Review of Economic Studies-1985-Sobel-557-73

8/7/2019 Review of Economic Studies-1985-Sobel-557-73

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Review of Economic Studies (1985) LII, 557-573

1985 The Socie ty for Economic Analysis Limited0034-6527/85/00390557$02.00

A Theory of CredibilityJOEL SOBEL

University of California, Sa n Diego

This paper presents models in which one agent must decide whether to t rust another, whosemotives are uncer ta in . Re li abi li ty can only be communica ted through act ions . In this con text , it

pays for people to bui ld a reputat ion based on rel iabl e behav iour ; someone becomes c redibl e byconsistently providing accurate and valuable informat ion or by performing useful services. Thetheory provides a jus ti fi ca tion for long-t erm arrangement s wi thou t b inding con tr ac ts . It alsod es cr ib es t ho se si tu at ion s where it pays an ag en t to cash in on his r ep ut at ion .

1. INTRODUCTION

Many decisions depend on trust ing someone. Consumers must decide whether to believeadvertisements, banks need to judge the reliability of loan applicants , and employersmust decide how much responsibil ity to delegate to their employees. This paper attemptsto model situations that arise when someone is uncertain about whether to trust the peoplehe deals with. In this context , reputations for reliability are valuable. An agent becomescredib le by cons is tently provid ing accurate , valuable information or by always ac tingresponsibly.

I analyse a series of simple, abstract models of information transmission. In thebasic model, the re are two agents, the Sende r (5) and the Receiver (R ) who are to playthe game a known, finite number of times. At each stage both players learn the value ofa parameter measuring the importance of that period 's play of the game ( that is, the valueof making a correct decision) and 5 obta in s payoff-relevant in fo rma tion; 5 then sendsa signal to R, who makes a decision tha t affects the welfare of both players. After bothplayers learn the consequences of R 's decision, the process repeats. At each stage Rmust dec ide what act ion to take. I n order to do that he must assess the credibility of 5.The difficulty arises because R is uncer ta in about S' preferences when play begins. Imodel this by assuming that with positive probabil ity 5 has ident ical preferences to R(5 is a friend) and with positive probabili ty 5 has completely opposed preferences to R(5 is an enemy). This uncerta inty creates a trade-off for enemy Senders. At each stage,providing honest information enhances S' reputation, but only at the expense of foregoingan oppor tuni ty for immedia te gain by duping R. The possibility tha t 5 is a friend lendscredibility to wha t he says, but this credibility may also be explo ited by an enemy S.

The princ ipal equil ibrium has the following form. In the basic model, where S hasno control over the impor tance of a given play, 5 typica lly conveys accurate informationfor the first several periods. R raises the probability he places on 5 being friendly af terreceiving accurate information if he believes tha t an enemy would have lied to him withpositive probabili ty. The more important the information, the more likely an enemy willattempt to deceive R. An enemy will eventually take advantage of R by misleading him.When this happens, the Sender loses all opportunities for deception in the future. Idescribe the basic model in Section 2. In Section 3, I analyse the one-stage version ofthe model. Sec tion 4 descr ibes equilibrium behaviour when the game lasts for more than

one period.557


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558 REVIEW OF ECONOMIC STUDIES

In Sections 5 and 6, I m od if y the b as ic game. In Se ctio n 5, 5 may r ed uc e the i mp or ta nc eo f the game in any period. This leads to a significant change from the earlier results:the specification of e qu il ib ri um now i nc lud es a r an ge of information levels in which Rtrusts no one. That is, in order to inf luence R, 5 cannot offer i nf or ma ti on in a c er ta inrange. I f 5 happens-to receive information in this range, then he must reduce its importancein order to maintain credibi li ty in future per iods .

R controls the importance of the game in Section 6. In th e principal equilibrium ofthis section, R rewards accurate information by increasing the importance of future plays.

The s ev en th s ect io n is a co ncl us ion .Recent papers by Kreps an d Wilson (1982b), Kreps , Milgrom and Roberts, and

Wilson (1982), and Milgrom and Roberts (1982) were th e first to p resent model s in whichplayers, like th e unfriendly sender in th is paper, exploit uncertainty and take short-termlosses in order build a reputat ion and make long- term gains . Wilson (1985) reviews thesepapers and other applications of the t yp e of reputation effects that I analyse. My paperowes much to this work.

2. TH E BASIC MODEL: THE PRINCIPAL-DOUBLE-AGENT PROBLEM

There is a finite s eq ue nc e of dates indexed (backwards) by t = T, T - 1 , . . . , 1. T hus, atdate t there are t per iods remaining . There are two players , 5 and R. There is a sequenceof independent, identically-distributed random variables {At}, each with differentiableprobabi li ty dis t ribut ion funct ion G(A) and density g(A) supported on [1-, .4], where1-


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SOBEL A THEORY OF CREDIBILITY 559

Assumption AI. U i ( . ) d ep en ds only on jy - MI an d A.

A ssu mp ti on Al has two consequences. First, it says t ha t the sin gle- per io d payoffsdo not depend on t. S eco nd , it is a symmetry assump tio n. I t says that utility dependsonly on the d is tance between y an d M. It allows a simplification of the analysis, in whichS' s two possible lies (communicating 1 when M = -1 an d communicating -1 when M = 1)

play analogous roles an d the probability of lying therefore need n ot d ep en d on M.

Assumption A2. V i ( . ) is s tr ic tly concave in y.

Assumption A3 . The so lu ti on to the p ro bl em:

max, Vi(y - M, A)

is y = M if i = R a n d y = - M if i = E.

Assumptions A2 an d A3 guarantee that R has a un iqu e best response to S's signalsan d makes it unnecessa ry for R to use mix ed strategies in eq ui lib rium. S up po se that ibelieves that the probabi li ty tha t M = 1 is r. Then his best action, call it yi(r, A), is thesolut ion to the problem:

max y rVi(y - 1 , A) - ( 1 - r) U'(y+ 1, A).

Assumptions A2 an d A3 guarantee that yl (r, A) is well defined for all r E [0, 1],A E

[ ~ ,.4], and i = R a n d E. Assumptions AI , A2, an d A3, combine to imply that

yR(O, A) == yE (1, A) == - 1 , v" (1/2, A) == yE (1/2, A) == 0, yR(1, A) == yE (0, A) == 1,/ ( r, A) + / ( 1 - r, A) == an d that the yl( . ) are st rict ly monotonic in r. R prefers higherac tions in [ -1 , 1] to lower if M = I , an d lower ac tions in [ -1 , 1] to higher if M = -1 .E's pre fe rences are the reverse. Thus, the interest s of E an d R are totally o pp os ed inthe sense that if M is known an d E weakly prefers an act ion u to another action v, then

R weakly prefers v to u.

