Strategyproof Sharing of Submodular Access Costs

Strategypro of Sharing of Submo dular Access Costs:

Budget Balance versus Eciency

Herv e Moulin

Department of Economics

Duke University

Durham, NC 27708-0097

[email protected]

Scott Shenker

Palo Alto Research Center

Xerox Corp oration

3333 Coyote Hill Drive

Palo Alto, CA 94304-1314

[email protected]

May 1996 1

Abstract

A given set of users share the submo dular cost of access to a network or, more generally,

the submo dular cost of any idiosyncratic binary go o d. We compare strategypro of mecha-

nisms that serve the ecient set of users but do not necessarily balance the budget with

those that exactly cover costs but are not necessarily ecient. Under the requirements of

individual rationality guaranteeing voluntary participation and consumer sovereignty an

agent will obtain access if his willingness to pay is high enough, we nd:

i a unique strategypro of and ecient mechanism a variant of the familiar pivotal mech-

anism dubb ed the marginal contribution mechanism MC

ii a whole class of strategypro of and budget-balanced mechanisms, each one corresp ond-

ing to a certain cost sharing formula; these mechanisms, unlike MC, are immune to manip-

ulations by coalitions

Within the second class, the mechanism asso ciated with the Shapley value cost sharing

formula is characterized by the prop erty that its worst welfare loss is minimal. We compare

the budget imbalances of MC with the welfare losses of the Shapley value mechanism. Ap-

plication of these metho ds to the case of a tree network without congestion is also discussed.

Stimulating conversations with Ra j Deb, Bhaskar Dutta, Wolfgang Pesendorfer, and William Thomson are

gratefully acknowledged. 2

1 The Problem and the Punchline

Sharing the cost of a service jointly pro duced for a given set of users is one of the most p opular

applications of co op erative game theory e.g.,Shubik [1962], Lo ehman and Whinston [1979],

Stran and Heaney [1989]. The standard mo del has a set N of users who can either have access

to the service or not; if the subset coalition S of N receives the service, the cost C S must b e

shared b etween the users in S . As the cost function C is not necessarily symmetrical in the di erent

users, we can think of \service to user i" as an idiosyncratic go o d di erent from \service to user

j". One canonical example of such a system, whose terminology we will use throughout the pap er

for convenience, is that of providing access to a network, where the network may b e distributing

water Okada et al. [1982], or telecommunication services Sharkey [1995], or multicast messages

Herzog et al. [1995], and so on; we discuss this network interpretation more thoroughly in Section

The literature on cost allo cation surveyed, for example, in Young [1985] and Moulin [1988,1995]

mostly addresses the two related questions of fairness which cost allo cations among users are

fair? and core stability which cost allo cations are not threatened by the secession of a coalition

standing alone to provide its own service?. In this pap er, we lo ok at the somewhat di erent

issue of incentive compatibility; we consider mechanisms that elicit from each user his/her will-

ingness to pay for access to the network, then decide who gets access and how the cost is shared.

Strategypro ofness is the minimal level of incentive compatibilitywe require, and to that end we

fo cus exclusively on strategypro of mechanisms, where truthful rep ort of one's willingness to pay

is always a dominant strategy.We also discuss another form of incentive compatibility, namely

group strategypro ofness where no coalition of users has an incentive to jointly misrep ort their

true willingness to pay.

There are other prop erties, b esides incentive compatibility, that such cost allo cation mechanisms

should have. In particular, one would naturally like the cost allo cations to b e ecient maximize

the total welfare and also budget balanced cover costs exactly. Unfortunately,even in problems

where preferences can b e represented by quasi-linear utilities as we assume in this pap er, there are

no strategypro of cost allo cation mechanisms that are b oth budget balanced and ecient Green,

Kohlb erg, and La ont [1976]. In this pap er we explore the eciency/budget-balance trade o

in the particular problem of access to a network as describ ed ab ove. We conclude that, in this

context, the strategypro of and budget balanced mechanisms are preferable to strategypro of and

ecient ones b ecause of their sup erior incentive compatibility prop erties and the greater normative

exibility they o er the mechanism designer. Accordingly, dropping the eciency requirementis

p otentially more useful than dropping budget balance in the access charge problem. Before laying

out our arguments, we describ e the kind of context for whicha choice b etween eciency and

budget balance makes sense.

A context where strategypro of and ecient hence budget-imbalanced mechanisms are often

prop osed e.g., Green and La ont [1979] is when a public authority elicits individual preferences

and implements an allo cation with the mandate of maximizing total welfare and the abilityto

absorb any de cit or surplus that may o ccur during the implementation of the mechanism the

public authority acts as a banker, covering the de cit or siphoning o the surplus without distorting 3

the incentives. The corresp onding set of strategypro of and ecient mechanisms under quasilinear

utilities is well known; they are usually called the Clarke-Groves mechanisms Clarke [1971], Groves

[1973].

A context where budget balance is inescapable is co op erative pro duction Israelsen [1980], Ro emer

[1986, 1989], Moulin [1987], Moulin and Shenker [1992], Fleurbaey and Maniquet [1994]; the users

of the service are in autarchy, so that covering costs is a feasibility constraint. The corresp onding

class of strategypro of but inecient mechanisms has not b een previously characterized in the

literature; our Theorem 1 o ers sucha characterization under a stronger incentive compatibil-

ity requirement than strategypro ofness i.e., ruling out pro table manipulations by coalitions of

agents; see Section 3.

In many realistic situations, b oth kinds of mechanisms ecient and budget imbalanced; budget

balanced and inecient are in fact available. A subset of memb ers of a club or a subset of

division managers in a rm, or any sub coalition in a larger community want to provide a certain

service, but the service will not b e o ered to the memb ership at large: e.g., the residents of a

certain neighborhood want to jointly establish a lo cal network of bicycle trails and the existing

highways prevent the extension of the network b eyond this particular neighb orho o d. This lo cal

pro ject mayormay not b e allowed to tap into the budget of the larger community; that is to say,

b oth kinds of mechanisms are conceivable and thus a normative comparison is called for.

We cast the fo cus of this pap er in the ab ove context. To supp ort our conclusion that strategypro of

and budget balanced mechanisms are preferable to strategypro of and ecient ones, we o er two

characterization results. First, we identify a particular strategypro of and ecient mechanism

called the marginal contribution mechanism and show that it is the only reasonable one in

this class: Prop osition 1 in Section 3 the meaning of reasonableness is given at the b eginning

of Section 3. Thus the mechanism designer has no exibility at all once he cho oses to imp ose

the eciency requirement; in particular he must treat all agents equally in the sense that if

two users imp ose equivalent costs then equal messages must result in equal allo cations. Second,

we uncover a rich class of budget balanced and strategypro of mechanisms; each one of them

corresp onds in a manner explained in Section 4 to a particular cost sharing formula satisfying a

certain monotonicity prop erty. This class gives the designer a fair amount of exibility, including

treating equals unequally by means of an asymmetric cost sharing formula typically, ranking the

agents in an arbitrary yet xed fashion and charging stand alone cost to the rst ranked agent

requesting service, incremental cost to the second ranked agent requesting service, and so on

or treating equals equally by using an equitable formula e.g., the Shapley value. Moreover,

every mechanism in this class satis es a much stronger incentive compatibility requirement than

strategypro ofness called group strategypro ofness; they are immune to joint misrep orts by coalitions

of any size whereas the marginal contribution mechanism is extremely vulnerable to suchmoves,

as explained at the end of Section 3. Theorem 1 in Section 4 characterizes the class of reasonable

group strategypro of and budget balanced mechanisms.

Within the ab ove class we single out the one asso ciated with the Shapley value formula, b ecause it

Yet another context of interest is that of a pro t maximizing rm supplying access to the network by means of

a strategypro of pricing mechanism. Although we do not discuss this context, it should b e clear from Prop osition

1 that a pro t maximizing rm will not use a strategypro of and ecient pricing mechanism. 4

generates the smallest p otential welfare loss the smallest deviation from eciency: Theorem 2 in

Section 5. Thus the paramount equitable solution of the cost sharing problem { the Shapley value

{ ends up b eing justi ed on incentive grounds, without invoking any consideration of equity not

even equal treatments of equals. In Section 6 we compare the maximal welfare loss of the Shapley

value mechanism with the maximal budget imbalance of the marginal contribution mechanism,

and nd that neither is systematically larger than the other.

