Soc Choice Welfare (2001) 18: 155±163

2001 9 9 9 9

The robustness of optimal organizational architectures: A note on hierarchies and polyarchies Ruth Ben-Yashar, Shmuel Nitzan Department of Economics, Bar-Ilan University, 52900 Ramat Gan, Israel Received: 4 January 1999/Accepted: 10 January 2000

Abstract. In this note we study the robustness of optimal organizational architectures, focusing on hierarchies and polyarchies. These two speci®c architectures are often applied in economic systems and have received considerable attention in the literature. It turns out that the application of these architectures usually involves ine½ciency, namely, the use of suboptimal organizational systems. This is demonstrated by proposing a measure of size robustness of optimal architectures and by analyzing the implications of its magnitude for hierarchies and polyarchies. 1 Introduction The architecture of an economic system speci®es ``how the constituent decision-making units are arranged together in a system and how the decisionmaking authority and ability is distributed within a system'', Sah and Stiglitz (1986). In economics the two frequently studied architectures that have received considerable attention in the literature are hierarchies,1 and polyarchies (for example, see Ben-Yashar 1993, Sah and Stiglitz 1985, 1986, 1988, Sah 1991, Koh 1992a, 1992b, 1994). A polyarchy is a system in which there are several decision makers who can undertake projects independently of one another. A project is chosen if it receives the support of at least one member of the organization. In contrast, the decision-making authority is more concentrated in a

1 Note that we use the term hierarchy in a di¨erent sense than Berg and Paroush (1998).

156

R. Ben-Yashar, S. Nitzan

hierarchy where one particular individual can undertake projects provided that he secures the support of all other decision makers. In other words, the choice of a project hinges on its receiving the support of all members of the organization. Under the reasonable assumption that the variety of architectures consists of all weighted quali®ed majority rules,2 these two architectures are just two of many possible plausible architectures. This paper studies the robustness of an optimal organizational architecture with respect to the size of the organization, focusing on hierarchies and polyarchies. We present a size robustness measure of optimal architectures and demonstrate that, in general, the magnitude of this measure is small. This implies, ®rstly, that a small change in the size of the organization does a¨ect the optimal architecture, and, secondly, that unless the size of the organization is very small a hierarchy or a polyarchy is not an optimal architecture. The optimal architecture depends on the relevant parameters: individual decisional skills and utilities, the frequency of good and bad projects and, in particular, the size of the organization. Absence of just few members of the organization and, in general, a slight variation in the size of the organization a¨ects the optimal architecture. It turns out that the use of a hierarchy or a polyarchy is almost always suboptimal. It is known that under some symmetry assumptions the optimal architecture is the simple majority rule, and that the optimality of hierarchy and polyarchy hinges on some asymmetry assumptions. There are four types of asymmetries, Ben-Yashar and Nitzan (1997): (i) the utilities associated with the two states of nature may di¨er, (ii) the priors of the two states of nature may di¨er, (iii) the individual decisional skills may depend on the state of nature, and (iv) the decisional skills may di¨er across individuals. When the four types of asymmetry favor acceptance (rejection) of a project, a polyarchy (a hierarchy) can be the optimal architecture, but not a hierarchy (a polyarchy). In this note we clarify that the required asymmetry for a hierarchy or a polyarchy to be optimal is very demanding. This implies that in the relevant literature and, apparently, in practice, attention is not focused on the right architectures. 2 The framework The economic system faces dichotomous choices, such as accepting or rejecting projects. n members (decision makers) independently determine whether to accept or reject a project so as to maximize the system's common utility function. A project can be good (1) or bad … 1† and a given member can accept (1) or reject … 1† the project. The decision matrix for each individual is therefore:

2 This assumption is plausible because the optimal collective decision rule is always a weighted quali®ed majority rule, as shown in Ben-Yashar and Nitzan (1997).

