HIAS-E-80

Variable Competence and Collective Performance:
Unanimity vs. Simple Majority Rule
Eyal Baharad(a) , Ruth Ben-Yashar(a) and Shmuel Nitzan(a),(b)(a) Department of Economics, Bar Ilan University, Ramat Gan 52900, Ierael(b) Hitotsubashi Institute for Advanced Study, Hitotsubashi University

November, 2018

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Variable Competence and Collective Performance:
Unanimity vs. Simple Majority Ruleby
Eyal Baharad a, Ruth Ben-Yashar a and Shmuel Nitzan b.c

Abstract
Under the unanimity rule, a single voter may alter a decision that is unanimously
accepted by all other voters. Under the simple majority rule, the impact of such avoter diminishes. This paper examines the marginal effect of competence on thecollective performance – the likelihood of reaching a correct decision. It is shown that
under the unanimity rule (simple majority rule), adding an incompetent voter to thegroup is inferior (superior) to giving up an existing competent voter. The negative
impact of an incompetent voter cannot (can) always be balanced by adding acompetent one when the decision mechanism is the unanimity (simple majority) rule.
Moreover, improving a single voter's competence may have a greater effect on thecollective performance under the simple majority rule relative to the unanimity rule.

Keywords: Unanimity rule, simple majority rule, voters' competence, collective
performance.

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

b Department of Economics, Bar Ilan University, Ramat Gan 52900, Israe and Hitotsubashi Institute forAdvanced Study, Hitotsubashi University.
c The third author is grateful to Hitotsubashi Institute for Advanced Study at Hitotsubashi University
for enabling the completion of this paper.

1. Introduction0Many collective decisions are based on the well-known simple majority or theunanimity rule. The former requires the support of a majority of the voters, while thelatter is more demanding: a certain alternative is selected only if supported by all
voters. Otherwise, the other alternative (possibly the status quo) is chosen.1
Unanimity rule and simple majority rule (SMR) are two polar examples of thequalified majority rules, under which a majority q, ½< q £ 1, of the voters, is requiredin order to approve a specific alternative. Somewhat surprisingly, these two polarcases are most common within the family of the qualified majority rules. Manyorganizations apply the unanimity rule for some specific decisions (usually sensitiveones) and the SMR for other issues. Some attempts have been made to bridge the gap
between these two decision mechanisms; the recent one that we are aware of is by
Bouton et al. (2018), who suggest a general framework of majority rule with vetopower.
We focus our attention on the special, common cases of SMR and theunanimity rule examining the marginal effect of competence when these two
mechanisms are applied. Voters' competence is a major factor that affects thelikelihood of reaching a correct decision (assuming one exists). Intuitively, under the
SMR every voter has a lesser impact on the collective outcome than under theunanimity rule (where every voter can change a decision that is unanimously acceptedby all other voters).
Competence can be marginally increased by adding to the group one or two
individuals or by increasing the skills of one of the existing group members. We
assume that the a single voter, referred to as incompetent , whose decisional skills areequal to ½ (will choose the single correct alternative out of two given ones with aprobability of ½) is added to a set of competent voters (each choosing the correct
1 Earlier studies of majority rule include Baharad and Ben-Yashar (2009) who studied the validity ofthe optimal qualified majority rule under subjective probabilities. Berend and Sapir (2005) showed thesuperiority of the simple majority rule when decisional skills are unknown. Ladha (1995) relaxed the
independence assumption under dichotomous choice. Nitzan and Paroush (1982) and Shapley andGrofman (1984) presented the conditions for the optimality of the majority rule. Studies focusing on
the unanimity (hierarchy) rule include Ben-Yashar and Danziger (2016), Ben-Yashar and Nitzan (1998,
2001), Sah and Stiglitz (1988).

