Electronic copy available at: http://ssrn.com/abstract=1013329

Centralization versus Decentralization as a Risk-Return Trade-off

Alessandra Arcuri Giuseppe Dari-Mattiacci

Amsterdam Center for Law & Economics Working Paper No. 2007-06

The complete Amsterdam Center for Law & Economics Working Paper Series is online at: http://ssrn.acle.nl

For information on the ACLE go to: http://www.acle.nl

Electronic copy available at: http://ssrn.com/abstract=1013329

Centralization versus Decentralization as aRisk-Return Trade-o¤�

Alessandra Arcuriy

Erasmus University RotterdamGiuseppe Dari-Mattiacciz

University of Amsterdam

February 11, 2009

Abstract

This paper characterizes the choice between centralization and de-centralization as a risk-return trade-o¤ and examines it in a model thatintegrates ideas from committee-decisionmaking and portfolio theories.Centralization, by pooling expertise, rarely yields erroneous decisions;however, when it fails, the consequences are global. In contrast, in adecentralized system, erroneous decisions are more frequent but theirconsequences are locally con�ned. We assess the relative desirability of(de-)centralization in various scenarios with independent versus interde-pendent risks. We further discuss the robustness of the model and therelevance of our results for policymaking.

Keywords : centralization, decentralization, federalism, Condorcet JuryTheorem, risk diversi�cation.

JEL classi�cation : D72, K00, K33.

�We are indebted to the editor, an anonymous referee, Britta Augsburg, Arnoud Boot,Roger Congleton, Bruno Frey, Edward Iacobucci, Jonathan R. Nash, Francesco Parisi,Margherita Saraceno, Jeroen van de Ven, Matthijs van Veelen, Bauke Visser, and Jose�envan Zeben for their invaluable suggestions on how to improve the analysis. We also thankSusan Russell for her skilled editorial assistance. We would also like to thank the participantsin the seminars at the University of Zurich and the University of Amsterdam, and the 2007 an-nual conferences of the International Society for New Institutional Economics at the Universityof Reykjavik, the Canadian Law and Economics Association at the University of Toronto andthe Midwestern Law and Economics Association at the University of Minnesota Law School.This paper was previously circulated as �Multilevel Governance and Risk Diversi�cation�.

yErasmus University Rotterdam, School of Law. Address: Burg. Oudlaan 50, 3062PARotterdam, The Netherlands. Email: arcuri@frg.eur.nl.

zUniversity of Amsterdam (ACLE, CSECLE and Tinbergen Institute). Address:Roetersstraat 11, 1018WB Amsterdam, The Netherlands Email: gdarimat@uva.nl. The �-nancial support provided by the NWO grant 016.075.332 is gratefully acknowledged.

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1 Introduction

The formation of law and policy in contemporary societies is often in�uencedby experts. Expert-driven law is justi�ed by the fact that experts are endowedwith specialized knowledge necessary to design e¤ective policies in technicallycomplex domains. These domains include a wide range of policy issues fromthe regulation of occupational health to the setting of food safety standards,environmental law and policy and the regulation of pharmaceuticals.1 An inter-esting question raised by the nature of expert-decisionmaking is how the level ofgovernance� centralized versus decentralized� a¤ects the decisions made. Thejuncture between expert-decisionmaking and the centralization versus decen-tralization debate is a somewhat unexplored territory in the �eld Law and Eco-nomics. The goal of this paper is to chart this area by examining the risk-returntrade-o¤s entailed by di¤erent levels of governance.Take, for example, the decision to authorize the marketing of Genetically

Modi�ed Organisms (GMOs). Assume that the process can take two forms:either a centralized or a local authority is in charge of the approval decision.Both authorities rely on the opinion of a given number of experts: If the de-cision is centralized, all experts sit together in one committee, whereas, if thedecision is decentralized, the same experts will be distributed among indepen-dent local committees. Assuming that there is a superior decision (for example,it is desirable to authorize the GMO in question), we show that centralization,by pooling expertise, rarely yields erroneous decisions (high returns); however,when it fails, the consequences are widespread (high risk). In contrast, in a de-centralized system, erroneous decisions are more frequent (low returns) but theirconsequences are locally con�ned (low risk). As a main insight, our analysis sug-gests that decentralization, implying the possibility that di¤erent jurisdictionsmake di¤erent decisions, works as a risk-diversi�cation device, thereby limitingsociety�s exposure to risk.The perspective introduced by this contribution pushes the centralization

versus decentralization debate towards a new frontier: It introduces risk asa dimension of the analysis and it emphasizes the role that decisionmakingprocesses may play for the generation of desirable outcomes. While our modelis built around the expert-decisionmaking case, the number of available expertscan be seen as a proxy for the resources devoted to the decisionmaking process.Thus, our conclusions can be generalized to any assembly or committee whosemembers hold imperfect knowledge about the issues at stake.

1 Institutions relying on technical, economic, and legal expertise can be found both at thelocal and at the central level. Examples of federal US institutions include: the OccupationalHealth and Safety Administration (OSHA), created in 1970, the Environmental ProtectionAgency (EPA), established in the same year and the Food and Drug Administration (FDA),which has existed for over a century. The establishment of administrative agencies and author-ities in Europe is more recent. To name just few examples: The European Food and SafetyAuthority (EFSA) created in 2002 and the European Medicines Agency (EMEA) establishedin 1995. The di¤erence in timing between US and Europe might be taken as evidence of aEuropean tendency to move to a higher level of centralization of administrative law closer tothe one already achieved in the US. These issues are further discussed in section 4.

2

After relating our analysis to the existing body of literature, we address thequestion concerning the optimal level of governance in section 2, in a model ofapproval (yes-or-no) decisions, such as in the GMO example. We let the optimallevel of governance depend on several parameters� the quality of the availableexpertise, the total amount of resources devoted to decisionmaking, the degree ofsociety�s risk-aversion and the characteristics of the risks involved (independentversus interdependent risks)� and analyze how changing circumstances a¤ectthe optimal level of governance.2 In section 3, we discuss the robustness ofour model in light of possible relaxations of our main assumptions. Finally, insection 4, we discuss policy implications, summarize our results and state ourconclusions.

