Cost Sharing under Increasing Returns: A Comparison of Simple Mechanisms

GAMES AND ECONOMIC BEHAVIOR13,225–251 (1996)ARTICLE NO. 0035

Cost Sharing under Increasing Returns: A Comparison ofSimple Mechanisms

Herve Moulin∗

Department of Economics, Duke University, Durham, North Carolina27708-0097

Received September 16, 1993

A technology with decreasing marginal costs is used by agents with equal rights. Eachagent demands a quantity of output and costs are divided by means of a fixed formula.Several such mechanisms are compared for the existence of Nash equilibrium demandprofiles and for the equity properties of these equilibria. Among three mechanisms, averagecost pricing, the Shapley–Shubik cost sharing, and serial cost-sharing, only the latter twopossess at least one Nash equilibrium on a reasonable domain of individual preferences.Only the serial cost sharing equilibria pass the equity tests of No Envy and Stand Alonecost.Journal of Economic LiteratureClassification Numbers: C72, D63.© 1996 Academic

Press, Inc.

1. INTRODUCTION

A given group of agents own in common a technology exhibiting increasingreturns to scale. This technology produces a single homogeneous output (a per-fectly divisible private good) at a monetary cost. The cost function is increasingand concave. We wish to set up a decentralized mechanism allowing the agentsto jointly use the technology.

The origin of this problem is clearly spelled out in the natural monopolyliterature (Sharkey, 1982; Baumolet al., 1982). Under increasing returns toscale, technical efficiency alone requires a single copy of the technology and thecompetition of two or more private firms (each one endowed with a technology)wipes out any possibility of profit (by the familiar contestability argument). The

∗ Numerous conversations with Scott Shenker are gratefully acknowledged. Thanks also to tworeferees and an Associate Editor of this Journal for their detailed criticisms of an earlier version.Supported by the NSF under Grant SES-9109005. The hospitality of GREQE during the preparationof the paper is gratefully acknowledged.

2250899-8256/96 $18.00

Copyright © 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

226 HERVE MOULIN

resulting monopolist must therefore be regulated for the benefit of the consumersas a whole.

The natural monopoly problem raises some notoriously thorny normative andincentive questions: since an ordinary price mechanism cannot be used, thereis no obvious way to share equitably the costs of production, nor is there anystraightforward decentralized mechanism to implement an efficient outcome.Both problems arise even in the simple case of a technology transforming asingle input into a single output (this is the context of this paper).

A familiar but simplistic answer to these difficulties is marginal cost pricing:the designer somehow finds an efficient level of production and charges the cor-responding marginal cost per unit of output to every consumer. This will generatethe correct level of demand but will fall short of covering costs: the remainingdeficit is then divided equally among the consumers (hence independently oftheir individual demands). This approach is defective on two accounts. First, itmay be impossible to reach an efficient outcome by means of such a “two-parts”tariff (Guesnerie, 1975; Vohra, 1990). Second, the basic incentive constraint ofvoluntary participation will be violated when an agent who does not want topurchase any output (at the proposed marginal cost price) must nevertheless payhis share of the deficit generated by other agents’ demands.

The recent literature on the natural monopoly does not offer a joint resolutionof its equity and incentives issues. Instead the two issues have been addressedseparately. One stream of literature ignores the incentives aspects altogether andseeks to define equitable (first best) solutions. Several tests of equity lead thediscussion of these solutions: the No Envy test, the Stand Alone test, and variouslower or upper bounds on individual welfare (these tests are informally discussedin Section 2 and formally defined in Section 6). See Mas-Colell (1980); Moulin(1987), (1990b), (1992); Vohra (1992); Maniquet (1994); and Fleurbaey andManiquet (1993). One unambiguous conclusion of that literature is that no (firstbest) solution will pass all equity tests at all reasonable profiles of preferences:typically the two tests No Envy and Stand Alone cannot be met by any (first best)solution; in some cases, even the No Envy test is incompatible with efficiency(Vohra, 1992). By contrast, in our second best world, the equilibrium of serialcost sharing always passes both tests (No Envy, Stand Alone) and more (theUnanimity test, Section 6).

Despite a vast literature on the abstract resolution of the implementation prob-lem (surveyed by Moore, 1992, and Palfrey, 1992), to the best of my knowledgethe literature does not address implementation of particular (first or second best)solutions of the natural monopoly problem. This paper takes one small steptoward the resolution of this mechanism design question.

Specifically, I assume that the set of users of the given (increasing returns)technology must choose an allocation mechanism independently of the character-istics of demand (that is, independently of the profile of individual preferences).I propose to call such mechanismsuniversalto distinguish them from theplan-ning mechanisms where the designer must know at least the aggregate demand

COST SHARING UNDER INCREASING RETURNS 227

in order to compute, say, a marginal cost pricing equilibrium (see, e.g., Brownet al., 1992) and from Bayesian mechanisms (where the designer may use sta-tistical information about preferences: see, e.g., Myerson, 1991). Examples ofuniversal mechanisms include all familiar voting rules (e.g., the Borda scoringmethod) as well as the Vickrey–Clarke–Groves pivotal mechanism for the pro-vision of public goods. In the natural monopoly context, universal mechanismsare especially compelling when only a small number of agents is concerned:think of division managers sharing a common facility (e.g., a copying machine)or of a network of college libraries sharing a database.

Surely the simplest universal mechanism is average cost pricing: each agenti independently selects his or her demandqi , all demands are served, and ev-eryone pays average cost (at the joint demand: so agenti paysqi · AC(

∑j qj ),

where AC is average cost). Thus each agent faces, for a given set of messagesby the other agents, a nonlinear price determined by these very messages. Thecelebrated tragedy of the common stresses that the noncooperative equilibriumof this mechanism involves inefficient underproduction of output. Of course, anymechanism where individual agents choose their consumption of output and payaccording to a mechanical formula typically has inefficient equilibrium alloca-tions, and hence the tragedy of the commons cannot be taken as an argumentagainst average cost pricing. However, we do offer two criticisms of average costpricing in this paper. The first one has to do with the possible nonexistence of aNash equilibrium outcome, even at a profile of “very regular” preferences (seeLemma 1). The second criticism is explained in the next paragraph.

An original feature of this paper is to introduce equity tests in the discussionof strategic mechanisms: we compare various mechanisms not only by theirequilibrium properties (when does a Nash equilibrium exist?) but also by thenormative properties of their equilibrium allocations: do they pass the No Envytest? the Stand Alone test? We note that an equilibrium of average cost pricingalways passes the Stand Alone test but often fails the No Envy test.

We argue instead for an alternative cost sharing formula,serial cost sharing,because (i) it possesses at least one Nash equilibrium for a reasonable classof preferences profiles and (ii) its equilibrium allocation always pass the NoEnvy test, the Stand Alone test (and more: see the Unanimity test in the nextsection). Serial cost sharing has been shown to possess remarkable incentiveand normative properties in the case ofdecreasing returns to scale (Moulin andShenker, 1992, 1994). The current paper confirms the outstanding features ofthis mechanism in the increasing returns case. Before introducing the formalmodel, we review and interpret the main findings of the paper.

