HIAS-E-61

A Note on the Possible Advantage of Size Flexibility
in Committees
Ruth Ben-Yashar
Department of Economics, Bar-Ilan University, Ramat Gan 52900, Israel

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

December 2017

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A note on the possible advantage of size flexibility in committees
byRuth Ben-Yashar a and Shmuel Nitzan b.c

Abstract
This note analyzes the possible advantage of size flexibility in decision-making bodies
facing an uncertain dichotomous choice. We find that under constant size variability,application of the unanimity rule might be more desirable than the simple majority
rule, yielding higher average performance. In contrast, and as is well known, the latter
rule is always the superior one, given a fixed number of decision makers withidentical decisional competence.

Keywords: obligatory or flexible committee size, unanimity rule, simple majority rule.
Classification Codes: D7.

a

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

b

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

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

1. Introduction
The size of decision-making bodies (court, committee, jury) is often fixed andmandatory. The objective of this note is to examine the plausibility of such arestriction under two commonly used collective decision rules: the simple majority
and the unanimity rules. Our main results establish that in the uncertain dichotomouschoice setting, such a restriction is advantageous under the simple majority rule, but it
is not plausible under the unanimity rule, especially in relatively large decisionmaking bodies that are naturally more vulnerable to nonattendance.
In countries where the jury system is applied, the collective decision making
rule is typically the unanimity rule. Unanimous support by all voters is required inorder to select a certain alternative: in our case conviction. Otherwise, the otheralternative (commonly the status quo or acquittal) is chosen.1 In contrast, in many
legal systems the collective decision making rule is the simple majority rule and thenumber of decision makers (the judges) is fixed.
The objective of this note is to show that in the former case the number ofjurors need not be fixed. Flexibility in the jury size might be advantageous, that is, theaverage probability of making a correct jury decision is higher provided that size
variability is constant. This result is not valid in decision-making bodies such as
courts, expert committees or board of managers where the applied decision rule is thesimple majority rule. Under this alternative decision-making setting, a fixedmandatory number of decision makers is superior because it yields a higher
probability of making the correct decision. Notice that practically, size flexibility
might also be more plausible in large decision-making bodies such as juries relative tocourts or relatively small expert committees. Furthermore, since the jury members are
not payed professionals like judges or experts, it might be more sensible to allow themincreased flexibility in attendance when fulfilling their voluntary decision-making
task.

1

Earlier studies focusing on the unanimity (hierarchy) rule include Ben-Yashar andDanziger (2016), Ben-Yashar and Nitzan (1998, 2001), Sah and Stiglitz (1988).
2. The model
Consider a group N n k with an odd number of members, n+k. The group confronts
two alternatives, 1 and -1, one of which is correct and therefore better for all voters.2
As is common in decision problems, the identity of the better alternative is unknown.
Every voter selects one of the two alternatives, and an aggregation mechanism isapplied to select the collectively preferred alternative. Each voter chooses the correct
alternative with probability p, which reflects his competence. We assume independent
decisional competencies and that ½<p<1 .By the unanimity rule, one of thealternatives (hereafter 1) is the selected outcome if and only if it is chosen by allvoters (otherwise the other alternative, -1, is chosen). Let 𝑇 denote the probability
that a group consisting of n voters associated with decisional competence p chooses
the correct alternative under the unanimity ruleTn

1 n 1p  1  (1  p) n .
22Let Tnk and Tnk denote the probability that groups consisting of n  k and n  kvoters , n  k >1, choose, respectively, the correct alternative under the unanimity rule.
3. Result
Assuming constant size variability, that is, the number of decision makers is larger orsmaller by k relative to n , we obtain the following result.
Theorem:If n-k>1 , then

2Tn k  Tnkt Tn
2Earlier studies of two alternative models include Baharad and Ben-Yashar (2009),Ben-Yashar and Nitzan (2014), Berend and Sapir (2005), Feld and Grofman (1984)
and Young (1988).