1.1 Relation to the literature

The question we address in this paper� whether decisions should be central-ized or decentralized� has long been the focus of scholarly attention from theconventional economic theory of federalism (Tiebout 1956; Stigler 1957) to thetheory of the �rm (Coase 1937) and the study of hierarchy (Sah and Stigliz1986).Advocates of centralization mainly base their claims on the detrimental con-

sequences of heterogeneity and lack of coordination that might result from de-centralized decisionmaking: most prominently, the externality problem.3 Oneinstantiation of the externality problem is the race to the bottom.4 In the 1933case Louis K. Liggett Co. v. Lee, (288 U.S. 517 [1933], 558-559), judge Brandeiswarned against the risk that di¤erent jurisdictions might ine¢ ciently lower theirstandards in order to attract individuals or �rms precisely on these grounds.The above opinion presents a di¤erent side of the coin from that given a

year earlier in New State Ice Company v. Liebmann (285 U.S. 262 [1932], 311),in which decentralization was described as a laboratory of democracy, insofaras jurisdictions use their discretion to develop solutions to common problemsthat can then spread to other states. This view on the pros of decentralizationformed the basis for later theories emphasizing that decentralization enhancesthe government�s proximity to local preferences and conditions (Tiebout 1956;

2Formal proofs of the propositions are available with the authors.3One interesting perspective on this topics can be found in Ellet (1839) who noted that

decentralized ownership of lines of transportation could result in excessive pricing, and hencesuboptimal use, due to what is known in di¤erent strands of literature as complementarymonopoly (Cournot [1838] 1897), double marginalization (Spengler 1950) or the anticommonsproblem (Heller 1998). Ellet advised the US government to centralize ownership in order tosolve this problem. Although from di¤erent perspectives, Bentham (1817) and Weber ([1925]1954) also prized the virtues of centralized decisionmaking by maintaining that codi�cationas opposed to common law reduces the uncertainty of the law and fosters economic and socialdevelopment.

4 In the speci�c �eld of environmental law, Oates and Schwab (1988) and Revesz (1992)cast doubts on the race-to-the-bottom hypothesis concerning environmental standards. To thesame e¤ect, by studying the history of the US Clean Air Act, Revesz (1996) argues that federalregulation (a form of centralized decisionmaking) does not solve problems of transboundaryexternalities.

3

Oates 1972), hence o¤ering citizens a way to choose the community which isclosest to their characteristics. In turn, the ability of citizens to move to otherstates or regions fosters a race to the top, inducing di¤erent jurisdictions to im-prove their production of goods and services (Breton 1996) such as, for example,their corporate laws (Romano 1985).5

Despite the expansion of the literature on the economics of federalism, nostudy has attempted to show the implications entailed by the risk dimensionsinherent to many decisions. We accordingly enrich this analytical frameworkby integrating �ndings from the literature on the Condorcet Jury Theorem(henceforth CJT; Condorcet [1785] 1994)6 with insights from portfolio theory(Markowitz 1952). From the CJT we take the idea that agencies with moreexperts are more accurate than agencies with fewer experts. Thus, according tothe CJT, centralization has the advantage that, by pooling experts, it yields theright decision more often than decentralization. In contrast, portfolio theorysuggests that decentralizing decisions allows for risk-diversi�cation, thus cast-ing a vote in favor of decentralization. There are many important contributionsthat apply the CJT to the study of decisionmaking by juries, committees andlegislative bodies.7 However, none of these contributions has employed the CJTto consider issues of centralization versus decentralization.

2 The optimal level of governance

In this section, we analyze binary yes-or-no decisions, such as the decisionwhether to approve a new GMO. The decision can be made at a centralizedor decentralized level. In the following sections, we build a model that allowsus to measure the performance of centralized versus decentralized decisionmak-ing in terms of returns and risk. We assume that, at the outset, it is notclear whether the GMO should be approved or banned.8 The right decision�

5Decentralization has also been associated with reduced rent-seeking (Brennan andBuchanan 1980), lower corruption (Fisman and Gatti 2002), improved accountability (Tom-masi and Weinschelbaum 2007), less strategic delegation (Besley and Coate 2003), broaderpolitical participation (Inman and Rubinfeld 2000), better communication between local divi-sions (Alonso, Dessein and Matouschek 2008), learning (Nelson and Winter 1982; March 1991)a feeling of participation among citizens (Frey and Stutzer 2002) and a terrorist-proof di¤usionof power (Frey 2004). In a more general spirit, Shumpeter (1934) and Hayek (1978) emphasizethe virtues of competition in fostering economic development and information acquisition.

6See also Grofman, Owen and Feld (1983); Black (1958); Grofman (1975 and 1978); Miller(1986); Young (1988); Ben-Yashar and Nitzan (1997).

7Kornhauser and Sager (1986); Grofman and Feld (1988); Estlund et al. (1989); Austen-Smith and Banks (1996); Feddersen and Pesendorfer (1998); McLennan (1998); Coughlan(2000); List and Goodin (2001); Levmore (2002); Dharmapala and McAdams (2003); Austen-Smith and Feddersen (2006); McGinnis and Rappaport (2008).

8This situation can be modeled by employing an information-aggregation model of approvaldecisions by a group of individuals with common preferences but diverse information (Edelman2002, pp. 333, 338-339). There are only two states of the world, A and B, with the sameprior probability of 50%. Only two decisions are possible: approve, �A, or ban, �B, the newGMO. The outcome � (�; �) depends on the state of the world and the decision taken, with��A; �A

�= �

�B; �B

�= G > 0 and �

�A; �B

�= �

�B; �A

�= 0. This formulation also implies an

assumption of symmetry in the cost and likelihood of errors on either side.

4

correctly approving or banning� yields a positive payo¤ of G, while the wrongdecision� banning a GMO that should be approved or approving a GMO thatshould be banned� yields a (normalized) payo¤ equal to zero.9 Thus, G is thevalue at stake.The decision is made by a regulatory agency relying on the opinion of a

committee of n identical experts, where n represents the amount of resourcesdevoted to decisionmaking. For the purpose of the analysis, we assume thatagencies always follow the advice of their committees and, hence, we use thewords committee and agency interchangeably. Reliance on expert committees isjusti�ed by the technical nature of the decisions. Laymen and politicians do notpossess the necessary knowledge to make an informed assessment: their choiceswould be guesses with a 50% probability of being correct.However, as is often the case, the state of the art is such that not even

experts are always able to pick the right outcome, although their assessmentsare better than those by laymen. Unlike laymen, each expert collects someinformation (a signal) about the problem. This information allows the expertto make an assessment of the situation� whether the GMO should be bannedor approved� which is correct in p > 50% of the cases. In this sense, decisionsby experts are better than decisions by laymen.10 The probability p indicatesthe quality of the scienti�c knowledge available and suggests that all expertspossess an identical level of expertise.When experts gather together, each of them brings in some information

on whether approving or banning is more desirable. After a possibly lengthydiscussion, the committee produces a deliberation that is then adopted by theregulatory agency. We are interested in the probability Pn (p) that a committeeof n experts with expertise p makes the right decision. This probability isin fact equal to the probability that a simple majority of the experts in thecommittee makes the right assessment.11 Intuitively, we expect the committeedecision to improve if there are more experts� who bring in more information�

9We do not imply that wrong decisions have no negative e¤ect. This formulation simplyindicates that the di¤erence between the outcome of a good decision and the outcome of abad decision is G.10 It is assumed that each expert i 2 f1; :::; ng independently receives a private signal �i 2

fa; bg, which is correlated with the state of the world. In particular, for each expert i, wehave Pr (�i = ajA) = Pr (�i = bjB) = p > 50%. Thus, p is the probability that an expert iscorrect in his assessment of the state of the world.11Experts have identical utility functions U (�; �), and derive disutility from wrong decisions:

U�A; �A

�= U

�B; �B

�= 0 and U

�A; �B

�= U

�B; �A

�= � 1

2. The committee takes two votes.

The �rst round is a non-binding communication round in which experts simultaneously revealtheir signals. In the second round, a formal vote determines the �nal decision. Since expertshave a common interest, it is a subgame perfect Nash equilibrium for the experts sincerely toreveal their signals in the communication round (Coughlan 2000, proposition 6). Given sincererevelation, in the formal voting round all experts vote for the alternative that collected morevotes in the �rst round, irrespective of the formal voting rule employed (Coughlan 2000, 382).Thus, this framework supports the CJT. Alternatively, with formal voting without previouscommunication, voting according to one�s signal results as a Nash equilibrium if the optimalvoting rule� in this case, a simple majority� is used (Ben-Yashar 2006). This framework alsosupports the CJT. See also McLennan (1998). For a recent contribution on voting behaviorin committees see Visser and Swank (2007).

5

or if the experts have better expertise� that is, if the information brought in ismore accurate. These results are known as the CJT. The probability that thecommittee makes the right decision can be expressed as follows:12

Pn (p) =nX

i=n+12

�n

i

�pi (1� p)n�i

where the right-hand side is simply the probability that more than half of theexperts receive a correct signal. The CJT enables us to rely on an array ofstandard results. In fact, we have Pn (p) > p, that is, the probability that acommittee makes the right decision is greater than the probability that a singleexpert makes the right decision; moreover, Pn (p) increases at a decreasing rateboth in n and in p and asymptotically approaches 1 as n grows to in�nity or papproaches 1. This means that the agency�s decision improves if more resourcesare devoted to the decisionmaking process, in terms of the number of expertsn, or if their expertise p improves.Given a total number N of available experts� a resource constraint imposed

on the decisionmaking process (Sah 1991, 68)� the question we address here iswhether decisions should be made at a centralized or decentralized level. Ata centralized level decisions are made by one regulatory agency with globaljurisdiction over the entire planet and pooling all N experts. Instead, at adecentralized level decisions are made by N di¤erent agencies, each having localjurisdiction in one of N (identical) regions in which the planet is divided13 andrelying on the advice of a single expert per agency. It is useful to begin theanalysis by posing the problem in such dichotomous terms; later on we willaccount for intermediate levels of decentralization, allowing each local agencyto rely on more than one expert.The fact that the state of the world cannot be perfectly observed gives rise

to two di¤erent problems. One is how to maximize the expected return onthe decision; the other is how to reduce the risk due to the variance of theoutcome.14 This trade-o¤ can be expressed by the following formulation for

12This formulation refers to n as being odd. If n is even and ties are randomly attributed toeither decision and there is an easy transformation rule to bring the analysis back to n beingodd: Pn (p) = Pn�1 (p) (Miller 1996, p. 175).13We use the notions of �planet� and �region� simply to denote the geographical unit of

analysis and its local subdivisions, respectively. Thus, depending on the context, �planet�might refer to the US or the EU as a whole and �region� to each of the (member) states, or�planet�might be the entire world and �region� a country, or else �planet�might indicate acounty and �region�one of its municipalities.14 It is easy to verify that the risks that we consider cannot always be ranked according

to stochastic dominance criteria. A mean-variance approach to the problem allows for moregeneral comparisons, which depend on the degree of individuals�aversion to risk.

6

society�s welfare:15

W = R� �V

where R is the expected return on a decision, V is its variance, and the risk-aversion index � > 0 measures individuals�willingness to accept lower returnsin exchange for less risk. The balance between expected returns and riskinessof decisions will be shown to depend on the level of governance at which thosedecisions are made. In the next subsections, we will discuss the issue of central-ization versus decentralization. Initially, we consider independent risks. Risksare said to be independent when a decision made by a local agency has no im-pact on other regions� that is, there are no externalities across regions. Risksare interdependent if the opposite holds true,16 a scenario that we will considerin section 2.3.

2.1 Centralization

When decisionmaking is centralized, all of the N experts are pooled togetherin one agency. Since the probability that a committee makes the right decisionincreases with the number of experts, employing all of the experts in one com-mittee (n = N) gives the most accurate decision possible, denoted by PN (p).For a decision with stakes G, the expected return and the variance of central-ization are easily calculated as follows:

R1 = PN (p)G

V1 = PN (p) (1� PN (p))G2

We will examine the performance of decentralized decisionmaking againstthis benchmark.17

2.2 Decentralization

Decentralized decisionmaking employs N agencies consisting of one expert each.In this scenario, the probability that the agency makes the right decision istrivially the same as the probability that the expert decides correctly, P1 (p) = p.

15 In turn, assuming that all individuals are identical, the social welfare function can besupported by a utilitarian summation of individual utility functions of the following form: Yj =1M(R� �V ), for j = 1; :::;M . It should not come as a surprise that, unlike individuals�utility

functions, experts�utility functions were assumed to be consistent with the maximization ofthe expected return but do not include concerns about the variance. Experts, in fact, operateat a local level, while the variance is a concern only if one takes a global perspective.16Note that local outcomes may be dependent on each other for other reasons. For instance,

local agencies might in�uence each other�s decisions. In this paper, we assume that agenciesdecide independently of each other.17Our model is based on an assumption of no decision costs. That is, increasing the size

of the committee does not trigger any additional cost due to more lengthy or cumbersomedecisionmaking processes. In a number of circumstances it might be more plausible to assumepositive decision costs that increase as the committee size increases. Should this be the case,the advantage of centralization in terms of better decisions will be constrained by concernsabout the costs of decisions.

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Because local outcomes are independent of each other, expected return andvariance of the decision are simple sums over the N local expected returns andvariances. Moreover, as each region is 1

N of the planet, the local payo¤ is GN ,

that is the stakes of the decision at the global level divided by the number ofregions. Accordingly, expected return and variance are as follows:

RN = pG

VN =1

Np (1� p)G2

Centralization has a clear advantage over decentralization: it yields largerexpected returns (R1 > RN ). This result occurs because the decision made bya centralized agency is more accurate than the decisions made by each of thelocal agencies (PN (p) > p). This is true for any level of expertise but is morepronounced for interior values of p. In fact, when p approaches 12 (complete lackof expertise) or 1 (very accurate expertise), PN (p) approaches p and hence thereturns to centralized and decentralized decisions are the same (see �gure 1).