2. OVERVIEW

The only exogenous data in our model result from a concave increasing costfunctionC (with C(0) = 0) viewed as the common property of a given group

228 HERVE MOULIN

of n agents. Each agent draws her preferences over output consumption andinput contribution from a domain (the same for everyone) defined by standardmicroeconomic assumptions (to be specified below).

We limit ourselves to a class of universal mechanisms that we callsimplemechanisms. In those, an individual message consists of a quantity of outputqi ;all demands are met and the mechanism computes the cost sharesxi , i = 1, . . . ,nas a function of the whole demand profileq1, . . . ,qn. The budget balance con-dition

∑i xi = C(

∑i qi )) and the anonymity property (or justicea priori: only

the messages, not the name of the agents should matter) are the only two formalrequirements limiting the choice of simple mechanisms.1 The simple mecha-nisms have the simplest functional form among those that do not preclude anyoutput distribution.

Our discussion is organized around four simple mechanisms. For a givendemand profileq1, . . . ,qn we writeq =∑i qi . In average cost pricing, agentipaysxi = (qi /q) ·C(q). In marginal cost pricing, xi = qi ·C′(q)+α, where the“fixed fee” α is the same for all agents (and is determined by budget balance).In the Shapley–Shubikmechanism,xi is the Shapley value of agenti in thecooperative game (with transferable utility) associating to each coalitionS itsStand AlonecostV(S) = C(

∑s qi ) (Shubik, 1962; Young, 1985). Inserial cost

sharing, the cost sharexi is given by formula (1) in Section 3; an alternativecharacterization is the property thatxi does not depend upon those demandsqj

(by other agents) that are bounded below byqi (as explained at the beginning ofSection 7: see property (13) and the discussion thereafter).

Given a cost sharing mechanism and a particular preference profile, we pos-tulate in standard game theoretic fashion that the noncooperative behavior of theparticipants will result in a choice of demandsqi , i = 1, 2 . . . ,n forming a Nashequilibrium of the corresponding strategic game if such equilibrium exists atall. We invoke the usual evolutive stories to explain how decentralized behaviorby agents mutually ignorant about each other’s preference converges toward aNash equilibrium (see the comments at the end of Section 5). So if no Nashequilibrium exists at some preference profile (a very real possibility, as we shallsoon see), we interpret it as a fundamental difficulty of the decentralized play ofour mechanism at this particular profile.2

Thus the first and foremost question about a mechanism is “On what domainof preferences does it guarantee the existence of at least one Nash equilibrium?”

1 We must interpret the allocation(xi ,qi ) as the actual consumption of agenti : no further trading ofinput and/or output between the agents is allowed. This may require close monitoring of the allocationas in the photocopying example.

2 Existence of an equilibrium in mixed strategies is of no help here because (i) a mixed strategy is animmensely more complicated object when pure strategies vary in the real line and (ii) no satisfactoryevolutive story can be invoked for mixed strategies.


This question conceptually precedes any discussion of fairness and/or efficiencyof equilibrium outcomes.

Our first critique of average cost pricing (and of marginal cost pricing, asdefined above) is that this mechanism fails to have any Nash equilibrium at allfor some very-well-behaved profile of preferences (in fact, for preferences rep-resented by quasilinear utility functions of the formui (qi )− xi ): see Lemma 1.On the other hand Theorem 1 uncovers a family of simple mechanisms (con-taining serial cost sharing, the Shapley–Shubik cost sharing mechanism, andmore) where at least one Nash equilibrium exists on a reasonably rich domain ofpreferences (namely monotonic, convex, and such that both goods are normal).This result is based on the fixed-point techniques in lattices (more precisely thesingle crossing property discussed by Milgrom and Shannon, 1994). By con-trast, in the decreasing returns case (when the cost functionC is convex) thefamiliar fixed-point techniques in convex sets (Nash’s theorem) show that manysimple mechanisms (among them average cost pricing) guarantee at least oneNash equilibrium on the domain of monotonic and convex preferences.3 On thisdomain I conjecture that, under increasing returns, there isnosimple mechanismguaranteeing at least one Nash equilibrium.

Next we turn to the normative properties of the equilibrium outcomes. Weknow that such outcomes will be inefficient in all but exceptional cases.4 We fo-cus on three familiar fairness tests that feed the discussion of first best solutions.The first one is the Stand Alone test (Faulhaber, 1975; Sharkey, 1982), whichrequires that no agent (or coalition of agents) be asked to pay more than the costof serving this agent (or coalition) alone:xi ≤ C(qi ) (or

∑i∈S xi ≤ C(

∑i∈S qi )).

If each agent or coalition has access to the technology, this test has a positiveinterpretation as part of a core stability property (or, equivalently, as a contesta-bility argument: if the test fails for a certain coalition then it would be profitablefor a firm to enter and serve this coalition).

If the Stand Alone test is the central normative property of the natural monopolyliterature, our second test, the No Envy property, is the most popular property ofthe distributive justice literature, where it conveys the idea of equal opportunityof consumption. A remarkably versatile concept (see the survey by Thomsonand Varian, 1985), it means in our context that no agenti strictly prefers anotheragent’s net trade(xj ,qj ) over her own net trade(xi ,qi ).5 Finally, the Unanim-ity test (the least familiar of the tree tests, see Moulin, 1990a, b), rules out the

3 One mechanism, serial cost sharing, is essentially characterized by the property that the Nashequilibrium is unique at all such preference profiles: Moulin and Shenker (1992).

4 The argument on p. 1023 of Moulin and Shenker (1992) is equally valid under increasing returns.5 We do not want agents to compare their net consumptions—i.e., input endowment minusxi —

because this would entail an ethical judgment over the initial distribution of endowments. We chooseinstead to incorporate those endowments in the preference profile and view the differences in individualpreferences as ethically neutral.

230 HERVE MOULIN

possibility that an agenti benefits from the difference between his preferencesand other agents’ preferences. In our context, this amounts to the inequalityxi ≥ C(nqi )/n: my cost share is at least what my fair share would be if everyother agent had the same demand as my own.

For first best (efficient) solutions of the natural monopoly problem, the StandAlone and No Envy tests are generally incompatible. This is easily checked evenin the simplest configuration of demands.6 But the equilibrium outcomes of oursimple mechanisms are inefficient; hence requiring them to pass all three testsat once may not be asking too much.

As it turns out, all Nash equilibrium outcomes of serial cost sharing passall three tests. On the other hand, the more familiar average cost pricing andShapley–Shubik mechanisms pass the Stand Alone test but fail No Envy (andUnanimity). I submit that this is a severe normative defect of those mechanisms:even though they offer the same strategic opportunities to all agentsex ante(thegame form is anonymous), their equilibrium outcome may not offer the sameconsumption opportunities to all agentsex post(the No Envy test may fail). Othermechanisms pass the No Envy and/or Unanimity tests, but fail Stand Alone: anexample is marginal cost pricing, charging the fixed fee to all agents, even thosewho consume no output.

