HIAS-E-59

The Eﬀect of Democratic Decision Making on
Investment in Reputation
Ruth Ben-Yashar
Department of Economics, BarIlan University, Ramat Gan 52900, Israel

Miriam Krausz
School of Economics, Ashkelon Academic College, 12 Yitzhak Ben-Tzvi, Ashkelon, Israel

Shmuel Nitzan
Department of Economics, BarIlan University, Ramat Gan 52900, Iseael
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

November 2017

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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The effect of democratic decision making on investment in reputation
by
Ruth Ben-Yashar a, Miriam Krausz b and Shmuel Nitzan c.dStudents wish to increase the probability of being admitted to a prestigious school.
Job candidates are interested in the probability of getting a desirable job. Defendants
are concerned about the probability of being acquitted. In all such binary settings, theprobability of the desirable outcome to individuals can be affected by their reputation.
Applying a standard uncertain dichotomous choice benchmark setting, we focus on
how the nature of the applied decision-making rule affects the individuals' incentive
to invest in improvement of their reputation. Our main results establish that ademocratic (non - democratic) decision-making system based on the simple majority(unanimity) rule ensures maximal (minimal) marginal productivity of reputation that
increases (decreases) with the size of the decision-making committee.
Keywords: decision making structure, investment in reputation.
Classification Codes: D7,a

Department of Economics, Bar Ilan University, Ramat Gan 52900, Israel.

b

School of Economics, Ashkelon Academic College, 12 Yitzhak Ben-Tzvi, Ashkelon,

Israel.
c

Department of Economics, Bar Ilan University, Ramat Gan 52900, Israe andHitotsubashi Institute for Advanced Study, Hitotsubashi University.
d

The third author is grateful to HIAS for enabling the completion of this paper.

01. Introduction
Group binary decision making is very significant in Economics, Medicine, Law andother disciplines. Examples include juries deciding whether to acquit or convict a
defendant, committees considering job candidates, projects or loans, boards ofmanagers choosing between two policies and medical experts deciding whether tocarry out a certain mode of treatment or procedure. In all of these settings theconcerned individual may have an incentive to invest resources in increasing theprobability of receiving a favorable outcome from the decision process. The defendant
wishes to be acquitted, the job candidate would like to be selected, the entrepreneur
would like to be approved for a loan, and so forth. These individuals are aware thatcertain environmental attributes enhance their reputation that positively affects theoutcome of the decision process. For example, the neighborhood in which anindividual grew up or lives in may be associated with high income, low crime andimpressive scholastic achievements. The school that an individual attended may beknown to produce successful graduates. In this paper we explore the role played by
the type of decision-making system faced by the individual in determining his
incentive to invest in reputation, focusing on two types of rules: the democratic simple
majority rule and the non-democratic unanimity rule.
For ease of exposition, we have chosen to focus on the case of a defendant
whose verdict is determined by a jury facing two states of nature (the defendant being
guilty or innocent) that have some given a-priory probability and two possible
decisions (conviction or acquittal). There are therefore four possible final decisions,two correct decisions (conviction of guilty and acquittal of innocent defendants) and
two incorrect ones (acquittal of guilty and conviction of innocent defendants). The
jury (committee) consists of several members (decision-makers) with the same
preferences who seek to maximize the probability to reach the correct decision. The
final decision is reached by the jury decision rule. Most of the studies in thisframework focus on the probability that the committee will choose the correct
decision.1 In this paper we focus on the defendant's point of view vis. the probability
1

These studies follow Condorcet’s (1785) approach. The Condorcet Jury Theorem defines thecontributions under which the majority rule is an efficient aggregation of all the information from all
the decision-makers. Various attempts have been made to generalize the theorem, for example, Nitzan
and Paroush (1982), Grofman et al. (1983), Shapley and Grofman (1984), Ben-Yashar and Nitzan1

