HIAS-E-21

Is Diversity in Capabilities Desirable When Adding
Decision Makers?
Ruth Ben-Yashar
Department of Economics, Bar Ilan University, Ramat Gan, Israel

Shmuel Nitzan
Department of Economics, Bar Ilan University, Ramat Gan, Israel
Hitotsubashi Institute for Advanced Study, Hitotsubashi University, Japan

March 16, 2016

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Is diversity in capabilities desirable when adding decision makers? +by
Ruth Ben-Yashar** and Shmuel Nitzan*
Abstract
When the benefit of making a correct decision is sufficiently high, even a slightincrease in the probability of making such a decision justifies an increase in thenumber of decision makers. Applying a standard uncertain dichotomous choicebenchmark setting, this study focuses on the relative desirability of two alternatives:
adding individuals with capabilities identical to the existing ones and adding identicalindividuals with mean-preserving capabilities that depend on the states of nature. Our
main result establishes that when the group applies the simple majority rule,variability in the capabilities of the new decision makers under the two states ofnature, which is commonly observed in various decision-making settings, is lessdesirable in terms of the probability of making the correct decision.

Keywords: decisional capabilities, group extension, asymmetry, homogeneity,
diversity, mean preservation
JEL Classification number: D81*Department of Economics, Bar Ilan University, Ramat Gan , Israel and Hitotsubashi
Institute of Advanced Studies, Hitotsubashi University.
**Department of Economics, Bar Ilan University, Ramat Gan , Israel+The authors are indebted to two anonymous referees and an associate editor for theirmost useful comments and suggestions.

1. Introduction
A sufficiently high benefit of making a correct decision may justify an increase in thenumber of decision makers that increases, even slightly, the probability of making
such a decision. Applying a standard uncertain dichotomous choice benchmark setting,with identical independent symmetric decisional capabilities that are invariant to thestate of nature, our objective is to compare the relative desirability of two alternatives:
adding individuals with symmetric capabilities identical to the existing ones andadding identical individuals with asymmetric mean-preserving capabilities that differ
in the two states of nature. Under the first alternative, the independent decisional
capabilities of the group members remain homogeneous and invariant to the state ofnature. Under the second strategy, the decisional skills of the new group members are
identical, but depend on the state of nature.1
Along time decision makers tend to become equally skilled due to theaccumulated common experience and the exposure to effective learning processes.
Skill uniformity can be changed, at least temporarily, by injecting “new blood” to thegroup, namely by adding new members. The added members can be of different
skills than the existing members , or even be identical to each other, but depend on
the state of nature. In the latter case, the probability of a correct decision is higher in
one state of nature than in the other.
According to Condorcet Jury Theorem (CJT), Condorcet (1785), if theidentical symmetric decision makers vote independently and each of them has thesame probability p > 1 2 of making the correct choice, then the likelihood of a correct
decision is increasing with the number of decision makers.2 This means that the first1

Sah (1991), Sah and Stiglitz (1988) relaxed the symmetry assumption with respect to the states ofnature and allowed the decisional skills of each voter to depend on the state of nature. Ben-Yashar andNitzan (1997) derived the optimal group decision rule under such asymmetric setting. Ben-Yashar(2014) reassesses the validity of the Condorcet Jury Theorem when voters are homogeneous and eachknows the correct decision with an average probability of more than a half. This paper shows thatlarger groups, in most cases, are less likely to reach a correct collective decision, even if the average
individual probability of making a correct decision exceeds one half.
2
CJT has previously been generalized in several other ways. Early expositions and generalizations
were proposed by Grofman et al. (1983), Feld and Grofman (1984), Nitzan and Paroush (1982), Young(1988) and Owen et al. (1989). Ladha (1995) relaxed the independence assumption. Austen-Smith andBanks (1996) and Ben-Yashar and Milchtaich (2007) generalized the setting to a strategic one. CJT canbe generalized to the case of heterogeneous voters. See, for example, Ben-Yashar and Zehavi (2011),
Ben-Yashar and Danziger (2011) and Berend and Paroush (1998). Baharad and Ben-Yashar (2009)
studied the validity of CJT under subjective probabilities. Dietrich and List (2013) presented a general1alternative is advantageous when the decision makers remain homogeneous andinvariant to the state of nature, but may differ in their judgments regarding the identityof that choice. In many decision-making contexts, the decisional capabilities
corresponding to the two possible states of nature often differ. Under the second
possible alternative that we consider, each of the two identical added members is
associated with two probabilities of voting correctly that correspond to the two
possible states of nature, such that their average is equal to the single probability ofmaking a correct decision that represents the ability of each of the existing groupmembers. The question is whether such mean-preserving variability in the capabilitiesof the added identical decision makers is more or less desirable than the firstalternative. It turns out that the first alternative is superior to the second one.
Note that another possible alternative is to add individuals with different
mean-preserving capabilities. In such a case, the decisional skills of the new groupmembers are different, yet still independent of the state of nature. Diversifying
capabilities in this sense is the best possibility in terms of its effect on the likelihoodof making a correct choice. Ben- Yashar and Paroush (2000) show that this meanpreserving alternative outperforms the addition of homogeneous decision makers, theskills of whom are identical to that of the group members. This paper's main result,according to which the addition of homogenous members outperforms the addition ofmembers whose skills are identical to the ones of the group members yet depend on
the state of nature, combined with Ben-Yashar and Paroush (2000) result, implies thatthe best alternative is adding heterogeneous members.
A possible application of our model is a panel of homogeneous (skill-wise)
judges. The judges are classified as "seniors" due to their expertise and experience.
When adding unexperienced judges, who are referred to as "juniors", they may obtain
different specializations such that different skills are associated with different statesof nature. Another application is a situation where the correct decision is obtained
after examining a number of tests or criteria. Every such criterion may identify thecorrect alternative with a given probability. One may perceive this probability as the"criterion’s skill". When it is possible to add some criteria, our result suggests that one
should add two new symmetric criteria of equal quality to the existing ones rather
analysis of proposition-wise judgment aggregation. For a recent survey of the literature inspired by CJT
see Nitzan and Paroush (2015).