The general model is defined in Section 3. In Section 4, three numericalexamples illustrate some of the main differences between the four basic mecha-nisms. Section 5 discusses the existence of a Nash equilibrium outcome in simplemechanisms. The three equity tests are formally introduced in Section 6, and arelated characterization of serial cost sharing is the subject of Section 7. Someconcluding comments are gathered in Section 8.

3. THE MODEL

Throughout the paper we are given a (one output) cost functionC with thefollowing properties:

C is concave, increasing,C(0) = 0 and lim∞C(q) = +∞.

6 Say that a large number of agents each demand one unit of the good and that their reservationprice generate the usual downward sloping demand curve. Callp the “competitive” price, namely thelowest price equal to the marginal cost of the corresponding demand (i.e., the lowest intersection of thedemand and marginal costs curve). Then efficiency requires that all agents be served with a reservationprice abovep and only those (call them active agents). No Envy requires that every active agent mustpay the same pricep′. Because of increasing returns,p′ must exceedp. By applying No Envy to anagent with reservation price immediately abovep and to one immediately belowp, we see that thisinactive agent (hence all inactive agents, by No Envy again) must payp′ − p. This contradicts theStand Alone test for the inactive agents (who, consumingqi = 0, should pay nothing:xi ≤ C(0) = 0).


A cost sharing mechanism amongn agents is a mappingζ from Rn+ into Rn

+,associating to each profile of demandsq = (q1, . . . ,qn), 0 ≤ qi , a profile ofnon-negative costsx = (x1, . . . , xn), and satisfying budget balance

∑i

xi = C

(∑i

qi

).

A simple mechanismis a cost sharing mechanism satisfying the anonymity prop-erty, namely for alli , j = 1, . . . ,n,

{qi = q′j ,qj = q′i ,qk = q′k all k 6= i, j }⇒ {ζi (q) = ζj (q

′) andζk(q) = ζk(q′) all k 6= i, j }.

Each agent is endowed with preferences overR2+. We consider two possible

domains of preferences, namelyL: preferences nonincreasing inxi , nondecreasing inqi , nonlocally satiated,

and decreasing on(C(q),q) for q large enough; moreover convex and continu-ous.L∗: the subdomain ofD containingbinormalpreferences (namely both goods

are normal): Ifz is in the demand set on a given budget line−p1x + p2q ≤ b,and ifb′ > b, then the demand set on the budget line−p1x+ p2q ≤ b′ containsat least a pointz′ such thatx′ ≤ x and q′ ≥ q.

For preferences represented by a differentiable utility functionu, the normalityproperty simply means that the MRS(∂u/∂x)/(∂u/∂q) (i.e., the slope(dq/dx)of the indifference contour) is nonincreasing inx and in q. For instance anadditively separable utility functionu(x,q) = a(q) − b(x) (with a concave,bconvex, and both increasing) represents a preference inL∗. Next we define fourbasic mechanisms.

Mechanism1: Average cost pricing(ACP).

For any q1, . . . ,qn: xi = ζi (q) = qi

qN· C(qN), where qN =

∑j

qj .

Mechanism2: Marginal cost pricing(MCP). Denote byC′(q) the right-handderivative ofC

for all q1, . . . ,qn : xi = ζi (q) = qi · C′(qN)+ 1

n(C(qN)− qN · C′(qN)).

Note that the deficitC(q)−qC′(q) is non-negative so that the above cost sharesare non-negative, too. The MCP mechanism divides equally the deficit amongall agents. Other divisions of the deficit are discussed in the literature (see,e.g., thealternative cost avoidedmethod in Young, 1985). The correspondingmechanisms do not have better strategic properties than MCP.

232 HERVE MOULIN

Mechanism3: the Shapley–Shubik mechanism(SS). Given a profile of de-mandsq (and a cost functionC′), consider the cooperative game (with sidepayments)V(S) = C(

∑S qi ). The SS cost sharing rule is the Shapley value of

this game, or

xi = ζi (q) = E[V(S∪ {i })− V(S)],

where the expectation bears on all orderings of{1, . . . ,n} andS is the (possiblyempty) coalition of agents precedingi in a random ordering.

Mechanism4: Serial cost sharing(SER). The formula is given for a pro-file of demandsq such thatq1 ≤ q2 ≤ · · · ≤ qn. The other cases follow byanonymity. Denote

q1 = nq1,

q2 = (n− 1)q2+ q1, . . . ,

qi = (n− i + 1)qi + qi−1+ · · · + q1, . . . ,

qn = qn + · · · + q1

ζ1 = C(q1)

n;

ζ2 = C(q2)

n− 1− C(q1)

n(n− 1);

ζi (q) = C(qi )

n− i + 1− C(qi−1)

(n− i + 2)(n− i + 1)− · · · − C(qi )

n(n− 1). (1)

Remark. At a demand profile such thatq1 is the smallest demand andqn thelargest one, we have the following inequalities:

x1(ACP) ≤ x1(SER) ≤ x1(MCP)

xn(MCP) ≤ xn(ACP); xn(SER) ≤ xn(ACP).

4. NUMERICAL EXAMPLES

We fix the following piecewise-linear cost function throughout this section:

C(q) = min{2q,

q

2+ 15

}. (2)

Marginal cost equals 2 up toq∗ = 10 and fails to12 afterwards.

In all numerical examples, we have two agents with quasilinear utilities ofthe formui = ui (qi )− xi (in particular, agents have unbounded reserves of the


transferable input called money). In the MCP mechanism, the jump in marginalcost atq∗ = 10 yields a discontinuous cost share. Sayq1 = 2. Then player 2’scost share isx2 = 2q2 as long asq2 < 8 but jumps down tox2 = 7.5+ (q2/2)whenq2 ≥ 8 (note that ifq1 = 8, thenx2 jumpsupatq2 = 2).

Consider the utility functions

u1(q1) = min

{q1,

q1

4+ 3

2

}, u2(q2) = min

{3q2,

2q2

7+ 19

}Efficiency requires thatq1 = 0, q2 = 7 (obtained by maximizingu1(q1) +u2(q2) − C(q1 + q2)). This pair is also the unique equilibrium demand profilein all three mechanisms ACP, SS, and SER. In all three mechanisms, agent 2covers all costs, yet(0, 7) is not an equilibirum profile of the MCP mechanism,because player 2 wishes to raise his demand toq′2 = 10: at that level he paysx′2 = (q′2/2) + 7.5 = 12.5 (because player 1 must pick up half of the fixedcost 15) as opposed tox2 = 14 if he demandsq2 = 7. Once player 2 demandsq′2 = 10, however, player 1 wishes to raise his demand toq′1 = 2 (he ends upwith a utility lossu1(2)− ((q′1/2)+ 7.5) = −6.5 anyway). Now givenq′1 = 2,player 2 wishes to demand onlyq′2 = 8 (he only wishes to make sure that player1 picks half of the fixed cost), to which player 1’s best reply isq′1 = 0 (wherebyhe has nothing to pay!) and so on. It turns out that the demand game generatedby the MCP mechanism has no Nash equilibrium whatsoever: the best replyfunctions are

br1(q2) = 0 if q2 < 10;= 2 if q2 ≥ 10

br2(q1) = 10− q1 if q1 ≤ 3;= 7 if q1 ≥ 3.