0.5 0.6 0.7 0.8 0.9 1.00.5

0.6

0.7

0.8

0.9

1.0

p

R

Figure 1 : Expected returns of decisions under centralization (dashed curve) versusdecentralization (solid line) with G = 1 and N = 9.

With respect to risk, however, there are two countervailing e¤ects. On theone hand, centralized decisions are more tightly clustered around the right deci-sion than decentralized ones� in fact, PN (p) (1� PN (p)) is less than p (1� p)�to the e¤ect that the variance tends to be lower under centralization. On theother hand, the outcome is the same for all regions in centralized decisionmak-ing while it may vary under decentralization. The possibility for di¤erent localdecisions realizes a spreading of the risk, which in turn allows for lower varianceunder decentralization� re�ected by the term 1

N . Which of these two e¤ectsprevails depends on the total number of experts N and their expertise p.

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0.5 0.6 0.7 0.8 0.9 1.00.00

0.05

0.10

0.15

0.20

0.25

p

V

Figure 2 : Variance of decisions under centralization (dashed curves) versusdecentralization (solid curves) with G = 1 and N = 9 (thick curves) or N = 15

(thin curves).

Figure 2 depicts the variance of decisions when there are N = 9 experts.With decentralization the variance is hill-shaped, while centralized decision-making yields a bell-shaped function. The �gure only depicts the halves ofthose functions that apply to our setting (p > 1

2 ). There is a level of expertise p̂at which the solid curves cross (p̂ is about 0:8 in the �gure). When the availableexpertise is poor (p < p̂), decentralization attains a lower variance; in contrast,with advanced expertise (p > p̂) centralization performs better. This resultsuggests that decentralization is a substitute for lack of expertise in terms ofreducing the riskiness of decisions. Is this also the case with respect to N?When N increases, the variance of decentralized decisions decreases because

there are more regions and hence spreading risk is easier; the variance of central-ized decisions also decreases because the accuracy PN (p) of decisions increases.Both (dashed) curves move downwards and to the left, crossing at a point p̂0 < p̂(p̂0 is about 0:76 in the �gure). Thus, centralization becomes desirable for abroader range of expertise, suggesting that decentralization is also a substitutefor a lack of experts as it is for a lack of expertise.We can now draw some implications for social welfare based on the choice

between centralization and decentralization. When the available expertise isadvanced, centralization is to be preferred as it yields more accurate and lessrisky decisions. With poor expertise, decisions under centralization, althoughmore accurate, are also more risky than under decentralization. The choice thendepends on how much weight is given to riskiness. In turn, this depends notonly on the risk-aversion index, which obviously makes decentralization moredesirable, but also on the stakes of the decisionmaking process G, which appearsas a simple term in the expected return but is squared in the calculus of thevariance. Hence, when the stakes are higher� that is, when the right decision isvery valuable compared to the wrong decision� decentralization is again more

9

desirable. Finally, the number of available experts in�uences this balance bylowering the threshold of expertise above which centralization dominates, thusundermining the scope of decentralization. The following proposition summa-rizes these results.

Proposition 1 With independent risks, if p � p̂, centralization is preferableto complete decentralization. If p < p̂, centralization is preferable for a lowdegree of risk-aversion � or small stakes G; otherwise complete decentralizationis preferable. The threshold p̂ decreases in the number of experts N .

2.3 Governance with interdependent risks

Although some risks might be independent, other risks are not. A bad decisionmade in one region might well a¤ect neighboring regions or even the entireplanet. It is not super�uous to stress once more that, although we are analyzingthe case of interdependency, this only refers to outcomes and not to decisions,which we still assume to be made independently in each region. There arevarious ways in which interdependencies could play a role in this analysis, butthere is something all formulations will have in common. When the outcomesare more dependent on one another, the advantage of decentralization in termsof risk-diversi�cation tends to fade away.To show that decentralization may still play a role, let us examine an extreme

scenario in which the decisions made by di¤erent agencies are aggregated toproduce a unique global outcome that applies to all regions; that is, the risk-diversi�cation advantage is completely lost. Let us de�ne interdependency tomean that if some regions make the right decision, this converts into a goodoutcome for the entire planet irrespective of the decisions other regions made.18

Note that in this case there is no advantage in diversifying risk as the outcomeis either good or bad for all regions. Accordingly, the problem in this scenario ismerely how to maximize the likelihood of a good outcome. In order to clarify therole played by interdependent risks, we will examine three stereotypical cases.

2.3.1 Best-shot risks

With best-shot risks, such as the risk that an endangered species will becomeextinct, local policies are perfect substitutes in the achievement of the globalpolicy and, thus, success at the local level results in success at the global level.Here, making the right decision in any of the regions means that the outcomewill be good for the whole planet, irrespective of whether other regions have

18Using terminology borrowed from reliability theory (Harrison 1965), the problem is one ofdetermining the reliability of an r-out-of-N system: the system works if at least r componentswork. The di¤erent types of risks analyzed below can be interpreted as follows: best-shotrisks describe a 1-out-of-N system (a parallel system), majority risks describe a N+1

2-out-of-N

system and weakest-link risks describe a N -out-of-N system (a series system). The probabilitythat at least r regions take the right decision is thus given by: Qr =

PNi=r

�Ni

�pi (1� p)N�i.

Accordingly, expected return and variance with interdependent risks are as follows: RN =QrG and VN = Qr (1�Qr)G2.

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also made the right decision. The formula governing these types of risks canbe written as 1 minus the probability that all regions make the wrong decision:Q1 = 1� (1� p)N > PN (p).The probability of success in best-shot risks is greater with decentralization

than with centralization. There are two forces at work. Although decentraliza-tion triggers a negative e¤ect, as its accuracy is less than that of a centralizedagency, the likelihood that all regions make the bad decision is small and turnsout to be less than the probability that the centralized agency makes the wrongdecision. Thus the likelihood of a good outcome can be improved by decentral-izing decisionmaking.

2.3.2 Majority risks

In the intermediate case of majority risks, success or failure depends on what themajority of the local agencies decide. The situation described corresponds to arisk that can be avoided if, at the local levels, there are more right decisions thanwrong ones. The probability of success in majority risks is the same whetherdecisions are centralized or decentralized. This is due to the fact that decisionsby local agencies in a decentralized system are aggregated in the same way asthe votes of individual experts in a centralized agency: QN+1

2= PN (p).19

2.3.3 Weakest-link risks

With weakest-link risks, such as the risk of a viral epidemic, local policies arestrict complements for the achievement of the global policy and, hence, localfailure results in global failure. Here, a wrong decision in any of the regionsentails a bad outcome for the whole planet. The formula governing these typesof risks can be written as the probability that all regions make the right decision:QN = p

N < PN (p).Here increasing decentralization has two negative e¤ects. The �rst negative

e¤ect is that a local agency is less accurate than a centralized one; the �rstnegative e¤ect is that for the outcome to be good all agencies have to make theright decision, which further lowers the odds that the outcome will be good. Thecombined e¤ect is that the likelihood of success decreases with decentralization.

