The Effect of Democratic Decision M Title Investment in Reputation. Author(s) BEN-YASHAR, Ruth; KRAUSZ, Miriam; N.

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1 The Effect of Democratic Decisio M Title Ivestmet i Reputatio Author(s) BEN-YASHAR, Ruth; KRAUSZ, Miriam; N Citatio Issue 07- Date Type Techical Report Text Versio publisher URL Right Hitotsubashi Uiversity Repository

HIAS-E-59 The Effect of Democratic Decisio Makig o Ivestmet i Reputatio Ruth Be-Yashar Departmet of Ecoomics, BarIla Uiversity, Ramat Ga 5900, Israel Miriam Krausz School of Ecoomics, Ashkelo

2 HIAS-E-59 The Effect of Democratic Decisio Makig o Ivestmet i Reputatio Ruth Be-Yashar Departmet of Ecoomics, BarIla Uiversity, Ramat Ga 5900, Israel Miriam Krausz School of Ecoomics, Ashkelo Academic College, Yitzhak Be-Tzvi, Ashkelo, Israel Shmuel Nitza Departmet of Ecoomics, BarIla Uiversity, Ramat Ga 5900, Iseael Hitotsubashi Istitute for Advaced Study, Hitotsubashi Uiversity November 07 Hitotsubashi Istitute for Advaced Study, Hitotsubashi Uiversity -, Naka, Kuitachi, Tokyo , Japa tel: HIAS discussio papers ca be dowloaded without charge from: All rights reserved.

3 The effect of democratic decisio makig o ivestmet i reputatio by b Ruth Be-Yashar a, Miriam Krausz ad Shmuel Nitza c.d Studets wish to icrease the probability of beig admitted to a prestigious school. Job cadidates are iterested i the probability of gettig a desirable ob. Defedats are cocered about the probability of beig acquitted. I all such biary settigs, the probability of the desirable outcome to idividuals ca be affected by their reputatio. Applyig a stadard ucertai dichotomous choice bechmark settig, we focus o how the ature of the applied decisio-makig rule affects the idividuals' icetive to ivest i improvemet of their reputatio. Our mai results establish that a democratic (o - democratic) decisio-makig system based o the simple maority (uaimity) rule esures maximal (miimal) margial productivity of reputatio that icreases (decreases) with the size of the decisio-makig committee. Keywords: decisio makig structure, ivestmet i reputatio. Classificatio Codes: D7, a Departmet of Ecoomics, Bar Ila Uiversity, Ramat Ga 5900, Israel. b School of Ecoomics, Ashkelo Academic College, Yitzhak Be-Tzvi, Ashkelo, Israel. c Departmet of Ecoomics, Bar Ila Uiversity, Ramat Ga 5900, Israe ad Hitotsubashi Istitute for Advaced Study, Hitotsubashi Uiversity. d The third author is grateful to HIAS for eablig the completio of this paper. 0

4 . Itroductio Group biary decisio makig is very sigificat i Ecoomics, Medicie, Law ad other disciplies. Examples iclude uries decidig whether to acquit or covict a defedat, committees cosiderig ob cadidates, proects or loas, boards of maagers choosig betwee two policies ad medical experts decidig whether to carry out a certai mode of treatmet or procedure. I all of these settigs the cocered idividual may have a icetive to ivest resources i icreasig the probability of receivig a favorable outcome from the decisio process. The defedat wishes to be acquitted, the ob cadidate would like to be selected, the etrepreeur would like to be approved for a loa, ad so forth. These idividuals are aware that certai evirometal attributes ehace their reputatio that positively affects the outcome of the decisio process. For example, the eighborhood i which a idividual grew up or lives i may be associated with high icome, low crime ad impressive scholastic achievemets. The school that a idividual atteded may be kow to produce successful graduates. I this paper we explore the role played by the type of decisio-makig system faced by the idividual i determiig his icetive to ivest i reputatio, focusig o two types of rules: the democratic simple maority rule ad the o-democratic uaimity rule. For ease of expositio, we have chose to focus o the case of a defedat whose verdict is determied by a ury facig two states of ature (the defedat beig guilty or iocet) that have some give a-priory probability ad two possible decisios (covictio or acquittal). There are therefore four possible fial decisios, two correct decisios (covictio of guilty ad acquittal of iocet defedats) ad two icorrect oes (acquittal of guilty ad covictio of iocet defedats). The ury (committee) cosists of several members (decisio-makers) with the same prefereces who seek to maximize the probability to reach the correct decisio. The fial decisio is reached by the ury decisio rule. Most of the studies i this framework focus o the probability that the committee will choose the correct decisio. I this paper we focus o the defedat's poit of view vis. the probability These studies follow Codorcet s (785) approach. The Codorcet Jury Theorem defies the cotributios uder which the maority rule is a efficiet aggregatio of all the iformatio from all the decisio-makers. Various attempts have bee made to geeralize the theorem, for example, Nitza ad Paroush (98), Grofma et al. (983), Shapley ad Grofma (984), Be-Yashar ad Nitza

