Collective decision making in hierarchies

Mathematical Social Sciences 35 (1998) 233–244

Collective decision making in hierarchies Sven Berg*, Jacob Paroush Department of Statistics, Lund University, PO-Box 6008, 220 07 Lund, Sweden Department of Economics, Bar-Ilan University, 52900 Ramat Gan, Israel Received 14 March 1997; received in revised form 19 August 1997; accepted 3 November 1997

Abstract This paper investigates properties of composite majority functions with the aid of recursive formulas in the context of collective decision making in multi-tier hierarchies. The effects of changes in system structure on collective competence are examined. In addition to the theoretical importance, the results given here are applicable in designing hierarchies where the cost in the form of competence loss has to be considered against the benefit of time-saving and other expenses.  1998 Elsevier Science B.V. Keywords: Collective decision making; Majority functions; Recursion relations; Hierarchies; Multi-tier voting systems

1. Introduction A tree is a connected graph that has no cycles. A rooted tree is a tree in which one node has been distinguished as the root. We are interested in decentralized decisionmaking where the decision makers are organized in a hierarchy that is representable by a rooted tree on which the end nodes (the lowest level) represent the individual decisionmakers (voters). The top of the hierarchy, ie the root of the tree, is there to close the system and the top is where the final decision is taken. Inner nodes are interpretable as ‘‘representatives’’ of lower level decision-makers or committees. This paper offers the necessary tools to choose the optimal structure of a hierarchy of collective voting. What we study here is a simple binary model of team decision making, where a team of individuals who share a common goal have different abilities to identify

*Corresponding author. Tel.: 46 46 2228904; fax; 46 46 2224220; e-mail; [email protected] 0165-4896 / 98 / $19.00  1998 Elsevier Science B.V. All rights reserved. PII S0165-4896( 97 )00047-4

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the best way to attain their mutual target. A hierarchy of collective decision for a given team is obtained by the specification of three elements. First, the number of levels or tiers of the hierarchy, i.e. the height of the hierarchy. Second, the number and size of the different groups on each tier and, third, the decision rule used within each group and among groups on each tier. In this paper we focus on a homogeneous, symmetric case. We first assume that the team members possess identical ability or competence p, where p is the individual’s likelihood to make the correct choice and, as a consequence, the optimal decision rule among individuals within each group is simple majority. Second, we will consider only hierarchies that exhibit a certain symmetry. Using a terminology borrowed from [16], we will call a hierarchy regular if (1) all the immediate successors of any unit are at the next lower level, and (2) at each level below the first, all members of the same level have the same number of immediate successors (i.e., in tree representation, nodes at the same level have the same degree). Hence, since all groups on the same tier have the same size, the optimal decision rule among groups on each tier is also simple majority. Thus the task of the designer of collective decision in the homogeneous, symmetric case is reduced to a choice of the number of tiers and the number and size of the different groups on each tier. More specifically, for a team with a given size N, the designer has to choose an ordered vector N 5 (N1 ,...,Nk ) such that N1 3 N2 3 ? ? ? 3 Nk 5 N. Two numbers are associated with each hierarchy, H, the likelihood that the collective decision is correct, say, M(H ) 5 M(N, p), and the cost of using the specific structure to reach a decision, say, C(H ) 5 C(N). The objectives of the designer are obviously to maximize the likelihood of correct choice and to minimize the cost. More specifically, a risk-neutral designer optimizes the target function BM(H ) 2 C(H ) over all possible hierarchies H that are available to a team of size N. B is the team’s net benefit from a correct decision so that BM(H ) is the team’s expected benefit for a correct choice. Recently, the first author of this paper introduced majority functions and studied their properties with the aid of recursive formulas (see [4]). The second author suggested extending some of Berg’s results obtained for two-stage decision making to a general multi-tier voting system and thus make it more easy to calculate and compare the functions M(H ) for different hierarchies H. The present paper is the joint product which studies the properties of the M(H ) functions. Essentially, the general framework of our analysis is the dichotomous choice model which has been studied intensively in the recent literature of social choice and decision theory (see, for instance, [1–3,5,8–14,19]). Note that the multi-tier voting systems that we study here are quite different from the hierarchies and decision making structures that are investigated in Sah and Stiglitz ( [17,18]), in [16], and in the book by [7]. The paper is organized as follows. Section 2 presents background material relating to majority functions and specifies the majority function M(H ) to be studied here. In Section 3 properties of multi-tier majority functions are studied. Section 4 discusses costs of hierarchies C(H ) and shows an example of how to apply the results of Section 3 to the problem posed at the outset of the introduction. Section 5, finally, provides a short summary.