To summarize, the result of the decisionmaking process heavily depends on thetype of risk. Decentralization has two countervailing e¤ects. On the one hand,it entails a lesser degree of accuracy in each of the regions, thus potentiallyreducing the expected return on the decision. On the other hand, it makes itmore likely that at least some regions will adopt the right decision. Whetherone e¤ect dominates the other depends on the proportion of regions that needto make the right decision for the global outcome to be good. The followingproposition follows directly from the previous analysis.

19Formally, QN+12

=PN

i=N+12

�Ni

�pi (1� p)N�i = PN (p).

11

Proposition 2 With best-shot risks, complete decentralization is preferable tocentralization; with majority risks, complete decentralization and centralizationare equivalent; �nally, with weakest-link risks, centralization is preferable.

2.4 Intermediate levels of governance

In the preceding subsections we have sought the optimal level of governance ina dichotomous fashion, only allowing for complete centralization or completedecentralization. In this section we extend the analysis to intermediate levelsof governance in between these two extremes. We thus employ a more generalframework, which encompasses centralization and complete decentralization asspecial cases. The planet is partitioned into k regions of equal size, so thatin each region decisions are independently made by a local agency. The Nexperts will be accordingly distributed among the k local committees, so thateach committee is composed of Nk experts. By varying the parameter k, onecan describe centralization (k = 1) and varying degrees of decentralization (1 <k � N). As the number of regions k grows, we have an increasing degree ofdecentralization and a decreasing number of experts per agency; the limit case(k = N) corresponds to complete decentralization, which was examined in theabove section.It is also possible in this framework to show that, with independent risks,

if expertise is better than a certain threshold (p > �p), making decisions at acentral level yields both less risk and greater accuracy. Therefore, centraliza-tion is preferable and the optimal partitioning of the planet would be in oneall-encompassing region (k� = 1). However, if expertise is poor (p < �p), thereis a trade-o¤ to be addressed. While decentralization progressively lowers ac-curacy, it also yields better risk spreading; thus, some level of decentralizationmight be optimal (k� < N), but not necessarily complete decentralization asin the dichotomous case of the previous section. As in the previous case, theoptimal balance between risk and expected returns and, hence, the optimal levelof decentralization k� depend on the parameters of the model. Finally, with in-terdependent risks, allowing for intermediate levels of decentralization does notchange the results obtained in the dichotomous case, since either centralizationor complete decentralization is optimal.

3 Discussion

In the previous sections, we have made a number of assumptions. Here wediscuss four of them, which have particular relevance for the interpretation andthe domain of application of our analysis.

3.1 Right and wrong decisions

We base our analysis on the idea that there is a �right�decision, one that mostbene�ts society. Such a right decision exists in those cases in which mankind

12

share a common goal, even if the right decision is generally not known ex ante�in fact, identifying the right decision ex ante is the problem we address in thispaper. This is a limitation of our approach, because individuals may di¤er intheir goals; in such cases, unanimous consent cannot be reached. Consequently,the notion of a �right decision�necessarily rests on a previous choice of a socialwelfare function that aggregates individual preferences.20 However, the formu-lation of a social welfare function is a well-known and more general problem inwelfare economics, which is not speci�c to our framework and we do not ad-dress it here. Thereby, our analysis is necessarily based on the assumption thatsociety�s goal has been set.

3.2 Decentralized knowledge

In the model we assume that scienti�c knowledge is exogenously given and doesnot depend on the level of governance. However, one could entertain the ideathat there might be some inherent informational advantages in going local. Ifthis is true, the expertise would increase with decentralized decisionmaking;in other words, p would become endogenous to the model. Let us considertwo plausible instances. The �rst is that local decisionmakers might betterknow the preferences of their constituencies. The main point is that what isconsidered an acceptable risk by one constituency may not be considered soby another.21 Second, even with homogenous preferences, local conditions orexposure to risk may vary. To illustrate, take two di¤erent standards commonlyused by regulatory food agencies: the acceptable daily intakes (ADI) and themaximum residue levels (MRLs) of potentially harmful compounds present infood. The optimal ADI may di¤er across regions because preferences concerningacceptable risks vary; with the same preferences, the optimal ADI standardsshould in theory be the same. In contrast, optimal MRLs might vary evenwith homogeneous preferences, because of local variations in food consumptionbehavior. Local experts are likely to better evaluate both local preferences andlocal conditions, such as exposure to risks, which translates into an increase in pwhen decentralizing. Ceteris paribus, decentralization becomes more desirable.In our model this would imply a larger p in the case of decentralization andquantitatively (but not qualitatively) bias the results away from centralization.

3.3 Centralized diversi�cation

Knowledge acquired by a centralized agency could be spread to local agen-cies or else the central government could directly implement di¤erent policiesin di¤erent regions. Apparently, in this way one could escape the risk-return

20Miller (1986) tackles this problem by considering as correct the alternative that wouldreceive a majority of votes ex post, when all information is available. See also Ladha (1992,620).21There are many examples of geographical di¤erences in risk perceptions and risk-aversion

towards food safety; see Schroeder et al. (2007) and Gaskell et al. (2004). For the seminalstudies on risk perceptions, see Slovic, Fishho¤ and Lichtenstein (1980) and Slovic (1987).

13

trade-o¤ and achieve the characteristic high returns of centralization and therisk-diversi�cation typical of decentralization. However, this scenario might beproblematic in some cases. In fact, assume that the centralized committee ofexperts has deliberated in favor of policy A. If this advice is transmitted tolocal agencies, the likely result is that all local agencies will implement policyA, thus risk-diversi�cation will not be achieved. Should a central agency try todiversify by implementing the superior policy A in some regions and the inferiorpolicy B in other regions, this approach would likely encounter strong politicalopposition of the not-in-my-backyard type. Again, once policy A is found to bethe superior policy, it will probably be implemented across the board.We concede that in some cases it is possible that a central agency might lo-

cally diversify its policies, but our main observation that decentralization gen-erates risk-diversi�cation remains qualitatively valid, even though the resultsmight be quantitatively a¤ected.

3.4 Individual mobility

The analysis of the previous sections is based on an implicit assumption thatindividuals care about global and not only local outcomes. This approach in turnimplies that individuals can move from one region to another should the wrongpolicy be implemented in their region.22 Instead, if individuals cannot moveor else they only care about local outcomes, the analysis changes. In this case,in fact, there would be no advantage in having di¤erent regions make di¤erentdecisions, because all individuals care about is the outcome in their own region.Paradoxically, when individuals are only concerned about local outcomes, theyfavor centralization, as this is the level of governance that guarantees the highestprobability of making the right decision overall.