In the next numerical examples, we illustrate the difference between the threemechanisms ACP, SS, and SER. Consider first aunanimousutility profile

ui (qi ) = 5√

qi , i = 1, 2.

Efficiency requires thatq1 = q2 = 25. The demand profile(25, 25) is a Nashequilibrium of both the SER and SS mechanisms: indeed given that the otherplayer demands 25, a player faces the actual marginal cost1

2 whenever he/shedemands 10 or more. Note that there is another Nash equilibrium (for bothmechanisms) atq1 = q2 = 1.56 (computed byu′i (qi ) = 2; note that givenqj < 5, playeri faces the marginal cost 2 for 0≤ qi ≤ 5) and this equilibriumis Pareto inferior to(25, 25): in Fig. 1 are drawn the opportunity sets in allthree mechanisms SER, SS, and ACP givenqj = 25 or qj = 1.56. In theACP mechanism, only the inferior equilibrium(1.56, 1.56) survives: there is noequilibrium(q1,q2) with q1+ q2 ≥ 10.7

7 Wheneverq1 = q2 ≥ 5, we haveu′i (qi ) < 1 anddxi /dqi > 1.

234 HERVE MOULIN

FIGURE 1

Our next example is one where only the serial equilibrium outcome is efficient.The functionsUi are concave and satisfy

u′1(3) = 2; u′1(8) = 12; u1(8)− u1(3) > 5.5

u′2(2) = 2; u′2(16) = 12; u2(16)− u2(2) > 11.5.

See Fig. 2. The efficient demand profile isq1 = 8, q2 = 16; it is an equilibriumof the serial mechanism (another equilibrium of SER isq′1 = 3, q′2 = 2 but thechoice ofu1, u2 makes this equilibrium Pareto inferior). Indeed givenq2 = 16,agent 1’s cost share isx1 = C(2q1)/2 for anyq1 (including q1 ≥ 16 becauseC(2q1)/2+ C(2.32)/2 = C(q1 + 32) for all q1), and hence agent 1 faces themarginal cost12 whenever 5≤ q1.

On the other hand(8, 16) is not an equilibrium of SS: givenq2 = 16, agent1’s cost share isx1 = (5/4)q1 for 0 ≤ q1 ≤ 10. Hence the equilibrium of SSis (q∗1, 16), whereq∗1 is defined byu′1(q

∗1) = 5

4 and it falls short of efficiency.The equilibrium (or equilibria) of the ACP mechanism is also inefficient in thisexample.

Finally we check that even SER may yield an inefficient equilibrium outcome


FIGURE 2

with a cost function like (1). Take

u1(q1) = 4√

q1

u′2(2) = 2; u′2(16) = 12; u2(16)− u2(2) > 17.5.

Then the unique equilibrium of SER isq1 = 1, q2 = 16. Indeedq1 = 1 reachesthe maximum ofu1(q1)− (C(2q1)/2).8 Moreover, givenq1 = 1, q2 ≥ 1, agent2’s utility is u2(q2)− (C(q2+ 1)− 2); the latter has two local maxima at 2 and16 but our choice ofu2 guarantees thatq2 = 16 is the global maximum. TheSER equilibrium is not efficient, however, since the marginal utilities are 2 and12, respectively. Of course, in this example, the SS and ACP mechanisms haveinefficient equilibria as well.

8 Note that we have two local maxima, atq1 = 1 andq′1 = 16, respectively.

236 HERVE MOULIN

In general, pick a continuous cost function consisting of two linear pieces(and decreasing marginal cost). Then for any profile of convex preferences, itcan be shown that (i) SER has at least one Nash equilibrium and (ii) SER has (atleast) one strong equilibrium and this equilibrium is also Pareto superior to allother Nash equilibria. The strong equilibrium is unique if preferences are strictlyconvex.

We omit the straightforward proof of these claims. In view of Lemma 2 below,statement (i) does not hold for an arbitrary concave cost function; the same istrue of statement (ii).

5. EXISTENCE OF A NASH EQUILIBRIUM

If the cost functionC is convex (decreasing returns to scale), all three9 mech-anisms ACP, SS, and SER have at least one Nash equilibrium outcome for everypreference profile inL.10 By contrast, under increasing returns, the ACP mech-anism may not have any equilibrium at all, even for a profile inL∗.

LEMMA 1. (i) There exists a concave cost function C made of two linearpieces(like the function(1)) and a (two person) preference profile inL∗ (infact a profile of quasilinear utilities) such that theACPmechanism has no Nashequilibrium. The same statement holds true for theMCP mechanism.

(ii) If C is differentiable and the function C′-AC is nonincreasing over aninterval [0, Q), then theACP mechanism has at least one Nash equilibriumfor every preference profile(with an arbitrary number of players) in L∗ (whenindividual demands are bounded above by Q/n).

Proof of Statement(i). The first example in Section 4 shows the claim forthe MCP mechanism. To prove the claim for the ACP mechanism, we use thecost functionC given by (2). We have two players. For a givenq2, we denote byOq2 the opportunity set of player 1, namely the curve

(q1, x1) ∈ Oq2 ⇔ x1 = q1

q1+ q2C(q1+ q2) = q1 · AC(q1+ q2).

On the curveOq2 we consider the two pointsE(q2) = (12, 12AC(12+ q2))

and F(q2) = (16, 16AC(16+ q2)). Then we choose the (quasilinear) utility ofplayer 1,u1(q1)−x1, in such a way that one of its indifference contours containsthe interval [E(14), F(14)], a vertical half-line starting atF(14) and analmosthorizontal half-line starting atE(14): See Fig. 3. By quasilinearity, all otherindifference contours are deduced by a horizontal translation.

9 Note that MCP is not defined in that case because the term(C(q)− qC′(q)) is negative.10 The equilibrium of SER is unique for every profile inL (Moulin and Shenker, 1992); that of ACP

is unique (utility-wise) for every profile inL∗ (Watts, 1992).


FIGURE 3

By construction, the best reply of player 1 atq2 = 14 contains 12 and 16.Indeed [E(14), F(14)] is a chord ofO14: see Fig. 3. We claim that the rest ofthe functionbr1 is as follows:

br1(q2) = 16 if q2 < 14; = 12 if 14< q2; br1(14) = {12, 16}. (3)

To see this, note that the slope of [E(q2), F(q2)] (expressed as a ratio1x/1q)is worth

σ(q2) = 16AC(16+ q2)− 12AC(12+ q2)

16− 12;

therefore

σ ′(q2) = 1

4

(16

16+ q2(C′ − AC)(16+ q2)− 12

12+ q2(C′ − AC)(12+ q2)

).