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2. Majority functions based on the binomial distribution We will need some background material relating to the concept of a majority function. The appropriate notation is introduced and some simple properties are listed here for ready reference. Consider a dichotomous choice situation in which a set of individuals, N 5 h1,...,nj, is required to select one of two options, denoted a and b; for example, ‘‘support’’ or ‘‘reject’’ a certain proposal, answer ‘‘yes’’ or ‘‘no’’ to a certain question ( [11]). In what follows, one of the a priori equally probable alternatives, a say, is assumed to be objectively correct. A voting profile, denoted x, is an element in h0, 1j n , where x i 5 1 and x i 5 0 are interpreted as votes cast by individual i for alternative a and b, respectively. Note that abstentions are not allowed. A decision rule d is a mapping: h0, 1j n → h0, 1j. We will confine ourselves to simple majority rule (and variants thereof), i.e., d(x) 5 1, alternative a preferred, if o x i . n / 2. We assume that the decision-makers are homogeneous with respect to their ability to identify the correct alternative, and for each individual i this ability (‘‘skill’’) is given by the probability Pr(x i 5 1) 5 p, p [ (0, 1). In the simplest dichotomous choice situation, the x i ’s are n i.i.d. bernoulli variables, and the sum x 5 o x i has a binomial distribution. We will write the binomial distribution as n b n (x; p) 5sxd p x (1 2 p)n2x , x 5 0, 1, . . . ,n,

p [ (0, 1).

(1)

Next we introduce a majority function, M(N; p), based on the binomial (1),

O N

M(N; p) 5

b N (x; p),

(2)

x 5n11

where in what follows the team size is assumed to be odd, N52n11, n51, 2,... For ready reference, we list some basic properties of the function (2). (Cf, e.g., [2]). Proposition 1. (i) The majority function defined by (2 ) satisfies M(N; p)512M(N; 12p), ;N, p, and p51 / 2 is a symmetry point, M(N; 1 / 2)51 / 2. (ii) The majority function satisfies the recursion relation M(N; p) 5 M(N 2 2; p) 1 ( p 2 1 / 2)b 2n (n; p),

(3)

for n51, 2, . . . , and p[(0, 1). (iii) The majority function has the alternative representation

Ob n

M(N; p) 5 p 1 ( p 2 1 / 2)

2n

(n ; p).

(4)

n 51

In terms of majority functions, the Condorcet jury theorem can be formulated as follows [6].

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Theorem 1. (Condorcet–Boland) The majority function defined by (2 ) is such that (i) M(N; p) is concave increasing in the interval (1 / 2, 1), convex increasing in (0, 1 / 2), while p51 / 2 is an inflexion point; (ii) p.1 / 2 ( p,1 / 2)⇒M(N; p).p (,p), M(N; 1 / 2)51 / 2; (iii) p.1 / 2 (,1 / 2)⇒M(N; p)↑1 (↓0) as N →`. The Condorcet jury theorem says that if each individual is somewhat more likely than not to make the ‘‘better’’ choice between some pair of alternatives—along some specified dimension—and each individual has the same probability of being correct in his choice, then under independent voting the probability that the group majority is correct is greater than the individual probability. Moreover, the probability of a correct decision increases towards a limiting value of 1, as the number of individuals increases. If on the other hand the probability of each voter voting correctly is less than a half, then the probability of a false decision increases indefinitely with the number of voters and tends to unity in the limit. Theorem 1 has been extended in various ways. [13], for example, prove a distributionfree version of the theorem. [10] and [2] discuss the effects on the theorem of correlated votes and dependency among voters, and [15] emphasizes the boundedness away from p51 / 2. We conclude this section with some preparatory material, which will be needed in the discussion of multi-tier majority functions. Consider now a hierarchic system, in which the decision-makers are divided into subgroups and the social ordering between any two alternatives is decided by simple majority over the subgroup orderings, which in turn are obtained by simple majority. To simplify the discussion somewhat, we will look at the symmetric situation where at each stage (tier) the groups (of groups or individuals) all have the same size, i.e. a regular hierarchy. Whatever the realism of such a situation, its principal effect is to simplify considerably notation and description. Note, however, that for two-stage decisionmaking, symmetry (subdivision into groups of equal size) pays in terms of collective competence (see [4]). Definition. A regular k-tier hierarchy can be unambiguously defined by the vector of the degrees, N5(N1 ,...,Nk ) for a k-tier hierarchy, where Ni is the i th level degree, or subgroup size. A k-stage composite majority function (k$3) is defined recursively by M(N; p) 5 M(N 2k ; M(Nk ; p)),