4 Summary and conclusions

The issue of whether to allocate regulatory powers to a central or local agenciesremains disputed in the context of European and American administrative lawas well as in the international arena, where an increasing number of internationalbodies is setting world-wide harmonized standards.Within the European context, the number of expert agencies has boomed

in the recent past: ten agencies were created in only �ve years (1990-1995).23

22Moving to another jurisdiction generates congestion costs� more individuals have to sharein the same pie� but allows individuals to overcome some of the e¤ects of a bad decision intheir jurisdiction by participating in the bene�t of a good decision in another jurisdiction.23These agencies are: the European Environmental Agency (EEA); the European Training

Foundation (EFT); the European Monitoring Centre of Drugs and Drug Addiction (EM-CDDA); the European Agency for the Evaluation of Medicinal Products, renamed in 2004European Medicines Agency (EMEA); the O¢ ce for Harmonization in the Internal Market(OHIM); the European Agency for Safety and Health at Work (EU-OSHA); the TranslationCentre for Bodies of the European Union (TC); the Community Plant Variety O¢ ce (CPVO);the European Foundation for the Improvement of Living and Working Conditions (EFILWC);and the European Centre for the Development of Vocational Training (CEDEFOP). These

14

The creation of several agencies in a relatively short period of time could beseen as an evidence of increasing centralization.24 An interesting example ofthe shift towards more centralization is provided by the regulation of pharma-ceuticals and the creation of the European Medicines Agency (EMEA). Beforethe establishment of the EMEA, pharmaceuticals were approved by nationalauthorities of various Member States. Under the new system, applications forthe authorization of marketing of innovative medicines must be submitted tothe EMEA.25 It is interesting to note that, like in our model, the new central-ized system relies on a wide number of experts� there are currently over fourthousand experts� pooled from the Member States.26

Notwithstanding the creation of several agencies, the allocation of regulatorypowers between European and Member States institutions is not always clear cutand the debate over whether to centralize or decentralize is still very intense.Insights from our model could help the European regulator assess the e¤ectsof granting more or less regulatory powers to existing (and future) regulatoryagencies. In this sense, notice that the jurisprudence of the European Court ofJustice, which has tended to allow more decentralization in the presence of highscienti�c uncertainty (Vos 1999, 47), is in line with our results.Our analysis may be similarly applied to the US context. While federal

agencies are an older phenomenon in the US, issues of risk-diversi�cation or ofadvantages generated by pooling experts never entered the debate on environ-mental federalism or other analyses of attribution of regulatory powers to federalversus national agencies (Revesz 1992, 1996 and 2000; Esty and Geradin 2000).Additionally, it is worth emphasizing that the harmonization of various typesof standards is also taking place at the international level where a number ofstandard-setting institutions are directly or indirectly gaining regulatory power,to the extent that some scholars speak of global administrative law.27 One il-

bodies are commonly referred to as European agencies, even if in the o¢ cial name they arecalled authorities or foundations. Today there are twenty-four agencies in Europe; while notall these agencies perform regulatory tasks, most have an important in�uence on the regulatoryprocess. For an analysis of the functions performed by these agencies see Chiti (2000).24The jargon used in some European legal texts may be confusing as decentralization is not

only used to denote the allocation of (regulatory) powers at the local level but also to denotethe functional division of powers (for example, the creation of specialized bodies to deal withspeci�c issues).25EMEA became operational in 1995. Under the previous system a process of �automatic�

mutual recognition was established by law; however, it never worked well in practice, leadingto a de facto highly decentralized system. It should be noted that under the new centralizedsystem, the decision is technically adopted by the Commission; despite that, the Commis-sion�s decision is generally in accordance with the Agency�s opinion. Additionally, it is worthremembering that the centralized authorization procedure is compulsory only for biotechno-logical medicines and it remains optional for other highly-technological medicines. Moreovera decentralized market authorization procedure remains; however, the EMEA still plays animportant role under this more decentralized system. For a more detailed description of theseprocesses see Vos (1999, 206-50).26EMEA operates through various technical committees, including the Committee for Medi-

cinal Products for Human Use (CHMP) and the Committee for Veterinary Medicinal Prod-ucts (CVMP), both counting on experts from the Member States. The CHMP, for example,is mainly composed by 54 members, two for each each of the 27 EU Member States.27For a seminal work in this area see Kingsbury, Krisch and Stewart (2005). Institutions

15

lustrative example is the Codex Alimentarius Commission, created in 1963 asa joint organ of the Food and Agriculture Organization (FAO) and the WorldHealth Organization (WHO). The food safety standards established by Codex�while mainly of a non-binding nature� are gradually acquiring more force dueto the interplay of risk law with WTO law (Charnovitz 2005; Masson-Matthee2007). If centralization of expert-driven law is also taking place at a global level,it is crucial to investigate whether and to what extent this process is desirable.Our model provides an analytical framework to begin research in this relativelyunexplored area.Furthermore, our analysis has bearing on a vast array of regulatory processes

that are performed by technical bodies. While our contribution is most directlyapplicable in the �eld of risk regulation (by which we refer to health, safety andenvironmental regulation), its results may be considered generally applicablebecause risk is virtually present in any decision and information is rarely perfect.What are the determinants in favor of more or less centralization of reg-

ulatory powers into a single agency? Cutting horizontally through di¤erentstrands of literature, we have argued that decentralization can function as arisk-diversi�cation device. Our analysis has re-conceptualized the notion of op-timal level of governance by unveiling a risk-return trade-o¤. Most prominently,we have characterized decentralization as a substitute for scienti�c expertise.We have shown that with independent risks� such as local pollution problems�

the choice between centralization and decentralization crucially depends on thelevel of scienti�c expertise available. If advanced expertise is available, central-ization guarantees both more accurate decisions and less risk. Instead, withpoor expertise, while centralization yields more accurate decisions, decentral-ization lowers risk. The balance is in favor of decentralization if the degreeof risk-aversion is su¢ ciently large and stakes are high. This result suggeststhat decentralization is a desirable solution for decisions that are important tosociety, a¤ected by serious scienti�c uncertainty and for which society is risk-averse. If resources devoted to the decisionmaking process are increased� anincrease in the number of available experts, in our model� the critical thresh-old for the expertise is reduced, making centralization desirable at lower levelsof expertise. Thus, a better-funded decisionmaking process is more e¢ cientlycentralized than a comparable process relying on less resources. We have alsoanalyzed interdependent risks, showing that decentralization plays a role in asubset of cases, namely, best-shot risks.Our model has considered a clean set of cases in which scienti�c uncer-

tainty plagues the decisionmaking process concerning balanced issues, where itis important to make the right decision but errors on either side have the same

functioning as global regulators include the Organisation for Economic Co-operation andDevelopment (OECD) network of committees, the World Trade Organization (WTO) com-mittees, the World Intellectual Property Rights Organization (WIPO), the World HealthOrganization (WHO), the International Standards Organization (ISO), and the InternationalLabour Organization (ILO). It is beyond the scope of this paper to discuss the extent towhich these institutions perform a role of centralized regulatory decisionmaking. For such ananalysis see the Global Administrative Law project: http://www.iilj.org/GAL/default.asp.