For t ≥ 10, we have(C′ − AC)(t) = −15/t , so that

σ ′(q2) > 0⇔ q2 < 192.

238 HERVE MOULIN

Therefore asq2 increases, the curveOq2 moves leftward and the slope of(E(q2),

F(q2)) increases: this yields the best reply function (3) for player 1: see Fig. 3.To complete the proof of statement (i), it remains to choose a (quasilinear)

utility function for player 2 guaranteeing

br2(12) < 14 and br2(16) > 14. (4)

This construction (for instance we can takebr2(12) = 12 andbr2(16) = 16) isomitted. Together, the two best reply functions (3), (4) preclude the existence ofa Nash equilibrium to the ACP mechanism.

The proof of statement (ii) in Lemma 1 is given immediately after the proofof Lemma 2.

A general result about the existence of Nash equilibria in simple mechanismscannot use the fixed point theorems from convex analysis because the best replycorrespondences are not convex valued. For a given choiceq−i of strategies bythe agents other thani , the opportunity set{(xi ,qi )/xi = ζi (qi ,q−i ),qi ≥ 0} istypically the graph of a concave function (with variableqi and valuexi ); henceeven a preference inL∗ may reach its maximum over this opportunity set in twoor more distinct points.

However, the fixed-point theorems from lattice theory (the descendants ofTarski’s theorem) can be fruitfully applied to our problem.

We introduce some notations. Ann-player normal form game where playeri ’s strategy isqi , 0 ≤ qi ≤ Q < +∞ will be denoted(π1, . . . , πn) withπi (q1, . . . ,qn) interpreted as the payoff to playeri whenever the strategic profileis q = (q1, . . . ,qn). The notation(q |i qi ) represents the demand profile withi th coordinateq′i and j th coordinateqj for all j 6= i . Thus the strategic profileqis a Nash equilibrium if we haveπi (q) ≥ πi (q|i q′i ) for all i and allq′i .

We say that the normal form game has the single crossing property if we havefor all i and all profilesq, q′

{qi ≤ q′i for all i and πi (q) ≤ πi (q|i q′i )} ⇒ {πi (q′|i qi ) ≤ πi (q

′)}and

{qi ≤ q′i for all i and πi (q) < πi (q|i q′i )} ⇒ {πi (q′|i qi ) < πi (q

′)}THEOREM(Milgrom and Shannon, 1994, Theorem 15).Let a normal form

game(π1, . . . , πn) be given such that for all i= 1, . . . ,n, (i) the payoff functionπi is continuous in q and(ii) πi satisfies the single crossing property. Then thegame possesses a largest Nash equilibrium q∗ and a smallest Nash equilibriumq∗ (so q∗ ≤ q∗). Moreover the strategies q∗i and q∗i are respectively the largestand the smallest serially undominated strategies of player i.

Note that in the above statement the strategy sets of all players must bebounded.


THEOREM1. Consider a simple mechanismζ among n agents satisfying

(a) ζ is continuous in q.(b) Cross Monotonicity: for all i , j , i 6= j , ζi (q) is nonincreasing in qj .

Note: this property implies thatζi is increasing in qi .(c) Complementarity: for all i , j , i 6= j the increment ofζi when qi increases

in nonincreasing in qj . Formally, for all q, q′,

q ≤ q′ ⇒ ζi (q|i q′i )− ζi (q) ≤ ζi (q′)− ζi (q

′|i qi ).

Then for any preference profile inL∗, the corresponding demand game satisfiesthe assumptions of the above theorem. Its largest Nash equilibrium is also Paretosuperior among all Nash equilibria.

When ζ is twice differentiable (as the ACP, SS, and SER mechanisms areif C itself is twice differentiable), the properties of Cross Monotonicity andComplementarity mean respectively∂ζi /∂qj ≤ 0 and∂2ζi /∂qi ∂qj ≤ 0.

Proof. Letu be a utility function representing a preference inL∗. Binormalityimplies the following property. Fixq1, q′1, x1, x′1, y1, y′1 such that

q1 < q′1, x′i ≤ y′i ; u(x1,q1) = u(x′1,q′1) and u(y1,q1) = u(y′1,q

′1). (5)

Then we must havey1− x1 ≥ y′1− x′1.Indeed, the(x1,q1)-indifference contour has equationx = f (q) and the

(y1,q1)-indifference contour has equationx = g(q), with f , g concave andincreasing on [q1,q′1]. Binormality impliesg′ (q) ≤ f ′+(q) for all q (whereg′

is the left derivative andf ′+ the right derivative), and henceg(q′1) − g(q1) ≤f (q′1)− f (q1) as desired.

We check thesingle crossingproperty. For allq, q′, q ≤ q′ we must showui (ζi (q|i q′i ),q′i ) ≥ ui (ζi (q),qi )⇒ ui (ζi (q′),q′i ) ≥ ui (ζi (q′|i qi ),qi ). Fix i = 1for convenience and the vectorsq ≤ q′ with q1 < q′1. Setx′1 = ζ1(q′), y′1 =ζ1(q|1q′1) and definex1, y1 as in (5). (note: ifx1 is not defined, neither isy1

and the SC property holds trivially). From Cross Monotonicity followsx′1 ≤ y′1,whence binormality implies

y1− x1 ≥ y′1− x′1 = ζ1(q|1q′1)− ζ1(q′). (6)

Assumeu1(ζ1(q|1q′1),q′1) ≥ u1(ζ1(q),q1). This impliesy1 ≤ ζ1(q). Combiningthis with (6) yields

x1 ≤ ζ1(q)− ζ1(q|1q′1)+ ζ1(q′).

Invoke Complementarity: the above inequality impliesx1 ≤ ζ1(q′|1q1); there-fore

u1(x′1,q′1) = u1(x1,q1) ≤ u1(ζ1(q

′|1q1),q1).

240 HERVE MOULIN

The other inequality of the SC property (involving strict inequalities) is provenby a similar argument. In order to apply the Milgrom–Shannon theorem we cuta bounded set of strategies for every player. This can be done because in thedomainL, utilities are decreasing on(q,C(q)) for q large enough. We omit thedetails.

LEMMA 2. Both theSSandSERmechanisms meet the properties of CrossMonotonicity and Complementarity. Hence the corresponding games have alargest and Pareto superior Nash equilibrium for all profile inL∗. However,both mechanisms may fail to have a Nash equilibrium for some(two person)profiles inL.

Proof. (a) The mechanism SS meets the assumptions of Theorem 1. Con-tinuity is clear. Cross Monotonicity and Complementarity of each termV(S∪{i })−V(S) follow at once from the concavity ofC, and they are both preservedby linear combinations.