Assumption A4 . Ui(yR(O, A) - M, A) == for all M and A.Assumption AS. If U E ( ) - 0 then IU E ( )1 is strictly inc reasing in A, for all M

an d y.

Assumption A4 is a normalization that guarantees that a total ly uninformative signal,which induces R to take the ac tion yR(1/2 , A), yields zero utili ty to E an d F. AssumptionA5 makes increases in A increase the importance of the game to E.

I make these restr ict ive assumptions mostly for convenience. A representat ive uti li tyfunction that satisfies Assumptions AI-AS is VR(y - M, A) == A ( 1 - (y - M)2).

Further, except that the principa l equ il ib ri a requi re compl icated mixed s trateg ies,the quali ta tive propert ies of equ ilib ria do not c han ge if the utility fu ncti on s are linear.F or ex amp le, the stage g ame co uld have the following payoff matrix.

R

y=

y =

E

-1 A , - A - A , A

1 - A , A A , - A

M=-I M = 1


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That is, R has two pure strategies available (y = -1 and y = 1) and always disagrees withE on which is best. Very little is lost by replacing the general u'i . ) by linear or quadraticutility functions.

3. TH E ONE-STAGE GAME

I begin the analysis with a discussion of the one-stage game. For a fixed prior Ph anequilibrium consists of the Sender's signalling rules, qi(nIM, A), and the Receiver's actionrule, y(n, A), which satisfy for i = E an d F, M = -1 an d 1, a nd all n EN:

Condition E 1. (a) qi( . )~

0 an d L n E N qi(nIM, A) = 1.(b) If qi(noIM, A) > 0 for some no E N, then no solves:

m a X n e N Ui(y(n, A) - M, A).

Condition E2. y(n, A) = yR(r(n, A), A) where r(n, A) is the probability that M = 1given n and A so that,

r(nIA) = PlqF (nil, A) + ( 1 - PI)qE (nil, A) .PI(qF (nil, A) + s" (nIO, A + ( 1 - PI)(qE (nil, A) + qE (nIO, A

if

Conditions El a nd E2 describe a Nash Equilibrium. The first condition says t ha tS' signalling rule is a probability distribution I an d that any signal 5 sends in equil ibriumis a best response to R's action rule. Condition E2 uses Bayes' Rule to compute R' sposterior distribution on M given his information: the value of A a nd the signal n; r( n, A )is the probabili ty tha t M = 1 given A and n. If 5 uses a signal t hat is unexpected, t ha tis, if 5 sends a signal n such that

then Bayes' Rule cannot be used to compute r(n, A). If R does not expect no inequil ibrium, it must be the case that no is not the unique maximizer of Ui(y(n, A ) - M, A).Since r(n, A) determines y(n, A), the specification of r( . ) cannot be made arbit rari ly inequilibr ium. Finally, Condition E2 guarantees that R' s action is a best response to thesignal.

Notice th at the equilibrium conditions hol d pointwise in A. To ease notat iona lclut ter, I suppress reference to A for the remainder of this section.

By the assumptions on preferences an d Condition E 1, it follows that at most twoactions are induced (signalled for with positive probability) in equilibrium. If Y ={y : y = y(n) for some n EN}, then Y c [ -1 , 1]and the maximization problem in ConditionEl is equivalent to picking y E Y to maximize Ui(y - M). This problem has a so lut ionif and only if Y has a least and greatest e lement; call these y an d y respectively. y an dy typically depend on the data of the problem, A an d Pl ' In order to satisfy Condi tionEl(b), 5 sends a signal tha t induces R to take action l ' if 5 = F and M = - l o r 5 = Eand M = 1, an d y if 5 = F an d M = 1 or if 5 = E and M = -1 . Without loss of generality,I restrict attention to signalling spaces t ha t have two elements, say N ={ -1 , I}, andassume that F weakly prefers to make signal i if M = i. Th at is, I assume that y( -1 ) = Yand y(l) = y. -


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T he se c om me nt s m ak e a c ompl ete c ha ra ct eri za ti on of e qu il ib ri a in t he on e-s ta gemodel routine.

Theorem 1. I f T = I , then there is an equilibrium in which y ( - I ) = y ( l ) = y R ( l / 2 ) .This is the only equilibrium when PI

~ ! .I f PI > t then there is exactly one other equilibrium;

in it, y( -1 ) = y R ( l - PI) an d y(l) = yR(PI).

The proof of T he ore m 1 is in the Appe ndix. T he no-transmiss ion equilibrium, inwhich R takes th e same action regardless of S' s signal, always exists. I f R chooses toignore S' s signal, then it is a best response for S to ma ke n on -in fo rm at iv e signals.However, if S is sufficiently reliable, then there is an equilibrium in which R uses th es ignal in order to d ete rmi ne his action. This equilibrium appears to be more sensiblethan th e first for two reasons . First, the no- tr ansmission equ il ib rium leads to an expectedpayoff of zero for both types of S an d for R, which is lower for all players (both typesof S a n d R) than the payoffs ava il ab le in th e other equil ibrium. Second, one expectsF' s signals to be informative in equilibrium. In what follows, I restrict attention toequ il ib ri a in which F sends different s ignals for different values of M. These equilibriainvolve no n-t riv ia l c om mu ni ca ti on whe rev er it is possible. As sumi ng that s ignal s a relabeled so that y( l )

~y( -1) , this means that q F (n IM ) = 1 if an d only if n = M. This type

of signal allows F to communicate the maximum amount of information to R. I call Fhonest if he uses this type of s igna ll ing rule; I intend to concent ra te on hones t equ il ib ri a:those equ il ib ri a in which F is honest.

Corollary 1. 1fT = 1, then there is a unique honest equilibrium in which y( -1 ) = y( 1) =yR(!)

i f P I ~ !andy(-1)=yR(1-PI) andy(l)=yR(PI) i f PI>!'

The proof of C orol lary 1 is in the Appe ndix.

When PI > t E always lies to R; for eit he r value of M, E an d F a lways makedifferent s ignals . Al though R knows that this is E's strategy, he is sufficiently confidentthat S is a f ri en d to t ak e different a ct io ns for different signals. W he n PI

~ ~ ,there is no

equi libr ium in which F is hones t an d E always lies. Given signall ing rules of this form,R would choose to take a higher action when he receives a lo we r signal, b ut then S'ssignalling strategy would no t be optimal. Ins tea d, when F is hones t an d PI < t E mustr an do mi ze to c ov er up the i nf or ma ti on in F' s s igna l. The s trategy presented in the proofof Corol la ry 1 makes r( - 1) = r( 1) =

~ .