(5)

where N5(N1 ,...,Nk ) defines a regular hierarchy of height k, and N 2k 5(N1 ,...,Nk 21 ). In particular, a two-stage composite majority function is defined by M(N; p) 5 M(N1 ; M(N2 ; p)),

(6)

where N5(N1 , N2 ) and M(N; p) is defined by (2). Note that Nk in (5) is the degree at the bottom of the tree structure, i.e. the size of each of the Nk 21 decision-making teams. The function (5) determines the probability of a

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correct decision in a regular k-tier hierarchy as a function of the individual skill probability p. In what follows, it is assumed that changes in hierarchical structure will not affect individual skill. Formula (6) is thus the majority function for decentralized decision making in N1 committees (juries etc) each of size N2 . Note that in general, M[(N1 , N2 ); p]±M[(N2 , N1 ); p], i.e. the order of the components is relevant. It is of special interest to study majority functions in the neighborhood of p51 / 2, and this for two reasons. First, for fairly high values of p, individual skill, almost any reasonable form of aggregation will lead to a very high collective skill. Second, majority functions are monotone, increasing and concave in the interval (1 / 2, 1). Therefore, it suffices to compare two majority functions near p51 / 2. For the majority function defined by (2) we have (≠ / ≠p)M(N; p) p 51 / 2 5Nb (N), and, more generally, for a k-tier k composite majority function (5) the derivative is the product (≠ / ≠p)M(N; p)5 i 51 Ni b (Ni ). The b -numbers are binomial probabilities

P

S

D

S D

N21 2n b (N) 5 b 2n (n; p) 5 (N 2 1) / 2 2 12N 5 n 2 22n ,

(7)

N52n11, n51, 2, . . . The following lemma gives a useful characterization of the numbers (7). Lemma. The b -numbers defined by (7 ) satisfy the inequality

P b(Ni) , b SP N D, k

k

i51

i 51

i

k 5 2, 3, . . .

and for all Ni odd $3. The ratio b ( coordinate, and asymptotically,

SP DYP k

lim b

i 51

P

k i51

(8) Ni ) /

P

k i51

b (Ni )) is increasing in each

k

Ni

i 51

b (Ni ) 5 (p / 2)(k21) / 2 ,

(9)

where in (9 ) at least one of the Ni tends to infinity. Proof. We indicate a proof for two levels, k52 in (8). First we note the recursion b (N1 12)b (N2 )5 b (N1 )b (N2 )N1 /(N1 11). Then with some algebraic manipulations it can be shown that

b ((N1 1 2)N2 ) 5 b (N1 N2 )(N1 N2 1 2N2 2 2)(N2 ,2) /(N1 N2 1 N2 2 1)(N2 ,2) . b (N1 N2 )N1 /(N1 1 1), where we have used generalized descending factorials, N (k,2) 5N(N22)? ? ?(N2(k21)2). Hence the denominator increases slightly faster than nominator and the assertion in the lemma holds. To prove (9) it suffices to note that by Stirling’s formula b (N)¯1 /Œ] p n as N →`, N52n11. j Incidentally, with the hierarchy regarded as a voting body using simple majority rule, the b -numbers defined by (7) are the Banzhaf voting power indices (nonnormalized). In view of this interpretation of the numbers, the lemma has an independent interest. A

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hierarchy entails a loss of voting power, both individual and collective, as measured by ] the Banzhaf index. With two levels, the factor Œ2 /p indicates a 20 per cent loss in voting power. As we shall see, there is a corresponding loss in collective competence.

3. Properties of composite majority functions We begin this section with a theorem relating to two-stage majority functions (6). It is given here for ready reference and without proof (for details, see [4]). Theorem 2. ( [4]) The majority function defined by (5 ) and (2 ) satisfies (i) for N1 .N2 and p[[1 / 2, 1] M[(N1 , N2 ); p] $ M[(N2 , N1 ); p]

(10)

with equality only for p51 / 2 and 1, and (ii) likewise for p[[1 / 2, 1] M(N1 N2 ; p) $ M[(N1 , N2 ); p].