The main restrictive assumption, maintained throughout most of the pap er, is that the cost func-

tion C is submo dular; that is, the marginal cost C S [ i C S of adding user i to the set of

users S is nonincreasing in S i.e., usage of the network by new agents can only reduce the cost

of including a given agent. Submo dular cost functions also called concave cost functions, as in

Shapley [1971] have many remarkable prop erties p ertaining to core stability see, for example,

Moulin [1988] Chapter 5. In the context of access to a network, a very natural family of submo d-

ular cost functions arise in tree networks without congestion Sharkey [1995]. Fix a tree from the

source to all p otential users each user sitting on a no de of the tree; a cost is assigned to each

link or edge of the tree. The cost of serving any coalition S is simply the sum of the costs of the

links irresp ective of the numb er of users \downstream" of each link needed to reach all memb ers

of S . An imp ortant recent example of such a problem is the allo cation of costs in multicast trans-

missions on the Internet Herzog et al. [1995]. Section 7 is devoted to the application of our two

outstanding mechanismstovarious tree networks.

2 The Mo del

Given are N , the set of users, and C , the nondecreasing, nonnegative and submo dular cost function:

C ;= 0 ; S T CS CT

CS\T+CS [T CS+ CT f or any S; T N

We denote by u , u 0, the amount user i is willing to pay for access to the service. A revelation

i i

mechanism elicits from each user his/her willingness to pay and, based on these rep orts, it cho oses

who gets access and how the costs are shared. We use the vectors x and q to denote the resulting

allo cation decisions; x is the cost share imputed to agent i, and q describ es whether or not agent

i i

i is given access q = 0 if agent i is not given access, and q = 1 if agent i is given access. When

i i

wewant to make the utility pro le dep endence of these allo cations explicit, we will write q u and

xu to denote the allo cations resulting at a particular pro le. We assume quasi-linear utilities, so

an agent's utility is simply given by u q x .We p ostulate that a \reasonable" mechanism must

i i i

satisfy the following three prop erties:

No Subsidy NS The cost shares are nonnegative: x 0 for all i.

Voluntary Participation VP The welfare level corresp onding to no access q = 0 at no cost

x = 0 is guaranteed to each user if they rep ort truthfully.

Following established, but somewhat confusing, notational practice, the vector u will denote the rep orted values

as well as the truthful values. When the two di er, we will make sp ecial note of it. 5

Consumer Sovereignty CS Every user has a message u guaranteeing that, regardless of the

other rep orted values u , he/she gets access q = 1.

i i

Voluntary participation is the standard individual rationality constraint; user i do es not prefer

the \status quo" q = x = 0 allo cation to the allo cation q ;x assigned by the mechanism. In

i i i i

view of No Subsidy, this implies that she pays nothing if she gets no service q =0x = 0.

i i

Consumer Sovereignty means that by rep orting a high enough willingness to pay,any user is

guaranteed to get the service. For instance, in the mechanisms discussed b elow, rep orting any

u C fig their stand alone cost ensures that q = 1 no matter what the other rep orted u

i i j

are. Note that we do not require that rep orting u = 0 ensures that q = 0; as long as x =0

i i i

we can have q =1even though u =0.IfCS is strictly increasing in S ,however, the later

i i

con guration u = 0 but q = 1 will not happ en in any of the mechanisms discussed b elow.

i i

The strategypro ofness literature for cost sharing problems discusses two classes of mechanisms,

corresp onding to the choice b etween budget balance and eciency. The rst class are often called

the Clarke-Groves CG mechanisms Clarke [1971], Groves [1973]; given the pro le of individual

rep orts, such a mechanism computes an ecient coalition S and assigns payments x , i =1;:::;n

to individual users in sucha way that truthful rep orting of one's willingness to pay is a dominant

strategy for each user. Requiring strategypro ofness and eciency of S precludes having a pro le

of payments that exactly covers the cost of S for all pro les Green, Kohlb erg, and La ont [1976];

the mechanism may run a de cit at certain pro les, namely x

pro les, namely x >CS , or b oth, but we know it cannot b e budget balanced, x = C S ,

N N

at all pro les.

In Section 3, we show that there is a unique CG mechanism satisfying NS and VP Prop osition 1,

and that this mechanism satis es CS as well. Denote by w S; u or simply w S if the underlying

utility pro le is unambiguous the stand alone surplus of coalition S given the technology C :

w S; u = max[u C T ] 1

T S

The unique CG mechanism meeting NS and VP works as follows: given the pro le of rep orts u ,

i 2 N , compute the largest ecient coalition S i.e., the largest solution of the program w N; u;

submo dularityof C ensures that this largest coalition is well-de ned. Then set x = 0 for each

i 62 S and assign the following payment to each i 2 S :

x = u w N; u wN i; u 2

i i

We call it the marginal contribution mechanism b ecause, if given access, agent i's net utility gain

u x is exactly equal to her marginal contribution w N; u wN i; u to the stand alone surplus

i i

function w and if not given access then her marginal contribution is zero, w N; u=wNi; u.

In the case of a symmetric cost function if all users are equivalent, C S dep ends only up on the

If u = 0 but C S + i >CSi for all S N , then i is not part of the ecient coalition. In the mechanisms

discussed b elow, the coalition served is always a subset of the ecient coalition, so u =0q = 0 whenever agent

i i

i imp oses strictly p ositive incremental costs.

We use the standard notation that z = z .

S i

i2S 6

cardinalityof S,so CS=cjSj for some function c this mechanism charges the marginal

cost cjS j cjS j1 to all users. While this is quite similar in form to the traditional pivotal

mechanism, there is an imp ortant di erence as well; the marginal contribution mechanism never

runs a budget surplus i.e, it either is budget balanced or runs a budget de cit, whereas the pivotal

mechanism never runs a budget de cit i.e, it either is budget balanced or runs a surplus.

The second class of strategypro of mechanisms sharing the cost of jointly pro duced go o ds, insists on

budget balance at all pro les but may fail to serve the ecient coalition of users; that is, it provides

service to a coalition S of users that may not maximize the rep orted welfare u C S maybe

less than w N; u. As we discuss b elow, the strategypro of and budget balanced mechanisms we

consider are derived from an underlying cost sharing method. A cost sharing metho d is a formula

asso ciating to each cost function C and to each coalition S of users an allo cation of the total

cost C S among these users in the form of nonnegative cost shares S . These cost shares must

satisfy the budget balancedness constraint S =CS.

Wenow explain why these cost sharing metho ds underlie the relevant strategypro of and budget

balanced mechanisms. Given a cost sharing metho d and a pro le u of willingness to pay, consider

the normal form demand game where each user decides whether or not to request service user

i's strategy choice is q =0orq = 1 and costs are shared among those who request service

i i

according to the cost sharing metho d . If it so happ ens that for all pro les u the demand game

has a unique -equilibrium where -equilibrium can b e either a Nash equilibrium or a strong

equilibrium then any -equilibrium outcome de nes a strategypro of revelation mechanism. That

is, this mechanism elicits u , i 2 N , and implements the unique -equilibrium outcome of the

rep orted demand game. This general fact is a classic result of the implementation literature that

hinges on the prop erty of the preference domain known as monotonic closedness: see Dasgupta,

Hammond, and Maskin [1979]. In the particular case of sharing the cost of a jointly pro duced

go o d, the general fact admits a converse statement. If we lo ok for group strategypro of mechanisms

satisfying NS, VP, and CS, then the only way to obtain these is bywayofa cross monotonic cost

sharing metho d of which the asso ciated demand game has a unique equilibrium at all pro les.