1alternative with a probability greater than ½). The set of voters (before adding newones) is homogeneous. Clearly, this is no longer the case after the set is expanded.
The homogeneity assumption means that all voters have the same competence, i.e.,they choose the correct alternative with a common probability.
Our first result establishes that eliminating a competent voter (possibly
because of retirement) is better (worse) than adding an incompetent voter (typically,an unexperienced voter). Put differently, giving a chance to an incompetent decisionmaking-trainee is not (is) justified, even when he is escorted by a competent decision
maker, when the decision mechanism is the unanimity rule (SMR). Our second result
ascertains that an incompetent voter may have a (non) significant effect on thelikelihood of reaching a correct decision, in the sense that his impact (can) cannot
always be balanced by adding a competent one when the decision mechanism is the(SMR) unanimity rule. Moreover, as shown in our third result, the impact ofimproving a single voter's competence on this likelihood may be more substantial
when the collective decision is made under the SMR (relative to the unanimity rule).
2. The model
Consider a case of two alternatives 1 and -1, one of which is correct and is thus betterfor all n=2k+1, k>0, voters.2 As is common in decision problems, the identity of thebetter alternative is unknown. Every voter selects one of the two alternatives, and an
aggregation mechanism is applied to select the collectively chosen alternative. The
prior probability that each alternative is correct is ½. 3 Each voter chooses the correct
alternative with probability p regardless of whether the correct alternative is 1 or -1. p
reflects the voter's competence. We assume that " j ¹ i , voter i's decision is
independent of voter j's decision and ½< p <1. The vector p n = ( p ,..., p ) is referred
to as the n voters' competence structure. Note that the homogeneity assumption meansthat the group of voters shares a common competence p.

2 Earlier studies of two-alternative models include Ben-Yashar and Paroush (2000), Dietrich and List

(2013) and Feld and Grofman (1984).
3 This assumption implies unbiasedness and is especially plausible when analyzing jury's decisions. It

is widely used; see, for example, Ben-Yashar and Danziger (2015) and Ben-Yashar and Nitzan (1997).

2

Let $ denote the collective performance, namely, the probability that the group will
choose the correct alternative. More specifically, $%&' ( p n )denotes the probability
that a group consisting of n voters associated with a competence vector p n will
choose the correct alternative under the SMR . Note that for the SMR, it is necessary()

for n to be an odd number . p SMR ( phn-2 ,½ , x) denotes the probability that a groupconsisting of n - 2 homogenous (h) voters associated with competence p, an
incompetent voter with a competence of ½ and a competent one with a competenceof x, will choose the correct alternative under the SMR.

()

Following Ben-Yashar and Paroush (2000), we present p SMR ( p hn-2 ,½ , x) as:$%&' (CDEFGH , 1I2 , xK) = 1I2 MN + C1 − C1 − 1I2K(1 − x)KB + C = 1I2 MN + 1I2 (M + 1)R + S (1)where

2k - 1ö k
æ 2k - 1ö k -1
÷÷ p
÷÷ p (1 - p )k -1(1- p )k , B = æççA = çç
è k -1 ø
è k ø

and

HUGV
W
HUGVGWS = ∑HUGVWXUYV ( W ) D (1 − D)

According to the unanimity rule, one of the alternatives (hereafter 1) is theselected outcome if and only if it is chosen by all n voters (otherwise the other( ) and p (p ) denote then1alternative, -1, is chosen). Under this rule, let p UNA
p

-1
UNA

n

probability of reaching a correct decision if alternative 1 and -1, respectively, is thecorrect one: 4

( )

( )

n
n
1
-1
p UNA
p = p n and p UNA
p = 1 - (1 - p) .

n

4 If the correct decision is 1 , the probability of a correct decision is equal to the probability that all themembers support this alternative. If the correct alternative is -1, then the probability of a correct
decision is equal to the probability that at least one member support -1.

3

( )

p UNA p n , the probability that a group consisting of n voters associated with acompetence vector p n will choose the correct alternative (each alternative withan equi-probable prior of 1/2) under the unanimity rule, is given by: 5( )p UNA P =
n

( )

( ).

n
n
1
-1
p UNA
p + p UNA
p

2

(2)

$def ((Dhℎ , ½)) denotes the performance of the group assuming that nhomogenous voters have the decisional skill p and one voter the probability of
½.