Hierarchies and polyarchies Decision

157

State of project Good

Bad

Accept

…1; 1†

…1; 1†

Reject

… 1; 1†

… 1; 1†

The individual's decision-making skills are represented by the probabilities p1 and p2 that he makes a correct decision under the two states of nature. …1 p1 † and …1 p2 † can be interpreted as type I and type II errors entailed in the member's decision. For simplicity, and without loss of generality with respect to the main point of this note, we assume that the decision makers who usually di¨er in their decisions are homogeneous in their decisional skills. We also assume that … p1 ‡ p2 †=2 > 1=2 which means that a member is more likely to accept a good project rather than a bad project, p1 > 1 p2 . The architecture of a system is represented by a decision rule, f, that transforms the pro®le of n individual decisions into a collective decision, acceptance (1) or rejection … 1† of a project. The utilities of the four possible outcomes are denoted B…1=1†, B… 1= 1†, B…1= 1† and B… 1=1†. B…1† and B… 1† denote, respectively, the positive net utility when the correct decision is 1 or 1, B…1† ˆ B…1=1† B… 1=1† and B… 1† ˆ B… 1= 1† B…1= 1†. a and …1 a† denote the apriori probabilities that a project is a good one (1) or a bad one … 1†. 3 The conditions for the optimality of an architecture By assumption, the variety of architectures is represented by all possible quali®ed majority rules. Formally,  1 N… 1† V nk f ˆ 1 otherwise where N… 1† is the number of decision makers who reject the project, n is the total number of decision makers, k is the proportion of decision makers out of the total number of decision makers necessary to decide 1 for the organization's decision to be 1 and kn is the number of decision makers who must decide 1 for the organization's decision to be 1. k or kn represents the decision rule f, or the architecture of the economic system. Ben-Yashar and Nitzan (1997) show that the optimal architecture k depends on the parameters: n; a; B…1†; B… 1†, p1 and p2 . Speci®cally, the optimal quali®ed majority rule is ^k, 2 3 n p1 …1 p1 † ln g ‡ d ‡ 6 2 p2 …1 p2 †7 ^k…n; g; d; p ; p † ˆ 1 61 ‡ 7 1 2 5 b1 ‡ b2 24 n 2 a B…1† p1 p2 where g ˆ ln , d ˆ ln , b1 ˆ ln and b2 ˆ ln . 1 a B… 1† 1 p1 1 p2

158

R. Ben-Yashar, S. Nitzan

We de®ne the size robustness measure (stability range) as the maximal permissible change in the size of the organization that does not alter the optimal architecture. To characterize the robustness of optimal architectures with respect to the organization's size, that is, to derive the size robustness measure, we introduce two preliminary results. Lemma 1 speci®es the optimality conditions for the various architectures that are represented by …^k
n†. This lemma is used when k
n is constant. Lemma 2 speci®es the optimality conditions for various architectures which are represented by ^k. This lemma is used when k is constant. Lemma 1. c

1 < ^kn U c , m ‡ …c

1†s < n U m ‡ cs

where mˆ

g‡d p1 ; ln 1 p2

sˆ

b1 ‡ b2 p1 : ln 1 p2

Note that c is a constant which determines the identity of the architecture. Proof. See Appendix A. The value of c determines the speci®c architecture. When c ˆ 1 the speci®c architecture is a hierarchy, which means that the collective decision is 1, only when all members decide 1.3 When c ˆ 2 the speci®c architecture is de®ned by 1 < kn U 2, which means that the collective decision is 1, if at least n 1 members decide 1. Note that when c is not a constant, Lemma 1 cannot be directly applied to obtain the size robustness measure. For example, when c ˆ n=2, in which case the implied architecture is the simple majority rule, it is clear that a change in n brings about a change in ^kn, although the implied architecture remains the simple majority rule.4 For these cases we need the following lemma.

3 kn ˆ 1 is a hierarchy, or more accurately, 0 < kn U 1. 4 Indirectly, Lemma 1 gives the size robustness measure also for c which is not a conn 1< stant. For example, when g ‡ d ˆ 0 and p1 ˆ p2 , it can be shown that 2 ^kn U n , n 2 < n U n which is always satis®ed. Or, when p ˆ p , it can be shown 1 2 2 n n that 1 < ^kn U , …b 1 ‡ b 2 † < …g ‡ d† U 0. These well known results imply that 2 2 under the assumption of symmetry or weak asymmetry the architecture implied by the simple majority rule is optimal.

Hierarchies and polyarchies

Lemma 2.

