Design of Price Mechanisms for Network Resource Allocation via

Price of Anarchy

Ying-Ju Chen1 and Jiawei Zhang1

Stern School of Business

New York University

{ychen0,jzhang}@stern.nyu.edu

February 2, 2006

Abstract

In this paper, we propose to use the concept of price of anarchy as a criterion in designing

price mechanisms for cost-sharing games. In particular, we investigate the network resource

allocation problem studied by Johari and Tsitsiklis [13], where users gather utilities by sending

flows through the communication network with quadratic cost structure. The search of optimal

price schemes is confined within the mechanisms characterized by a set of commonly adopted

axioms in cost-sharing games. We show that, if the information of users’ utilities is not available,

the optimal price mechanism yields a lower bound 4(3 − 2√

2) ≈ 0.686 of efficiency. Thus the

popular marginal cost pricing with efficiency loss less than 13 ≈ 33.3% [13] is nearly optimal.

We further allow the contingency on users’ preferences in the single-link problem. The efficiency

is shown to be achieved if users possess linear utilities, and the efficiency loss is no more than19 ≈ 11.1% when the single-crossing condition is satisfied among users’ preferences. In the

single-link problem, the case with linear utilities achieves the worst-case efficiency loss when

price mechanisms do not depend on users’ preferences, but becomes the best scenario while

such information is available.

Keywords: congestion pricing, resource allocation, price of anarchy

1Mailing Address: 44 West 4th street, KMC 8th floor, New York, NY 10012, USA.

1 Introduction

We consider a communication network and a set of users who require service from links in the

network. Each user gathers utility by sending her flow through the network, and the congestion on

each link depends on the total traffic flow over it. Since each user neglects the negative externality

she brings to others while requesting the communication rates, the equilibrium behavior in general

differs from what would be under the centralized control.

In order to close the gap between the user equilibrium and the centralized solution, congestion

pricing has been proposed, which treats the network as a collection of scarce resources, and allocates

these resources through a price mechanism; see, e.g., Falkner et al. [6], Kunniyur and Srikant [19],

and Srikant [35]. If the price mechanism is allowed to depend on users’ identities, then the central-

ized solution can be achieved by applying the Vickrey-Clarke-Groves (VCG) mechanism. However,

such an ideal scenario may not occur in practice, and hence the network designer has to charge a

common (uniform) price among the users (e.g., Tauman [37]). The common price, also referred to

as the “per-unit” tariff mechanism (Kumar et al. [18]), eliminates the incentive of splitting orders

or redistributing quantities among users (Friedman and Moulin [8] and Tauman [37]).

The “proportionally-fair” price mechanism proposed by Kelly et al. [15] is an example of the

common price mechanism. Despite its simplicity, Kelly et al. show that the centralized solution

can be achieved when users are price takers. However, this remarkable result does not hold if price

anticipation exists (Johari and Tsitsiklis [13]). Being aware of this, Johari and Tsitsiklis [13] adopt

the marginal cost pricing to investigate the efficiency loss of network resource allocation.

A natural question one would ask is that whether there exists a mechanism for network re-

source allocation better than marginal cost pricing. In fact, the network resource allocation falls

into a broad category of “cost-sharing” games, where users share the joint output that is contributed

from each individual (Young [42]). In the economics literature, many price mechanisms have been

proposed for allocating joint costs, whereas marginal cost pricing is merely one of these alterna-

tives. Other popular price mechanisms include Aumann-Shapley pricing, average cost pricing, and

Ramsey pricing (see Tauman [37]).

In this paper, we try to answer the above question to some extent by studying the design of

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“optimal” price mechanisms. To this end, two issues need to be addressed: (1) How to compare

given mechanisms, i.e., what is the criterion based on which one mechanism is better than another?

(2) What family of mechanisms should we choose from? One has to consider the normality and

implementability of the price mechanisms. Our proposal for addressing these two issues is as follows.

The criterion of comparing mechanisms we consider is the concept of “price of anarchy” (Kout-

soupias and Papadimitriou [16]), which can be informally stated as the worst case efficiency loss

of the equilibrium under a given mechanism. Its central idea is to provide a lower bound for the

ratio of aggregate payoff in any Nash equilibrium over that of the centralized solution (the system

optimal level). Price of anarchy has been applied in transportation problems (Roughgarden and

Tardos [29], Stier Moses et al. [25], Fleischer et al. [7], and Karakostas and Kolliopoulos [14]),

resource allocation of network bandwidths (Johari and Tsitsiklis [12]), network designs (Anshele-

vich [1]), supply chain management (Golany and Rothblum [10]), and allocation of divisible goods

(Yang and Hajek [41]). This notion coincides with the concept of the worst case ratio or the per-

formance guarantee of approximation algorithms for NP-hard problems. Comparing performance

guarantees is one way to compare two approximation algorithms for the same problem, and this

motivates us to rank price mechanisms by their individual prices of anarchy: a price mechanism

is said to be better than another if it provides a smaller price of anarchy. Note that this does not

imply the “dominance” between two mechanisms for every instance.