of his acquittal.
The probability of acquittal can be affected by the relevant evidence which is amajor criterion for making a decision, as well as the number and skills of thecommittee members, the applied decision rule and the defendant’s reputation. As
noted, reputation consists of environmental factors such as the defendant's
background, socio-economic status, residence neighborhood, education etc. and isgeneral information. In a fair trial we would prefer the jury not to be biased by suchinformation. We would like the jury to make a decision based on information
pertaining purely to the crime for which the defendant is on trial. Nonetheless,defendants wish to increase the probability of being acquitted. They cannot change
the evidence and they cannot control the size and the quality of the committee.
However, they can invest in their reputation, which consists of the environmental
factors, in order to affect the a-priory probability of innocence.2 The novelty of ourmodel is that we analyze the effect of the defendant's reputation on the outcome of thedecision-making process
We focus on reputation, assuming that it is represented in the decision-making
model by the a-priory probability of innocence. The question we ask is how the natureof the decision-making rule used by the court affects the incentive to invest in
reputation. Our results clarify how the decision-making rule applied by the jury andthe size of the decision-making group affects the benefit of reputation – the marginalincrease in the probability of receiving an acquittal.3
More specifically, assuming that the a-priory probability of innocence is equal
to one half, we show that the probability of a decision that is favorable for the
defendant is equal to one half under the simple majority rule, regardless of the size
and the quality of the decision-making committee. From the defendant's point of view,(1997), Berend and Paroush (1998), Ben- Yashar and Danziger (2011) and Dietrich and List (2013).
Discussion of extreme decision rules can be found in Sah and Stiglitz (1986) and Ben-Yashar andNitzan (2001).
2

Some papers discuss the significance of the prior on a correct decision in committees, see for example
Ben-Yashar and Nitzan (2014).
3

In the literature on strategic deliberation, the optimal amount of information acquisition is discussed
in Persico (2004). Gerardi and Yariv (2008), show that the incentive to acquire information depends onthe committee's decision rule. Also discussed is the effect of payoffs to decision makers' on thecommittee's behavior (e.g., Levy, 2007 and Midjord, et al. 2017).

2

this is tantamount to tossing a coin. Under the unanimity rule, we show that thedefendant is in a worse position. In this case, even though the committee has no bias
in favor or against the defendant, still, his probability of acquittal is lower than one
half. We explore the possibility of the defendant affecting his a-priory probability of
innocence, i.e., investing in enhancing his reputation with the purpose of increasing
the probability of a favorable decision. Since this process is costly for the individual,
the individual's incentive to invest in reputation crucially depends on the marginal
productivity of his investment. Marginal productivity is defined as the marginalincrease in the probability of receiving a desired outcome from the jury. Our results
establish that a decision-making system based on the democratic simple majority ruleensures maximal marginal productivity of reputation. It therefore yields the largest
incentive to invest in reputation4. Moreover, the marginal productivity of investing in
reputation increases with the size of the committee. Conversely, the decision-making
system based on the non-democratic unanimity rule results in minimal productivity of
reputation which decreases with the size of the committee.
2. The model
An n=2k+1member jury decides whether to acquit (1) or convict (-1) a defendant. The
defendant is either innocent (1) or guilty (-1). The final decision is based on theindividual jurors’ decisions. There are therefore two possible correct decisions (1/1)(acquitting an innocent defendant), and (-1/-1) (convicting a guilty defendant) and twocorresponding incorrect ones ((-1/1) and (1/-1)). Let D be the a-priori probability thata defendant is innocent. If D=½, the probabilities that defendants are guilty or
innocent are equal. Let xi denote the juror i’s decision, xi=1 and xi=-1 then represent
acquittal and conviction, respectively. The vector x=(x1,…,xn) describes the decision
profile of n jurors making decisions simultaneously. Each juror chooses the correct
alternative with probability 0.5<p< 1, 1 if the defendant is innocent (1), and -1 if the
defendant is guilty (-1). Note that (1-p) is the probability that the juror chooses theincorrect alternative: -1 if the defendant is innocent (1) and 1 if the defendant is guilty(-1). The final decision regarding the defendant is made using a decision rule f,4

Reputation has an important role in decisions concerning individuals in a variety of settings. For
example, reputation incentives and the return to reputation in a career model in Bar-Isaac and Deb(2014). The value of reputation information is discussed in Bolton, Ockenfels and Ebeling (2011).
Seminal reputation models include Kreps and Wilson (1982) Bayesian updating model in which the"reputation effect" is demonstrated.