2

than adding two new mean–preserving, identical, asymmetric criteria that are superior
in one state and inferior in the other state relative to the existing ones. For example, incritical medical, environmental and earthquake or tsunami preventive diagnostic
decisions, classic distinction is symmetric, namely, independent of the state of nature.
This is the case with cancer tests based on biopsy and with non-invasive procedures
such as ultrasound, MRI and CT. The reason that these diagnostic means are
symmetric is that they do not focus on a particular attribute of the disease but take into
account all or most of the attributes of the disease. The new studies are concernedwith the detection of a disease by taking into account one of its attributes’ -- tests that
focus on early detection, less invasive tests and on lower cost tests. In this caseasymmetric capabilities are commonly observed so the probability of a correct
decision is higher in one state of nature than in the other. This is the case with genebased diagnostic systems for early detection of certain types of cancer. The same istrue for the PSA test for the early detection of prostate cancer. These systems are
typically more reliable in diagnosing a patient who is sick than in making a correct
diagnosis when the patient is healthy. The reason is that cancer has several possible
origins and the common diagnostic means mentioned above focus on one of these, so
the probability of a correct decision is higher in one state of nature (the patient having
cancer) than in the other (the patient not having cancer).
2. The model
Let the decision-making group consists of n = 2k + 1 members, k being a positive
integer. 3

Each individual selects one of two alternatives. The two possible equi-probable states of nature are denoted by 1 and 2. A voter chooses the correct
alternative with probability p1 ( p 2 ) in state of nature 1 (2). We assume that p1 >1-

p 2 ; that is, a voter is more likely to decide 1 in state 1 than in state 2, which implies
that the average of the voter's probabilities in the two states of nature exceeds 1 2 .
Note that all individuals share the same objective of selecting the correct alternative. 4
Individual probabilities of voting for the correct alternative are not statistically3
Since we consider a choice between two alternatives using the simple majority rule, we require an odd
number of decision makers. The rule is not defined otherwise.
4
Within our framework the objective is to make the right decision, which is equivalent to maximizing
the expected payoff when symmetric payoffs are assumed.

3

correlated and the simple majority rule is used to aggregate votes to determine thecollective decision.
According to the common version of CJT, group members are assumed to behomogeneous and symmetric , i.e., possess identical decisional skills that are equal in
the two states of nature, p, such that 1 / 2 < p = p1 = p2 . Let π ( p, n) denote theprobability of the group to decide correctly under the simple majority rule. Formally:⎛ 2k + 1⎞ i⎟⎟ p (1 − p )2 k +1−i .
ii = k +1⎝⎠
2 k +1π ( p, n ) = ∑ ⎜⎜The probability of reaching a correct decision under simple majority rule is equal to
the sum of the probabilities of the majority making a correct bdecision. That is, whenat least k + 1 members support the same correct alternative.
By isolating two members from the original group, following Ben-yashar andParoush (2000), we can rewrite π ( p, n) as: π ( p, n) = p 2 A + (1 − (1 − p) 2 ) B + C , where⎛ 2k − 1⎞ k −1
⎛ 2k − 1⎞ k⎟⎟ p (1 − p )k , B = ⎜⎜⎟⎟ p (1 − p )k −1 andA = ⎜⎜
k
−
1
k
⎝
⎠
⎝
⎠