We apply the axiomatic approach in our selection of the potential price mechanisms, which

coincides with the main theme of the economics literature on cost-sharing games. The axiomatic

characterization provides a normative justification in purely economics terms for mechanisms that

are considered desirable. In particular, our candidate mechanisms for network resource allocation

are characterized by four axioms (rescaling, additivity, positivity, and weak consistency) proposed

by Samet and Tauman [30]. This family of price mechanisms include the popular marginal cost

pricing, Aumann-Shapley pricing, and average cost pricing (Watts [40]).

Also, the price mechanisms considered in this paper can be classified into two categories,

depending on whether the information of users’ preferences is available. If the central planner has

no access to users’ preferences, then the price mechanisms cannot be contingent on their utilities.

If such information is available, then the planner is able to utilize this advantage in designing

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mechanisms (examples include Ramsey pricing (Tauman [37]), traffic tolls on peak-hour congestion

and gasoline (Parry [28]), and externality-reducing taxation (Sheshinski [33])). However, we assume

that even in the second case, i.e., the central planner knows the preference profiles of these users,

she is unable to either identify the users or trace back the orders from a specific user. This setting

coincides with the literature on nonlinear pricing, where a monopolist’s goal is to extract the

maximum rent from players that have private types (Laffont and Martimort [20]).

We make similar assumptions on model characteristics as those in Johari and Tsitsiklis [13].

In particular, we assume that the utility functions of users are concave and non-decreasing, and

the congestion of each link is represented by a cost which is a convex and quadratic function of

the flow over the link. Note that this is equivalent to the affine marginal latency assumption in

Roughgarden and Tardos [29]. Based on these assumptions, we obtain the following results:

• If the information of users’ preferences is not available, i.e., the price mechanism does not

depend on the utility functions, then the smallest achievable price of anarchy is 4(3−2√

2) ≈31.4%. The marginal cost pricing studied in Johari and Tsitsiklis [13] falls into this category,

and that the price of anarchy 1/3 ≈ 33.3% implies it is “nearly optimal” even with users’ price

anticipation. Our result may justify why the marginal cost pricing is adopted in cost-sharing

games.1 It is also worth mentioning that the approach through which we derive the price of

anarchy is quite different from that in [13]. We further provide tight bounds contingent on

the number of users.

• Suppose that the information of users’ preferences is available. In the single-link case, if the

utility functions are all linear, then there exist a continuum of efficient mechanisms satisfying

the four axioms of [30], i.e., the efficiency loss is zero. If the utility functions satisfy the

single-crossing (sorting) property,2 then the smallest price of anarchy that can be achieved is

strictly between 0 and 1/9 ≈ 11.1%. In particular, there are examples for which no efficient

mechanism exists.1Marginal cost pricing is proposed for highway and airport transportation (Shepherd [32]), and energy resource

allocation such as water and electricity power, see Chambouleyron [5] and Lavigne et al. [21]. It is also adopted

as a price mechanism for consumption goods (Samet and Tauman [30]), and in welfare economics to achieve social

optimum with price-taking players (Mas-Collel et al. [22]).2We defer the exact definition to Section 5. This is a common regularity condition in the economics literature.

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In the single-link problem, when the planner has no access to users’ preferences, the worst case

efficiency loss is achieved when the utility functions are linear. However, linear cases become the best

scenario while allowing the price mechanism contingent on users’ references. This strict contrast of

efficiency demonstrates the value of information for such price mechanism design problems.

The rest of this paper is organized as follows. In Section 2, we describe in details the network

resource allocation problem. In Section 3, we introduce the four axioms, and we then focus on the

single-link case and characterize the price mechanisms that satisfy those axioms. Section 4 provides

the efficiency analysis when the central planner has no access to users’ preferences. We first obtain

the optimal price mechanism and its associated efficiency lower bound for the single-link problems

with linear utilities, and then propose to use the same parameters for unit prices per link in the

general network problem with concave utilities. We prove that the efficiency bound can still be

established in the general problem, and hence our proposed price mechanism is indeed optimal. In

Section 5 we allow the dependence on users’ preferences, and obtain the efficiency loss for single-link

problems. Finally, the conclusion is drawn in Section 6.

