HIAS-E-78

Skill, Value and Remuneration in Committees
Ruth Ben-Yashar(a) and Shmuel Nitzan(a),(b)(a) Department of Economics, Bar Ilan University, Ramat Gan 52900, Israel(b) Hitotsubashi Institute for Advanced Study, Hitotsubashi University, Tokyo, Japan

November, 2018

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Ecclesiastes Chapter 7, 1, "A good name is better than precious oil"

Skill, Value and Remuneration in Committees
byRuth Ben-Yashar a and Shmuel Nitzan b.

Abstract
Relatively modest difference in individual decisional skills may warrant substantial
inequality in personal remunerations. This implies that the return on investment indecisional skills may be very high. These claims are illustrated applying the uncertain
dichotomous choice setting while clarifying the complex relationship between theindividual skills, the individual optimal decisional weights and the skill-dependent
values, i.e., the individual probabilities to be pivotal.

Keywords: decisional skills, optimal decisional weights, skill-dependent power,remuneration.
Classification Codes: D7.

a

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

b

Department of Economics, Bar Ilan University, Ramat Gan 52900, Israel andHitotsubashi Institute for Advanced Study, Hitotsubashi University, Tokyo, Japan .

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1. Introduction
In standard neoclassical economics, heterogeneous inputs deserve differential
payments. This basic principle applies to decision makers who are members of agroup that makes collective decisions under uncertainty. In this context, however, anindividual's contribution to the collective decision depends on his and the other groupmembers' decisional skills that determine the optimal group decision rule. Given thisoptimal rule, the group members' probabilities of being pivotal and unequivocallycontribute to the collective decision can be calculated and, in turn, the warrantedremunerations.
This letter presents a new notion of an individual's skill-dependent value(power), i.e., the probability that he is pivotal, taking into account not only the relative
number of the decision profiles enabling the decision maker to be pivotal, but also thedifferent probabilities of these profiles. Applying the basic uncertain dichotomouschoice setting, the skill-dependent value is calculated under the optimal groupdecision rule. It is shown that even modest heterogeneity in individual decisional
skills may warrant substantial inequality in personal remuneration of the groupmembers. In other words, one should not underestimate the effect of relatively small
variations in the decisional skills of the group members. Such variations may result inan optimal weighted majority rule that gives rise to very unequal warranted
remunerations. This finding sheds new light on the old and still ongoing debate
regarding the possibility of justifying extreme unequal payments within groups ofprofessionals of seemingly inconsequential diverse personal qualifications. It implies
that, the marginal benefit of investment in skill can be very high.

2. The model
Consider an n-member committee that confronts two alternatives, 1 and -1, one ofwhich is correct and therefore better for all decision makers. The two alternatives are
symmetric i.e., the priors of the two states of nature determining whether analternative is correct or incorrect are equal and the net benefits of a correct decision
under the two states of nature are equal. The identity of the better alternative isunknown. Every decision maker selects one of the two alternatives. We denote by xi
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individual i’s decision, where xi =1 or xi =-1. Decision maker i chooses the correct
alternative with probability pi, which reflects his decisional skill (competence). We
assume independent decisional competencies and that ½<pi<1. With no loss ofgenerality, it is assumed that i<j implies that pi≥ pj, that is, committee members areordered by their skills. Hence, x1 is the decision of the most competent committee
member. The collective decision is based on the decisions of the individuals and ismade by a decisive aggregation rule f that assigns 1 or -1 to any decision profile (x1,x2,…, xn). 1
3. The optimal decision rule
The collective decision rule that optimizes the decision making process, i.e.,
maximizes the collective probability of making a correct decision, is a weighted
majority rule (WMR), Nitzan and Paroush (1982) and Shapley and Grofman (1984).
Specifically, the optimal aggregation rule is sign (w*1 x1+ … +w*n xn) , where w*i =ln[pi ⁄ (1 – pi)] and sign(m)=1, if m>0 and -1 otherwise.
In committees, the number of the potentially optimal (efficient and, therefore,economically relevant) weighted majority rules is surprisingly small. In particular, incommittees of three, four and five-members there exist, respectively, only two, threeand seven such rules. The systems of weights "# =(w1, w2, w3, w4 ,w5),i={%, ', (, ), *, +, ,}, such that the weights are ordered by skill (that is, w1≥ "/ ≥"0 ≥ "1 ≥ "2 ), defining these seven different weighted majority rules for a fivemember group are:fa (the expert rule) defined by wa = (1,0,0,0,0).
fb defined by wb = (1,1,1,0,0).
fc defined by wc = (2,1,1,1,0).
fd (simple majority rule) defined by wd = (1,1,1,1,1).
fe defined by we = (3,1,1,1,1).
ff defined by wf = (2,2,1,1,1).
fg defined by wg = (3,2,2,1,1).

