by

)

and

aCeltics

game

(denotedby). Eachofthemhasstrategies. If they both play the same strategy,thentheymeetatthecorrespondingplace,andtheyendupatdifferentplaces if their strategies do not match. You are asked to find a pair of utilityfunctions to represent their preferences, assuming that they are expected utilitymaximizers.Alicespreferences: Sheprefersanymeetingtonotmeeting,andsheisindiffer-encetowardswheretheyendupiftheydonotmeet. SheisindifferentbetweenasituationinwhichshewillmeetBobat, or , or ,eachwithprobability1/3,and a situation in which she meets Bob at with probability 1/2 and does notmeetBobwithprobability1/2. IfshebelievesthatBobgoestoBostonCommonswith probability and to the Celtics game with probability 1 , she weakly preferstogotoBostonCommonsifandonlyif13.Bobs preferences: IfhegoestotheCelticsgame,he is indifferentwhereAlicegoes. IfhegoestoAquariumorBostoncommons,thenheprefersanymeetingtonotmeeting,andhe is indifferenttowardswheretheyendup inthecasetheydonotmeet. He is indifferentbetweenplaying,, and ifhebelievesthatAlicemaychooseanyofherstrategieswithequalprobabilities.

(a) Assuming that they are expected utility maximizers,find a pair of utilityfunctions : {}2 Rand : {}2 RthatrepresentthepreferencesofAliceandBobonthelotteriesover{}2.Solution: Alicesutilityfunctionisdeterminedasfollows. Sincesheisindif-ferent between any () with6=,byTheorem2.2,onecannormalizeher payoff for any such strategy profile to () = 0. Moreover, sinceshe prefers meeting to not meeting, () 0 for all {}.By Theorem 2.2, one can also set () = 1 by a normalization. Theindifferenceconditioninthequestioncanthenbewrittenas

1 1 1 1() + ( ) + () = ()

3 3 3 2


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24 CHAPTER2. DECISIONTHEORYThelastpreferenceinthequestionalsoleadsto

1 2( ) = ()

3 3Form the last equality, ( ) = 2, and from the previous displayed equal-ity,() = 6.Bobs utility function can be obtained similarly, by setting () = 0 for any distinct when {}. The first and the last indifferenceconditionsalsoimplythat()0,andhenceonecanset() = 1 for all {} by the first indifference. The last indifference thenimpliesthat

1 1() = ( ) =() = 1

3 3yielding() = ( ) = 3.

(b) Findanotherrepresentationofthesamepreferences.Solution: ByTheorem2.2,wecanfindanotherpairofutility functionsbydoublingallpayoffs.

(c) Findapairofutilityfunctionsthatyieldthesamepreferenceas anddoesamongthesureoutcomesbutdonotrepresentthepreferencesabove.Solution: Take () = 60 and() =( ) = 30 whilekeep-ingallotherpayoffsasbefore. ByTheorem2.1,thepreferencesamongsureoutcomesdonotchange,butthepreferencesamongsomelotterieschangebyTheorem2.2.

3. [Homework1,2011]Inthisquestionyouareaskedtopriceasimplifiedversionofmortgage-backedsecurities. Abanker lendsmoneytohomeowners,whereeachhomeowner signs a mortgage contract. According to the mortgage contract, thehomeowner is to pay the lender 1 million dollar, but he may go bankrupt withprobability, in which case there will be no payment. There is also an investorwho can buy a contract in which case he would receive the payment from thehomeownerwhohassignedthecontract. Theutilityfunctionofthe investor isgivenby() = exp(), where isthenetchangeinhiswealth.

(a) Howmuchistheinvestorwillingtopayforamortgagecontract?


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252.5. EXERCISESWITHSOLUTIONSolution: Hepaysaprice ifandonlyif[( )] (0),i.e.,

(1)exp ((1 ))exp((0 )) 1That is,

1

ln(+ (1 )exp ())where isthemaximumwillingtopay.