5 of his acquittal. The probability of acquittal ca be affected by the relevat evidece which is a maor criterio for makig a decisio, as well as the umber ad skills of the committee members, the applied decisio rule ad the defedat s reputatio. As oted, reputatio cosists of evirometal factors such as the defedat's backgroud, socio-ecoomic status, residece eighborhood, educatio etc. ad is geeral iformatio. I a fair trial we would prefer the ury ot to be biased by such iformatio. We would like the ury to make a decisio based o iformatio pertaiig purely to the crime for which the defedat is o trial. Noetheless, defedats wish to icrease the probability of beig acquitted. They caot chage the evidece ad they caot cotrol the size ad the quality of the committee. However, they ca ivest i their reputatio, which cosists of the evirometal factors, i order to affect the a-priory probability of iocece. The ovelty of our model is that we aalyze the effect of the defedat's reputatio o the outcome of the decisio-makig process We focus o reputatio, assumig that it is represeted i the decisio-makig model by the a-priory probability of iocece. The questio we ask is how the ature of the decisio-makig rule used by the court affects the icetive to ivest i reputatio. Our results clarify how the decisio-makig rule applied by the ury ad the size of the decisio-makig group affects the beefit of reputatio the margial icrease i the probability of receivig a acquittal. 3 More specifically, assumig that the a-priory probability of iocece is equal to oe half, we show that the probability of a decisio that is favorable for the defedat is equal to oe half uder the simple maority rule, regardless of the size ad the quality of the decisio-makig committee. From the defedat's poit of view, (997), Bered ad Paroush (998), Be- Yashar ad Daziger (0) ad Dietrich ad List (03). Discussio of extreme decisio rules ca be foud i Sah ad Stiglitz (986) ad Be-Yashar ad Nitza (00). Some papers discuss the sigificace of the prior o a correct decisio i committees, see for example Be-Yashar ad Nitza (04). 3 I the literature o strategic deliberatio, the optimal amout of iformatio acquisitio is discussed i Persico (004). Gerardi ad Yariv (008), show that the icetive to acquire iformatio depeds o the committee's decisio rule. Also discussed is the effect of payoffs to decisio makers' o the committee's behavior (e.g., Levy, 007 ad Midord, et al. 07).

6 this is tatamout to tossig a coi. Uder the uaimity rule, we show that the defedat is i a worse positio. I this case, eve though the committee has o bias i favor or agaist the defedat, still, his probability of acquittal is lower tha oe half. We explore the possibility of the defedat affectig his a-priory probability of iocece, i.e., ivestig i ehacig his reputatio with the purpose of icreasig the probability of a favorable decisio. Sice this process is costly for the idividual, the idividual's icetive to ivest i reputatio crucially depeds o the margial productivity of his ivestmet. Margial productivity is defied as the margial icrease i the probability of receivig a desired outcome from the ury. Our results establish that a decisio-makig system based o the democratic simple maority rule esures maximal margial productivity of reputatio. It therefore yields the largest icetive to ivest i reputatio 4. Moreover, the margial productivity of ivestig i reputatio icreases with the size of the committee. Coversely, the decisio-makig system based o the o-democratic uaimity rule results i miimal productivity of reputatio which decreases with the size of the committee.. The model A =k+member ury decides whether to acquit () or covict (-) a defedat. The defedat is either iocet () or guilty (-). The fial decisio is based o the idividual urors decisios. There are therefore two possible correct decisios (/) (acquittig a iocet defedat), ad (-/-) (covictig a guilty defedat) ad two correspodig icorrect oes ((-/) ad (/-)). Let be the a-priori probability that a defedat is iocet. If =½, the probabilities that defedats are guilty or iocet are equal. Let xi deote the uror i s decisio, xi= ad xi=- the represet acquittal ad covictio, respectively. The vector x=(x,,x) describes the decisio profile of urors makig decisios simultaeously. Each uror chooses the correct alterative with probability 0.5<p<, if the defedat is iocet (), ad - if the defedat is guilty (-). Note that (-p) is the probability that the uror chooses the icorrect alterative: - if the defedat is iocet () ad if the defedat is guilty (-). The fial decisio regardig the defedat is made usig a decisio rule f, 4 Reputatio has a importat role i decisios cocerig idividuals i a variety of settigs. For example, reputatio icetives ad the retur to reputatio i a career model i Bar-Isaac ad Deb (04). The value of reputatio iformatio is discussed i Bolto, Ockefels ad Ebelig (0). Semial reputatio models iclude Kreps ad Wilso (98) Bayesia updatig model i which the "reputatio effect" is demostrated. 3