In our problem, the prop ertyof cross monotonicity of the cost sharing metho d means that user

i's cost share cannot increase when the set of users expands

The pivotal mechanism assigns the net utility u x = w N w N i to user i, where w S = max [u

i i T S T

jT j

C T ] see, for example, Green and La ont [1979].

jN j

Moreover, the pivotal mechanism fails VP.

This route has b een intro duced much more recently: Shenker [1992,1995], Moulin and Shenker [1992], Moulin

[1994,1996], Deb and Razzolini [1995ab], Ohseto [1995], Serizawa [1994].

In the language of co op erative game theory, a cost sharing metho d is a \value" de ned for all reduced games

S; C .

In general, quasi-linear utilities do not yield a monotonically closed domain of individual preferences. In our

particular problem, given NS and VP,we can restrict agent i's consumption set to the union of 0; 0 no service at

no cost and the half-line 1;x service at some nonnegative cost. On this choice set, quasi-linear utilities form

a monotonically closed domain.

Two technically very di erent formulations of this statement are in Shenker [1992,1995] for the case where

the demand q is a real numb er and Moulin [1996] for the case where q is an integer, as we consider here.

i i

In some of the literature, the cross monotonicity prop erty is known as population monotonicity, suchasin

Sprumont [1990]. 7

S T; i 2 S T S 3

i i

recall that S is user i's cost share when S is the set of users who get access. Under cross

monotonicity, the demand game has nondecreasing b est reply functions if q = 1 is a b est reply

when a coalition S has access, then q = 1 remains a b est reply when any sup erset of S gets access

hence the demand game p ossesses a Pareto sup erior Nash equilibrium. This equilibrium can b e

computed by the simultaneous Cournot tatonnement starting at N ; that is to say the equilibrium

outcome S ; u is the limit of the following inclusion monotonic sequence:

S = N ; S = fiju S g 4

0 t+1 i i t

It turns out that the outcome S ; u is the welfarewise-unique strong equilibrium there maybe

other equilibrium q 's but they pro duce the same welfare of the demand game N; ; u, hence

the corresp onding revelation mechanism eliciting the pro le u, then implementing S ; u, and

sharing the cost C S ; u according to is strategypro of. It is group strategypro of as well. The

relevantcharacterization here is Theorem 2 in Moulin [1996], stating that every group strategypro of

revelation mechanism satisfying Budget Balance costs are exactly covered, NS, VP, and CS

obtains as ab ove as the strong equilibrium outcome of a cross monotonic cost sharing metho d

this result is stated here as Theorem 1 in Section 4.

Prop osition 1 characterizing the MC mechanism and Theorem 1 yield the rst contribution of

this pap er; together, they show that the combination of budget balance and strategypro ofness

allows greater exibility to the designer and stronger incentive prop erties than the combination of

eciency and strategypro ofness. The former combination, in particular, allows for more exibility

than the latter; in the former case, agent i's cost share can b e anyvalue b etween her stand alone

cost C fig and her incremental cost C S C S i where S is the coalition of users actually

served, whereas in the latter case her cost share is entirely determined. See the discussion in

Section 4.

Wenow turn to the second contribution of the pap er. Within the class of cross monotonic cost

sharing metho ds, we show that the Shapley value Shapley [1953] is characterized by the prop erty

that the worst welfare loss of its asso ciated revelation mechanism is minimal Theorem 2 in Section

5. In other words, if we denote by ; u the welfare loss pro duced by a mechanism at a pro le

u,so

C S ; u 5 ; u=wN; u u

S;u

then what we show is that the maximum of ; uover all u 2< is smallest when the metho d

is the Shapley value formula. Note that the Shapley value formula is cross monotonic when costs

are submo dular see Sprumont [1990], so it is indeed in the class of metho ds we are considering.

Thus, somewhat surprisingly, this minimax prop erty { minimizing the maximal welfare loss {

leads to a characterization of the Shapley value over the class of submo dular cost functions

based exclusively on incentive compatibility and eciency considerations.

Another characterization of the Shapley value that uses no considerations of equity is based on its p otential

prop ertyi.e., the fact that cost shares are the derivatives of a p otential function: Hart and Mas-Colell [1989].

Actually, the pro of of our Theorem 2 uses the p otential prop ertyaswell; see Section 5. 8

Finally,we devote Sections 6 and 7 to a quantitative comparison of our two outstanding strate-

gypro of mechanisms, namely marginal contribution 2 and the mechanism asso ciated with the

Shapley value. Both mechanisms have a shortfall { a budget de cit for the marginal contribution

mechanism and an unrealized welfare gain for the mechanism based on the Shapley value { and

these can b e quantitatively compared. Even though the nature of these shortfalls are quite dif-

ferent, the relative sizes the shortfalls incurred in our two mechanisms is still an interesting p oint

of comparison as discussed in Section 1. We compute explicitly Prop osition 2 the maximal

welfare loss of the Shapley value mechanism namely the quantity = sup ; u where

u2<

; u is de ned by 5 and is taken to b e the Shapley value mechanism and the maximal

budget imbalance of the marginal contribution mechanism, namely

= sup [C S u x u] 6

u2<

where xuisgiven by 2. The two formulae are, resp ectively,

=[ CN i] n 1C N 7

i2N

3 2

jS j 1!jN jjSj!

5 4

C S C N 8 =

jN j!

S N

For any cost function C , is smaller than if n = 2 or 3. For larger values of n, the comparison

can go either way but in many simple examples exceeds .Inanyevent, the ratio is b ounded

away from zero and in nity indep endently of the submo dular cost function C Corollary 3 to

Prop osition 2.

Section 7 is devoted to the sp ecial case of cost sharing of a tree network when cost is additive

along the links of the network and there are no congestion e ects i.e., the cost of each link is

indep endent of the numb er of users of the link. We compute and foravariety of simple tree

structures, and con rm the ndings of Section 6 that for small n, is smaller than but the

inequality is often reversed for larger values of n.However, certain tree structures yield < for

any size.

We conclude in Section 8 by raising some additional issues for future consideration. These issues

include the average case rather than worst-case analysis of shortfalls, other comparative asp ects

such as the computational complexity of the mechanisms, and the consideration of sup ermo dular

rather than submo dular cost functions.

3 The Marginal Contribution Mechanism

Given a xed cost function C ,a revelation mechanism is a mapping M asso ciating to all pro les

a coalition S u N equivalently expressed by the binary inclusion vector q u with u 2<

q u = 1 for all i 2 S u and q u = 0 for all i 62 S u and a vector xuof nonnegative

i i 9

monetary comp ensations. Budget balance is not required, so x u C S u can b e nonzero at

some pro les. In addition to the No Subsidy prop ertyxu0 for all i, all uwenow write the

other two prop erties intro duced in Section 2 as follows:

Voluntary Participation VP: for all pro les u, all agents i

u q u x u 0

i i i

Consumer Sovereignty CS: for all i 2 N there is a value u such that, for all u , S u

i i

contains i where u is the N i pro jection of u.

Wesay that M is strategyproof if for all pro les u, all agents i, and all \misrep orts" u ,wehave

0 0

uquxuuqu ;u x u ;u

i i i i i i i i

i i

In order to de ne our rst mechanism, we state without pro of two easy and imp ortant consequences

of the submo dularityof C for the surplus function w S; u and the maximizing coalitions.

Fact 1:For any u, w ;u is sup ermo dular: w S; u+wT; u wS [ T; u+wS \T; u for any

S; T N .

Fact 2:Ifanytwo coalitions S and T are ecienti.e., u C S = u CT= wN; u

S T

then so is S [ T ;we denote by S u the largest ecient coalition at u, and denote by q u the

corresp onding inclusion vector.

The marginal contribution MC mechanism picks the coalition S u=S u and charges the cost

shares x u given by formula 2.

Prop osition 1 Consider a strategyproof revelation mechanism M selecting an ecient al location

not necessarily the largest at al l pro les, meeting NS and VP. Then M is welfareequivalent to

MC: for al l u,all i

uquxu=uq ux u

i i i i

i i

Conversely, the MC mechanism meets NS and VP as wel l as CS and is strategyproof.