L

8 > > > > > > 1 < vU0 > > > > > > > <

g‡d b1 ‡ b2

and

0 , B d ˆ 1 @L

p1 1 p2 when L ˆ b1 ‡ b2 p1 ln 1 p2 when L > b1 ‡ b2 p1 ln 1 p2 when L < b1 ‡ b2 ln

1 ^ n > > > > > > > > > > > > : …v ‡ 1†d < n U v
d

where

vˆ

159

ln

p1

1

1 p2 C A: b1 ‡ b2

Note that L is a constant that determines the identity of the architecture. Proof. See Appendix B. The value of L determines the speci®c architecture. When L ˆ 1 the speci®c architecture is a polyarchy, which means that the collective decision is 1 if at least one member decides 1.5 Assuming that p1 ˆ p2 , L ˆ 1=2 implies that …1=2† …1=n† < ^k U …1=2† i¨ 1 < v U 0. That is, under the assumption of symmetry or weak asymmetry the architecture implied by the simple majority rule is the optimal decision rule. Note that when L is not constant, Lemma 2 cannot be directly applied to obtain the size robustness measure. For example, when L ˆ 1=n, in which case the implied architecture is a hierarchy, it is clear that a change in n brings about a change in ^k, although the implied architecture remains a hierarchy. From Lemma 1 and Lemma 2 one can derive a size robustness measure. Proposition 1. The size robustness measure for architectures that are repreb ‡b sented by a constant ^kn is s ˆ 1 p 2 . For architectures which are represented 1 ln 1 p2 p1  ln 1 p2 ^ .6 by a constant k, this measure is jdj ˆ 1 L b ‡b 1

5 k ˆ 1 is a polyarchy, or more accurately, 1

2

1 < k U 1. n

p1 1 p2 which are optimal 6 Except for architectures which are represented by L ˆ b1 ‡ b2 for any size of the organization provided that 1 < …g ‡ d†=…b 1 ‡ b 2 † U 0. ln

160

R. Ben-Yashar, S. Nitzan

The proposed measures s and jdj are size robustness measures since a change in n, such that Dn > s or Dn > jdj, respectively a¨ects the optimal architecture ^kn or ^k. The signi®cance of Proposition 1 stems from the observation that organizations frequently experience changes in size resulting, for example, from the absence of members or from the requirement that the organization increases its size, due to internal or external pressure to democratize its decision-making process. 4 Hierarchy and polyarchy The necessary and su½cient condition for the optimality of a hierarchy can be derived using Lemma 1: 0 < ^kn U 1 , m < n U m ‡ s g‡d p1 < n U ln 1 p2

,

g‡d b1 ‡ b2 p1 ‡ p1 ln ln 1 p2 1 p2

The necessary and su½cient condition for the optimality of a polyarchy can be derived using Lemma 2:7 1

1 ^ < k U 1 , vd U n < …v ‡ 1†d n ,

g‡d g‡d b1 ‡ b2 p2 U n < p2 ‡ p2 ln ln ln 1 p1 1 p1 1 p1

The size robustness measure for a hierarchy is therefore equal to b ‡b b ‡b s h ˆ 1 p 2 . For a polyarchy this measure is equal to d p ˆ 1 p 2 . 1 2 ln ln 1 p2 1 p1 We thus obtain, Proposition 2. p1 ˆ p2 ) s h ˆ d p ˆ 2. This result means that increasing or decreasing the size of the organization by one member may not change the optimal architecture (hierarchy or polyarchy), but an increase or a decrease of size by at least two members certainly changes the optimal architecture (hierarchy or polyarchy). In general, the size robustness measure is small. Hence, the optimality of hierarchies or polyarchies can be expected only for a limited range of sizes of