(11)

It is useful to have the majority function properties listed under Proposition 1 extended to the k-tier situation. Proposition 2. (i) For the k-stage composite majority function (5 ), p51 / 2 is a symmetry point, M(N; p)512M(N: q), p512q, ; p[(0, 1). (ii) The k2stage composite majority function is concave increasing (convex decreasing) for p.(,)1 / 2. (iii) The k 2 stage composite majority function satisfies the recursions M(N; p) 5 M

F

G

Nk 21

N 2k ; p 1 ( p 2 1 / 2)

Ob

2n

(n ; p) .

1

(12)

and N 1 21

M(N; p) 5 M(N 21 ; p) 1 [M(N21 ; p) 2 1 / 2]

Ob

2n

(n ; M(N 21 ; p).

(13)

1

Making frequent use of the properties listed under Proposition 2 and of results proved elsewhere, we will now present a series of inequalities characterizing the multi-tier majority function (5). To this end, we will need some additional notation: N 2i is the (k21)-vector with component Ni removed, i51, . . . ,k, while N i 3j is the (k21)-vector with Ni , Nj replaced by Ni 3Nj . These two vectors define new hierarchies in which, respectively, one level is removed, and two levels are amalgamated into one. The first of these inequalities tell us, not very surprisingly, that we have monotonicity in the number of layers, ie the height of the hierarchy.

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Theorem 3. For p$1 / 2, the k-tier (k$2) composite majority function (5 ) satisfies the inequalities M(N; p) $ M(N 2k ; p) $ M(N1 ; p) $ p,

(14)

M(N; p) $ M(N 21 ; p) $ M(Nk ; p) $ p,

(15)

and

with equality only for p51 / 2 or 1. More generally, for p$1 / 2 and for any i51, . . . ,k we have M(N; p) $ M(N 2i ; p) $ p,

(16)

and equality only for p51 / 2 or 1. Proof. The inequalities (14) and (15) follow directly from Proposition 2, and formulas (12) and (13), respectively. For example, M[(N1 , . . . ,N 2 k); p] 5 M[(N1 , . . . ,Nk 21 ; M(Nk ; p)] $ M(N k21 ; p) by Proposition 2, formula (12). To prove (16), we first have M[(N1 , . . . ,Ni ,Ni 11 , . . . Nk ); p] 5 M[(N1 , . . . ,Ni 21 ); M[(Ni ,Ni 11 , . . . ,Nk ); p]], which by (13) and (15) is $ M((N1 , . . . ,Ni 21 ); M(Ni 11 , . . . ,Nk ); p)) 5 M(N 2i ; p). j The next theorem shows what happens when the height of the hierarchy is reduced, or the shape of the hierarchy is changed, but the total number of decision makers is kept constant. Theorem 4. (i) For p$1 / 2 the k-tier majority function (5 ) satisfies the inequalities M(N i 3i 11 ; p) $ M(N; p) $ p with equalities only for p51 / 2 or 1. In particular, with N5 M(N; p) $ M(N; p) $ p,

(17)

P

k i

Ni , we have (18)

for p$1 / 2, with equality only for p51 / 2 or 1. (ii) With N* being the ordered vector, N*5(N(1) , . . . ,N(k ) ), N(1 ) $ N( 2) $ ? ? ? $ N(k ) , the k-tier majority function (5) satisfies the inequalities, M(N*; p) $ M(N; p),

(19)

with equality only for p51 / 2 or 1. Proof. Obviously, the theorem is a straightforward generalization of Theorem 2. j The following theorem is useful in that it gives two bounds for the multi-layer

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majority function (5) expressed in terms of the simpler, single-level majority function (2). Theorem 5. For a given vector of degrees N5(N1 ,...,Nk ), let N1 be the smallest degree, k k N1 ±Ni , i52, . . . ,k, and let N and N21 be the products 1 Ni and 2 Ni , respectively. We than have that the k 2 tier majority function (5 ), k$2, satisfies the inequalities

P

P

M(N; p) $ M(N; p) $ M(N21 ; p) $ p

(20)

¯ for p$1 / 2, with equality only for p51 / 2 or 1. In particular, if all Ni are equal to N, ¯ ¯ N5(N,...,N ), then M(N¯ k ; p) $ M(N; p) $ M(N¯ k 21 ; p) $ p.