The one-s tage game is s imilar to the information t ransmiss ion model s of Green an dStokey (1981a) an d Crawford an d Sobel (1982). These models have more generaluncertainty about t he s ta te of nature ( M ) , but no uncertainty about preferences. Green

an d Stokey (1981 b) an d Holmstrom (1980) analyze similar one-s tage models, but allowthe u ni nfo rme d p lay er to c ommit himself to an a ction rule.

4. THE MULTI-STAGE GAME

When there is m or e than on e stage, E somet imes wants to supply use fu l information toR at first in order to t ake advan tage of his reputat ion later. No w I b uil d on th e analysisof the one-s tage game to descr ibe equil ibria of th e f initely-repeated game. As in Section3, I restrict a tt enti on to signalling spaces that ha ve two el eme nt s, 1'1 = { - I , l}, and Iidentify th e signal n = - 1 with the lower a ct io n i nd uc ed . I c ha ra ct er iz e o nl y th e honestequ ili bria . C or ol la ry 1 pr ov id es a jus tifica tion for this in the o ne -p er io d g am e; a s im il ar


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result, Corollary 2, holds for the finitely-repeated game. Also, I assume tha t the stateM = -1 and M = 1 lead to symmetric behaviour so tha t, in part icular, the probabili tythat E an d F send the same signal does not de pen d on M. This assumption actually isa straightforward consequence of the symmetry assumptions on util ity functions (for aproof, see Sobel, 1983).

With these simplifications, an honest equilibrium consists of signal ling rules for 5,

denoted by q ~ ( A ,p) for i = E an d F, which represent the probability that i tells the truth(reports n = M ) given A with t periods remaining an d P is the probability that R believes5 = F, and act ion rules for R, denoted by Yt( n, A, p) . Given the symmetry assumptions,an honest signal induces an act ion that differs in absolute value from M by z, (h, A, p ) ==

1 - Yl(l, A, p) == 1 + Yt(O, A, p), while a d ishonest signal induces an ac tion tha t differs inabsolute value from M by Zt(d, A, p) == 1 + Yt(l, A, p) == 1 - YI(O, A, p). In an honest equilibrium,

q ~ (. ), Yl( . ), and Zt( . ) must satisfy:

Condition EO'.

{

o i f / = d o r p = Op t - l ( l , A , p ) = p / [ p + ( l - p ) q ~ ( A , p ) ]i f l = h andp=O.

Condition E I'. (a) 0~ q ~ (

. )~

1(b) q ; ( . ) = 1(c) I f

q ~ ( A , p O ,then

W ~ ( h , A , p ) ~W ~ ( d , A , p ) ,an d if 1>

q:(A, p) > 0, then W:(d, A, p)~

W:(h, A, p) whereV ~ ( p ,

A ) == 0,V:(p) is the expected value of

V ~ ( p ,A), W:( l, A , p) ==

Ui(Z t( l, A , p), A ) + V:-t(Pt-t(l , A, p.

Condition E2'. Yt(n, A, p) = yR(r t(n, A , p), A) where r.in, A, p) is the probabili tytha t a signal is honest given A so t hat 'len A, p) = p + ( l - P

) q ~ ( A ,p).

Given PT, Condition EO' defines a sequence of probabilities, PT, PT-h . . . , Ph whereif At is the real ized value of A in period t, an d II = d or h depending on whether 5 isdi shones t or hones t in per iod t, then Pl-t = P,-t(l" A" P,),

The functionP t ~ l (

. ) defined in Condition EO' is an expression for the probabi li tyt ha t 5 = F given S' signalling rule. When It = d, P,-I( . ) = 0 since in an honest equilibriumR believes that F always is honest. Fur ther, once R discovers a lie he remains certainthat 5 is an enemy regardless of what ha pp en s in the remaining periods. The secondline in Condition EO' follows from Bayes' Rule since P + ( 1 - P

) q ~ ( A ,p) is the probability

of an honest signal. The probabi li ty Pt summarizes all of the information that R needsto know about 5 when t periods remain. In general, strategies may de pen d on more

detailed descriptions of the history t ha n the sequence PT, PT-h . . . ,Pl. However, thisinformation is not relevant to R in an honest equilibrium. To see why, I argue inductively.By Corollary 1, there is a unique honest equilibrium when T = 1; PI completely determinesequil ibrium strategies . Therefore, all equil ibrium strategies for the final period do notd epe nd on p as t information ot her t ha n Pl ' Similarly, Theorem 2 shows tha t an hones tequilibrium for any t-period game is determined uniquely by P,. Conditioning on historiesmay be used to select from a number of equilibrium outcomes for the remainder of thegame", but since honesty is enough to predict definite outcomes, information not includedin Pt plays no role in the t-period equilibrium. Following Corollary 2, I discuss anexample of an equil ibrium in which strategies depend on more general descriptions ofhistory than the posterior probability that S is F.


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Condition El ' describes S' s ignal ling rule. Condi tion El ' (a) says that S' signallingrule is a probabi li ty dis tr ibution over s ignals , Condition El'(b) says that F is honest, an dCondition El'(c) guarantees that S best resp on ds to R's action rule. Condition E2' usesBayes' Rule to compute R' posterior distribution on M given his information: the valueof A; the signal n; an d the probability that S = F, p. Bayes' Rule determines the probabilitythat S is telling the truth, r t ( . ) . Notice that Condition E2' implies that R's period-t

payoff determines his action in period t. This is because his payoff with t periods remainingdepends on YI( . ) only in p eri od t. Thus, in order to best r esp on d to S's signal, R mustbehave myopically.

I must define equilibrium strategies for all p, not simply for the equilibrium realizationsPT, PT-l> . . . , p ; This is because only by defining these strategies can the players computepayoffs associated with non-equilibrium behaviour. The definition of equilibrium guarantees that cont inuat ions of hones t equ il ib ri a are also hones t equi libr ia. Moreover, once Slies the relat ionship between R a n d S stops in that S' s ignals are not informative an dR' act ions do no t depend on S' s signal.

LetV ~ ( p ,

A) be t he expected value fun ctio n for R when t periods remain an d Rbelieves that S = F with probability p. The corresponding value funct ions , V:(p, A) fori = E and F, have b een defined in C on di ti on E2'.

Theorem 2 character izes the equil ibrium.