Pro of: Tocheck that MC satis es NS, rst pick a pro le u and an agent i in an ecient coalition,

i 2 S u; nonnegativityof x u amounts to w N; u u + wN i; u, which follows from

w N; u=u CS u u + u C S u i u + w N i; u

i i

S u S ui

VP is a straightforward consequence of w N i; u w N; u. CS follows from the fact due to

submo dularity that, for a given i,ifu Cfig then u C S + i u C S for all S N

i S +i S

and all u 2< .

Conversely,cho ose a mechanism M as in the premises of the rst statement. By the standard

characterization result of CG mechanisms e.g., Green and La ont [1979] or Moulin [1988] the 10

prop erty of strategypro ofness combined with the eciency of S u implies the following form for

xu:

x u=uquwN; u h u

i i i i i

Consider an arbitrary N i pro le u and denote by u its completion to an N -pro le such

0 0

that u =0.Wemust have x u =0 by NS and VP, as noted at the b eginning of Section 2

and the ab ove equation yields

0 0

h u =wN; u =wN i; u =wN i; u

i i

for any pro le u with N i pro jection u .Thus, the mechanism M takes the same form as MC,

except for the fact that S u can b e any ecient coalition not necessarily the largest one as in

MC. In order to check that the two mechanisms are welfare equivalent, we need only to consider

an agent i in S u S u b ecause b oth mechanisms give the precisely same allo cation to every

agentinSu or outside of S u. For any agent i in S u S u, wehavewN; u=wNi; u

hence she receives from the two mechanisms the q ;x allo cations of 0; 0 and 1;u resp ectively,

i i i

which are welfare equivalent. 2

Manipulating the marginal contribution mechanism

Strategypro ofness do es not prevent all forms of manipulation of the mechanism, and indeed there

are stronger forms of incentive compatibility to consider. Group strategyproofness expresses the

prop erty that no coalition of users will nd it pro table to jointly misrep ort their willingness to

0 0

pay.To make this precise, x any coalition T N and anytwo pro les u; u such that u = u

0 0 0

for all j 62 T . Denote byq; x and q ;x the allo cations implementedat u and u resp ectively.

Group strategypro ofness requires that if the inequality

0 0

u q x u q x

i i i i

i i

holds for all i 2 T then it must b e an equality for all i 2 T as well.

The marginal contribution metho d MC is not group strategypro of. To see this, consider a three

agent problem with a symmetric cost function

C S = cjSj w her e c1=3;c2=5; and c3=6

The utility pro le is not symmetric:

u =4;u =u =1:2

1 2 3

Straightforward computations yield S u=f1g,wN; u=1,wN1;u=0,andwN2;u=

wN 3;u = 1. Hence, MC yields the net utility1;0;0. However, we see that the coalition

0 0 0

f2; 3g can successfully manipulate by rep orting u =4;4;4 we = u = 4 b ecause at pro le u

2 3

0 0 0 0

have S u =N,wN; u =6,andwNi; u = 3, so x =46 3 = 1. At this pro le, agents

2; 3 end up with q =1, x = 1, and a p ositive net utility; naturally, the outcome is inecient but

i i 11

all agents including agent 1 b ene t from the misrep ort by agents 2; 3. The banker is the loser,

as he must comp ensate for 3 units of budget de cit.

In the ab ove example the manipulation by coalition f23g requires reliable co ordination b etween

these two users, b ecause agent1would su er a net loss if she were to rep ort u = 4 while agent3

is still rep orting truthfully he would end up paying 1:8 for the service.

However, there is a sp ecial kind of manipulation again a violation of group strategypro ofness

that do es not require any such co ordination b etween agents and to which the MC mechanism is

always vulnerable. Fix a pro le u the true pro le and a willingness to pay u for agent i such

0 0 0 13

that u

i i

i i i

Fact 1: S u S u

0 0

Fact 2: no agent j , j 6= i,ishurt: w N; u wN j; u w N; u wN j; u

0 0

Fact 3:ifi2S u then S u =S u and x u= x u

i i

Facts 2 and 3 tell us that if an agent i is served at pro le u, rep orting a higher willingness to pay

is a matter of indi erence to her, and it can never hurt any other agent. To see that in some cases

it can strictly help other agents, consider the following example with two agents:

C 1 = C 2=6; C12 = 8 ; u = u =5

1 2

Truthful rep orts yield S u= f12g and cost shares x u=x u = 3. By raising his willingness

1 2

to payto u = 7, agent 1 do es not a ect his allo cation and reduces agent 2's cost share to

x u ;u = 2. Once again, the banker is the sucker!

2 1

Since MC is the unique CG mechanism that satis es NS and VP, there is no way to prevent

manipulation by coalitions without abandoning the goal of eciency.Wenow turn to the class of

mechanisms that do just this.

4 Cross Monotonic Cost Sharing Metho ds

Recall that a cost sharing metho d is a mapping asso ciating to each coalition S avector S ;i 2

Sofnonnegative cost shares such that S =CS. Assuming that the metho d is cross

monotonic prop erty 3 then the algorithm de ned in formula 4 de nes an inclusion monotonic

13 0

TocheckFact 1, denote S = S u, T = S u and compute

0 0 0

u C S \ T u C S C S [ T C T C S C S \ T u u u C T u C S [ T

S \T S S T

S T T S [T

implying, by de nition of T , S [ T = T . Similar reasoning establishes S u; N j S u; N for all j .To prove

0 0

Fact 2, namely w N j; u w N j; u w N; u wN; u, we note that b oth sides of the inequality are

0 0

nonnegative and we distinguish three cases. If S N j; u contains i,sodoS N; u, S N; u , and S N j; u

0 0

and b oth terms of the inequality equal u u .IfSNj; u do es not contain i, neither do es S N j; u

and the left hand term is zero. If i 62 S N j; u but i 2 S N j; u , call u the smallest value such that

S N j; u ;u contains i. Because S N; u ;u contains i as well, the righthand term in the inequalityis

i i

0 0

at least u u whereas the lefthand term equals u u . The easy pro of of Fact 3 is omitted.

i i i i 12

~ ~

sequence in N , with limit S ; u or S u if, from the context, the underlying cost sharing metho d

is unambiguous. We asso ciate to the cost sharing metho d the following revelation mechanism

denoted by RM :

~ ~ ~

S u= S; u; q u= 1 and x u= S; u if i 2 S ; u; q u= 0and x u= 0other w ise

i i i i i

The prop erty of group strategypro ofness has b een de ned in the previous section after the pro of

of Prop osition 1.

Theorem 1 For any cross monotonic cost sharing method , the mechanism RM de ned

by 9 is budget balanced, meets NS, VP, CS, and is group strategyproof. Conversely, pick a

revelation mechanism M satisfying budget balance, NS, VP, CS, and group strategyproofness.

Then there exists a cross monotonic cost sharing method such that RM is welfareequivalent

to M .

The prop osition is identical to Theorem 2 in Moulin [1996]. In the sp ecial case of an excludable

public go o d C S = 1 for all nonempty S , Deb and Razzolini [1995a,b] and Ohseto [1995] provide

similar yet di erentcharacterization results.

We do not provide a pro of of the \converse" statement in Prop osition 1; however, the \direct"

statement is easy to prove and we rep eat its pro of here.

Pro of of direct p ortion of Theorem 1:We x a cross monotonic metho d and check that

RM is group stategypro of all other statements are trivial.

Let u b e the true pro le of willingness to pay and S = S ; u. Assume a certain coalition T

0 0 0 0 00 0

manipulates at u by u where u = u for all j 62 T and set S = S ; u . Finally, set S = S [ S

0 00

and denote by q resp ectively, q and q the vector of 0 and 1 corresp onding to S resp ectively,

0 00

S and S . Check rst

00 00 0 0

for all i 2 T : u q S u q S 10

i i i i

i i

00 0 0 0 00

This is clear if i 62 S implying i 62 S . If i 2 S this follows from S S and cross monotonicity.