2 The Model

Our model is identical to that of Johari and Tsitsiklis [13], where n ≥ 2 users request rates in a

general network. Let I be the set of users. User i ∈ I can choose an amount of flow qi from her

position to the desired destinations through a number of paths. Let Γi be the set of paths available

to user i, and define path p ∈ Γi if path p is available to user i. Without loss of generality, we

assume each path is uniquely assigned to a user. If two distinct users happen to use the same path,

we shall duplicate the path and rename them separately. We further use Γ =⋃

i∈I Γi to denote

the entire set of paths. For user i, we have qi =∑

p∈Γiqip, where qip denotes the flow over path p.

Let E be the set of links, and e = 1, ..., |E| denote the indices of the links in the network. A link e

satisfies e ∈ p if path p goes through link e. We use yie to denote the total flow that user i sends

via link e, and from the flow conservation yie =∑

p:p∈Γi,e∈p qip, ∀i ∈ I, ∀e ∈ E.

The aggregate cost in a single link depends merely on the aggregate traffic flow that runs

over it, and the cost function is assumed to be quadratic. That is, the aggregate cost in link e is

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Ce(Qe) = beQe + 12aeQ

2e, ∀e ∈ E, where constants be ≥ 0, ae > 0 and Qe ≡

∑i yie is the aggregate

traffic flow on link e. We let Q ≡ (Q1, ..., Q|E|) ∈ R|E|+ represent the traffic flows in this network,

where Rm+ denotes the set of all m-dimensional nonnegative vectors and R+ ≡ R1

+. Let Ui(qi)

denote the payoff function of user i’s while sending flow qi, and assume that Ui(·) is nonnegative,

increasing, and concave, ∀i ∈ I. From the fact that Ui(qi) depends on user i’s total traffic flow

qi, ∀i ∈ I, users do not discriminate against any path. When users possess linear utility functions,

we define Ui(qi) = uiqi, ∀i ∈ I, where ui’s are nonnegative constants, and u1 = maxi∈I ui without

loss of generality.

In the single-link problem, we suppress the subscripts associated with the link, i.e., C(Q) =

bQ + 12aQ2, where Q =

∑i∈I qi.

2.1 Centralized system

The centralized solution is the allocation {qip}’s that solves the optimization problem:

maxn∑

i=1

Ui(qi)−|E|∑

e=1

(beQe +12aeQ

2e),

s.t.,∑

p∈Γi

qip = qi, ∀i ∈ I,

∑p:e∈p

qip = yie, ∀i ∈ I, ∀e ∈ E,

n∑

i=1

yie = Qe, ∀e ∈ E,

qip, qi, Qe ≥ 0 ∀i ∈ I, ∀e ∈ E, ∀p ∈ Γi,

(1)

or equivalently,

max∑n

i=1 Ui(∑

p∈Γiqip)−

∑|E|e=1

(be

∑ni=1

∑p:e∈p qip + 1

2ae(∑n

i=1

∑p:e∈p qip)2

)

s.t. qip ≥ 0, ∀i ∈ I, p ∈ Γi

We in the sequel let {q∗ip}’s and {Q∗e}’s denote respectively the planned flows on paths and

links in the centralized solution, and let Π∗ denote the aggregate payoff under this allocation. To

eliminate trivial cases where {q∗ip = 0, ∀p ∈ Γi, ∀i ∈ I} is system optimal, we make the following

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assumption. ∀m ∈ N , a vector ξ ∈ Rm is called a subgradient of a concave function f : Rm → R

at x ∈ Rm if f(x) ≥ f(x′) + ξ(x

′ − x)T , ∀x′ ∈ Rm, where yT is the transpose of vector y.

Assumption 1. There exist i ∈ I and p ∈ Γi such that ξ −∑e∈p be > 0, where ξ is a subgradient

of Ui(·) at 0.

Now we derive the centralized solution for a special case, namely the single-link problem when

users possess linear utility functions. This simple case not only is illustrative but will be recalled

intensively for our analysis later on.

In this case, Assumption 1 implies that u1 > b. The central planner’s problem becomes

maxq{∑

i uiqi−(bQ+ 12aQ2) : s.t. qi ≥ 0, ∀i ∈ I}. Since all quantities have the same impact on the

aggregate cost bQ+ 12aQ2, it is always optimal to allocate q1 = Q and qi = 0, ∀i 6= 1. The centralized

solution boils down to a quadratic maximization problem maxq1≥0{u1q1 − bq1 − 12aq2

1}. Applying

the first-order condition, we obtain that the centralized solution should be q∗1 = Q∗ = u1−ba , which

yields the aggregate payoff Π∗ = (u1−b)2

2a .

2.2 User equilibrium

Now we introduce the decentralized system. The price mechanism P(Q) ≡ (P1(Q), ..., P|E|(Q))

specifies the common price charged per link, independent of the path it belongs to and users’

identity, and hence the payoff of user i is Ui(qi)−∑

p∈Γiqip

∑e∈p Pe(Q).