3

whereby the function f assigns 1 or -1 (acquittal or conviction) to a decision profile xin ^ 1,1` . That is, f : ^ 1,1` o ^ 1,1`. Supposing a qualified majority rule is used,nn

N (1) t q ½¾ ,where, N(1) is the number of jurors who decide to acquit
otherwise¿

­1
®
¯ 1

f

the defendant , N(-1) is the number of jurors who decide to convict the defendant , n is
the total number of jurors and an integer q is the quota required of jurors deciding toacquit that will lead to a verdict of acquittal. Hereafter q represents the decision rule5.
Note that q=k+1 represents the simple majority rule and q=n represents the unanimity
rule, whereby only a unanimous verdict will lead to acquittal.
3. Results
In this section we examine the probability of a jury acquitting the defendant, and theeffect of changes in the a-priory probability of innocence on the probability ofacquittal. Let us denote the probabilities that an n-member jury acquits an innocent or
guilty defendant by T 1 : 1 , and T 1 : 1 , respectively. Hence, the probability toacquit a defendant is denoted by Pr(1 : q) where Pr(1 : q) DT 1 : 1  1  D T 1 : 1 .
Formally,
n

§n·

j q

© ¹

¦ ¨¨ j ¸¸ p 1  p

T 1:1

j

n j

n

§n·

j q

© ¹

¦ ¨¨ j ¸¸ 1  p p, and T 1 : 1

j

n j

. Hence,nw Pr 1 : q
wD

T 1 : 1  T 1 : 1

where ' j§n· j¨¨ ¸¸ p 1  p n  j  1  p j p n  j .
© j¹

¦'

j

j q

^

`

Since
(a) j !

n, ' j ! 0 .6
2(b) ja, ' a' na .7

5

Using qualified majority rules is a plausible assumption in a model that assumes decision makers
with homogenous skills (see Ben Yashar and Nitzan, 1997).
6p' j !0 
1 p

p j
p n j!
 j !n  j  j ! n . (Note that under the model's assumptions, p>1/2 and hence
2
1 p
1 p

!1 ).

4

If q !

nnw Pr 1 : q, then by (a),! 0 . If q  , then by (b) , we know that,
2
2
wD

n

n

¦' j

¦ ' . Since n-q+1>n/2, this last term is positive by (a) and therefore,j n  q 1j q

jw Pr 1 : q! 0.
wD

Hence, the probability to acquit a defendant increases with the a-priory
probability that he is innocent.
We now focus on two types of decision rules. The democratic simple majority
rule and the non-democratic unanimity rule. The following theorems show that thedecision rule used to aggregate the decisions of the committee members is of crucial
importance in determining the magnitude of the marginal effect of the a-priory
probability of innocence on the probability to acquit.
Theorem 1:
Under the simple majority rule:(a) Given equal a-priory probabilities that a defendant is guilty or innocent, there
is no bias in favor of acquittal or conviction, i,e,. Pr(1 : k  1)

1,
2

regardless of the size of the committee and the skills of its members.
(b) An increase in the a-priory probability of innocence produces the largest
increase in the probability to acquit, compared to all other qualified decisionrules.
(c) The increase in the probability to acquit resulting from an increase in the apriory probability of innocence, increases with the size of the committee.
Proof:(a) According to the simple majority rule, q=k+1

­1
®
¯ 1

f

N (1) t k  1½¾
otherwise ¿

­1
®
¯ 1

Hence T 1 : 1 = T  1 : 1

7 '

a

>

n
na
a na
p a 1 p 1 p
p
a

@ and

and

' na

N (1) d k ½¾otherwise¿

­1
®
¯ 1

otherwise ½¾
N (1) t k  1 ¿

T 1 : 1

1  T  1 : 1 , which implies that>@ . Hence,

n
a
na a
p n  a 1 p  1 p
p
na

5

'a

' n a .

Pr(1 : k  1)

1
T 1 : 1  1  T  1 : 1212.

(b) See Appendix A.
(c)w Pr 1 : k  1wD

2T(1:1)-1. Based on the Condorcet Jury Theorem

wT 1 : 1! 0 , hence,wn

w Pr 1 : k  1
wD
! 0.
wn

Q.E.D
According to Theorem 1, when facing a committee that uses the simple
majority rule, the defendant can expect that given equal a-priory probabilities that heis guilty or innocent, the probability of acquittal under the simple majority rule is ½,
regardless of whether the defendant is in fact innocent. Moreover, under these
conditions the probability to acquit a defendant is independent of the number of jury
members and their skills. This can be considered a desirable state from the jury's pointof view, because it implies that there is no bias in favor or against the defendant- thedecision is made solely on the basis of the merit of the relevant criteria. However, as
shown, under the simple majority rule the defendant can expect to be acquitted orconvicted with equal probability, the equivalent of tossing a coin. Should thedefendant decide to increase the a-priory probability of innocence by investing in
reputation, he can expect the largest marginal increase in the probability of being
acquitted. That is, the simple majority rule induces the maximal increase in theprobability to acquit a defendant in response to a marginal change in the a-priory
probability that the defendant is innocent. It therefore induces the strongest incentive
to invest in reputation. Finally, the marginal productivity of reputation increases withthe size of the committee.
We now turn to case of the non-democratic unanimity rule to investigate thedefendant's incentive to invest in reputation when the jury applies this rule.
Theorem 2:
Under the unanimity rule:6