7 whereby the fuctio f assigs or - (acquittal or covictio) to a decisio profile x i,. That is, f :,,. Supposig a qualified maority rule is used, f N() q,where, N() is the umber of urors who decide to acquit otherwise the defedat, N(-) is the umber of urors who decide to covict the defedat, is the total umber of urors ad a iteger q is the quota required of urors decidig to acquit that will lead to a verdict of acquittal. Hereafter q represets the decisio rule 5. Note that q=k+ represets the simple maority rule ad q= represets the uaimity rule, whereby oly a uaimous verdict will lead to acquittal. 3. Results I this sectio we examie the probability of a ury acquittig the defedat, ad the effect of chages i the a-priory probability of iocece o the probability of acquittal. Let us deote the probabilities that a -member ury acquits a iocet or guilty defedat by T :, ad T : acquit a defedat is deoted by : ) Formally, T q, respectively. Hece, the probability to Pr( q where Pr( : q) T: T : : p p, ad T : p Pr : q T : T: q where p p p Sice (a), 0. 6 p. (b) a,. 7 a a. p. Hece, q 5 Usig qualified maority rules is a plausible assumptio i a model that assumes decisio makers with homogeous skills (see Be Yashar ad Nitza, 997). 6 p p 0 p ). p p p. (Note that uder the model's assumptios, p>/ ad hece 4

8 If q, the by (a), Pr : q 0. If q, the by (b), we kow that, q Pr : q q 0.. Sice -q+>/, this last term is positive by (a) ad therefore, Hece, the probability to acquit a defedat icreases with the a-priory probability that he is iocet. We ow focus o two types of decisio rules. The democratic simple maority rule ad the o-democratic uaimity rule. The followig theorems show that the decisio rule used to aggregate the decisios of the committee members is of crucial importace i determiig the magitude of the margial effect of the a-priory probability of iocece o the probability to acquit. Theorem : Uder the simple maority rule: (a) Give equal a-priory probabilities that a defedat is guilty or iocet, there is o bias i favor of acquittal or covictio, i,e,. Pr( : regardless of the size of the committee ad the skills of its members. k ), (b) A icrease i the a-priory probability of iocece produces the largest icrease i the probability to acquit, compared to all other qualified decisio rules. (c) The icrease i the probability to acquit resultig from a icrease i the a- priory probability of iocece, icreases with the size of the committee. Proof: (a) Accordig to the simple maority rule, q=k+ f N() k otherwise N( ) k otherwise Hece T : = T : ad T: T: otherwise N( ) k, which implies that 7 a a a a a p p p p a ad p a a p p a a 5 a p a. Hece, a a.