Theorem 2. There is a u ni qu e honest e qui libri um. In it, fo r i = E, F, and R, V:(p) = 0fo r P

~2- 1 and is str ictly positive an d in cr ea sing in p and t fo r p > 1 1 There is a f un ct io n

Ht(p), increas ing in p and t, s uch that E never lies in period t if A < Ht(p). The probabilitythat S 's signal is honest and the degree to which R believes S i nc rea se s in p and t.

Before discussing T heo rem 2, I p re sen t a justification for c on cen tr at in g on h on es tequilibria.

Corollary 2. I f the honest equilibrium is played when s < t, then R an d both types o fS prefer F to be honest in period t than an y other equilibrium behaviour.

I prove Theorem 2 an d C ol lo rar y 2 in t he Ap pen dix .Corol la ry 2 just ifies the hones t equ il ib rium induct ively. If all of the players bel ieve

t ha t the h on es t e qu il ib riu m will be p la ye d for s-stage games, t hen they cannot do b et te rthan play honestly in an (s + l l-stage game. As Co ro llar y 1 establ ishes the p ro mi ne nc eof the h onest equilibrium for the one-stage game, players expect it to be played. Thus,they have no reason not to play it in longer games. On the other hand, Corol la ry 2 doesno t prove that the hones t equ il ib rium Pareto dominates all other equilibria of the game.

This is n ot true when T > 1. F or ex amp le, let T = 2 an d suppose that F uses a signall ingrule when t = 1 t ha t d ep en ds on R's action when t = 2. In par ti cu la r, suppose tha t Fusesan h on es t signalling rule when t = 1 if an d only if Y2( n, A, p) = n for n = 0 and 1, an dotherwise F conveys no i nf or ma ti on so that q i (n IM) = ! for n = 0 an d 1 an d M = - 1

an d 1. I f F ch ooses to convey no i nf or ma ti on , t he n the only eq ui li br iu m strategy for Rwhen t = 1 is YI(n) = yRO) for n = 0 an d 1. Therefore, unless R takes F' s most preferredact ion when t = 2, the no- transmission equ il ib rium is se lected when t = 1. I f P2 is c loseto 1, then R prefers to set yi n, A, p) = n rather than sacrifice a posi tive payoff whent = 1. With appropriate signal ling rules for E, these strategies then comprise a sequentialequ il ib rium (see Kreps and Wilson ( I982a. F prefers this equilibrium to the h on esteq ui li br iu m b ec au se he i nd uces his most p ref er red o ut co me in both per iods . Provided


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that E lies with posi tive probabi li ty in th e honest equil ibrium, he to o does bet te r s incea lie cau ses R to take E's most pre fe rred act ion. Not ice that in this equil ibrium R doesnot make his myopic b est response to t he signal wh en t = 2, because E will lie to himwith posi tive probabi li ty. This is possible because F conditions his t = 1 signal on R' sact ion. I cou ld res tr ict a tt en tion to h is to ri es summarized by the probabi li ti es Pt when Icharacterized honest equilibria because Pt uniquely determines equilibrium strategies. In

general, however, conditioning on earlier strategies can be u sed to mak e selectio ns f ro mmultiple equilibria. This is what happens in the example. I have chosen to restricta ttent ion to honest equil ibria because Corol laries 1 an d 2 suggest that F's threat to rever tto the no- transmission equ il ib rium is no t plausible.

In th e honest equilibrium, E does on e of three things, depending on the value of A.I f A is small relative to th e number of periods remaining an d E's current reputat ion(A < Ht(p, E always tells the truth because he expects to have a more attractiveopportunity for deception in the future. Therefore , when A is sufficiently low, R is willingto believe anything that 8 says, bu t for this reason honesty does not improve 8's reputation.E always lies with posi tive probabi li ty if his cur rent reputat ion has no fu tu re value.

Fo r larger values of A, E lies with positive probability. I f P an d A are sufficientlylarge, then E lies with probabi li ty one. This type of behav iour only occurs if P ~ ! sinceotherwise R would no t t rust the signal . For some values of A, E uses a mixed strategy.I f 8 does tell the truth in these situations, then th e fact that R thought that E might lieleads R to feel more confident about 8 in the future. Wit ho ut f ur th er as su mp ti on s aboutpreferences, I cannot guarantee that increases in A increase the equilibrium probabilityof lying. Th is is b eca us e an i ncrease in A may change yR(p, A) so as to make lying lesscostly to E. However, if yR( . ) d oes n ot depend on A, increases in A always make lyingmore att ract ive to E, an d the equilibrium probability of lying inc reases as a resu lt.

T he or em 2 sh ed s s ome light on how t he re la ti on sh ip b etw een S a n d R depends onthe reputat ion of S, P, an d the number of periods remaining, t. Increasing P is beneficialto R because it increases his chance of d ea li ng with a friend. I ncr ea si ng p is beneficialto S because R receives more informative signals (honesty is more l ikely) so the payoffsassociated with honesty (for F) and dishonesty (for E ) increase. When more per iodsremain or wh en p is hi gh er, the value of pretending to be a friend is high to an EnemySender.

The variable importance level A provides an incen tive for Enemy Senders to supp lyhonest information. I f

~= A, so t hat t he importance of e ac h stage g ame d oes no t vary

over time, then E signals hones tly with pos itive probabi li ty in some per iods . However,E is n ever h on est with probability one. Here is why. E tells the truth if he expects tohave an o pp or tu ni ty to g ai n m or e by lying in the future. This c an happen for two reasons.Either t her e is a ch an ce that future oppor tuni ti es are more valuable (higher A), or that

R will be more trusting in the future (higher reputation). The first cannot happen if~

= A an d th e second cannot happen unless R has some reason to beli eve that 8 is morelikely to be a friend, which in turn only happens if R expects E to lie with positiveprobability and then receives an ho nest signal. Thus, the role of a var iab le A in them od el is to p ro vi de si tu at io ns in wh ich E always tells the t ru th .

R pre fe rs to dea l with a single Sender for T per iods rathe r than T Senders separately.This is because repea ted p lays allow him to bet te r eva luate the usefulness of S' informat ion. Long- te rm arrangements help to moderate the inefficiencies caused by incompleteinformation.

Theorem 3. In the hones t equ ilib ri um,V ~ ( p )~

t V f ( p ) fo r all t and p.


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(1)

Proof Th e proof requires two observations . Th e first step is to o bs er ve that if 0 < Pan d p < 1, then

p V ~ ( p ,A)

~ V ~ ( p p ,A).