(a) Given equal a-priory probabilities that a defendant is guilty or innocent, there
is an implicit bias against acquittal, i.e., Pr(1 : n) 1, which depends on the2

size of the committee.
(b) An increase in the a-priory probability of innocence produces the smallest
increase in the probability to acquit, compared to all qualified decision rules.
(c) The increase in the probability to acquit resulting from an increase in the apriory probability of innocence decreases with the size of the committee.
Proof:
(a)

­1
®
¯ 1

f

N (1) t q ½¾
otherwise¿n  N (1) t q ½¾
otherwise ¿

­1
®
¯ 1

­1
®
¯ 1

otherwise ½¾
N (1) ! n  q ¿If q>k+1 ( n-q<k+1 ) then T 1 : 1  T  1 : 1 .Thus,

Pr(1 : n)

1
T 1 : 1  T 1 : 1
2

1
T 1 : 1  1  T  1 : 1 , which is less than ½.
2(b)See Appendix A.
(c) Under the unanimity rule, q=n,w Pr 1 : nwDn

T(1:1)- T(1:-1) = p n  1  p .

w Pr 1 : n
wD
Hence
wn

p n1  1  p

n 1

n

 pn  1 p .
nSince p p  1  1  p
n

n

§ p ·
§ p ·
p
¸¸ !
¸¸
 1  1  p { ¨¨{ ¨¨©1 p ¹ 1 p ©1 p ¹

n 1

!1,

w Pr 1 : n
wD
0
wn

Q.E.D
Theorem 2 states that given equal a-priory probabilities that a defendant is
guilty or innocent, when the decision rule is the non-democratic unanimity rule, theprobability of acquittal is less than ½, and is dependent on the number of jurors andon the probability each juror chooses the correct alternative. From the defendant's
point of view, the probability of being acquitted is extremely low. Should thedefendant wish to invest in reputation, thereby increasing the a-priory probability of7

innocence, he can expect the smallest increase in the probability of being acquitted
under the unanimity rule in comparison to all other qualified decision rules. The
marginal increase in the probability to be acquitted in response to an increase in the apriory probability of innocence, decreases with the size of the jury. Hence, defendantsfacing a jury that bases its decisions on the unanimity rule have the lowest incentive
to invest in reputation when the jury is large.
4. Conclusions
Our note examines the impact of the democratic nature of the decision-making rule on
the incentive to invest in reputation. Such investment increases the a-priory
probability of innocence and, in turn, the probability of the acquittal decision. Since
investment in reputation is costly, the effectiveness of such an investment is
economically significant. Our results imply that the democratic nature of the decisionmaking rule determines both the incentive to invest in reputation as well as the effectof the size of the committee on this incentive. The largest incentive to invest in
reputation exists in a large jury that applies the simple majority rule. On the other
hand, the defendant will have to invest the largest amount of resources in reputation inorder to achieve a target increase in the probability of gaining a favorable decision
when facing a large jury that applies the extreme non-democratic decision rule.
To our knowledge, our model is the first to include in a collective-decisionmodel the issue of the incentive of the agent affected by the collective decision toinvest in reputation under alternative decision-making systems. Our results areapplicable to all settings of binary decision making where the concerned individuals
can influence the collective decision by investing in their reputation. These results
give defendants and candidates a strategic decision tool for planning the investment of
personal resources in reputation. In our setting, it is the combination of the type of theapplied collective decision rule, democratic or non-democratic, with the size of thedecision-making body, large or small8, that determines the marginal productivity ofinvestment in reputation, viz., the incentive to invest in reputation.