(b) The mechanism SER meets the assumptions of Theorem 1. Continuityfollows upon noticing that the formulas yieldζi = ζi+1 wheneverqi = qi+1. Tocheck the last two properties observe that∂ζi /∂qj = 0 if qi < qj . Hence it isenough to look at the caseqj < qi . Formula (1) implies (assuming differentia-bility of C to simplify) after some rearrangement

∂ζi

∂qj=

j+1∑k=i

C′(qk)− C′(qk−1)

n− k+ 1,

where each term in the summation is nonpositive becauseqk is nondecreasingin k in C′ is nonincreasing. This proves Cross Monotonicity. Taking derivativesone more time yields

∂2ζ1

∂qi ∂qj= C′′(qi ) ≤ 0.

The conscientious reader will complete the argument for the case whereC is notdifferentiable everywhere (or consult Moulin and Shenker, 1992).

(c) On the domainL, both SS and SER may have no Nash equilibrium at all.We prove the claim forn = 2 and the SS mechanism. Adapting our exampleto SER andn ≥ 3 is straightforward. We fix a strictly concave functionC andconstruct a profile(u1, u2) with the best reply functions

br1(q2) = [3, 4] if q2 < 2

= [ 12, 1

] ∪ [3, 4] if q2 = 2

= [ 12, 1

]if q2 > 2

(7)


and

br2(q1) =[

12, 1

]if q1 < 2

= [ 12, 1

] ∪ [3, 4] if q1 = 2

= [3, 4] if q1 > 2.

(8)

The construction ofu1 takes place in two steps. Denote byζq2 the cost share ofplayer 1 for a given choiceq2 by player 2:

ζq2(q1) = 12C(q1+ q2)+ 1

2C(q1)− 12C(q2).

Note that the graphs ofζq2 are all concave and do not overlap (see Fig. 4). We callδ a lower bound on the derivative ofAC(q) = C(q)/q on [0, 4]. This bound canbe taken positive by the strict concavity ofC. We define first the parametrizedfunctionsαq2 as

αq2(q1) = ζq2(12)− ( 1

2 − q1) · AC(0) if 0≤ q1 ≤ 12

= ζq2(q1) if 12 ≤ q1 ≤ 1

= (ζq2(3)− ζq2(1) · (q1−1)2 + ζq2(1)) if 1≤ q1 ≤ 3

= ζq2(q1) if 3≤ q1 ≤ 4

= ζq2(4) if 4≤ q1.

Thusαq2 coincides withζq2 in the two intervals [12, 1] and [3, 4] and is linearon each of the other three intervals. Thereforeαq2 ≤ ζq2 andαq2 is concave forall q2. Moreover, for any givenq1, the termαq2(q1) is continuous and strictlydecreasing inq2. Hence the family of the graphs ofαq2 forms the indifferencecontours of a preferenceu0 in L (of course after chopping off the part of thegraph whereαq2(q1) is negative: see Fig. 4).

Now we define two modifications ofαq2, denotedβq2 andγq2, yielding respec-tively the indifference contours ofu1 andu2,

if q2 ≤ 2, βq2(q1) = αq2(q1)− δ9(2− q2) · (3− q1)+ for all q1

if q2 ≥ 2, βq2(q1) = αq2(q1)− δ9θ(q2)− (q1− 1)+ if 0≤ q1 ≤ 4

= αq2(4)− δ3 if 4≤ q1,

where(x)+ = max{x, 0} andθ is the functionθ(x) = min{(x − 2)+, 1}. Onechecks at once thatβq2 is concave and increasing w.r.t.q1 (the latter becaused/dq1[αq2(q1)] is bounded below byδ/2) and thatβq2(q1) is continuous andstrictly decreasing inq2 (the latter becaused/dq2[αq2(q1)] is bounded below byδ/2). Thus the family of the graphs ofβq2 forms the indifference contours ofa preferenceu1 in L. Note thatβq2 ≤ αq2 ≤ ζq2 for all q1. If q2 ≤ 2, these

242 HERVE MOULIN

FIGURE 4

inequalities are equalities only if 3≤ q1 ≤ 4; if q2 = 2, these are equalities for12 ≤ q1 ≤ 1 and3 ≤ q1 ≤ 4; if q2 > 2 they are equalities for12 ≤ q1 ≤ 1 (seeFig. 4). This shows (7). The functionsγq1 are defined next:

if q1 ≤ 2, γq1(q2) = αq1(q2)− δ9(2− q1) · (q2− 1)+ if 0≤ q2 ≤ 4

= αq1(4)− δ3(2− q1) if 4≤ q2

if q1 ≥ 2, γq1(q2) = αq1(q2)− δ9θ(q1) · (3− q2)+ for all q2.

The functionsγq1(q2) are similarly concave and increasing inq2 and continuousand decreasing inq1. They represent a preferenceu2 in L for which the bestreply correspondence is given by (8).

Proof of Statement(ii ) of Lemma1. The mechanism ACP meets all assump-tions of Theorem 1 except perhaps Complementarity. However, Complementar-ity holds if (C′ − AC) is nonincreasing (we omit the straightforward compu-tations). AsC is concave and nondecreasing,(C′ − AC) can be nonincreasingonly on a bounded interval [0, Q] (unlessC is linear).


The two results (Theorem 1 and Lemma 2) reveal an important differencebetween our two domains of individual preferences. If the domain isL∗, wehave many simple mechanisms where Nash equilibrium behavior, although notunambiguous (because at some profiles we may have several equilibrium out-comes), has a Pareto superior selection (and where Cournot tatonnement startingfrom “very high” demands is sure to converge: see Milgrom and Roberts, 1991).If the domain isL (Giffen goods are possible), then I do not know of any simplemechanism guaranteeing the existence of at least one Nash equilibrium.11

6. THREE EQUITY TESTS (IN EQUILIBRIUM)

The Stand Alone test is the most familiar equity property of the naturalmonopoly literature (e.g., Sharkey, 1982). The test says that an agent (or coalitionof agents) should not pay more than the cost of serving his or her own demand.

for all S⊆ {1, . . . ,n},∑i∈S

xi ≤ C

(∑i∈S

qi

).

LEMMA 3. If a simple mechanismζ satisfies Cross Monotonicity(see Theo-rem1) then every allocation(xi ,qi ) i = 1, . . . ,n passes the Stand Alone test.

Proof. Let ζ be a cross monotonic mechanism. Then we have for allq

∀i = 1, . . . ,n, qi = 0⇒ ζi (q) = 0. (9)

Indeed ifqi = 0, Cross Monotonicity impliesζi (q) ≤ ζi (0) = 0. Hence property(9) becauseζi is non-negative. Next fix an arbitraryq and a coalitionS. Defineq∗ by q∗i = qi if i ∈ S andq∗i = 0 if i 6∈ S. Property (9) and budget balanceimply

∑i∈S ζi (q∗) = C(

∑i∈S qi ). On the other hand, Cross Monotonicity yields

ζi (q) ≤ ζi (q∗) for all i ∈ S, whence the conclusion follows by summing theseinequalities overi ∈ S.