When pp~ ~ ,

(1) fol lows since V~

(pp, A ) = O. Otherwise,

p V ~ ( p ,A) = ppUR(yR(p, A) - 1 , A) + p ( l - p) U R( _yR(p, A), A)

> ppUR(yR(pp, A) - 1 , A) + p ( 1 - p) U R( _yR(pp, A), A)

> ppUR(yR(pp, A) - 1 , A) + ( 1 - pp ) U R( _yR(pp, A), A)

=V ~ ( p p ,

A).

Th e first inequality follows from th e definition of yR( . ), th e second inequality followsbecause U R( _yR(pp), A) < 0 whenever p p > t an d the equat ions fol low from th e definition of

V ~ (. ) an d because in t he h on est , one-period equilibrium if th e probability of S

being E is less than on e half, E is a lways d ishonest .T he s ec on d s tep is to n ot e t ha t for some r E [p, 1],

V ~ ( p ,A ) ~ V ~ ( p ,A ) + r V ~ _ I ( p lr). (2)To obtain (2), note that

V ~ ( p ,A) = rUR(yR(r, A ) - I , A) + (1 - r) U R( - yR(r, A), A) +

r V ~ _ l(p i r) ,

where r is the probabi li ty that R receives an honest si gn al in period t. Thus, s ince Rreceives an honest signal more often in th e first of t periods than in a single period (whenE is a lways dishonest or no information is transmitted), (2) is e st ab li shed .

Th e proof now fol lows f rom induc tion . The theorem is true when t = 1, an d if itholds for s - 1 periods, then

V ~ ( p ,A ) ~V ~ ( p ,A ) + r V ~ _ l ( p l r )

~ V ~ ( p , A ) + ( s - l ) r V ~ ( p l r ) ~V ~ ( p , A ) + ( s - l ) V ~ ( p (3)

where th e first inequality fol lows from (2 ), t he s ec on d from t he induc tion hypothesis,an d th e third fro m (1). Tak in g e xp ec te d val ue s of (3) establishes th e theorem. II

While Theorem 3 indicates th e advantages of d ea li ng with a si ngl e S, it is a re la ti ve lyweak resul t. It w ou ld be of interest to s ho w t ha t

(4)

However, (4) will no t hold without further a ss um pt io ns . First, t hu s fa r I have assumedthat E and R have total ly opposed p re fe re nc es in a ny si ng le period, but that t he y may

d iffe r in their assessments of a s it ua ti on 's importance. If E and R do differ in theirordering of A, t he n t he re may be situations in which (4) does not hold. To see this,suppose that i nc reas ing A inc reases IV Ic( )1,b ut th at i nc reas ing A first dec reases , thenincreases, IU R ( )1,so that R loses very l it tl e when he is duped for intermediate valuesof A. As th e length of th e hor izon inc reases , it is more likely that E will wait fo r a highvalue of A an d c au se m or e d am ag e to R when he is dishonest. In this case, ex ante, Rw ou ld p re fe r to s ho rt en t he m ax im um feasible length of t he par tner sh ip and use tw oi nfo rma nt s for s hor t p eri ods ra the r t ha n one for a long period. This p ro bl em c an ber ul ed o ut by assuming that E an d R have opposing i nt er es ts in an extended sense , forexample, if U E (y - M, A) == - VR(y - M, A) so that E an d R a lways agree on the importance of a d ec is io n and di sa gr ee on t he appropriate action.


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Assuming that U E (y - M, A) == - UR(y - M, A) does not appear to be enough toprove (4), however. S ince R is risk averse, he is willing to pay a p re miu m to discover if5 is an enemy. However, because commitment is not possible , he cannot do this by beingmore credulous in early periods an d thereby inducing an unfriendly Sender to be dishonestearly. On the o ther hand, a shorter horizon allows fewer opportunities for deception an dR may prefer to limit the length of attachments so that the first pe riod s will p ro vide

stronger tests of S's motives. When u'; . ) is l inear, (4) is satisfied, indicat ing that riskavers ion may p la y a role in discouraging long-term associations if co mmitme nt is n otpossible . When

~= A, (4) is also satisfied. This suggests t ha t p ar t of the problem with

long-term arrangements is the increased possibi li ty that E could deceive R on a veryimportant matter.

Many of the a ss umptions m ade in this section are n ot really necessary. I ma de thesymmetry assumption A3 for exposition convenience. Aside from the fact that theprobability of being h one st will d ep en d on M, no results will be altered by relaxing theassumption. The distribution of A an d the util ity funct ions can depend on t: in particular,int roducing d iscounting presents no problem. More general types of preferences for Emay be allowed. If there are only two states of the world, preferences will either betotally opposed or in agreement. If it is common knowledge that E's preferences areidentical to R's periodically, th en the results do not ch an ge ; the re is complete tr ust inthose periods. On the other hand, if the nature of the uncertainty about M is morecomplicated, then there are more possibilities for diversity of preferences. In such avariation it mig ht be possible to analyse the effect t ha t increasing the degree by whichpreferences differ has on the character is tics of equilibrium.

The results re main basically u nc ha ng ed if R observes A only after he makes adecis ion. There typical ly is an interval of values of A such that E always tells the truth,followed by an interval in which mixed strategies are used, and then by an interval inwhich E always lies. Changing the payoff structure could make more or less randomizationnecessary. I f U R ( ) were not s tr ictly concave, then R would need to randomize. If, forexample, R observed the value of A with noise (or, for some reason, was uncer ta in aboutthe value of A that entered S' utility funct ion) , randomization by 5 would probably notbe necessary. What is nee de d is a way by which R places non-degenerate probability onE telling the truth. This can be do ne in a variety of ways.

Another modification is worth noting. I f R can commit himse lf to an ac tio n rule ina given p er io d, but is u na bl e to c ommit h ims elf to ignoring in for ma tio n in the futur e,the equilibrium changes somewhat. R would like to take more drast ic (be more credulous)ac tions in the early per iods . This increases the probabi lity that E is dishonest, an d thusallows R to be more confident t ha t in for ma tio n p rov id ed in future p eriods is reliable.Formally R's payoff c an be divided into two par ts , the future e xpected utility an d the

utility he expects from the cur rent per iod' s decision. Since he is unable to make commitments, R must use the current period's information optimally. This maximizes th econtribution of the current per iod to expected utility. However, the future considerationargues for R to increase the amount of dishonest behaviour in the present per iod in orderto be more confident, after he receives reliable information, th at he is dealing with afriend. I f R could make commitments , this effect would cause him to be more credulousin the current per iod in order to be more confident in the future . However, it will typical lynot be optimal for E to lie with probability o ne in p eriod T, an d the analysis of thissection captures the qualitative properties of the equilibria with commitment.