2 k −1 2k − 1⎛⎞ i⎟⎟ p (1 − p )2 k −1−iC = ∑ ⎜⎜ii = k +1 ⎝⎠Suppose now that the above two members decide correctly. Hence, in order to obtaina majority, k-1 out of 2k-1 are required to decide correctly. This requirement is
described in term A. When at least one of the two members decide correctly, then amajority is obtained if k out of 2k-1 decide correctly, which is reflected in B. Finally,
term C presents the sum of probabilities for obtaining a majority of at least k+1 out of
the N-1 members. The probability that the group chooses the correct alternative
without the two members (i.e., a group of 2k − 1 members) is B+C.
Therefore, adding two homogeneous members improves the performance of the groupiff
p 2 A + (1 − (1 − p) 2 ) B + C > B + C ⇔ p 2 A − (1 − p) 2 B > 0.

This condition is always satisfied since

(1)

B
p
and, indeed, by CJT, under=
A 1− psymmetric homogeneous valuable independent decisional skills the marginalextension of a group is always advantageous.

4

When each of the two identical added members is associated with two
probabilities p1 and p2 of voting correctly under the two possible states of nature 1
and 2, respectively, the performance of the existing homogeneous group is improved
iff

(

2

)

2

0.5 p1 A + (1 − (1 − p1 ) 2 ) B + C + p2 A + (1 − (1 − p2 ) 2 ) B + C > B + C ⇔((p + p )A − ((1 − p ) + (1 − p ) )B)0.5 > 0 ⇔
2

1

2

2

2

2

1

2

p12 + p 22
p.
<
1 − p (1 − p1 )2 + (1 − p 2 )2(2)

3. Result
The increase in the collective probability of making the correct choice due to theaddition of two identical members that are equally skilled to the existing members isdenoted by Δ hs (the index hs refers to homogeneous skills). By (1), Δ hs

is given by:2

Δ hs = p 2 A − (1− p ) B .
The increase in the collective probability of making the correct choice due tothe expansion of the existing homogeneous group by two identical members withasymmetric skills that depend on the state of nature is denoted by Δ has (the index has
refers to homogeneous asymmetric state-dependent skills). By (2), Δ has is given by:

(

)

(

)

2
2
Δ has = 0.5 A p12 + p22 − 0.5B (1 − p1 ) + (1 − p2 ) .

To compare the marginal effects of group extension we have considered above,
suppose that the addition of asymmetric members is mean-preserving. That is, theaverage decisional skills of the added identical asymmetric members is equal to thedecisional skill of the existing homogeneous symmetric members, p =Let Δ hts = p1 p2 A − (1 − p1 )(1 − p2 ) B.

p1 + p2.
2

Δ hts represents the increase in thecollective probability of making the correct choice due to the extension of the existinghomogeneous group by two different members, one with probability p1 in the two5states of nature and the other with probability p2 in the two states of nature (here theindex hts refers to heterogeneous skills)..5
Formallyp1 p2 A + (1 − (1 − p1 )(1 − p2 ))B + C − (B + C ) ≡ p1 p2 A − (1 − p1 )(1 − p2 ) B

This expression is always positive (Ben-Yashar and Paroush (2000)) provided that the
average skill of the added members is at least equal to the homogeneous skill of theexisting group members.
Suppose that the addition of members is mean-preserving. That is, the averagedecisional skills of the added different members or of the added identical asymmetricmembers are equal to the ones of the existing homogeneous symmetric members, thatis, p =

p1 + p2.
2

Theorem 1:
If p =

p1 + p2, then Δ hts > Δ hs > Δ has .
2

Proof:

Δ hs > Δ has ⇔
2

(

)

(

2

2

p 2 A − (1 − p ) B > 0.5 A p12 + p22 − 0.5B (1 − p1 ) + (1 − p2 )⎛(1 − p1 )2 + (1 − p2 )2
p 2 + p22 ⎞ ⎛⎟⎟ − B⎜⎜ (1 − p )2 −A⎜⎜ p 2 − 1
2 ⎠ ⎝
2
⎝

)⇔

⎞
⎟>0⇔
⎟
⎠

⎛
⎞
p 2 + p22 + 2 p1 p2A⎜⎜ p 2 − 1+ p1 p2 ⎟⎟ −2⎝⎠⎛
(1 − p1 )2 + (1 − p2 )2 + 2(1 − p1 )(1 − p2 ) + (1 − p )(1 − p ) ⎞⎟ > 0 ⇔
2B⎜⎜ (1 − p ) −
1
2 ⎟
2
⎝
⎠

5

Note that in this case the probabilities p1and p2 are of different individuals, whereas in (2) the

different probabilities p1and p2 represent the skills of a single individual under two states of nature.