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See Austen-Smith and Banks (1996), Ben-Yashar and Nitzan (1997), Dietrich and List (2013), Young(1988) and recently, Nitzan and Paroush (2018).

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Notice that, for a five-member group, any possible system of optimal weights is
equivalent to one of the above seven systems of weights.
4. The standard power of committee members
A plausible measure of the value of a decision maker is his probability of determining
the collective decision, i.e., the probability of being pivotal. If all decision profiles areequally likely, as usually assumed in the literature, an individual's power is thefraction of all his swing decisions2. For a five-member committee decision rules, thepossible standard power distributions are listed in Table 1. In a similar way, it ispossible to derive the power distributions for 3 and 4-member committees. For
example, in the case of the WMR defined by (3,1,1,1,1), the most competent groupmember, individual 1, is pivotal in 28 decision profiles and each of the remaining
members is pivotal in 4 profiles. After normalization, we therefore get that thestandard power distribution in this case is (63.64,9.09,9.09,9.09,9.09).

Table 1: Weights and standard power
weights

standard power(normalized)

wa = (1,0,0,0,0)(100, 0, 0, 0,0)

wb = (1,1,1,0,0)(33.33,33.33,33.33,0,0)

wc = (2,1,1,1,0)(50,16.67,16.67,16.67,0)

wd = (1,1,1,1,1)(20,20,20,20,20)

we = (3,1,1,1,1)(63.64,9.09,9.09,9.09,9.09)

wf = (2,2,1,1,1)(28.57,28.57,14.29,14.29,14.29)

wg = (3,2,2,1,1)(38.46,23.08,23.08,7.69,7.69)

2

In the context of simple games, the same idea motivates the power index proposed by Banzhaf (1965).

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Table 1 clarifies that different decisional weights result in standard power inequality.

5. The skill-dependent power of committee members
The identity of the optimal rule and, in turn, of the group members' power/value
hinges on their skills. First, the skills determine the identity of the optimal weightedmajority rule. Second, given the group members' optimal weights, the skills determine
the probability of an individual to be pivotal, i.e., the probability of the decision
profiles of the other (n-1) members that enable him to determine the collectivedecision.
To illustrate and clarify the meaning of skill-dependent power, consider thesimplest setting of a three-member group, n=3, where the skills of the group members
are given by (0.85 0.8 0.6). In this case, the optimal weighted majority rule is simple
majority.3 Under this rule, every decision maker is pivotal in the two situations wherethe votes of the other two group members are divided. However, now taking into
account the information about decisional skills, the probability of each of the threedifferently skilled committee members to be pivotal is no longer the same. It depends
on the decisional skills of the other two members as follows:
Individual 1' power: 0.8*0.4+0.2*0.6=0.44
Individual 2' power: 0.85*0.4+0.15*0.6=0.43
Individual 3' power: 0.85*0.2+0.15*0.8=0.29
So the normalized distribution of skill-dependent power becomes (0.379, 0.3706,0.25).
Table 2 illustrates seven possible skill profiles that result in all the possible
optimal weighted majority rules and their corresponding distributions of skilldependent power/value.

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The only other potentially optimal but inferior rule is the expert rule.

5Table 2: Skills, weights and power
Decisional skills

Optimal weights

optimal weights

standard power(p1,…,p5)(w*1,…,w*5)(Normalized)

skill-dependent power(Normalized)(0.8,0.7,0.6,0.5,0.5)(1.386,0.847,0.405,0,0)"% = (1,0,0,0,0)(100,0,0,0,0)(100,0,0,0,0)(0.8,0.75,0.6,0.5,0.5)(1.386,1.098,0.405,0,0)"' = (1,1,1,0,0)(33.33,33.33,33.33,0,0)(36.29,35.48,28.22,0,0)(0.8,0.75,0.6,0.54,0.5)(1.386,1.098,0.405,0.157,0)"( = (2,1,1,1,0)(50,16.67,16.67,16.67,0)(56.16,16.75,13.66,13.42,0)(0.8,0.7,0.7,0.7,0.64)(1.386,0.847,0.847,0.847,0.574)") = (1,1,1,1,1)(20,20,20,20,20)(21.73,19.89,19.89,19.89,18.6)(0.8,0,7,0.6,0.54,0.54)(1.386,0.847,0.405,0.157,0.157)"* = (3,1,1,1,1)(63.64,9.09,9.09,9.09,9.09)(69.57,8.38,7.41,7.31,7.31)(0.8,0.75,0.6,0.6,0.58)(1.386,1.098,0.405,0.405,0.322)"+ = (2,2,1,1,1)(28.57,28.57,14.29,14.29,14.29)(31.8,30.78,12.5,12.5,12.4)(0.8,0.75,0.6,0.54,0.54)(1.386,1.098,0.405,0.157,0.157)", = (3,2,2,1,1)(38.46,23.08,23.08,7.69,7.69)(43.36,24.24,19.61,6.38,6.38)