00 0

Lastly if i 2 S S implying i 2 S then u S =0because the manipulation prop erty

i i

0 0 00

requires u q S u q S and so, by cross monotonicity, u S 0 as desired.

i i i i i i i

Combined with the manipulation assumption, formula 10 implies

00 00

for all i 2 T : u q S u q S 11

i i i i i

with at least one strict inequality.We claim that 11 holds true as well for i 62 T . This is clear

00 00

if i 2 S implying i 2 S by cross monotonicity,orif i 62 S in which case b oth sides of the

In that pap er, individual preferences are \classical", namely monotonic strictly in money, continuous and

convex; this domain is generally larger than that allowed by quasi-linear utilities. However, in our problem, NS and

VP imply that x =0ifq = 0 hence agent i's consumption set is the union of 0; 0 q =0, x = 0 and of the half

i i i i

line 1;x , x 0. On this set, a classical preference is entirely describ ed by a simple number u the willingness

i i i

to pay such that 0; 0 is indi erentto1;u . Hence the classical domain and that of preferences represented by

quasi-linear utilities coincide. 13

00 0 0 0

inequality are zero; if i 2 S S implying i 2 S observe that u S 0 b ecause q is

i i

0 0 00

Nash equilibrium at u and u = u soby cross monotonicity u S 0 as desired.

i i i

Wehave proven that 11 holds for all i with at least one strict inequality. Therefore S S is

nonempty or 11 would b e an equality for all i and there is a largest integer t such that S S

where the sequence S is derived by 4 at u. Pick an agent i in S \ S S ; wehave

t t t+1

00 00 00

u <S S u q S < 0

i i t i i i

But this contradicts 11 as q is a Nash equilibrium at u. 2

The family of revelation mechanisms uncovered by Theorem 1 leaves a fair degree of exibilityto

the mechanism designer. Supp ose rst the agents have equal right to the use of the technology

C , so that the prop erty \equal treatment of equals" is deemed desirable: if the function C is

symmetrical in i and j i.e., C S + i=CS+j for all S N fi; j g, then the cost shares

of agents i and j should b e symmetrical as well. Then the designer can cho ose from a variety

of formulae, including the Shapley value Shapley [1953] , the egalitarian solution Dutta and

17 18

Ray [1989] , convex combinations of these, and more. For instance, in the case of two users

N = f12g, these twovalues are

1 1

Shapley 12 = C 12 + C 1 C 2; f1; 2g= C12 + C 2 C 1

1 2

2 2

C 12 C 12

Egalitarian 12 = medf ;C1;C12 C 2g; 12 = medf ;C2;C12 C 1g

1 2

2 2

where medfg is the median op erator.

Another p ossibility is that the users have unequal claims on the technology C p erhaps b ecause

of seniority, earlier contributions to its development, and so on and that the designer wishes to

cho ose a cost sharing formula that fairly re ects this fact. For simplicity, supp ose that the users

are ordered as f1; 2;:::;ng and that their claims are lexicographic: agent 1's claim outweighs any

other claim; agent 2's claim outweighs those of any agent3;4;:::;n, and so on. A consequence of

Theorem 1 is that this hierarchy of claims determines entirely the choice of our designer within the

class de ned by NS, VP, CS, and group strategypro ofness. Indeed, in any cross monotonic cost

sharing metho d, agent1must pay out at least C S C S 1 where S is the coalition of users who

get the service; this follows from S S 1 = C S 1 and S =CS S.

S 1 S 1 1 S1

Thus, agent 1's sup erior claim entitles him to this minimal cost share C S C S 1. Similarly,

agent 2's cost share must b e at least C S 1 C S f1; 2gby rep eating the ab ove argument,

and so on. The resulting cost sharing metho d is the familiar hierarchical stand alone metho d

often called the marginal contribution metho d where the agent i with the largest index in S

i.e., the agent with the least claim to the resources pays her stand alone cost C fig, the agent

Note that is is a strong version of the general idea that cost shares should re ect the inherent symmetries of

the cost structure.

For a pro of that this value is cross monotonic when C is submo dular, see, for example, Sprumont [1990].

The egalitarian solution of the co op erative game N; c picks a pro le of cost shares x within the stand alone

core of this game x C S .

See Sprumont [1990] for a systematic discussion of the cross monotonicity prop erty. Note that the nucleolus

Schmeidler [1969], another familiar value for co op erative games, is not cross monotonic see Somnez [1993]. 14

with the next largest index j pays C fij g C fj g, and so on. This metho d is cross monotonic

given a submo dular cost function recall that the Shapley value is but the arithmetic average of

such metho ds over all p ossible ordering of N .

5 The Shapley Value Mechanism

We turn to the outstanding p osition of the Shapley value within the class of cross monotonic cost

sharing metho ds. Given such a metho d ,we denote by the maximal welfare loss of the

mechanism RM de ned by 9;

= sup ; u = sup [w N; u u C S ; u]

S;u

n n

u2< u2<

+ +

We write as the Shapley value cost sharing metho d

jTj!jS j jTj 1!

S = [C T [ i C T ] f or al l S and i 2 S

jS j!

T Si

We refer to the corresp onding revelation mechanism RM asSH.

Theorem 2 Among al l mechanisms RM derivedfrom cross monotonic cost sharing methods,

SH has the uniquely smal lest maximal eciency loss:

< for all 6=

Pro of: The pro of is divided into 5 steps. In the rst three steps we assume a given cross monotonic

cost sharing metho d and a utility pro le u.We write S = S ; u and T = S u.

Step 1: S T

Assume, to the contrary, that S T is nonempty. Observe that for all i 2 S ,wehaveu S

i i

by construction of S as in 4. Hence, we compute:

u C S [ T u C T = u + C T C S [ T S +CT S [T

S [T T S T S T S[T

= S S[T + C T S [ T 0

ST ST T

But this is a contradiction b ecause T is the largest ecient coalition.

The inclusion just established implies in particular the following expression for the eciency loss:

; u=wN; u wS ; u;u

Monderer and Shapley [1993] establish a seemingly unrelated prop erty of the demand game asso ciated with

an arbitrary cost function C not necessarily submo dular, not even nondecreasing and an arbitrary cost sharing

metho d. They show that the demand game is a \p otential game" if and only if the metho d is the Shapley value.

Interestingly, our pro of relies on the representation of the Shapley value byapotential function due to Hart and

Mas-Colell [1989]. 15

To see this, apply Step 1 to the restriction of the game to S ; u.

Step 2: For al l R N , R \ S = ;, there exists i 2 R such that u <S[R

i i

This follows from the de nition of S ;ifwehave for all i 2 R, S [ R u then every coalition S

i i t

in the sequence de ned by 4 must contain S [ R by cross monotonicity and an obvious induction

argument, hence a contradiction.

Step 3: Bounding ; u

We represent an ordering of N = f1; 2; 3;::: ;ng by a p ermutation of N , with the interpretation

that 1 is the rst in the ordering, 2 is second, and so forth. We write E k; = fij i

kg the set of users ranked k or higher by the p ermutation .We set:

= max E i; C N 12

i=1

where the maximum is taken over all p ermutations of N .

Wenow show that ; u for the arbitrary pro le u xed at the b eginning of the pro of. If

S = T ,wehave ; u = 0 and the inequality follows from the nonnegativityof a consequence

of cross monotonicity. From now on, assume S 6= T and so, by Step 1, wemust have S T .

We lab el arbitrarily the agents in N T as f1; 2;:::;kg and use cross monotonicity to see that

C N C T N and:

N T

C N C T N f1;:::;i 1g

i=1

Next we lab el the elements in T S such that T S = fk +1;k +2;:::;k +k g.We apply Step 2

rep eatedly, rst to R = T S to pick an agent lab eled k +1 in R , then to R = T S [fk +1gto

1 1 2

pick an agent k +2 in R , and so on, with agent k + i selected in R = T S [fk +1;::: ;k+i1g

2 i

such that

0 0

u < T ;:::;u < T fk +1;:::;k + i 1g;:::;u < S[fk +k g

k +1 k+1 k+i k+i k+k k+k

Finally,we lab el arbitrarily the agents in S as fk + k +1;:::;ng and rep eatedly use cross

monotonicity to show:

S fk +k +1;:::;i 1g C S

i=k +k +1

Summing the ab ove inequalities yields

C N C T + u + CS E i;

i=1

where is the ordering i= i, i =1;:::;n. The lefthand term in the inequalityabove equals

u C T u C S + C N = ; u+CN

T S 16

hence the desired conclusion ; u follows.