(21)

Proof. To prove the middle inequality in (20), it suffices to look at the case k53 and to show that ≠ ≠ ] M[(N1 , N2 , N3 ); p] p51 / 2 . ] M(N2 N3 ; p) p51 / 2 . ≠p ≠p Equivalently, in terms of b -numbers (7), we must show that N1 N2 N3 b (N1 )b (N2 )b (N3 ) . N2 N3 b (N2 N3 ). This inequality follows, however, directly from the Lemma and the fact that N1 b (N1 )$ 3 / 2. With majority function concavity and monotonicity, the inequality implies M[(N1 , N2 N3 ); p]$ M(N2 N3 ; p) for p[[1 / 2, 1]. j Numerical example. With N5(3, 3, 5) we have M(45; 0.6)50.91355 and M(15; 0.6)50.78690; and also M[(3, 3, 5); 0.6]50.85667, in accordance with Theorem 5. We conclude the section with a multi-tier version of the Condorcet jury theorem. Theorem 6. Condorcet’ s jury theorem for multi-tier majority functions. Let N1 , N2 , . . . ,Nk , . . . be a sequence of odd real numbers, all $3. For each k, M[(N1 , . . . ,Nk );|p] is the multi-tier majority function defined by (11 ). (i) The multi-tier majority function is monotonous in the number of layers k, i.e. for p$1 / 2 M[(N1 , . . . ,Nk , Nk 11 ); p] $ M[(N1 , . . . ,Nk ); p] $ p,

(22)

with equality only for p51 / 2 or 1. (ii) As the number of layers in the hierarchy goes to infinity, we have for p.1 / 2 lim k →` M[(N1 , . . . ,Nk ; p] 5 1

(23)

Proof. Part (i) is already treated above. For part (ii) we refer to formula (20) and apply k k the deMoivre–Laplace theorem to the majority function M( 2 Ni ; p), noting that 2 Ni →` as k→`. j

P

P

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4. Costs and structures of hierarchies In the light of Eq. (18) it must be clear that there is always a loss of collective competence in the erection of a hierarchy. Moreover, this loss is an increasing function of the height of the hierarchy as can be seen by Eq. (17). Thus the sole justification for the use of hierarchies in collective decision making is apparently the cost saving. The case is somewhat analogous to sampling. Sampling theory shows that sampling in clusters or in stages entails a loss of efficiency in terms of increased variance in estimates, but costs saved in such a design may admit a larger effective sample size. Although it is not within the scope of this paper to develop in detail empirically realistic cost functions for decision hierarchies, we will discuss briefly a possible cost model. The cost of collective decision making is composed of several components. For example, secretarial costs to coordinate time and place for a meeting, or the shadow cost of time necessary to discuss the issue, to present the individuals’ positions, or to express their opinions. Thus, it is quite realistic to assume that the cost of collective decision making is an increasing function of team’s size N and the number of groups on each tier of the hierarchy. One way of specifying a cost function, C(Hk ), associated with a k-tier hierarchy would be to specify a cost per branching point and level of the tree structure defining the hierarchy. To exemplify, for a two-tier hierarchy, N1 teams with N2 members each, and linear cost in both variables, we would have C(H2 ) 5 N1 c 1 1 N1 N2 c 2 5 N1 c 1 1 Nc 2 , N 5 N1 N2 .

(24)

For a single-level hierarchy, analogously C(H )5N1 N2 c5Nc. Realistically, we might assume higher per individual cost in the latter case, i.e. that c.c 2 . This would be the case, for example, with individual cost proportional to team size, or at least increasing with team size. The cost function (24) can be extended; for example, with higher level costs small and negligible a reasonable extension could be C(Hk ) 5 Nk 21 c k21 1 Nc k , where c k 21 is the per team cost and c k is the individual team member’s cost. Let us assume that collective benefit is proportional to the probability of a correct decision, say BM(H ), where M(H ) is the majority function associated with the hierarchy. Then in the two-tier case the utility differential relevant to the constructor of the decision making hierarchy would be BM(H2 ) 2 C(H2 ) 2 [BM(H ) 2 C(H )] 5 B[M((N1 , N2 ); p) 2 M(N; p)] 1 [N(c 2 c 2 ) 2 c 1 N1 ].