5.1. Modest skill heterogeneity may warrant very unequal remunerations
Table 2 provides an unequivocal positive answer to the question "can unequal pay bejustified given modest variability in the decisional skills of the committee members".
For example, in row five, the skills (0.8,0,7,0.6,0.54,0.54) give rise to the optimal
weights (w*1,…,w*5) = (1.386,0.847,0.405, 0.157, 0.157) that are equivalent to theoptimal normalized system of weights "4 = (3,1,1,1,1). These weights justify thenormalized standard power distribution (63.64,9.09,9.09,9.09,9.09) or the normalized
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skill-dependent power distribution (69.57,8.38,7.41,7.31,7.31). In this case, thedifference between the skills of individuals 1 and 2 is just 0.1, however 1 deserves apay which is seven times larger than that of 2 and of the other members according to
the standard power and a pay which is more than 8 (9) times larger than that ofindividual 2 (3, 4 or 5) according to the skill-dependent power. The effect of using
skill-dependent power instead of standard power intensifies the power inequality and,in turn, the warranted payment inequality. In row four, the optimal decision rule
corresponding to the skills (0.8,0.7,0.7,0.7,0.64) is the simple majority defined byequal decisional weights. In turn, the decision makers are assigned equal optimal
decisional weights and therefore the standard power distribution is uniform.
Nevertheless, the skill-dependent power distribution is (21.73,19.89,19.89,19.89,18.6)
reflecting the difference in the individual skills. To sum up, the application of skilldependent power justifies increased payment inequality.

5.2 More on the relationship between skills and payment
Table 2 suggests that investment in individual decisional competence can yield
substantial marginal benefit. For example, consider the situation in the first row of thetable where the decisional skills are (0.8,0.7,0.6,0.5,0.5) and the optimal rule is theexpert rule fa and, in turn, all payment should be given to individual 1. If in thissituation the decisional skill of individual 2 increases from 0.7 to 0.75, see the secondrow, he now deserves 35.48% of the total payment, because the optimal decision rulechanges from fa to fb. To further illustrate the possible substantial benefit ofinvestment in skills, suppose that initially all group members share the same skill 0.51.
If one skill changes to 0.535, that is the vector of competencies becomes (0.535,0.51,0.51,0.51,0.51), the optimal rule becomes fe defined by the weights (3,1,1,1,1). If
the same skill changes from 0.51 to 0.54, then the optimal rule becomes fa defined by
the weights (1,0,0,0,0). In the former case the justified share in the total payment of7the group member whose skill has increased changes from 20% to 63.71%. In thelatter case, his share increases from 20% to 100%.
Notice that in this section, we have focused on the optimal decision rule and,therefore, on the optimal decisional weights which determine by the committees skills.
In general, the skill-dependent power is valid when the decision rule is not optimal.
Of course, the assignment of decisional weights is not necessarily based on
qualifications. Clearly, once weights are based on alternative criteria, such as gender,
nepotism or randomness, the existence of unequal contributions and, in turn,payments is inefficient. In particular, women’s limited access to organizational power
structures may result in inefficiency as well as explain male-female unequalcontributions and, in turn, wages.

6. Conclusion
In the context of collective decision making, the justified distribution ofremunerations can be calculated. It hinges on the applied decision rule, which in oursetting is optimally determined on the basis of the group members' decisional skills.
Variability in decisional competencies may result in interesting and surprising value
distributions that sometimes justify extremely unequal remunerations.

References
Austen-Smith, D. and J. Banks (1996), "Information aggregation, rationality and theCondorcet jury theorem", American Political Science Review, 90(1), 34-45.
Banzhaf, J.F., (1965), "Weighted Voting Doesn't Work: A MathematicalAnalysis", Rutgers Law Review, 19 (2), 317–343.
Ben-Yashar, R. and S. Nitzan (1997), "The optimal decision rule for fixed sizecommittees in dichotomous choice situations - The general result", International
Economic Review, 38(1), 175-187.
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Dietrich, F. and C. List (2013), "Propositionwise judgment aggregation: the generalcase", Social Choice and Welfare, 40, 1067-1095.
Nitzan, S. and J. Paroush (1982), "Optimal decision rules in uncertain dichotomouschoice situations", International Economic Review, 23, 289-297.
Nitzan, S. and J. Paroush (2018), "Aggregation of Information by Binary VotingRules", Chapter 7 in The Oxford Handbook of Public Choice, Volume 1, edited byR.D. Congleton,B.N. Grofman and S.Voigt, Oxford University Press.
Shapley, L. and B. Grofman (1984), "Optimizing group judgmental accuracy in thepresence of interdependence", Public Choice, 43, 329-343.
Young, P. (1988), "Condorcet's Theory of Voting", American Political Science Review,
82(4), 1231-1244.

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