(

)

Let p UNA ( phn-2 ,½ , x) denote the probability that a group consisting of n - 2
homogenous voters associated with a competence of p, an incompetent voter
with a competence of ½ and a competent one with a competence of x, will choose
the correct alternative under the unanimity rule. Note that for the unanimity rule,
it is not necessary for n to be an odd number .
3. Giving up an existing competent voter or adding an incompetent voter
Eliminating a competent voter (possibly because of retirement) is better (worse) thanadding an incompetent voter (typically, an unexperienced voter). Put differently,
giving a chance to an incompetent decision-making-trainee is not (is) justified, evenwhen he is escorted by a competent decision maker, when the decision mechanism isthe unanimity rule (SMR). Under the simple majority rule with an odd number ofvoters, a meaningful increase in size requires the addition of at least two voters.
Hence, under this rule, adding two unskilled voters to a group of experienced skilled
voters is better than eliminating two of the existing skilled voters.

5 Note that under the unanimity rule alternative 1 is selected if it is supported by all voters. Alternative-1 is selected if supported by at least one voter. Such asymmetry requires the distinction between( ) and p (p ). Under the SMR, however, alternative 1 (or -1) is selected if supported by asimple majority of the voters. Such symmetry implies p(p )= p (p ), which we denote asp (p ).
1

p UNA p

n

-1
UNA

n

1
SMR

n

SMR

4

n

-1
SMR

n

Theorem 1:$noN (D2p−1 ) > $noN (CD2pℎ , ½ K).
$%&' (DHUGV ) < $%&' (CDEHUYV , ½ , ½K)

Proof:The proof for the unanimity rule:
VV

V

V

V

V

$def (CDEHU , ½ K) = H H (D)HU + H (1 − H (1 − D)HU )=H + s ((D)HU − (1 − D)HU )
V

V

V

V

$def (DHUGV ) = H (D)HUGV + H (1 − (1 − D)HUGV ) = H + H ((D)HUGV − (1 − D)HUGV )Hence,V(D)HUGV − (1 − D)HUGV > ((D)HU − (1 − D)HU ) ≡H
11(D)HUGV t1 − D u > (1 − D)HUGV t1 − (1 − D) u ≡22w(VGw)HUGV >xy

VG (VGw)
x
y

VG w

.

(3)

If k=1, (3) is valid since:
w
VGw

>

x
y

VG (VGw)
x
VG w
y

V

V

≡ D − H DH > 1 − D − H (1 − D)H ≡ 2(2D − 1) > DH − (1 − D)H ≡(3) is also valid for k>1 , since 2(2D − 1) > (2D − 1) ≡ 2D − 1 > 0, which is true.
in the left hand size becomes larger and the right side of (3) is not changed.
The proof for SMR:
A straightforward application of Ben-Yashar and Zahavi (2011), implies that adding
two incompetent voters to a group of competent voters is better than eliminating twoof the existing competent voters.6
Q.E.D
Theorem 1 can be considered as a justification of supporting retirement of a single
competent voter relative to preserving all competent voters and giving a chance to asingle incompetent new voter, when the decision mechanism is the unanimity rule.
More precisely, reducing the group size by eliminating one of the competent voters is
6 Ben-Yashar and Zahavi (2011) showed that adding a pair of competent and incompetent voters to agroup that consists of either competent or incompetent voters, provided that at least one of the existinggroup voters is incompetent, increases the probability that the group makes the correct decision. Under
the unanimity rule, the result is preserved when comparing the case of adding two incompetent votersrelative to eliminating two of the existing competent voters, see Appendix A. But we have chosen topresent the more interesting case of adding just one competent voter relative to the case of eliminating
a competent voter.

5superior to increasing the group by a single incompetent voter. In contrast, under thesimple majority rule, adding two incompetent voters to a group of competent voters is
better than eliminating two of the existing competent voters.

The case against

retirement is therefore plausible under simple majority rule: adding two new
inexperienced incompetent voters while preserving, instead of eliminating, two of theexisting experienced competent voters enhances the performance of the group.
4. Balancing the impact of an incompetent voter
Let us now consider the case of balancing the impact of an incompetent voter.
We examine the case where two voters are added to a set of n-2 homogenous
voters. One of the added voters is associated with competence that is equal to ½,
a voter referred to as an incompetent one. The other added voter is associatedwith competence x, who is intended to balance the incompetent voter, in thesense that the collective probability for reaching a correct decision is not reduced
due to the addition of the two voters. We present the minimal requiredcompetence of the latter voter, when the two applied voting mechanisms are the
SMR and the unanimity rules.7
Theorem 2:

p SMR (( phn-2 ,½ , x) ) ³ p SMR ( phn-2 ) Û x ³ p
p UNA (( p hn - 2 ,½ , x) ) ³ p UNA ( p hn - 2 ) Û

x +1 æ p ö
÷³ç
2 - x çè 1 - p ÷øn-2

Proof:

By adding the two voters when the applied decision mechanism is the SMR, we
obtain that, using equation (1):p SMR æç ( phn - 2 ,½ , x) ö÷= 0.5MN + R − 0.5(1 − M)R + S
èø
7 The case of adding a member (a skilled voter with p>½) to an existing set of voters, was discussed in

the literature (see, e.g., Feld and Grofman, 1984 or Karotkin and Paroush, 2003). Our study, like somerecent studies (see, e.g., Ben-Yashar and Zahavi, 2011), considers the special case of adding a nonskilled voter (p=½).

6

$%&' (DEFGH ) = B+C
Hence,
0.5 MN + 0.5R(1 + M) + S ≥ R + S ≡0.5 xA - 0.5(1 - x) B ³ 0 Û
Since

B
p
=
,
A 1- px
B³
1- x Ax ³ p.

By adding the two voters when the applied mechanism is the unanimity rule, we
obtain that, using equation (2):p n-2 0.5 x + 1 - 0.5(1 - x)(1 - p ) n-2 p n-2 + 1 - (1 - p) n-2³
Û22
p n-2 (0.5 x - 1) ³ (1 - p) n-2 (0.5(1 - x) - 1) Û
p n-2 0.5( x - 2) ³ -(1 - p) n-2 0.5((1 + x) Û
p n-2
1+ x
£
n-2
2- x
(1 - p)from which we obtain that
æ p ö
çç
÷÷
è1- p ø

n-2

£

x +1.
2- x

(4)
Q.E.D.

Under the SMR, it is always possible to counterbalance the incompetent voter byadding one that is at least as competent as the voters in the group. It can be verified
that under the unanimity rule, however, there are many cases where such balancing is
impossible. Sufficient condition for such impossibility is obtained from (4), since
1 /( n - 2 )ÉYV
M cannot exceed 1 and HGÉ ÑhÖÜáàáâ äÑãℎ M, p > 2(e.g., when p>2/3 and1 /( n - 2 )1+ 2

n>1,when p>0.56 and n>4, or when the group is large enough).

5. The impact of improving a single voter's competence7

Our last objective is to examine the impact of improving a single voter's
competence, when the applied rules are the SMR and the unanimity rule.

( )The first derivative of p SMR p n with respect to pi (that is, the competence of a
specific voter i) is obtained from (1), assuming that all the group voters arehomogeneous. Recall that equation (1) views the probability of a correct collective
decision by relating to the probabilities of two specific voters and to the probabilitiesof the remaining voters. The terms A,B and C do not depend on the probabilities of
the two voters. Hence, when all voters are equally skilled and we change thecompetence of one of the specific voters i, equation (1) takes the form:
DÑ D N + (1 − C1 − DÑ K(1 − D)) R + S and, therefore,( ) = pA + (1 - p)B¶p SMR p

n

¶pi

,

which is positive, independent of pi .

( )By taking the first derivative of p UNA p n with respect to pi , we obtain:( ) = 1 ( p + (1 - p) ) .

¶p UNA p

n

n-1

¶pi

n-1

2

which is positive as well, independent of pi .
The following theorem reveals the condition under which changes in pi are
more influential under one rule relative to the other:
Theorem 3: Changes in pi are more effective under SMR than under theunanimity rule when:( ) > ¶p (P ) if æç p ö÷ < 2 × æç 2k - 1ö÷ + 4 × æç 2k - 1ö÷ - 1¶p SMR P
¶pi

n

¶pi

2

k

n

UNA

ç1- p ÷èø

ç k -1 ÷èø8ç k -1 ÷èøProof:( ) > ¶p (P ) Û pA+(1- p)B> 1 ( p + (1 - p) )¶p SMR P
¶pi

n

n

n -1

UNA

¶pi

n -1

2

Since n=2k+1, we obtain:

æ 2k - 1ö k
1
÷÷ × p (1 - p )k > p 2 k + (1 - p) 2 k
2 × çç
2
è k -1 ø()