6⎛ 2 p +p 2⎞
2
A⎜ p − 1+ p1 p2 ⎟ −
⎜
⎟
2
⎝
⎠

(

)

⎛
(2 − ( p1 + p2 ))2 + (1 − p )(1 − p ) ⎞⎟ > 0 ⇔2B⎜ (1 − p ) −
1
2
⎜
⎟
2
⎝
⎠

⎛
⎞
4 p2
A⎜⎜ p 2 −+ p1 p2 ⎟⎟ −2⎝⎠⎛(2 − 2 p )2 + (1 − p )(1 − p ) ⎞⎟ > 0 ⇔2B⎜ (1 − p ) −
1
2
⎜
⎟
2
A⎝− p 2 + p1 p2 −2B − (1 − p ) + (1 − p1 )(⎠1 − p2 ) > 0 ⇔

(

) (

)

2

p1 p2 A − B(1 − p1 )(1 − p2 ) > − B(1 − p ) + Ap 2 .

(4)

By Ben-yashar and Paroush (2000) Δ hts > Δ hs which implies that2

p1 p2 A − (1 − p1 )(1 − p2 ) B > p 2 A − (1 − p ) B , which is the same as (4).
This completes the proof.

Q.E.D

4. Discussion
The result of Ben-Yashar and Paroush (2000), namely the superiority of marginal
group extension by different individuals with symmetric mean-preserving decisional
skills, Δ hts > Δ hs , is due to the so called “dimensionally strict monotonicity” of thefunction specifying the collective probability of making a correct choice using thesimple majority rule. By this property, the effect of the improved individual skill isstronger than the effect of the reduced individual skill. On the other hand, our result,

Δ hs > Δ has namely the relative inferiority of marginal group extension by identicalindividuals with asymmetric mean-preserving decisional skills that depend on thestate of nature, is due to the fact that the marginal change in the collective probabilityof making a correct choice using the simple majority rule is declining with respect tothe decisional quality of the two added members. Hence, the positive effect of the7

improved individual skills in one state of nature is more than counter balanced by thenegative effect of the reduced individual skills in the other state of nature.6
The relevance of the assumption of symmetric prior (i.e. symmetricalternatives) is due to the unbiasedness required from the final decisions, which isessential in courts decisions. Clearly, if the environment is not perfectly symmetric, it
is possible to obtain a different result. For instance, suppose that the a- priori
probability of one state is relatively high. In such a case, the addition of decision
makers who are highly skilled in this state, would clearly be advantageous. In thiscase, the inequality is reversed, i.e.. Δ hs < Δ has .7
Our results are not robust to the addition of members with different skills.
Suppose that the two states of nature are characterized by similar parameters, then it is
optimal that the two members balance each other, in the sense that one is more skilled
in one state and the other in the other state. As a special case, consider adding two
different individuals whose skills depend on the state of nature, one with probabilityof p+d in state 1 and of p-d in state 2, and the other member with probability p-d instate 1 and p+d in state 2, which is still mean-preserving. This case is equivalent(probability wise) to adding two heterogeneous individuals; yet, as we have shown, it
is advantageous. That is,0.5(( p + d )( p − d ) A + (1 − (1 − p − d )(1 − p + d ))B + C + ( p − d )( p + d ) A + (1 − (1 − p + d )(1 − p − d ))B + C ) −( p + d )( p − d ) A + (1 − (1 − p − d )(1 − p + d ))B.
which is equal to Δ hts .

References
6

Notice that this economic intuition is not valid for explaining the former result of Ben-Yashar andParoush (2000). The reason is that in their case, in each state of nature the added decisional capabilitiesof the two individuals are different. Since the positive marginal effect requires the use of both of thesedifferent capabilities, it is meaningless to resort to the notion of declining marginal productivity.
7

To analyze, let a be the apriory probability of state 1 . In this case

(

) (

2

2

)

Δhas = A ap12 + (1 − a) p22 − B a(1 − p1 ) + (1 − a)(1 − p2 ) .
If a is close to 1 and p1 is higher than p2 , then Δ hs < Δ has .Alternatively, one can add aparameter q such that p1 = p + q and p2 = p − q ,and look for the optimal value of q. When theprior is completely symmetrical one can obtain q = 0 ,i.e, symmetric skills. As far as a priori probability
stay away from half than the optimum value of q is considerably greater than 0.

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