In the model an enemy has only one opportunity to mislead R. Afterwards, hisr ep uta ti on falls a nd R believes nothing that he says. While he is providing useful


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information, S' s reputation increases through time. Several simple modifications of themodel would alter these results. I f S did not receive per fect information , then R couldnot be cer ta in that a misleading signal was del iberate or accidental. In this case, mistakeswould cause F's reputation to fall, and E may be able to deceive R at regular intervals.I t is worth a dd in g that if R had the opt ion of replacing S with a no th er p la ye r whosereputa tion was the same as S' s original r epu tatio n, he would do so in the pres ent mo de l

only if S is an enemy who has just lied. If S can make mistakes, then there is a positiveprobabil ity tha t a f riend will be replaced.

5. REDUCING AGENTS

Theorem 2 showed th at if A, is sufficiently low E will send the same signal as F withprobability one. This is costly to E ; he makes shor t- te rm losses because R is able tomake a well-informed decision, but does not improve his reputation. I t is only theexpec ta tion tha t futu re values will be greater than the cur rent one tha t makes his strategyoptimal. However, if E was able to manipulate A" he might choose to provide lessvaluab le information in these s ituations. Continu ing the analogy with spying, a doubleagent might convey accurate, but less than comprehensive, information in order to lowerhis losses when decep tion is not justified. To allow for this poss ibility, in this section itis assumed tha t S observes the value of A" again drawn from the dis tr ibut ion descr ibedby G( . ), an d then selects a reduced value at E

[ ~ ,At] to report to R, thereby lowering

the poten tial impor tance of his services to R. Next he sends a message to R about Mas before, an d R chooses an action. R learns the reduced value at when S reports it;this is the value t ha t enters his utility function. R never learns the true value of A" Ico ntinu e to make all of the o th er assumptions of Section 2.

This section presents an informal description of the equ il ib rium for this model. Thedeta ils are in Sobel (1983). For the reduc ing model, the strategies of the players include

signall ing rules for S an d decision rules for R, as in Section 2. In addition, S must havea rule tha t tells them what level of impor tance a to report given t, p, and A" RevaluatesS' s reputation on the basis of S' s report, a, so that an Enemy Sender cannot systematicallyreduce the importance of the information that he provides without lowering R' s t rust inhim.

There always exists an honest equil ibrium with reducing. It involves both types ofS reduc ing A to

~in every period. This equilibrium can be su pp orte d if R refuses to

trust anyone who'offers information A >~ ;

consequen tly, no one offers it. The reducedequil ibrium with importance 1- in each period can be analysed as in Section 4. Theequilibrium is extreme because R never is able to gain access to any important information.I t might be h ope d that the re are o th er equilibria t ha t b ot h types of S a n d R prefe r to

this one. In par ti cu la r, just as it was re aso nable to expec t F to pu t as much informationas possible about M into his signal (tha t is, to be honest), it seems r eas ona ble to expectF never to reduce the importance of this information. I t turns ou t that this is not possiblein general.

Theorem 4. I f U E ( y - M , A ) = U F ( y - M + 1 , A ) , U E ( y - M , A ) = O , an d T > l ,then in every honest equilibrium there exists a nondegenerate interval o f importance levelsA that neither type of S provides in some period. Moreover, the reputation o f S cannot bemonotonically increasing in the level o f A provided.

For a proof of Theorem 4 see Sobel (1983).


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The message of Theorem 4 is that the ability to manipulate the importance of asituation adds a new inefficiency to the p ro bl em of contracting with an agent who h asuncertain motives. In order to avoid placing confidence in an agent who provides accurate,bu t u ni mp ort an t, info rmation , R must be wary of an 8 who reports low values of A.Theorem 4 s ta tes that th is is no t consistent with accepting certain values of A. A typicalequil ibrium involves 8 reducing to A those values of A that would cause E to be h on est

with probability one. F mu st also reduce these values, for otherwise an enemy 8 co ul dobtain a good reputat ion at low cost by imi tat ing F. In order to persuade F n ot to r ed uc eA, R must con jecture that only unreliable agents provide certain levels of A. This meansthat reputat ion cannot general ly inc rease with the level of information provided.

The assumption that U E (y - M, A) == U F (y - M + 1, A) guarantees that increasingA inc reases impor tance for both R a n d 5. The assumption that U

E(y - M, A) == 0

guarantees that there are some levels of importance so low that E would pre fe r to a lwaystell the truth in the hope of making a big gain tomorrow, than lying with pos it iveprobability.

This type of model app li es to economic s ituat ions . Fo r example , let the informedplayer, 5, be in charge of market research for R. 5 learns two things about R's product.He learns th e appeal of the item, measu red by A. 5 also learns whether the productworks (o r ca n be produced profi tably) , measured by M. R wan ts to m ark et t he productif M = 1, but does no t w an t to m ar ket it if M = -1 . In thi s contex t, a ssume tha t a f ri endlySender's preferences agree with R's, bu t an Enemy Sender, never wants R to m ar ket t heproduct.' The results of th is section a pp ly if I assu me that R l ear ns t he value of M afterhe makes his marke ting deci sion a nd t ha t 5 is able to reduce that size of A by claimingthat only a < A p eo pl e might use the pro duct. In this case, R observes a bu t no t A. Forthi s model , the resu lt s ind icate that E p ro vi des reliable i nf or ma ti on for a p er io d whilewaiting for an o pp ort un ity to convince R no t to pro duce when A is l arge an d M = 1.Moreover, the larger is A, the more costly it is for E to r ec om me nd p ro du ct io n. C on -

seq uen tly, t he ability to re du ce the size of projects is att ract ive to E. Theorem 4 s tatesthat in some situations even F must reduce A in order to maintain credibi li ty. A part icularform of e qu il ib ri um b eh av io ur involves R mar ket in g o nly t ho se p ro du ct s t ha t at tr ac t alarge enough group of people. R does this no t because of any economic of scale inproduction, but b ec au se if R believes rep or ts of low values of A t hen he allows En emyS end ers to deceive him at low cost.

The model of this sec tion has the property that th e level of information (magnitudeof A) an d n ot j us t its precision (rela tionship to M ) convey information to R about S'smotives. W hen 8 has con tro l over the level of i nf orm at io n, R m us t be wary of Senderswho do no t provide important information; otherwise, E will take advantage of him bysupplying important i nf or mat io n o nl y w he n he i nt en ds to lie.