Remark1. At every Nash equilibriumq of a cross monotonic mechanism, astronger version of the Stand Alone test holds true, namely

ui (ζi (q),qi ) ≥ maxq≥0

ui (C(q),q). (10)

This says that no agent can improve his welfare by being the sole user of thetechnology. But the corresponding coalitional version of inequality (10) (definingthe familiar Stand Alone Core, see Scarf, 1986) does not hold.

11 I conjecture that none exists: should we drop the anonymity requirement in the definition of asimple mechanism, a dictatorial mechanism would do.

244 HERVE MOULIN

Recall that out of the four mechanisms introduced in Section 4, only MCPfails Cross Monotonicity. As the example in Section 3 shows, MCP fails eventhe (weak) property (9).

Our second test is the No Envy property, perhaps the single most importantconcept of the microeconomic literature on distributive justice. In the cooper-ative production model presently discussed, it can take two forms dependingon whether agents compare their input/output trade via the mechanism or theirnet consumption of these two goods (endowment minus input/output trade). Seethe discussion of these two variants in Thomson and Varian (1985) and Vohra(1992). We retain the first interpretation because we study the fairness of themechanism without any concern for the inequality of initial endowments (justlike bargaining theory seeks a fair division of the surplus without questioningthe fairness of the disagreement point). The No Envy test states that

for all i , j = 1, . . . ,n, ui (xi ,qi ) ≥ ui (xj ,qj ).

The No Envy test interprets common ownership as equal opportunity: everyagent gets the best of all the actual allocations (the actual opportunities at thisparticular equilibrium). Any allocation resulting from a nonlinear pricing scheme(see, e.g., Guesnerie, 1975; Vohra, 1990) must be envy-free. And vice versa:an envy-free outcome can be interpreted as a nonlinear pricing equilibrium,provided the nonlinear price is chosen after we learn of our particular outcome.12

Our third (and last) equity test is the Unanimity test:

for all i = 1, . . . ,n, xi ≥ C(nqi )

n. (11)

Thus an agent must pay at least his fair share in the hypothetical situationwhere all individual demands coincide with his. This inequality is compatiblewith budget balance becauseC is concave.

The Unanimity test has been proposed (Moulin, 1990b; 1992) as a normativeconsequence of the fact that differences in individual demands should not beardifferently on different agents (whereas ifxi >

C(nqi )

n and xj <C(nqj )

n , agenti suffers—and agentj benefits—from the fact that other agents’ demands aredifferent from his own).

The next lemma states a property of a mechanismζ , guaranteeing that allNash equilibrium outcomes pass the No Envy and the Unanimity tests. Figure 5provides the intuition for the lemma by showing four configurations of the op-portunity sets in a two person mechanism. In three of these configurations a Nashequilibrium outcome with envy is depicted. The only configuration without envyis that wherex1 ≥ ζ2(q1,q1) and x2 ≥ ζ1(q2,q2).

12 Of course the simple mechanism that we call marginal cost pricing does not necessarily result ina nonlinear pricing equilibrium (because the opportunity set faced by one agent is determined by the


FIGURE 5

LEMMA 4. Let ζ be a simple mechanism such that

for all q, all i , j, ζi (q) ≥ ζi (q| j qi ). (12)

Then every Nash equilibrium(at an arbitrary profile inL) is envy-free. Moreoverthe Unanimity test holds true at every demand profile.

Proof. Fix a profileu in L and a Nash equilibriumq. Fix also two agentsi ,j . The equilibrium property and property (12) yield

uj (ζj (q),qj ) ≥ uj (ζj (q| j qi ),qi )

ζj (q|i qi ) = ζi (q| j qi ) ≤ ζj (q)⇒ uj (ζj (q),qj ) ≥ uj (ζi (q),qi ),

which says that agentj does not envy agenti .On the other hand, inequality (11) follows from repeated applications of

anonymity and property (12).

strategic choice of the other agents). Still, the Nash equilibrium outcomes of the MCP mechanism arealways envy-free with only two agents (as explained below).

246 HERVE MOULIN

Which simple mechanisms do meet property (12)? Marginal cost pricing meets(12) for two person problems(n = 2). There (12) amounts to

12[C(q1+ q2)+ (q1− q2) · C′(q1+ q2)] ≥ 1

2C(2q1),

which is a consequence of the concavity ofC. A similar computation shows thatMCP fails (12) ifn ≥ 3 (and an example of Nash equilibrium with envy is easilyconstructed). On the other hand, MCP passes the Unanimity test for alln (asinequality (11) reduces toC(qN)− C(nqi ) ≥ (nqi − qN) · C′(qN)).

The SER satisfies (12) for anyn. Consider a demand profileq such thatq1 ≤ · · · ≤ qn and two agentsi , j . If qi ≤ qj notice thatqj does not enter formula(1): thereforeζi (q) remains unchanged ifqj changes in arbitrary fashion as longas it remains bounded below byqi . This implies inequality (12) whenqi ≤ qj .If qj ≤ qi , then Cross Monotonicity implies (12).

On the other hand, both ACP and SS fail the Unanimity test and have enviousequilibrium outcomes for anyn ≥ 2. The second statement follows from Fig. 5and the observation that the opportunity sets of both mechanisms do not cross(they are “strictly” cross monotonic). The first statement is true for any demandprofile of distinct demands providedC is strictly concave.

SER is not the only example of a simple mechanism satisfying property (12).Another such mechanism is thedecreasing serial mechanism(deFrutos, 1992)defined by the same formulas (1) at a demand profile such thatq1 ≥ q2 ≥ · · · ≥qn. Equivalently this is the only simple mechanism where agenti ’s cost sharedoes not depend uponqj as long asqj remains boundedaboveby qi . Thus forn = 2 this mechanism is defined byx1 = C(2q1)/2,x2 = C(q1+q2)−C(2q1)/2if q1 ≥ q2. Checking property (12) forn = 2 is immediate, and the general caseis hardly more difficult (we omit the details). Notice that the decreasing serialmechanism is disqualified for the same normative reason as MCP, namely thefact that an agent demanding no output must pay a positive share of the costincurred by the other agents’ demand (namely, a violation of property (9)). Withn = 2 andq2 = 0< q1, agent 2 must payC(q1)− C(2q1)/2.