16

impact. Future research might enrich this framework by considering di¤erentlikelihoods and costs for type I and type II errors, introducing some form ofstatus-quo bias, considering regulatory capture, and including learning explic-itly in the model.

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21

Appendix (not in published article)

Here we provide a formal proof of propositions 1 and 2 in the paper and of theresults concerning intermediate levels of governance. The following results willbe used in the proofs. From Mood (1950, p. 235):

Pn (p) =nX

i=n+12

�n

i

�pi (1� p)n�i

= n

�n� 1n�12

�Z p

0

xn�12 (1� x)

n�12 dx (1)

with the following �rst and second derivatives:

P0

n (p) = n

�n� 1n�12

�pn�12 (1� p)

n�12 > 0

P00

n (p) =1� 2p2

n (n� 1)�n� 1n�12

�pn�32 (1� p)

n�32

=(n� 1) (1� 2p)2p (1� p) P

0

n (p) < 0

This formulation refers to n odd. If n is even, ties are randomly attributedto either decision and there is an easy transformation rule to bring the analysisback to n odd: Pn (p) = Pn�1 (p) (Miller, 1996, p. 175). From Boland (1989, p.181):

1� Pn (p) = Pn (1� p) (2)

Finally, from Berg (1997, p. 564):

Pnk (p) > Pk (Pn (p)) (3)

Remark 3 The variance of a decision Vk decreases in p. For k = N , Vkdecreases at an increasing rate; for 1 � k < N , Vk decreases at an initiallyincreasing and then decreasing rate.

Proof. Since k cannot take all integer values between 1 and N , let us �rstde�ne a set of feasible odd values for k and n � N

k . Let T = ft1; :::; tT g be anon-empty set of T prime numbers greater than 2 and let Tn and Tk be twosubsets of T such that Tn [ Tk = T and Tn \ Tk = ?. Let N =

Qi ti 2 T,

n =Qi ti 2 Tn if Tn 6= ? (n = 1 otherwise), and k =

Qi ti 2 Tk if Tk 6= ?

(k = 1 otherwise). Note that nk = N .Let us now consider 1 � k < N . The variance is

Vk =1

kPn (p) (1� Pn (p))G2

22

withV

0

k =1

k(1� 2Pn (p))P

0

n (p)G2 < 0

and, using (1),

V00

k =1

k

h(1� 2Pn (p))P

00

n (p)� 2P02

n (p)iG2

=1

k

P0

n (p)

2p (1� p) (� � �)G2

where:

� = (2p� 1) (2Pn (p)� 1) (n� 1) > 0� = 4p (1� p)P

0

n (p) > 0

Note that sign�V

00

k

�= sign (� � �); since � is zero at p = 1

2 and increases

in p, while � is zero at p = 1 and decreases in p, V00

k changes sign from neg-ative to positive as p increases. Thus Vk decreases in p at a rate that is ini-tially increasing and then decreasing, as in �gure 2.2. Instead, for k = Nwe have VN = 1

N p (1� p)G2, which is a strictly concave function of p with

V0

N =1N (1� 2p)G

2 < 0 and V00

N = � 2NG

2 < 0.

Lemma 4 There exists a unique p̂k :�Vk = VN and 1

2 < p̂k < 1.

Proof. Vk = VN implies

1

kPn (p)

2 � 1

kPn (p) +

1

Np (1� p) = 0 (4)

and

Pn (p) =1

2+1

2

r1� 4

np (1� p) � Fn (p) (5)

Note that p̂k is a solution to (4) i¤ it satis�es the condition in (5). It hasbeen shown above that Pn (p) is a concave function of p and it is easy to showthat Fn (p) is a convex function of p:

F 0n (p) =2p� 1

nq

1n (n� 4p (1� p))

> 0

F 00n (p) = 2(n� 1)

q1n (n� 4p (1� p))

(n� 4p (1� p))2> 0

Thus, Pn (p) crosses Fn (p) at most twice. Note that Pn (p) crosses Fn (p)from above at p = 1, since we have Pn (1) = Fn (1) = 1 and 0 = P

0

n (1) <F

0

n (1) =1n . Moreover, Pn

�12

�< Fn

�12

�. Thus, there exists one and only

23

one p̂k such that Pn (p) < Fn (p) for 12 < p < p̂ and Pn (p) > Fn (p) for

p̂k < p < 1. Since the discriminant is positive, the left-hand side of (4) isnegative (Vk > VN ) for p < p̂k and positive (Vk < VN ) for p > p̂k. It followsthat any variance function Vk for a level of decentralization 1 � k < N (note thatk = 1 corresponds to centralization) crosses the variance function VN (k = N ,complete decentralization) from above at p̂k.

Remark 5 p̂k increases in k and decreases in N .

Proof. First note from (5) that p̂k is a function of n = Nk . Thus p̂k increases in k

(for a given N) and decreases in N (for a given k) i¤ p̂k decreases in n, which weneed to show here. We have calculated the value of p̂k using Pn (p)�Fn (p) = 0for n taking odd values between 3 and 35. The plot is shown below. Suchvalues of n cover situations with N � 35 and any feasible 1 � k � N . Scienti�ccommittees are likely to be small in size, hence this simulations are a su¢ cientproof of our results. We conjecture that these results are also valid for any kand N greater than 35.

0.75 0.80 0.85 0.90 0.95 1.00

­0.001

0.000

0.001

0.002

0.003

0.004

0.005

0.006

p

P(p)­F(p)

n=3n=35

Figure 4: Values of p̂k for n between 3 and 35.

The �gure above shows the results of our numerical calculations.

Proof of proposition 1. Let p̂ � p̂1. From lemma 4 it follows that V1 > VNfor p < p̂ and V1 < VN for p > p̂. Thus, for p > p̂, we have R1 > RNand V1 < VN ; hence, social welfare in (??) is maximized by centralization. Incontrast, for p < p̂, we have R1 > RN and V1 > VN . In this case, social welfareis maximized by centralization if the weight given to the variance is small, that

24

is if � or G are small. Otherwise complete decentralization is desirable. Fromremark 5 it follows that p̂ decreases in N .

Proof of proposition 2. With best-shot risks, the inequality 1� (1� p)N >PN (p) can be rewritten as:

(1� p)N < 1� PN (p) (6)

Using (2), we can write:

1� PN (p) = PN (1� p)

= (1� p)N +N�1Xi=N+1

2

�N

i

�pN�i (1� p)i > (1� p)N

which shows that (6) always holds true. The claims concerning majority risksand weakest-link risks are self-evident.