Step 4: sup n ; u=

u2<

In view of Step 3, it is enough to show that for any ordering of N ,wehave

sup ; u ; = E i; C N

u2<

i=1

We pick an arbitrary ordering , denoted for simplicity i= i, and a number ,0<<1.

Consider the utility pro le

u =1Ei; for all i 13

i i

Denote by S the inclusion monotonic sequence 4. Note that S ; uistypically empty; it will

b e emptyifS is p ositive for all S .However, if some of the cost shares S are zero, wemay

i i

have S ; u 6= ;.

Wenow prove ; u ; 1 C N . As the choice of and were arbitrary this will

conclude the pro of of Step 4. We shall use the following fact: for all S , all i 2 S

S =0 [CS= CSi and S = S i for all j 2 S i] 14

i j j

To see this, sum up the inequalities S S i and invoke the monotonicity of the cost

j j

function.

If S = ; wehave

; u=wN; u u C N = ; 1 C N

and so we are done. However, as we noted ab ove, S need not b e empty.Wenow show that this

do es not alter our conclusion. De ne A = fi 2 N ji 2 S g, where the S are de ned in 4. If

i i

A = ;, then S = ; and we are done. Supp ose A 6= ; and let i b e its smallest element. For all

j smaller than i wehavej 62 S hence j 62 S ;thus, S E i; and the assumption i 2 S

j i1 i1 i

implies

1 E i; = u S E i;

i i i i1 i

Hence u = 0 and E i; = 0. Nowwe can \eliminate" user i and consider the reduced problem

i i

over N i with the ordering induced by ; let denote this induced ordering. In view of 14,

wehaveN; =Ni; noting N must b e zero, hence C N =CNi. Similarly,

denoting by u the N i pro le obtained by removing u ,wehave, by 14:

i i

for all j < i; Ej; =0 Ej; = E j;

i j j i

where E j; are the successors of j in N i. Therefore, the pro le u is de ned in exactly

i i

the same way namely by prop erty 13 in the reduced problem as u was in the original problem.

Consider again the sequence S ; u de ned in 4. Dropping the dep endence on and u tem-

p orarily,we denote by S T the t'th term in the sequence de ned by the program 4 with the

0 0

starting p oint S T = T. Note that in general T T S T S T ; thus, denoting the limit

0 t t

0 0 0 0

~ ~ ~ ~ ~

of these sequences by S T and S T , wehave ST 6=ST,T T \ST 6=;. 17

Applying this, and 14, to the two sequences S N and S N i, we see that either S N =

t t

~ ~ ~

SNiorSN=SN i[fig. Therefore

~ ~

w N; u=wN i; u ; w S ; u;u= wS; u ;u

i i i

implying ; u= ; u . We can now rep eat the ab ove argument for the reduced problem,

and use an obvious induction argument to arrive at a reduced problem where indeed A = ;. This

completes the pro of of Step 4.

Step 5:

Wehave proven =. It remains to show the inequality < when is the Shapley

value and is another cost sharing metho d. First, one checks easily:

1 0

X X

1 jSj 1!n jSj!

A @

C S C N 15 ; =

n! n!

S N

where the lefthand sum is taken over all p ermutations of N . Note that the righthand sum is the

p otential function P N of Hart and Mas-Colell [1989] who proved that the Shapley value metho d

can b e written as:

jTj 1!jS j jTj!

S =PSPSi f or al l i; al l S; w ith P S = C T 16

jS j!

T S

We show b elow, by induction on jN j, that the Shapley value is characterized by the prop erty that

; is indep endentof . In view of 15 this will imply the desired conclusion, namely

=PNCN and >PNCN for all ; 6=

This claim is obvious for n = 1 there is only one cost sharing metho d and very easy to check for

0 0

n = 2. Assume it holds for all N such that jN jn1 and consider N such that jN j = n. Fix

an ordering such that 1 = i, and denote by the induced ordering on N i.

Check rst that ; is indep endentof by computing:

N+ ; +CN iCN ;=

=PNPN i + P N i C N i + C N i C N = P N C N

Conversely, assume that is a metho d such that ; is indep endentof . Check that ;

is indep endentof this is obvious from the de nition of ; , therefore, for any given such

that 1 = i,we compute:

P N C N =; = N+; +CN iCN = N +PN iCN

i i i

so N =PNPN i. The conclusion = then follows from the p otential formula for

the Shapley value 16. 2 18

6 The Maximal Losses of MC and SH

The maximal budget de cit of MC, de ned by 6, can b e computed with the help of 2 which

de nes the cost allo cations of MC:

= sup n 1w N; u wN i; u 17

u2<

+ i2N

On the other hand, the maximal welfare loss of the Shapley value mechanism SH is de ned as

h i

= sup w N; u u C S ;u

S ;u

u2<

where S ;u is the equilibrium demand computed according to 4.

Prop osition 2 For any nondecreasing and submodular cost function C such that C ;= 0,we

have:

= C N i n 1C N 18

i2N

0 1

jSj 1!n jSj!

@ A

= P N C N = C S C N 19

S N

Pro of: The formula 19 giving has b een established in Step 5 of the pro of of Theorem 2. Wenow

prove 18. Check rst a consequence of submo dularity: the quantity C S i n 1C S

i2S

is nondecreasing in S using the convention that S i = S if i 62 S . We omit the straightforward

pro of. Then x a pro le u and set S = S u. Compute

X X

n 1w N; u wN i; u n 1u C S u C S i

S S i

i2N i2N

X X

= C S i n 1C S C N i n 1C N

i2N i2N

In view of 17 this shows C N i n 1C N . The converse inequality follows

i2N

bycho osing a pro le with each u large enough so that S u= N and S u =N i for all i.

i i

This completes the pro of. 2

We derive four corollaries of Prop osition 2. Corollary 1 says that with 2 or 3 users, never

exceeds ; SH has a smaller shortfall than MC for these small systems. Corollary 2 gives an

intuitive formula for and in the case of symmetric cost functions. Corollary 3 computes the

largest and smallest ratios ; while the maximal budget imbalance of MC can b e as muchas n

times the maximal welfare loss of SH, the latter can only b e as much as log n times the former.

Finally, Corollary 4 computes the worst ratio of loss to total cost, for b oth mechanisms; it can b e

as large as log n for SH but never exceeds 1 for MC. 19

Corollary 1 In two agent problems jN j =2, we have

= = C 1 + C 2 C 12

2 2

In three agent problems jN j =3, we have

s s 2s

1 2 3

= + s 2s = 20

2 3

3 6 3

where s = C S

S N ;jS j=t

Corollary 2 If the cost function takes the form C S = cjSjwhere c is concave on f0; 1; 2;:::;ng

and c0=0, then

= cn cn 1 21

n n

ci cn 1

= 22

n n i n

i=1

Corollary 3 Given n, the number of users, we have

C C

= n ; sup = f n sup

C C

where the supremum is taken over al l nondecreasing submodular cost functions such that C ;= 0,

0 n 1

where we use the convention that =1, and where f is the function f n .Inparticular,

i=2

0 i

C = 0 if and only if C =0.

Corollary 4 Given n, the number of users, we have

C C

sup =1 ; sup = f n

C N C N

C C

with the same notational conventions as in Corol lary 3.

Pro of of the Corollaries:

Corol lary 1: The formulae for ; for the cases n =2;3 follow at once from Prop osition 2.

3 1

Inequality 20 amounts to 2s +8s 5s and follows from the combination of s s + s

1 3 2 2 3 1

2 2

and s s b oth of which are implied by submo dularity.