Proof:
1 n k1( p  p n  k  2  (1  p) n  k  (1  p) n k ) t ( p n  1  (1  p) n ) 42p n  k  p n  k  2  (1  p) n  k  (1  p) n k t 2 p n  2  2(1  p) n (1  p) n  (1  p) n  k  (1  p) n  (1  p) n  k t p n (1  p k )  p n (1  p  k ) (1  p) n (1  (1  p) k )  (1  p) n (1  (1  p)  k t p n (1  p k )  p n (1  p  k ) (1  p) n (1  (1  p) k  1  (1  p) k ) t p n (1  p k  1  p k ) 2(1  p) k  (1  p) 2 k  1p n 2 p k  p 2k  1t(1  p) k
(1  p) npk1  (1  p) k
(1  p) k2t

pn
1  pk(1  p) n
pk

2§ 1  (1  p) k · § p ·¨¨¸¸ d ¨¨¸¸k© 1 p¹ ©1  p ¹2

§ 1  (1  p) k · § p ·¨¨¸¸ d ¨¨¸¸k© 1 p¹ ©1  p ¹22

nk



nk

22

§ 1  (1  p) k · § 1  p · § p ·¸¸ ¨¨¸¸ d ¨¨¸¸ ¨¨k© 1 p¹ © p ¹ ©1  p ¹§ 1  (1  p) k 1  p · § p ·¨¨¸ d¨¸kp ¸¹ ¨© 1  p ¸¹
© 1 pnk 2

nk 2

Since the right hand side term is no less than 1 if n-k>1, to complete the proof weneed to show that the left hand side term is not larger than 1. That is, we need to provethat

2§ 1  (1  p ) k 1  p ·1  (1  p ) k 1  p¨¨¸d1
d 1  1  p  (1  p ) k 1 d p  p k 1 kk¸p ¹p1 p© 1 pp k 1  (1  p ) k 1 d 2 p  1

This inequality is proved by induction.
If k=1, p 2  (1  p) 2 d 2 p  1  2 p  1 d 2 p  1, which establishes the inequality forthis case.
If k=2 , p 3  (1  p) 3 d 2 p  1 (2 p  1)( p 2  p(1  p)  (1  p) 2 ) d 2 p  1  p  1  2 p  p 2 d 1  p 2  1  p d 1, which establishes the inequality for this case.
By induction, assume that for some kp k  (1  p) k d 2 p  1  p k d 2 p  1  (1  p) k
For k+1 we prove thatp k 1  (1  p) k 1 d 2 p  1  pp k  (1  p)(1  p) k d 2 p  1  p k d2 p  1 (1  p) k 1
.
ppSince2 p  1 (1  p) k 1
t 2 p  1  (1  p) k  (2 p  1)(1  p) t (1  p) k ( p  1  p)  1 t (1  p) k 1pp
, the proof is complete.
Q.E.D.
Note that, if n-k=1, which implies that one of the groups consists of just onemember, the result is not valid.
Under simple majority, the result is reversed. The probability of making acorrect decision under the simple majority rule increases with the number of decision
makers, but in a decreasing rate. Therefore, in this case, a fixed group size is superior
to constant variable group size because performance in the former case is larger than
the average performance in the latter case, see Baharad, Ben-Yashar and Patal (2017).

4. Conclusion
Our results imply that the rigidity of the decision-making systems resorting to theunanimity rule (notably the jury system) is questionable. That is, mandatory
participation is disadvantageous because it reduces performance relative to a flexible
system that allows constant size variability. In contrast, flexible decision-making
systems resorting to the majority rule are dubious. That is, non-mandatory attendanceis disadvantageous because it reduces performance relative to a rigid system that does
not allow constant size variability.
An alternative “dual” application of the results is the following. If the size ofthe group is fixed, then it makes sense to apply the simple majority rule because it
results in a higher performance. However, under constant size variability, applying theunanimity rule might be more desirable than applying the simple majority rule, which
is clearly the superior rule given a fixed number of decision makers with identicaldecisional competence. In other words, it is useful to recognize that under variable
size flexibility a modified version of Condorcet Jury Theorem need not be valid.

References
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Majority Rule to Probabilities Distortion”, Public Choice 139, 53-59.
Baharad, E. , Ben-Yashar, R. and Patal, T. (2017): “On the Merit of Non-Specializatio
in the Context of Majority Voting”, mimeo.
Ben-Yashar, R., and Danziger, L., (2016). “The Unanimity Rule and ExtremelyAsymmetric Committees”, Journal of Mathematical Economics 64, 107112.10.
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Ben-Yashar, R. and Nitzan, S. (2014): “On the Significance of the Prior of a Correct
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Sah, R.K. and Stiglitz, J.E. (1988): “Committees, Hierarchies and Polyarchies”, The
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Young, H.P. (1988): “Condorcet’s Theory of Voting“, American Political Science
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