6. A LOAN MODEL

The m od el of thi s sec tion does no t involve direct information transmission, but shareswith e ar li er mo del s t he idea of uncertainty about motives. I will call two players theBorrower (B) an d the Lender (L). The game repea ts a fixed, finite number of times. Atany stage, L d ecid es on an amount A E [0, 1] to lend to B. B, taking M as given, decideswhether to invest (1) or to d ef au lt (D). T he re are two t yp e of B, E an d F, an d L assignsthe ini tial probabil i ty PT that B = F. The payoffs for a single p la y of t he g ame are givenby the u ti li ty funct ions Ui(A,j) . In this sect io n, I assu me that

UF(A, D) = UL(A, D) = UE(A, 1)= - A


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an dVF(A, 1) = U1(A, 1) = VE(A, D) = A.

k=O, . . . , t - l ,

Th e payoffs suggest that E cannot produce anything with the lo an , bu t is able to takethe money. F cannot costlessly default, but he is able to i nve st th e m on ey an d make aprofit, which is shared with L. T he two typ es of borrowers differ only in their abi li ty toproduce income from a loan. Fo r example, if a f ri endly borrower ca n always triple th evalue of th e loan, while an enemy bor rower cannot increase th e loan's value, thenpreferences similar to th e ones used arise provided that the lender requires that anon-defaulting borrower returns $2 for each $1 borrowed."

Informally an equilibrium consists of loans, At(p), probabilities that B = F, P an dprobabilities of defaulting, d;(p, A) for i = E an d F, such that reputations ar e updatedby B aye s' Rule when possible, L best responds and B puts positive probability only onstrategies that maximize ut il ity. In thi s def in it ion, L is able to commit himself to a valueAt in e ac h p er io d, but cannot make commitments about future loans . I t turns o ut t ha twhen u'i . ) is l inea r, thi s res tr ic tion does not influence L's payoffs.

Theorem S. The equilibrium to the loan model is

{2 k: 2 k:

~P

~2- k - I ,

Ar(p)= 2 t > p ~ 0 ,

d ;( P, A) == 0,

~ l A > A r ( P ) ,or i fA=Ar(p )

a n d p ~ !

i f A = A r( p) and P < ~

The associated value functions are

V::(p)={02k

2 - k ~ p ~ 2 - kI , k=O, . . . , t - l ,

2 - r > p ~ 0 ,

V ;( p ) = {ot- L . : I 2 i 2- k~

P~

2 k - \ k = 0, . . . , t - 1,

2 r ~ p ~ 0

an dV ~ ( p )

== pV;(p) - ( 1 - p)V ~ ( p ) .

Th e proof is a routine verification, I omit it.In equilibrium, L is u nw il li ng to loa n the full amount to B unless he is confident

that B is a friend. Th e level of confidence does no t d ep en d o n th e number of periodsremaining; until full confidence is reached (p

~!), th e amount of loan increases with

each successful loan. Also, the value of th e game is increasing in t and p for all playersand if p > 0, then lim T _0 0

V ~ (p) > 0, so that if there is some chance that B is a friend,

there ar e T's large enough so that it is worthwhile for L to make loans. Notice that Fnever defaults, and E is indifferent between default ing or invest ing in th e first period.When L ca n commit h imsel f to At in period t, d

~(p, At ) is no t uniquely determined

although it cannot be equal to on e (the largest feasible amount) unless L c an c om mi thimself to th e entire future sequence of loans. On the o th er h an d, if d

~(p, At (p =

P / (I - p) when p~

t the L's prior on the probabi li ty of default is ! and he is ind iffe renta bo ut a mo un ts t ha t he lends at t

~in particular, he is willing to lend At(p) an d he does

not need to be able to commit himself to do so. As a final remark about th e equilibrium,


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570 REVIEW O F E CO NO MIC STUDIES

note that L wo uld p refer to have T o pp ort un ities to lend to a given B, rather than tomake T in di vi du al l oan s to different Borro wers ( each with p ro bab il it y PT of beingfriendly). This is consi st en t with Theorem 3 an d underlines a basic idea of the analysis:Long-term arrangements are of value when there is uncer ta inty about preferences becausepast t ransactions provide relevant informat ion to agents.

The model of this section is much to o simple to give useful insights into cred it

market s. More general p re fe rences can be a llowed without des troy ing the flavour of th eresul ts. Yet it seems more impor tant to extend the analysi s to a marke t setting , and allowt he L en de r to select a r ep ay me nt sc hed ul e ( th at is, an in terest rate).

7. CONCLUSION

If an agent is uncer ta in about the motives of someone upon whom he mus t depend, e ithe rto p ro vi de i nf or mat io n or ma ke d ecision s, then the extent to which he trusts the otherwill be based on the p ar tn er 's ear li er actions. T hu s t he re is an in cen tive for an e nem y tobehave like a friend in order to increase his future oppor tuni ti es , and for partnerships

to last until someone cashes in. I have tried to make this point with two simple models.Techn ical extensions, such as more general uncer ta in ty, cou ld ad d further insights.Final ly, it should be emphasized that in my models credibi li ty can only be established

through act ions that inf luence pas t payoffs. The abi lity to make commi tments or to tak eactions that inf luence future payoffs would al so help to determine an agent 's rel iabil ity.

APPENDIX

Proof o f Corollary 1. I f F is hones t an d

ifP I ~ !

if PI >!

an d qE ( 1 - M I M ) = 1 - qE ( M I M ) for M = -1 an d 1, then S b est r es po nd s to the ac ti onrules given in t he statemen t of the corol lary an d r (n) = max (Ph !), so that R's actionrules best re sp on d to these signals. Fu rt her, if F is honest, then y( -1 ) < y(1) providedPI >!, so that th e no-transmission equilibrium cannot be an h on es t eq ui li bri um if PI >!.The co ro lla ry now follows from T he or em 1. II

Proof o f Theorem 1. If S uses the strategies qi (nIM) = ! for i = E an d F, n = - 1an d 1, and M = -1 an d 1, then r( n) = ! for n = -1 , so tha t Condi tion E2 is satisfied wheny( -1 ) = y ( l ) = yR(!). In addition, if y( -1 ) = y ( l ) = yR(!), t he n these signalling rulessatisfy E1. If y( -1 ) < y ( l ) in eq ui lib riu m, t he n by E l the equil ibrium signall ing rulesmu st be