Lemma 6 There exists a unique p̂hk :�Vk = Vh, with 1 � k < h < N and 1

2 < p̂hk < 1

.

Proof. Vk = Vh implies

Pn (p) =1

2+1

2

r1� 4m

nPm (p) (1� Pm (p)) � Fn (Pm (p))

where Nk � n > m � Nh . To prove the lemma, let us rewrite Pn (p) as a function

of Pm (p): G (Pm (p)) � Pn (p) and formulate the problem in a similar fashionas in lemma 4, which we will then be able to apply. Using @G

@p = P0

n (p) we have

@G (Pm (p))

@p=

@G (Pm (p))

@Pm (p)P

0

m (p) = P0

n (p)

) @G (Pm (p))

@Pm (p)=P

0

n (p)

P 0m (p)

> 0

Likewise, using @2G@p2 = P

00

n (p), we have

@2G (Pm (p))

@p2=

@2G (Pm (p))

@P 2m (p)P

0

m (p) +@G (Pm (p))

@Pm (p)P

00

m (p) = P00

n (p)

) @2G (Pm (p))

@P 2m (p)=P

00

n (p)�P0n(p)

P 0m(p)

P00

m (p)

P 0m (p)

=1� 2p2p (1� p)

P0

n (p)

P 0m (p)

(n�m) < 0

25

Thus, G (Pm (p)) is a concave function of Pm (p). Moreover, we have G (1) =Fn (1), G

0(1) < F

0

n (1) and G�12

�< Fn

�12

�, which all follow from straightfor-

ward calculations. Let us now remark that we can calculate G0(1) as follows

limp!1

G0(Pm (p)) = lim

p!1

P0

n (p)

P 0m (p)

= limp!1

n�n�1n�12

�m�m�1m�12

�pn�m2 (1� p)n�m2 = 0

It follows, as in lemma 4, that there exists a unique value of Pm (p) suchthat G (Pm (p)) = FN

k(Pm (p)). Since PN

h(p) is monotone increasing in p, there

exists a unique p̂hk that satis�es the former condition.

Lemma 7 If Vh and Vk, with h > k, cross at p̂hk , then Vg, with g > h, crossesVk at p̂

gk and Vh at p̂

gh with p̂

hk > p̂

gk > p̂

gh.

Proof. The proof follows trivially from lemma 4 and remark 5. Note that thisresult implies that Vk decreases in k for p < p̂ and any 1 � k � N .

Proposition 8 With independent risks, if p � p̂ the optimal level of governanceis k� = 1 (centralization). If p < p̂ the optimal level of governance k� increasestowards decentralization in the degree of risk aversion � and the stakes G, anddecreases towards centralization in the number of experts N and in the expertisep.

Proof of proposition 8. It follows from lemma 7 that, the variance functionV1 crosses the variance function VN from above at p̂ after having crossed all othervariance functions Vk with 1 < k < N . For p � p̂, the lowest variance function isV1; thus, centralization is optimal (k� = 1) for any levels of the parameters, sinceit yields both greater expected returns and lower variance than any other levelof k. It also follows from lemma 7 that all variance functions Vk cross VN fromabove after p̂. Thus, for p < p̂ the lowest variance function is VN . In this case,centralization maximizes the expected return, while complete decentralizationminimizes the variance. The expected return Rk decreases in k; from lemma 7we know that also the variance Vk decreases in k if p < p̂. Thus, by reducingk we have greater expected returns at the price of greater risk and, vice versa,by increasing k we have lower risk at the price of lower expected returns. Thesocially optimal level of k depends on the weight given to Vk in terms of �and G; hence, k� increases in � and G. Finally, we know that for any k andfor p � p̂, the variance Vk decreases in p and� as a direct consequence of fromlemma 7� in N . It follows that when p or N increases, the same expected returncorresponds to lower variance and hence k can be further increased to attaingreater expected returns. Thus, k� increases in p and N .

26

Proposition 9 The results of proposition 2 also apply to the case of interme-diate levels of governance

Proof. If we allow for intermediate levels of governance, the probability of agood outcome in the case of k agencies is

Qr (p; k) =kXi=r

�N

i

�PN

k(p)

i�1� PN

k(p)�k�i

(7)

With best-shot risks, (7) becomes

Q1 (p; k) = 1��1� PN

k(p)�k

The level of k that maximizes Q1 (p; k) is k� = N , complete decentralization.

This is the case if 1��1� PN

k(p)�k< 1�

�1� PN

N(p)�N. By letting n = N

k and

rearranging the previous inequality can be reduced to (1� Pn (p)) > (1� p)n,which is always satis�ed by virtue of (??). Thus, even when intermediate levelsof decentralization are feasible, it remains optimal to decentralize governancecompletely. With majority risks, (7) becomes

Q k+12(p; k) =

kXi= k+1

2

�k

i

�PN

k(p)

i�1� PN

k(p)�k�i

(8)

Note that Q k+12(p; k) in (8) can be written as Pk

�PN

k(p)�, which is the

same as the probability that a committee of k members takes the right decision,where the probability that each member of the committee votes for the rightdecision is given by PN

k(p). This is in turn the probability that a committee of

Nk experts takes the right decision, given a probability p that each expert takesthe right decision. From the perspective of calculating the probability thata good outcome results, majority risks are analogous to an indirect majoritysystem.The inequality in (3) suggests that one (direct majority) committee of nk

members has a larger probability to take the right decision that a two-step pro-cedure in which �rst k di¤erent committees of n members decide independentlyand then they each send a delegate to a an assembly of the k delegates, whovote again and take the �nal decision. Applying this result to our setting im-

plies Qk�p;

Nk +1

2

�< Q1

�p; N+12

�= PN (p), that is, any intermediate level of

decentralization achieves a lower probability of a good outcome than completedecentralization and centralization. With weakest-link risks, (7) becomes

Qk (p; k) = PNk(p)

k

It is easy to see that by reducing k to 1 the former probability improves,suggesting that centralization fares better than any level of decentralization.

27

References

[1] Berg, S. (1997), �Indirect Voting Systems: Banzhaf Numbers, MajorityFunctions and Collective Competence,� 13 European Journal of PoliticalEconomy, 557-573.

[2] Boland P. J. (1989), �Majority Systems and the Condorcet Jury Theorem,�38 The Statistician, 181-189.

[3] Miller, N. R. (1986), �Information, Electorates, and Democracy: Some Ex-tensions and Interpretations of the Condorcet Jury Theorem,� in Grofman,B. and Owen, G. (eds.), Information Pooling and Group Decision Making,Greenwich, CT: JAI Press.

[4] Mood, A. M. (1950), Introduction to the Theory of Statistics, New York:McGraw-Hill.

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