The following table summarizes the equity properties of (the equilibriumallocations of) our four mechanisms:

ACP MCP SS SER

Stand Alone test Y N Y Y

No Envy test N n = 2:Y N Y

n ≥ 3:N

Unanimity test N Y N Y


7. CHARACTERIZATION OF SERIAL COST SHARING

Our three equity tests are at the heart of the normative discussion of (first best)solutions of the cooperative production problem (all three properties generalizeto technologies with multiple inputs and/or multiple outputs). An importantobservation is that in general there is no first best solution passing both the StandAlone and the No Envy tests.13 Therefore it is quite remarkable that one simplemechanism, namely serial cost sharing, always produces equilibrium outcomespassing all three tests (and, of course, inefficient).14

Is serial cost sharing the only simple mechanism of which the equilibriumoutcomes always (that is, for all profiles inL) meet our three tests? Surely it isthe only such mechanism satisfying Cross Monotonicity and property (12). Forthe combination of these two properties implies at once

{qi ≤ qj ⇒ ζi (q) = ζi (q| j qi )} for all q, all i , j . (13)

This says thatζi does not depend onqj as long asqj remains bounded below byqi . Combined with anonymity and budget balance, this forces serial cost sharing.Indeed supposeq1 ≤ q2 ≤ · · · ≤ qn and consider agent 1. Repeated applicationsof (13) give

ζ1(q) = ζ1(q1,q1, . . . ,q1) = C(nq1)

n,

then

ζ2(q) = ζ2(q1,q2, . . . ,q2) = C(q1+ (n− 1)q2)− ζ1(q)

n− 1,

and so on, until we get inductively the whole formula (1).Yet this is not a satisfactory characterization result because property (12) has

no clear ethical justification. We want a result using directly the No Envy test.Note that Cross Monotonicity, a stronger requirement than Stand Alone, doeshave ethical meaning: when one agent raises her demand, thereby lowering bothaverage and marginal costs, no one else should be hurt; more utilization of thetechnology can only produce positive externalities.

THEOREM2. (a)Serial cost sharing is the only continuous, cross monotonic,simple mechanism of which all Nash equilibrium outcomes, for all profiles inL∗, pass the No Envy test.

13 This is true for preference profiles inL∗—or even quasilinear preferences: see footnote 7 andMoulin (1990a). See also Vohra (1992) showing a two person example—with preferences inL∗—whereno efficient and envy-free outcome exists.

14 Notice that this holds true even with continuous and monotonic but not necessarily convexpreferences.

248 HERVE MOULIN

(b) There are other continuous simple mechanisms of which all Nash equilib-rium outcomes, for all profiles inL (and not just inL∗), pass the No Envy test,as well as the Stand Alone and the Unanimity test.

Proof of Statement(a). By the argument preceding Theorem 2 it is enoughto show that a simple mechanism satisfying Cross Monotonicity and No Envy inequilibrium must satisfy property (12) as well. Take a cross monotonic mecha-nismζ and suppose there is a demand profileq and two agentsi , j , i 6= j , suchthat

qi < qj andζi (q) < ζi (q∗), where q∗ = (q| j qi ).

Cross Monotonicity and budget balance imply

for all qj ≥ qi {for all k 6= j : ζk(q| j qj ) ≤ ζk(q∗)}

and

{ζj (q| j qj ) ≥ C(qj + qN\ j )−

∑k 6= j

ζk(q∗)

}

This says that on [qi ,+∞), the opportunity setOq− j stays to the right of the graphof the concave increasing functionC(qj+qN\ j )−

∑k 6= j ζk(q∗). Moreover, the set

Oq− j stays to the left of the graph ofC (by Stand Alone): see Fig. 6. Thereforewe can find a numberqj , qj ≥ qj and a preferenceuj in L∗ of which thelower contour at(ζj (q| j qj ),qj ) containsOq− j but does not contain(ζi (q), qi )

(remember(ζi (q∗), qi ) is in Oq− j ). By Cross Monotonicity this implies

uj (ζi (q| j qj ), qi ) ≥ uj (ζi (q), qi ) > uj (ζj (q| j qj ),qj ).

Figure 6 shows a possible construction ofuj using continuity ofζj . It remainsto complete the profileu (in Ln

∗) by utilitiesuk, k 6= j , such that(ζk(q| j qj ), qk)

maximizesuk on agentk’s opportunity set at(q| j qj ) (using an indifferencecontour with a right angle at that point). In this way we construct a profile withan equilibrium at(q| j qj ) where agentj envies agenti .

Proof of Statement(b). In the case of two agents(n = 2), we constructanother simple mechanism meeting all three tests in equilibrium. In the case oftwo agents, the Unanimity test and property (12) coincide, and therefore all wehave to do is to find a simple mechanismζ such that

C(2qi )

2≤ ζi (q1,q2) ≤ C(qi ) for i = 1, 2. (14)

There are many such mechanisms, even under the additional requirement thatζi

should increase withqi . The following mechanism can be checked to be the best


FIGURE 6

mechanism for the agent with the largest demand among those satisfying (14).For allq2 ≥ 0, defineα(q2) ∈ [0,q2] as the smallest solution of the equation

C(q2+ x)− C(x) = 12C(2q2).

This is well defined because the left-hand side is nonincreasing inx, andC(2q)/2≤C(q). Then define our mechanism as follows for allq such thatq1 ≤ q2 (useanonymity to define it forq2 ≤ q1):

if 0≤ q1 ≤ α(q2);ζ1 = C(q1)

ζ2 = C(q1+ q2)− C(q1)

if α(q2) ≤ q1 ≤ q2;ζ1 = C(q1+ q2)− C(2q2)

2

ζ2 = C(2q2)

2 .

Check thatζ is well defined (check forq such thatq1 = α(q2) or q1 = q2)and continuous; moreoverα is nondecreasing andζi is nondecreasing inqi , andfinally ζ satisfies (14).

250 HERVE MOULIN

8. CONCLUDING COMMENTS

Two earlier characterizations of serial cost sharing can be compared to The-orem 2. The first one uses the fact that it has a unique Nash equilibrium whenthe cost function is convex (decreasing returns to scale) and for any preferenceprofile inL: Moulin and Shenker (1992). See also deFrutos (1992) for a similarcharacterization involving both the increasing and decreasing versions of serialcost sharing (defined in the last paragraph of Section 6). The second one is purelyaxiomatic and relies on the linearity of formula (1) w.r.t. the cost functionC:Moulin and Shenker (1994). The present result uses normative properties of theequilibrium outcomes (the No Envy test) as well as normative properties of thecost sharing formula itself (Cross Monotonicity).

In the cooperative production problem, the dual of the cost sharing approachis surplus sharing: each agent sends an input contributionxi and the total output(F(

∑i xi ), whereF is the production function) is divided according to a certain

formula (e.g., Israelsen, 1980). Most cost sharing mechanisms are unambigu-ously transformed into surplus sharing mechanisms: out of our four mechanisms(ACP, MCP, SS, and SER) only MCP is hard to transform.15

The characterization result (Theorem 2) is easily adapted to the surplus sharingcontext: we only need to reverse the Cross Monotonicity axiom (agenti ’s outputshareqi is nondecreasing in agentj ’s input contributionxj , j 6= i ). Theorem 1,on the other hand, has no obvious counterpart in the surplus sharing context:the combination of Cross Monotonicity and (the reverse of) Complementarity(namely∂2qi /∂xi ∂xj ≥ 0) does not imply that the corresponding games havestrategic complementarities onL∗.

A more satisfactory characterization result would not make use of the CMaxiom. I conjecture that statement (a) of Theorem 2 still holds if we replace CMby the property that the mechanism has at least one Nash equilibrium for allprofiles inL∗.

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Cost Sharing under Increasing Returns: A Comparison of Simple Mechanisms