1 2

Corol lary 2: This follows directly from the formulae for and equations 18 and 19 resp ec-

tively.

Corol lary 3: Note that b oth C and C are symmetrical and additive functions of C ;ifC

obtains from C by p ermuting the agents according to ,wehave C = C and C = C. 20

C C

C is the cost function obtained by symmetrizing C ,wehave as Therefore, if C = =

n! C

well. Thus the suprema in the statement of Corollary 3 can b e computed by restricting attention

to symmetric cost functions where C S = cjSj as in Corollary 2. In view of Corollary 2 we are

left with the task of computing the b ounds of the ratio

i=1

ncn 1 n 1cn

over all nondecreasing concave functions c on f0; 1;:::;ng such that c0=0.

Fixing cn and cn 1 such that the denominator in 23 is p ositive, we reach the supremum

of 23 by taking ci= cnnicn cn 1 for i =1;:::;n 2. Then the ratio

23 is computed to b e f n. On the other hand, the minimum of 23 is reached by taking

i 1

ci= cn1, i =1;:::;n 2, and for this choice the ratio 23 is computed to b e .

n1 n

To conclude the pro of it remains to observe that the function c is linear if and only if C =0,

~ ~

and if and only if C = 0, where C is the symmetrized form of C .

Corol lary 4:To prove that sup = 1, note that C C N holds by monotonicityof C.

C N

Then consider the cost function C S = 1 for all S 6= ;, and observe that C =C N. To

C C

prove that sup = f n, note that C f nC N follows from the upp er b ounds on

C N C

and computed ab ove. Checking the inequality C =fnC N is straightforward.

C N

7 Application to Cost Allo cation on a Tree Network

Anumb er of cost allo cation problems in the transp ortation and communication industries are

commonly mo deled with the help of a tree network Sharkey [1995] provides an excellent survey

of the relevant game theoretic literature. Some numb er of users are lo cated at the no des of a

xed distribution tree. In the communications context, whichwe will use as our guide in this

section, the origin of this tree is interpreted as a source sending a signal to the users; access to

the network represents the ability to read the signal, b e it cable TV, or Internet services, or some

other medium.

In the most common mo del, the cost of using a given link an edge connecting two no des of the

tree is indep endent of the numb er of users sharing this link, and the total cost of op erating a

certain tree is the sum of the cost of its links. Therefore, the minimal cost of serving an arbitrary

coalition of users is a submo dular function b ecause the addition of new users reduces or leaves

unchanged the set of links needed to graft a given user onto the distribution tree.

This represents an assumption of no congestion e ects. This is an accurate mo del when lo oking at multicast

transmissions Herzog et al. [1995], the construction of networks where the exp ense is in laying the b er, not in

the amount of b er, and many other examples. It is not a go o d mo del for congestion-prone facilities.

We do not consider the option of changing the design of the distribution tree when memb ers are not included;

that is, denying service to a memb er never results in utilizing an additional link. In our mo del, the set of links

needed to service a particular agent is xed, and the inclusion of other memb ers only a ects what other links are

needed. Minimal cost spanning trees are an example of a network where this condition fails e.g., adding a member 21 c0

c1 c0

c1 c4 c1 c2 c3 c4 c2 c2 c3

BushLinear Binary

Figure 1: Three kinds of trees, each illustrated with n =4.

Wenow compare the maximal losses and of our two mechanisms MC and SH for a number of

simple spanning trees. A general formula for obtains at once. Consider a tree T describ ed by the

function T N , where T S is the set of links needed to service the coalition S . Let D b e the set

of links followed by only one user; D = [ T N T N i. Then wehave =CTCD.

i2N

This general formula is applied in the next result to three simple kinds of trees. The function

n 1

f n= intro duced in the previous section plays a central role in the computation of in

i=2

f n

these simple trees. Note that converges to 1 as n grows large, and that the rst ten values of

log n

f are as follows:

n 2 3 4 5 6 7 8 9 10

f n .5 .833 1.083 1.283 1.450 1.593 1.718 1.829 1.929

Table 1: The value of f n for n 10 three signi cant gures.

Consider three trees depicted in Figure 7, a \bush" tree, a \linear" tree, and a \binary" tree.

Prop osition 3 For the bush tree we have

= c ; = f nc

0 0

For the linear tree we have

n1 n

X X

= c ; = f ic

i ni+1

i=1 i=2

For the binary tree of depth l ,so n=2 , we have

l1 l1

X X

k k lk

2 c ; = 2 f 2 c =

k k

k =0 k =0

Pro of: To prove this, we need only check the computations of . Recall from Step 5 in the pro of

of Theorem 2 that for an arbitrary ordering of N wehave

= E i; C N 24

i=1

may result in some link no longer b eing needed and may result in a non submo dular cost function: see Sharkey

[1995] page 724. 22

Pick an arbitrary link of the tree from no de a to no de b and an ordering of N in which all the

descendants of b app ear last. The cost of the link ab is shared b etween all the descendants of b;if

there are p such descendants that cost app ears with the factor f p in the righthand sum of 24.

This technique yields easily the formulae for in Prop osition 3. 2

Note that the argument in the pro of also gives more general formulae for . Consider a tree with

l levels and a splitting factor of k at level i and link cost c .

i i

1 0

l i l1

Y Y X

A @

f k k = c

j j i

j =i+1 j =1 i=1

For this same example, wehave

1 0

i l1

Y X

A @

= C N nC k = C

l j i

j =1 i=1

Note that the maximal pro t i.e., minimal revenue imbalance arises when l = 1 and k = n.

Thus, maximizing pro t leads to no sharing b etween agents, b ecause sharing makes it harder to

extract revenue from the agents.

Prop osition 3 con rms the general fact that for small numb ers of users is smaller than , but

the comparison is reversed as the numb er of users grows. For instance, in the bush tree, the linear

tree, and the binary tree with constant link cost, one checks that increases with n; this ratio

exceeds 1 for n 4 in the case of the bush tree, for n 6 in the linear and binary trees. In the

bush and linear trees, the ratio grows asymptotically as log n; in the binary tree the ratio go es to

a constant 1:118.

8 Concluding Comments

Wenow brie y address two additional asp ects of our problem.

8.1 Sup ermo dular Cost Functions

Another relevant context is when the cost function is sup ermo dular rather than submo dular.

This o ccurs, for example, in tree networks with concave congestion i.e., the cost of the link

is a concave function of the numb er of users \downstream". For sup ermo dular costs, the MC

mechanism remains the canonical CG mechanism, as Prop osition 1 and its pro of remain valid

word for word. However, the computation of its maximal budget imbalance b ecomes hard and

the imbalance can b e a surplus or a de cit. On the other hand, the demand game asso ciated

with the Shapley value cost sharing metho d do es not, in general, have a unique strong equilibrium

the b est reply functions are decreasing, so multiple noncomparable equilibria are p ossible, hence

it do es not result in a strategypro of revelation mechanism. The small class of cost sharing

metho ds that do result in a strategypro of mechanisms b ecause their demand games always have 23

a unique strong equilibrium has b een characterized in Moulin [1996]. It reduces essentially to

the family of hierarchical stand alone metho ds discussed in Section 4, where users pay their

incremental costs according to a xed ordering of N : N =CE;i C E ;i 1 for

some ordering . The criterion of minimizing the maximal welfare loss do es not seem to select

a clearly identi able metho d in this family. Thus, in the sup ermo dular case, the combination

of group strategypro ofness and budget balance proscrib es equal treatment of equals, whereas

the combination of strategypro ofness and eciency as in the submo dular case prescrib es equal

treatment of equals. While the former group strategypro ofness and budget balance still has

sup erior incentive prop erties in the sup ermo dular case, it no longer has claim to greater exibility.

8.2 Incentives for Adopting AlternativeTechnologies

The shortfalls in the two candidate mechanisms, SH and MC, lead to interesting incentive questions

in the choice of technology. Assume that the facility has committed to using either MC or SH.