8

A candidate may be applying for a job at a large corporation or conversely at a small start-up

company. Academic program candidates may be facing a large university or a small college.
Entrepreneurs can apply for a loan either at a large bank or at a small bank. Assuming that the size ofthe decision making committee is correlated (positively or negatively) with the size of the organization,

8

References
Bar-Isaac H., and Deb. J., 2014. "What is a good reputation? Career concerns withheterogeneous audiences", International Journal of Industrial Organization 34, 44–50
Ben-Yashar, R. and Danziger. L., 2011. “On the optimal allocation of committee
members“, Journal of Mathematical Economics, 47,440-447.
Ben-Yashsr, R., Krausz M. and Nitzan S., 2017. "Government loan guarantees and thecredit decision-making structure", Canadian Journal of Economics, forthcoming.
Ben-Yashar, R. and Nitzan, S. 2014. "On the significance of the prior of a correct
decision in committees". Theory and Decision, 76, 317-327.
Ben-Yashar, R., and Nitzan, S., 2001. “The Robustness of Optimal Organizational
Architectures: A Note on Hierarchies and Polyarchies”, Social Choice and Welfare 18,
155-163.
Ben-Yashar, R. and Nitzan, S., 1997. “The optimal decision rule for fixed-sizecommittees in dichotomous choice situations: The general result”, International
Economic Review 38, 175–186.
Berend, D., & Paroush, J. 1998. "When is Condorcet’s Jury Theorem valid?" Social
Choice and Welfare, 15, 481–488.
Bolton G. E. ,A. Ockenfels, F. Ebeling., 2011 "Information value and externalities in
reputation building", International Journal of Industrial Organization 29, 23–3.
Condorcet, N.C. de. 1785. "Essai sur l’Application de l’Analyse à la Probabilité desDécisions Rendues à la Pluralité des Voix", Paris. See I. McLean and F. Hewitt,
translators, 1994.
Dietrich, F. and List, C. 2013. “Proposition-wise judgment aggregation: The generalcase”, Social Choice and Welfare 40, 1067-1095.

Gerardi, D., Yariv L., 2008. "Information acquisition in committees" Games andEconomic Behavior, 62, 436–459.
Grofman, B., Owen, G., Feld, S.L., 1983. "Thirteen theorems in search of the truth".
Theory and Decision 15, 261-278.
which may not always be the case, yields clear cut implications regarding the dependence of the
incentive to invest in reputation on the size of the organization.

9

Kreps D. M., and Wilson R., 1982. "Reputation and Imperfect Information" Journalof Economic Theory 27. 253-279.
Levy, G., 2007. " Decision Making in Committees: Transparency, Reputation,
and Voting Rules", The American Economic Review Vol. 97 No. 1, 150-168.
Midjord R., Barraquerb T.R., Valasekc J., 2017. "Voting in large committees withdisesteem payoffs: A‘state of the art’ model", Games and Economic Behavior 104
430–443.
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Persico, N., 2004, "Committee Design with Endogenous Information", Review of
Economic Studies, 71, 165–191.
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10Appendix A
We show that the effect of a change in the a-priory probability of innocence on theprobability to acquit a defendant is maximal at the simple majority rule and isminimal at the unanimity rule.9

w Pr 1 : k  1 w Pr 1 : k  1  i!
wDwDw Pr 1 : k  1  i
wDand§ w Pr 1 : k  1  i ·
w¨
¸
wD
©
¹  0 , where i is a positive integer.
wi

Proof:
Recall that
where ' jandw Pr 1 : q
wD

T 1 : 1  T 1 : 1

j

^

`

§n· j
¨¨ ¸¸ p 1  p n  j  1  p j p n  j .
© j¹

n, 'j ! 0.
2(b) ja, ' aBy (a) ¦ ' j !
j k 1¦'j q(a) j !

nn' na .

n¦ ' , hence,j k 1ijw Pr 1 : k  1 w Pr 1 : k  1  i!
, where i is a positive integer.
wD
wD
9The proof is based on part of the proof of Theorem 2 in Ben-Yashar, Krausz and Nitzan (2017). Their
paper focuses on a banking model in which the government uses a guarantee as an instrument for credit
inducement and for affecting the bank’s decision-making system i.e., its degree of centralization, bias
towards approval of loans and reliance on objective loan-specific information.

11

By (b) above, if q 
w Pr 1 : qwDnnthen ¦ ' j
2
j q

n

¦ ' . Therefore,j n  q 1jw Pr 1 : n  q  1.
wD

Specifically, this is true when q = k 1  i . Substituting n

2k  1 yields,n  k 1 i 1 k 1 i .

Hence,

w Pr 1 : k  1  iwDnw Pr 1 : k  1  i. Furthermore, ¦ ' jwD
j k 1 i§ w Pr 1 : k  1  i ·
w¨
¸
wD
¹ 0.
Therefore, ©
widecreases with i.

Q.E.D.

12