(b) Now suppose that the banker can form "mortgage-backed securities" bypooling all the mortgage contracts and dividing them equally. A mortgagebackedsecurityyields1ofthetotalpaymentsbythehomeowners,i.e.,ifhomeownersgobankrupt,asecuritypays( )millionsdollar. Assumethathomeownersbankruptcyarestochasticallyindependentfromeachother.How much is the investor willing to pay for a mortgage-backed security?Assumingthatislargefindanapproximatevalueforthepriceheiswillingto pay. [Hint: for large , approximately, the average payment is normallydistributed with mean 1(milliondollars)andvariance(1). If is normallydistributed withmeanand variance 2, theexpectedvalueof exp()isexp

2

12 .] Howmuchmorecanthebankerraisebycreating mortgage-backed securities? (Use the approximate values for large.)Solution: Writing forthenumberofcombinationsoutof,theprob-ability that there are bankruptcies is (1). If he pays foramortgage-backedsecurity, hisnetrevenue inthecaseof bankruptcies is1 . Hence,hisexpectedpayoffis

X exp((1 ))(1)

=0Heiswillingtopayiftheaboveamount isatleast1, the payofffrom0.Therefore,heiswillingtopayatmost

!X = 1 1ln exp()(1)

=0Forlarge,

1 (1) =1 ln(exp((+(1)(2))))=1

2
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26 CHAPTER2. DECISIONTHEORYNotethathe isaskingadiscountof(1)2fromtheexpectedpayoffagainst the risk, and behaves approximately risk neutral for large . The bankergetsanextrarevenueof fromcreatingmortgage-backedse-curities. (Checkthat

0.)

(c) Answerpart(b)byassuming insteadthat the homeowners bankruptcy areperfectlycorrelated: withprobabilityallhomeownersgobankruptandwithprobability 1 none of themgo bankrupt. Briefly compare your answersforparts(b)and(c).Solution: Withperfectcorrelation,amortgage-backedsecurityisequivalenttoonecontract,andhenceheiswillingtopayatmost. Ingeneral,whenthereisapositivecorrelationbetweenthebankruptciesofdifferenthomeown-ers

(e.g.

due

to

macroeconomic

conditions),

the

value

of

mortgage

backed

securities will be less than what it would have been under independence.Therefore, mortgage backsecuritiesthatare pricedunder the erroneous as-sumptionofindependencewouldbeover-priced.

2.6 Exercises1. [Homework1, 2000]ConsideradecisionmakerwithVonNeumannandMorgen-

strenutilityfunctionwith() = ( 1)2. CheckwhetherthefollowingVNMutilityfunctionscanrepresentthisdecisionmakerspreferences. (Providethede-tails.)

(a) :7 1;(b) :7 ( 1)4;(c) :7 ( 1)2;(d) :7 2 ( 1)

2 1

2. [Homework1,2004]Whichofthefollowingpairsofgamesarestrategicallyequiv-alent,i.e.,canbetakenastwodifferentrepresentationsofthesamedecisionprob-lem?


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272.6. EXERCISES(a)

L R L RT TB B

2,2 4,03,3 1,0

-6,4 0,0-3,6 -9,0

(b)L R L R

T TB B

2,2 4,03,3 1,0

4,4 16,09,9 1,0

(c)L R L R

T TB B

2,2 4,03,3 1,0

4,2 2,03,3 1,0

3. [Homework1,2001]Wehavetwodates: 0and1.Wehaveasecuritythatpaysasingledividend, atdate1. Thedividendmaybeeither$100, or$50, or$0, eachwith probability 1/3. Finally, we have a risk-neutral agent with a lot of money.(Theagentwilllearntheamountofthedividendatthebeginningofdate1.)

(a) Anagentisaskedtodecidewhethertobuythesecurityornotatdate0. Ifhedecidestobuy,heneedstopayforthesecurityonlyatdate1(notimmediatelyat date 0). What is the highest price at which the risk-neutral agent iswilling

to

buy

this

security?

(b) Nowconsideranoptionthatgivestheholdertheright(butnotobligation)

to buy this security at a strike price at date 1 after the agent learnstheamountofthedividend. Iftheagentbuysthisoption,whatwouldbetheagentsutilityasafunctionoftheamountofthedividend?

(c) Anagentisaskedtodecidewhethertobuythisoptionornotatdate0.Ifhedecidestobuy,heneedstopayfortheoptiononlyatdate1(notimmediatelyat date 0). What is the highest price at which the risk-neutral agent iswillingtobuythisoption?

4. [Homework1,2001]Take=R,thesetofrealnumbers,asthesetofalternatives.Definearelationon by

12 forall.


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28 CHAPTER2. DECISIONTHEORY(a)Isapreferencerelation? (Provideaproof.)(b)Definetherelationsandby

[and6 ]and

[and]respectively. Istransitive? Istransitive? Proveyourclaims.

(c)Wouldbeapreferencerelationifwehad=N, where N={0 1 2 }isthesetofallnaturalnumbers?


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14.12Economic Applications of Game Theory

Fall 2012

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