9 Pr( : k ) T : T: (b) See Appedix A. Pr: k (c) T(:)-. Based o the Codorcet Jury Theorem T : 0, hece, Pr : k 0.. Q.E.D Accordig to Theorem, whe facig a committee that uses the simple maority rule, the defedat ca expect that give equal a-priory probabilities that he is guilty or iocet, the probability of acquittal uder the simple maority rule is ½, regardless of whether the defedat is i fact iocet. Moreover, uder these coditios the probability to acquit a defedat is idepedet of the umber of ury members ad their skills. This ca be cosidered a desirable state from the ury's poit of view, because it implies that there is o bias i favor or agaist the defedat- the decisio is made solely o the basis of the merit of the relevat criteria. However, as show, uder the simple maority rule the defedat ca expect to be acquitted or covicted with equal probability, the equivalet of tossig a coi. Should the defedat decide to icrease the a-priory probability of iocece by ivestig i reputatio, he ca expect the largest margial icrease i the probability of beig acquitted. That is, the simple maority rule iduces the maximal icrease i the probability to acquit a defedat i respose to a margial chage i the a-priory probability that the defedat is iocet. It therefore iduces the strogest icetive to ivest i reputatio. Fially, the margial productivity of reputatio icreases with the size of the committee. We ow tur to case of the o-democratic uaimity rule to ivestigate the defedat's icetive to ivest i reputatio whe the ury applies this rule. Theorem : Uder the uaimity rule: 6

10 (a) Give equal a-priory probabilities that a defedat is guilty or iocet, there is a implicit bias agaist acquittal, i.e., size of the committee. Pr( : ), which depeds o the (b) A icrease i the a-priory probability of iocece produces the smallest icrease i the probability to acquit, compared to all qualified decisio rules. (c) The icrease i the probability to acquit resultig from a icrease i the a- priory probability of iocece decreases with the size of the committee. Proof: (a) f N() q otherwise If q>k+ ( -q<k+ ) the T : T: Pr( : N( ) q otherwise 7.Thus, otherwise N( ) q ) T : T: T : T:, which is less tha ½. (b)see Appedix A. (c) Uder the uaimity rule, q=, Pr: T(:)- T(:-) = p p. Hece Pr : p p p p p p p p, p p p Sice p p p Pr : 0. Q.E.D Theorem states that give equal a-priory probabilities that a defedat is guilty or iocet, whe the decisio rule is the o-democratic uaimity rule, the probability of acquittal is less tha ½, ad is depedet o the umber of urors ad o the probability each uror chooses the correct alterative. From the defedat's poit of view, the probability of beig acquitted is extremely low. Should the defedat wish to ivest i reputatio, thereby icreasig the a-priory probability of

11 iocece, he ca expect the smallest icrease i the probability of beig acquitted uder the uaimity rule i compariso to all other qualified decisio rules. The margial icrease i the probability to be acquitted i respose to a icrease i the a- priory probability of iocece, decreases with the size of the ury. Hece, defedats facig a ury that bases its decisios o the uaimity rule have the lowest icetive to ivest i reputatio whe the ury is large. 4. Coclusios Our ote examies the impact of the democratic ature of the decisio-makig rule o the icetive to ivest i reputatio. Such ivestmet icreases the a-priory probability of iocece ad, i tur, the probability of the acquittal decisio. Sice ivestmet i reputatio is costly, the effectiveess of such a ivestmet is ecoomically sigificat. Our results imply that the democratic ature of the decisiomakig rule determies both the icetive to ivest i reputatio as well as the effect of the size of the committee o this icetive. The largest icetive to ivest i reputatio exists i a large ury that applies the simple maority rule. O the other had, the defedat will have to ivest the largest amout of resources i reputatio i order to achieve a target icrease i the probability of gaiig a favorable decisio whe facig a large ury that applies the extreme o-democratic decisio rule. To our kowledge, our model is the first to iclude i a collective-decisio model the issue of the icetive of the aget affected by the collective decisio to ivest i reputatio uder alterative decisio-makig systems. Our results are applicable to all settigs of biary decisio makig where the cocered idividuals ca ifluece the collective decisio by ivestig i their reputatio. These results give defedats ad cadidates a strategic decisio tool for plaig the ivestmet of persoal resources i reputatio. I our settig, it is the combiatio of the type of the applied collective decisio rule, democratic or o-democratic, with the size of the decisio-makig body, large or small 8, that determies the margial productivity of ivestmet i reputatio, viz., the icetive to ivest i reputatio. 8 A cadidate may be applyig for a ob at a large corporatio or coversely at a small start-up compay. Academic program cadidates may be facig a large uiversity or a small college. Etrepreeurs ca apply for a loa either at a large bak or at a small bak. Assumig that the size of the decisio makig committee is correlated (positively or egatively) with the size of the orgaizatio, 8