p1 1 p2 7 Note that polyarchy is represented by L ˆ 1 > . b1 ‡ b2 ln

Hierarchies and polyarchies

161

the organization. In particular, in a neutral environment, where, a ˆ 1=2 and B…1† ˆ B… 1†, m ˆ v ˆ 0. Hence, in light of Proposition 2, if p1 ˆ p2 hierarchy and polyarchy cannot be the optimal architectures for n > 2 (the simple majority rule is the optimal architecture). More generally, the size robustness measure is small enough even when p1 0 p2 . Using simulations we obtain that if jp1 p2 j < 1=2 and n > 5, the optimal architecture cannot be a hierarchy or a polyarchy. This strengthens the conclusion that the common application of hierarchies and polyarchies usually involves ine½ciency, namely, the use of suboptimal organizational systems. 5 Summary In this note we show that, in general, slight variations in the relevant parameters and, in particular, the frequently observed small changes in the organization's size a¨ect the optimal architecture. Although we focus on robustness with respect to the size of the organization, similar results can be obtained when the robustness of optimal architectures is analyzed with respect to the other relevant parameters. The use of a hierarchy or a polyarchy is almost always suboptimal.8 It is known that under symmetry assumptions the optimal decision rule is the simple majority rule, and that the optimality of a hierarchy or a polyarchy hinges on an asymmetry assumption. In this note we clarify that the required asymmetry is very demanding. Hence, in general, these two rules are not optimal. Furthermore, while the optimality of the simple majority rule holds for certain parameters regardless of the size of the organization, the optimality of a hierarchy or a polyarchy holds only for a very limited size range. The optimality of these rules, under the rare circumstances when they do turn out to be optimal, is highly sensitive to small changes in the size of the organization and to changes in the other parameters that are relevant to the design of optimal architectures. This implies that in the relevant literature and, apparently, in practice, attention is not focused on the right architectures. Appendix A Proof of Lemma 1. It can be shown that ln

p1

^k
n ˆ g ‡ d ‡ n 1 p2 b1 ‡ b2 b1 ‡ b2 8 Fedderson and Pesendorfer (1998) obtain a related result regarding the optimality of a hierarchy in the di¨erent context of strategic voting. Speci®cally, they demonstrate that in certain environments the unanimity rule is never optimal.

162

R. Ben-Yashar, S. Nitzan

Hence,

p1 g‡d 1 p2 ^k
n U c , n
Uc , n U cs ‡ m b1 ‡ b2 b1 ‡ b2 p1 ln g‡d ^k
n > c 1 , n
1 p2 > c 1 , n > …c b1 ‡ b2 b1 ‡ b2 ln

1†s ‡ m Q.E.D.

Appendix B Proof of Lemma 2 It can be shown that p1 ln g ‡ d 1 p2 ^k ˆ ‡ n…b 1 ‡ b2 † b1 ‡ b2 Hence,

0

^k U L ,

g‡d B U @L …b1 ‡ b2 †

p1 1 1 p2 C An b1 ‡ b2

ln

0 ^k > L Therefore,

L

1 g‡d B , ‡ 1 > @L n b1 ‡ b2 8 > > > > > > 1 < vU0 > > > > > > > <

p1 1 1 p2 C An b1 ‡ b2

ln

1 ^ n > > > > > > > > > > > > : …v ‡ 1†d < n U v
d

ln

p1

1 p2 b1 ‡ b2 p1 ln 1 p2 when L > b1 ‡ b2 p1 ln 1 p2 when L < b1 ‡ b2

when L ˆ

Q.E.D.

References Ben-Yashar R (1993) Organizational quality and optimality of the decision-making structure. Ph.D. Dissertation, Bar-Ilan University Ben-Yashar R, Nitzan S (1997) The optimal decision rule for ®xed-size committees in dichotomous choice situations: The general result. Int Econ Rev 38(1): 175±186

Hierarchies and polyarchies

163

Berg S, Paroush J (1998) Collective decision making in hierarchies. Math Soc Sci 35: 233±244 Feddersen T, Pesedorfer W (1998) Convicting the innocent: The inferiority of unanimous jury verdicts under strategic voting. Am Pol Sci Rev 92: 23±35 Koh WTH (1992a) Variable evaluation costs and the design of fallible hierarchies and polyarchies. Econ Letters 38: 313±318 Koh WTH (1992b) Human fallibility and sequential decision making: Hierarchy versus polyarchy. J Econ Behav Organiz 18: 317±345 Koh WTH (1994) Fallibility and sequential decision making. J Inst Theoret Econ 150(2): 362±374 Sah RK (1991) Fallibility in human organizations and political systems. J Econ Perspectives 5: 67±88 Sah RK, Stiglitz JE (1985) The theory of economic organizations, human fallibility and economic organization. Am Econ Rev Pap Proc 75: 292±297 Sah RK, Stiglitz JE (1986) The architecture of economic systems: Hierarchies and polyarchies. Am Econ Rev 76: 716±727 Sah RK, Stiglitz JE (1988) Committees, hierarchies and polyarchies. Econ J 98: 451± 470