F (n IM) = { 1 if n = Mq 0 if n -:j:: M

and q E ( n I M ) = I - qF(nIM) . Therefore, r (1)=PI an d r ( - 1 ) = I - P h so that y ( - I ) =y R ( l - PI), an d y(1) = yR(PI). Thus, an equil ibrium with y( -1 ) < y(1) exists if and onlyif y R ( l - PI ) < yR(PI), which occurs exact ly when P I > ! . The theorem follows because ify ( -1 ) = y( l ) , then y( - 1 ) = y ( l ) = yR(!). II


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Proof o f Theorem 2. Fix p and A and suppose that the theorem holds for alls = t - 1 , . . . ,0. Set ting

V ~ ( p ,A) == for i = E, F, an d R, it suffices to prove that the theorem

holds fo r t. I f q is th e probability that E is h on es t, then Yj, t he a ct io n induced whenn = j, r, th e probability that M = n, v, th e probability that S = F given M = n, an d z( 1),the absolute value of th e difference between M an d th e action induced by an honest (if1 = h) or dishonest (if 1 = d) signal are simple functions of q. Specifically, it must be that

an d

r = p + (1 - p ) q,

v = p j [p + (1 - p) q ],

Y-l = y R ( l - r, A) an d YI = yR(r, A ),

z ( d ) = l + Y I an d z ( h ) = l - Y I '

(Al)

(A2)

(A3)

(A4)

Final ly, let Wi(l , q) denote th e v alu e to i of an honest (i f 1 = h) or d ishones t (i f 1 = d)signal when q is t he probabi li ty tha t E tells th e truth. That is, for i = E and F let

Wi( h, q) = Ui(z (h ) , A ) + Vr-I(V)an d

Wi(d, q) = Ui(z (d) , A ) + Vr-I(O) = Ui(z (d) , A).

The express ion for Wi(d, q) follows because the probabi li ty S = F is zero a fte r a lie an dbecause Vr - I (0) = by the induction hypothesis. W i ( . ) depends on q since q determinesz( . ) and v through (Al)-(A4).

The probabil i ty q is an e qu il ib ri um v al ue if an d o nl y if

(A5)

if q < 1, then(A6)

and if q > 0, then

(A7)

(A5) guarantees that honesty is a best response for F, an d (A6) an d (A7) guarantee thatq is a best response for E.

I no w prove that there exists a unique q E [0, 1] that sat is fies (A5), (A6), an d (A7).First, note that if (A6) and (A7) are sat is fied then z(h) :2 zed) and therefore, since U F ( )is decreasing in z fo r z > 0, (A5) holds whenever both (A6) an d (A7) hold. To see this,

assume z(h zed). Then WE(z(h) , q WE(z(d ) , q). This fol lows from the definitionof WE ( . ) because U E ( ) is i nc re as in g in z for z

~0. Thus, (A6) impli es that q = 1.

However, if q = 1, then (Al)-(A4) imply that

0= z( h) < z( d) = 2, (A8)

which con trad ict s z( h)~

z( d) . There fo re , it suffices to show that there exi st s a unique qthat sat isfi es (A6) and (A7). However, this follows easily since WE(z(d) , q )WE (z( h), q) is s tr ic tly dec reas ing in q.

Next, I verify that V:() has the desired properties. If p~

2- r, then when q =( l -2p ) j [2 ( l -p ) ] , (Al)-(A4) imply that r= l , v:22- t + I , an d z ( d ) = z(h)= 1. Thus, s inceV:_I(v)=O for v:22- r + 1 by the induction hypothesis and U i ( l , A ) = O by (A3),


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572 REVIEW O F E CO NO MIC STUDIES

WE(z(d), q) = WF(z(h), q) = o. Therefore, if p~

2-', thenV ~ ( p )

= o. SinceV ~ _ I ( P )

> 0for p > 2- '+ \ WE (z(d), q) - WE (z(h), q) > 0 when q = ( 1 - 2p ) / [2(1- P )]. It follows thatif p > 2- '+ \ q > ( 1 - 2p ) / [2(1- p)] an d therefore r > ! so that

V ~ ( p )> 0 for i = E, F, an d

R. To show thatV ~ ( p )

is s tr ic tly increasing when p > 2-', I first s ho w that equilibriumvalues of r and v increase in p. To see thi s, observe that if E is indifferent between beinghonest and being d is hon es t for a given value of p, an increase in p, keeping z(d) an d

z( h) constant, causes E to tell th e t ruth with probabi li ty one, because V ~ _ I( . ) is increasingby t he induction hypothesis; to prevent this, z( d) must increase (and z( h) must decrease),which makes lying more attractive. H en ce , to satisfy (A6) an d (A7) v increases. That

V ~ ( p )is s tri ct ly inc reas ing in p for i = E an d F an d p > 2-' now follows d irec tly f rom

th e definition ofV ~ (

. ) . Finally,

V ~ ( p ,A) =

r V ~ _ I ( v )+ rUR(z,(h), A) + ( 1 - r) UR(z(h), A)

so thatV ~ (

. ) is increas ing in r and v. A similar argument shows that if t increases, thenso do r and v, and therefore

V ~ ( p )increases in t fo r i = E, F, an d R. ,

It r emains to estab li sh the propert ie s of th e function H,(p). Let H,(p) be def ined tosatisfy

u" (2, A) +V ~ _ I ( P )

- U E (0, A) = 0 (A8)

or H,(p) = A if th e right-hand side of (A8) is no t e qua l to zero for all A. By assumption,U E (2, A) - -U E (0, A) is strictly d ec re as in g in A, so H,(p) is well defined. Also, th emonotonicity of U E (2, A) - U E (0, A) in A, (A8), an d the induction hypothesis guaranteethat H,(p) is increasing in t an d p. It fol lows from (A6) that q = 1 whenever A < H,(p). II

Proof o f Corollary 2. The proof is by induc tion . If the hypotheses of the corol laryare true for s = t - 1 , . . . ,0 , then the unique equ il ib rium is character ized in Theorem 2.Assume that th e corollary is tr ue for s < t and, in order to reach a con trad ic tion , assume

that for s om e A, F induces two act ions ,WI

an d w2

,

when M = 1 in period t. It followst ha t, if Wi leads to reputat ion Vi,

U R ( w l - l , A)+ V;-t(v l ) = U R( w 2 _ 1, A)+ V;_t(v 2).

If WI> w 2 , then U R ( WI - 1 , A) > U R ( w 2 - 1 , A), an d so V;_I (VI) < V;_I( v 2 ) . The induct ion hypothesis then implies that VI < v 2 an d

V ~ _ I ( V I )

Documents

Review of Economic Studies-1985-Sobel-557-73