12 Refereces Bar-Isaac H., ad Deb. J., 04. "What is a good reputatio? Career cocers with heterogeeous audieces", Iteratioal Joural of Idustrial Orgaizatio 34, Be-Yashar, R. ad Daziger. L., 0. O the optimal allocatio of committee members, Joural of Mathematical Ecoomics, 47, Be-Yashsr, R., Krausz M. ad Nitza S., 07. "Govermet loa guaratees ad the credit decisio-makig structure", Caadia Joural of Ecoomics, forthcomig. Be-Yashar, R. ad Nitza, S. 04. "O the sigificace of the prior of a correct decisio i committees". Theory ad Decisio, 76, Be-Yashar, R., ad Nitza, S., 00. The Robustess of Optimal Orgaizatioal Architectures: A Note o Hierarchies ad Polyarchies, Social Choice ad Welfare 8, Be-Yashar, R. ad Nitza, S., 997. The optimal decisio rule for fixed-size committees i dichotomous choice situatios: The geeral result, Iteratioal Ecoomic Review 38, Bered, D., & Paroush, J "Whe is Codorcet s Jury Theorem valid?" Social Choice ad Welfare, 5, Bolto G. E.,A. Ockefels, F. Ebelig., 0 "Iformatio value ad exteralities i reputatio buildig", Iteratioal Joural of Idustrial Orgaizatio 9, 3 3. Codorcet, N.C. de "Essai sur l Applicatio de l Aalyse à la Probabilité des Décisios Redues à la Pluralité des Voix", Paris. See I. McLea ad F. Hewitt, traslators, 994. Dietrich, F. ad List, C. 03. Propositio-wise udgmet aggregatio: The geeral case, Social Choice ad Welfare 40, Gerardi, D., Yariv L., 008. "Iformatio acquisitio i committees" Games ad Ecoomic Behavior, 6, Grofma, B., Owe, G., Feld, S.L., 983. "Thirtee theorems i search of the truth". Theory ad Decisio 5, which may ot always be the case, yields clear cut implicatios regardig the depedece of the icetive to ivest i reputatio o the size of the orgaizatio. 9

13 Kreps D. M., ad Wilso R., 98. "Reputatio ad Imperfect Iformatio" Joural of Ecoomic Theory Levy, G., 007. " Decisio Makig i Committees: Trasparecy, Reputatio, ad Votig Rules", The America Ecoomic Review Vol. 97 No., Midord R., Barraquerb T.R., Valasekc J., 07. "Votig i large committees with disesteem payoffs: A state of the art model", Games ad Ecoomic Behavior Nitza, S. ad Paroush, J., 98. Optimal decisio rules i ucertai dichotomous choice situatio, Iteratioal Ecoomic Review 3, Persico, N., 004, "Committee Desig with Edogeous Iformatio", Review of Ecoomic Studies, 7, Sah, R.K. ad J. Stiglitz,986. "The architecture of ecoomic systems: Hierarchies ad polyarchies", America Ecoomic Review, 76, Shapley, L. ad Grofma, B., 984. Optimizig group udgmetal accuracy i the presece of iterdepedecies, Public Choice 43,

14 Appedix A We show that the effect of a chage i the a-priory probability of iocece o the probability to acquit a defedat is maximal at the simple maority rule ad is miimal at the uaimity rule. 9 : k Pr: k i Pr: k i Pr Pr : k i i Proof: Recall that Pr : q 0, where i is a positive iteger. T : T: where p p p ad (a), 0. By (a) (b) a,. k q p. a ki a, hece, : k Pr: k i Pr ad, where i is a positive iteger. 9 The proof is based o part of the proof of Theorem i Be-Yashar, Krausz ad Nitza (07). Their paper focuses o a bakig model i which the govermet uses a guaratee as a istrumet for credit iducemet ad for affectig the bak s decisio-makig system i.e., its degree of cetralizatio, bias towards approval of loas ad reliace o obective loa-specific iformatio.

15 By (b) above, if q the : q Pr: q q Pr. q Specifically, this is true whe q = k i k i. Hece, Pr Therefore, : k i Pr: k i Pr : k i i. Therefore, k i. Substitutig k yields,. Furthermore, k i decreases with i. 0. Q.E.D.

Monopoly vs. Competition in Light of Extraction Norms. Abstract

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