Hence,

æ p ö
÷÷
0 >çç
è 1- p ø2kk
æ 2k -1ö æ p ö÷÷×çç÷÷ +1 Û- 4×çç
è k -1 ø è 1- p ø22k

æ 2 k - 1ö
æ 2 k - 1ö
æ 2 k - 1ö
æ 2k - 1öæ p ö
÷÷ - 4çç÷÷ - 1 < çç÷÷ + 4çç÷÷ - 1÷÷ < 2 × çç
2 × çç
k
1
k
1
k
1
k
1
1
p
è
ø
è
ø
è
ø
è
ø
è
ø
k

æ
ö
The left hand side condition is satisfied since, assuming p>½, çç p ÷÷ > 1 andè1- p ø2

æ 2k - 1öæ 2k - 1ö÷÷ - 4çç÷÷ - 1 < 1. Hence, the right hand side condition is identical to2 × çç
è k -1 øè k -1 ø
the one in the theorem.8 Thus, the proof is completed.
Q.E.D.
Theorem 3 implies that changes in pi are more effective under the SMR when p issmall enough. In particular, this greater effectiveness is valid for k=1 when p<0.78,
for k=2,3 when p<0.77 and for k=4 when p<0.75. This relates to jury's selection; jury
applies the unanimity rule, under which a minor increase in relatively small p will notmake a major change. Under committees (i.e., small number of members), however,
8 Let u= CHUGVK. Then,

UGV

2 × u - 4u 2 - 1 < 1 Û 2 × u - 1 < 4u 2 - 1 Û
2u - 1 <(2u - 1) × (2u + 1) Û (2u - 1) < (2u + 1) ,which is always satisfied.

9where the SMR is applied, a similar change in p would contribute significantly to thelikelihood of reaching a correct decision. Theorem 3 characterizes the cases where
improving decisional skills is useful.
6. Concluding remarks
This paper examines the marginal impact of competence on the collective likelihoodof reaching a correct decision, when the decision mechanisms are the unanimity ruleand the SMR. We first proved that, under the unanimity rule, increasing ahomogeneous group of competent voters by a single incompetent voter is inferior toeliminating one of the existing competent group voters. This means that adding an
incompetent voter accompanied by a competent one is disadvantageous. This is not
the case under the simple majority rule, where the enlargement of a homogeneousgroup of capable voters by two balanced pairs of competent and incompetent votersenhances the performance of the group. In other words, under the simple majority rule,
there exists an incentive to give a chance to incompetent voters-trainees as long as
they are escorted by competent ones. Furthermore, we proved that the negative impactof an incompetent voter on the likelihood of reaching a correct decision cannot (can)always be balanced by adding a competent one when the decision mechanism is theunanimity rule (SMR). Moreover, the impact of improving a single voter's
competence on this likelihood may be more substantial when the collective decision ismade under the SMR (relative to the unanimity rule).

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11

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Appendix A$def (DHUGV ) > $def (CDEHUYV , ½ , ½K).
Proof:
1
1 1$def (DHUGV ) = (D)HUGV + − (1 − (1 − D)HUGV ) =
2
2 2
1 1
+ ((D)HUGV − (1 − D)HUGV )2 2= $def (CDEHUYV , ½ , ½K)
11
1 11(D)HUYV + + t1 − (1 − D)HUYV u)24
2 2
4
Hence,
1(D)HUGV − (1 − D)HUGV > ((D)HUYV − (1 − D)HUYV ) ⇔41
1(D)HUGV t1 − DH u > (1 − D)HUGV t1 − (1 − D)H u ⇔441
HD HUGV 1 − 4 (1 − D)
>
.
1 H
1−D
1 − 4D(*) is valid since, if k=1,
w
VGw

>

x
ë

VG (VGw)y
x
VG wy
ë

V

(∗)

V

≡ D − s Dí > 1 − D − s (1 − D)í ≡ 4(2D − 1) > Dí − (1 − D)í

(*) is also ⇔ 4 > DH + D − DH + 1 − 2D + DH ≡ 0 > −3 − D + DH , äℎÑÖℎ Ñâ ãÜìá.
valid for k>1 , since in the left hand size becomes larger and the right side of (*) is notchanged.
Q.E.D.

12