(25)

Clearly, with team cost c 1 relatively small in relation to individual cost, what matters in (25) is the difference in individual cost, c2c 2 , in relation to the differential in collective skill. If this difference is big enough, it becomes worthwhile to construct a hierarchy. More generally, as stated in the introduction the objective is to maximize the

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target function M(H )2c(H ) over a class of hierarchies. The theory developed in Section 2 Section 3 is obviously of relevance for this task. So far we have analyzed the case where all the team members possess identical ability. For the more realistic case of heterogeneous skills we add the following remarks. [11] and [19] prove that within the dichotomous choice model, the set of all efficient decision rules is identical to the set of all weighted majority rules where the weights are proportional to the logarithm of the individual’s competence odds, i.e. ln( pi /(12pi )). Hierarchies, in general, are not equivalent to weighted majority rules and therefore they are not efficient in terms of the team’s collective competence, which, of course, is entirely compatible with Theorems 3 and 4. For instance, there is not a weighted majority rule available to a team of nine members that is equivalent to the following simple two-tier hierarchy: (ABC, DEF, GHI). The decision rule of this hierarchy is not neutral (in the strong sense), while every weighted majority rule is strongly neutral (see [11]). Note, however, that if we allow overlapping membership in hierarchies, we can obtain hierarchies that are equivalent to weighted majority rules. For example, the two-tier hierarchy (AABCD, AABCE, AACDE, AABDE, ABCDE) is equivalent to the one-tier team (ABCDE), where the decision rule assigns identical weights to B, C, D and E, and triple the weight to A. Similarly, the two-tier hierarchy (AABCD, AABDE, ABBCE, ABCDE, ABCDE) is equivalent to the one-tier team (ABCDE), where the decision rule assigns identical weights to C, D and E and doubles the weights of A and B. Finally, the two-tier hierarchy (ABCDE, ABCDF, ABCEF, ABDEF, ACDEF ) is equivalent to the one-tier team (ABCDEF ), where the decision rule assigns identical weight to B, C, D, E and F and doubles the weight of A.

5. Summary Taking our cue from Berg’s (1997) study, this work continues the investigation of majority functions, extending previous results and adding new results in the context of collective decision making in multi-tier hierarchies. We obtain new and extended results in the form of four theorems. In addition to their elegance, these results turn out to also be of practical use in designing decision making hierarchies, in which it is necessary to balance the benefit of cost saving against loss of competence. This work leaves room for further investigation of hierarchies in the nonhomogeneous case, and especially hierarchies with overlapping membership.

Acknowledgements Berg is grateful to Bo Larsson for stimulating discussions and perceptive remarks. The authors acknowledge valuable remarks from the journal’s referees. Financial support

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from the Swedish Council for Research in the Humanities and Social Sciences is also gratefully acknowledged.

Appendix 1 In order to clarify the idea of cost saving by designing hierarchies, let us illustrate with the analogy of organizing conferences. Suppose there are 45 papers to be presented in a conference and that a presentation of each paper takes an hour (including discussion), and suppose that one cannot squeeze more then five papers in one day (the rest of the day is needed for refreshments, rest and entertainment). Thus, a conference of this type would last 9 days. However, the conference organizer can ‘‘save’’ time be setting a two-tier hierarchy, i.e. by running three sessions simultaneously. In this case the length of the conference can be reduced to three days. The two-tier conference (3, 15) saves six days in comparison to the one tier (45), and this arrangement might be preferable even with the additional expenses involved by using three lecture halls instead of one. The analogy to the inefficiency is the inconvenience imposed on participants that cannot go to two interesting lectures given at the same time in two different sessions. Suppose now that a three-day conference is too long for very busy participants, then the conference organizer can set up a three-tier conference (3, 3, 5) by renting three hotels where each is equipped with 3 lecture halls. Assume that transportation among hotels is very bad, so there is a severe inconvenience. Each participant has to choose in advance only one hotel and select the suitable lectures from only 3 sessions, but the total length of the conference is reduced to a single day. If the organizers uses the ‘‘counting’’ cost function it will be Hierarchy H (45) (3, 15) (3, 3, 5)

O

c(H ) 5 c Ni c 3 45 c 3 18 c 3 11

where c is the shadow cost of time of one hour. The cost of hotels, lecture halls etc., should be added to the basic ‘‘counting’’ cost. In order to find the ideal structure of the conference, the organizer would weigh the cost saving with the inconvenience of the participants that results from the shrinkage of their choice set. Of course, an efficient organizer would do the utmost to classify the papers and allocate them to hotels and sessions by similarity of content in order to reduce the inconvenience of the participants. The above analogy clarifies that the architecture of hierarchies could be very useful in designing collective decisions. An expensive time period of elaboration, explanation and justification of votes, especially where accountability is needed, can be saved by classifying voters into groups and subgroups in a form of hierarchy before structuring the voting process. The actual voting is done within the groups simultaneously. The sacrifice of efficiency can still be worthwhile when compared against the cost saving.

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