What happ ens when a facility currently using a technology with cost function C is given the

~ ~

chance to adopt a b etter technology C , where C S C S for all S N .

Consider the case where the cost functions C and C are b oth symmetric. Then C S = cjSj and

C S =c ~jSj as in Corollary 2.

In the SH mechanism, the budget is always balanced, and so the comparison b etween technologies

cjSj

rests strictly on the welfare comparison. The Shapley cost shares take the simple form S =

jSj

~cjSj

and S = . For any given utility pro le u, the resulting equilibrium with the new

jSj

technology will have higher or equal welfare compared with the welfare asso ciated with the old

technology b ecause all cost shares are lower with the new technology, and so the newer equilibrium

will b e at least as large and so have at least as great total willingness-to-pay, and will havelower

costs. Thus, with the SH mechanisms, the incentives p oint unambiguously towards adoption of

cheap er technologies in the symmetric case.

In the MC mechanism, the answer is less clear. For any given utility pro le u, the welfare is at least

as great under the new technology.However, the budget imbalance may b e larger with the new

technology. Consider, for example, the case wherec ~s= cs for all s

cn 1. The revenue R raised by the MC mechanism is given by R = jS jcjS j cjS j1

where S is the ecient coalition. Then, for any u such that S = N in the old technology, the

budget loss cost minus revenue is higher under the new technology, with the di erence b eing

1 ncn cn 1. When lo oking only at the loss at a given utility pro le, this p erverse

incentive to not adopt cheap er technologies can even o ccur when the new technology has merely

scaled down costs,c ~s=cs for some 0 <<1. However, if one compares instead the worst

case budget imbalance for a given technology, then this p erverse incentive do es not o ccur when

the costs are simply scaled down since scales with C but can o ccur when the functional form

ofc ~ exhibits much smaller marginal costs prop ortional to the total cost of including additional

users. Thus, technologies suchasmulticast transmissions which do not change the stand alone

costs but rather increase the extent to which the resource is shared, may lead to increased rather

than decreased losses. 24

9 References

Clarke, E. H., 1971. \Multipart Pricing of Public Go o ds," Public Choice, 11, 17-33.

Dasgupta, P., P. Hammond, and E. Maskin, 1979. \The Implementation of So cial Choice Rules,"

Review of Economics Studies, 46, 185-216.

Deb, R. and L. Razzolini, 1995a. \Voluntary Cost Sharing for an Excludable Public Pro ject,"

Mimeo, Southern Metho dist University.

Deb, R. and L. Razzolini, 1995b. \Auction Like Mechanisms for Pricing Excludable Public Go o ds,"

Mimeo, Southern Metho dist University.

Dutta, B. and D. Ray, 1989. \A Concept of Egalitarianism under Participation Constraints,"

Econometrica, 57, 615-635.

Fleurbaey, M., and F. Maniquet, 1993. \ Fair Allo cation with Unequal Pro duction Skills: The No

Envy Approach to Comp ensation," Mathematical Social Sciences, forthcoming.

Green, J., E. Kohlb erg, and J. J. La ont, 1976. \Partial Equilibrium Approach to the Free Rider

Problem," Journal of Public Economics, 6, 375-394.

Green, J. and J. J. La ont, 1979. Incentives in Public Decision Making,inStudies in Public

Economics,Vol. 1, Amsterdam: North-Holland.

Groves, T., 1973. \Incentives in Teams," Econometrica, 41, 617-663.

Hart, S. and A. Mas-Colell, 1989. \Potential, Value, and Consistency," Econometrica, 57, 589-614.

Herzog, S., S. Shenker, and D. Estrin, 1995. \Sharing the `Cost' of Multicast Trees: an axiomatic

analysis," Pro ceedings of Sigcomm '95 and forthcoming in Transactions on Networking.

Israelsen, D., 1980. \Collectives, Communes, and Incentives," Journal of Comparative Economics,

4, 99-124.

Lo ehman, E. and A. Whinston, 1974. \An Axiomatic Approach to Cost Allo cation for Public

Investment," Public Finance Quarterly, 2, 236-251.

Maskin, E., 1985. \The Theory of Implementation in Nash Equilibrium" in Hurwicz, L., D.

Schmeidler, and H. Sonnenschein eds. Social Goals and Social Organizations, Cambridge: Cam-

bridge University Press.

Monderer, D., and L. S. Shapley, 1993. \ Potential Games," mimeo, Technion and UCLA and

forthcoming in Games and Economic Behavior.

Moulin, H., 1987. \A Core Selection for Pricing a Single Output Monop oly," Rand Journal of

Economics, 18, 397-407.

Moulin, H., 1988. Axioms of Cooperative Decision Making, Monographs of the Econometric So ci-

ety, Cambridge: Cambridge University Press.

Moulin, H., 1994. \Serial Cost-Sharing of Excludable Public Go o ds," Review of Economics Stud- 25

ies, 61, 305-325.

Moulin, H., 1995. Cooperative Microeconomics: a game theoretic introduction, Princeton: Prince-

ton University Press and Prentice-Hall.

Moulin, H., 1996. \Incremental Cost Sharing; characterization by strategypro ofness," mimeo,

Duke University.

Moulin, H., and S. Shenker, 1992. \Serial Cost Sharing," Econometrica, 50, 1009-1039.

Okada, N., T. Hashimoto, and P.Young, 1982. \Cost Allo cation in Water Resources Develop-

ment," Water Resources Research, 18, 361-373.

Ohseto, S., 1995. \Strategy-Pro of Cost Sharing for an Excludable Public Pro ject," mimeo, Rit-

sumeikan University.

Ro emer, J., 1971. \The Mismarriage of Bargaining Theory and Distributive Justice," Ethics, 97,

88-110.

Ro emer, J., 1989. \A Public Ownership Resolution of the Tragedy of the Commons," Social

Philosophy and Policy, 6, 74-92.

Schmeidler, D., 1969. \The Nucleolus of a Characteristic Function Game," SIAM Journal of

Applied Mathematics, 17, 1163-1170 .

Serizawa, S., 1994. \Strategy-Pro of and Individually Rational So cial Choice Functions for Public

Go o ds Economies," mimeo, Osaka University.

Shapley, L. S., 1953. \A Value for n-Person Games" in Kuhn, H., and W. Tucker eds. Contri-

butions to the Theory of Games, Princeton: Princeton University Press.

Shapley, L. S., 1971. \Core of Convex Games," International Journal of Game Theory, 1, 11-26.

Sharkey, W., 1995. \Network Mo dels in Economics" Chapter 9 in Network Routing, M. O. Ball

et al. eds., Handb o ok in Op erations Research and Management Science, Vol. 8, Amsterdam:

North-Holland.

Shenker, S., 1992. \On the Strategy-Pro of and Smo oth Allo cation of Private Go o ds in a Pro duc-

tion Economy," mimeo, XeroxPARC.

Shenker, S., 1995. \Making Greed Work in Networks: a game theoretic analysis of switch service

disciplines," Transactions on Networking, 3, 819-831.

Shubik, M., 1962. \Incentives, Decentralized Controls, the Assignment of Joint Costs and Internal

Pricing," Management Science, 8, 325-343.

Sprumont, Y., 1990. \Population Monotonic Allo cation Schemes for Co op erative Games with

Transferable Utility," Games and Economic Behavior, 2, 378-394.

Sprumont, Y., 1991. \The Division Problem with Single Peaked Preferences: a characterization

of the uniform rule," Econometrica, 59-2, 509-520. 26

Sprumont, Y., 1994. \Strategypro of Collective Choice in Economic and Political Environments,"

Canadian Journal of Economics, forthcoming.

Somnez, T., 1993. \Population Monotonicity of the Nucleolus on a Class of Public Go o d Prob-

lems," mimeo, UniversityofRochester.

Stran, P. D., and J. P. Heaney, 1981. \Game Theory and the Tennessee Valley Authority,"

International Journal of Game Theory, 10, 35-43.

Young, H. P., ed. 1985. Cost Allocation: methods, principles, applications, Amsterdam: North-

Holland. 27