Given the price mechanism, each user maximizes her own payoff by choosing flows {qip, p ∈Γi}’s. The equilibrium concept we adopt here is Nash equilibrium, and we focus on pure-strategy

equilibria throughout this paper. In a Nash equilibrium, each user’s strategy is her best response

given all others’ strategies (Fudenberg and Tirole [9]).

More specifically, let us assume that {qip(P)}’s is a Nash equilibrium given the mechanism

P, and denote (qi, q−i(P)) as the flow profile while replacing user i’s strategies in {qip(P)}’s by

{qip}p∈Γi . The definition of Nash equilibrium requires that :

{qip(P)} ∈ argmax{qip} {Ui(∑

p∈Γi

qip)−∑

p∈Γi

qip

∑e∈p

Pe((qi, q−i(P))) : s.t. qip ≥ 0}, ∀i ∈ I. (2)

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Let Π({qip(P)}) =∑

i∈I Ui(qi(P))−∑e∈E [beQe(P) + 1

2aeQ2e(P)] be the aggregate payoff in a

Nash equilibrium, where qi(P) =∑

p∈Γiqip(P), ∀i ∈ I, and Qe =

∑i∈I

∑p:e∈p,p∈Γi

qip(P), ∀e ∈ E.

Note that there may exist multiple equilibria given a price mechanism. Therefore, we denote

Π(P) = min{qip(P)}

Π({qip(P)}).

Then, The worst case efficiency loss under price mechanism P is defined as

min{Ui(·)}

Π(P)Π∗

,

while the price of anarchy is defined as

1− min{Ui(·)}

Π(P)Π∗

.

Note that Ui(qi) ≥ 0, ∀qi ≥ 0, ∀i ∈ I ensures that the social surplus in the centralized solution is

always nonnegative, and therefore the efficiency is well-defined.

From Eq. (2), users’ equilibrium behavior depends crucially on the price mechanism P. Dif-

ferent choices of price mechanisms lead to different equilibrium behavior as well as different payoff

profiles. Hence in the next section we will focus on the choice of price mechanisms.

3 Price mechanisms and axiomatic approach

We in this section first introduce the four axioms that characterize our candidate mechanisms, and

then focus on the single-link problems to obtain the specific price schemes that satisfy these axioms.

In the end we discuss briefly the existence and uniqueness of Nash equilibrium given such price

mechanisms.

3.1 Axiomatic approach

In this section, we will first give a short overview of axiomatic approach in cost-sharing games, and

then describe the four axioms that characterize our candidate price mechanisms.

The axiomatic approach of cost sharing problems was first introduced by Aumann and Shapley,

where they consider the values of cooperative nonatomic games, see Tauman [37] for an extensive

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survey. Since then the attention of economists has been put mainly in axiomatic characterization

of existing allocation rules (e.g. Mirman and Tauman [24], and Samet and Tauman [30]), and the

development of new sharing rules by pre-selecting a set of “desirable” axioms, see Friedman and

Moulin [8], Moulin and Shenker [27], Sprumont [34], and Thomson and Zhou [38].

Rules proposed for such problems can be divided into two major classes. The first class, usually

denoted as cost-sharing rules, requires the cost allocation be break-even. Cost-sharing rules include

Aumann-Shapley pricing, equal split rule, Shapley-Shubik price, and serial cost sharing (Moulin

and Shenker [27] and Watts [40]).

The second class of rules–the price mechanisms–relax the break-even assumption, and users are

charged a common unit price (Tauman [37]). Samet and Tauman [30] find four axioms (rescaling,

weak consistency, additivity, and positivity)-and with other modification-that characterize respec-

tively the marginal cost pricing and Aumann-Shapley price, which is a natural extension of average

cost pricing to multiple-output problems. These price mechanisms are widely adopted due to either

the simplicity or the “fairness” concern. For example, the applications of Aumann-Shapley pricing

range from the phone call assignment, water and electricity resource allocation, and waiting time

scheduling for congested servers, see Billera and Heath [2], Castano-Pardo and Garcia-Diaz [4],

Haviv [11], and Samet et al. [31]. Average cost sharing scheme is observed in computer network

(Moulin and Shenker [26]), pollution cleanup issue among factories (Watts [39]), and queueing sys-

tems (e.g., the FCFS discipline of M/M/1 model, Friedman and Moulin [8]). Please see [5], [13],

[21], [22], [30], and [32] for applications of marginal cost pricing.

Since these price mechanisms have been widely adopted as welfare coordination, incentive

compatible, and demand-compatible schemes (Tauman [37]), they shall be natural candidates of

our optimal design for our network allocation problem. The definitions and interpretations of these

axioms are stated in the following.

Note that these axioms are defined for general cost structure C(Q) ≡ C(Q1, ..., Qm) for

arbitrary cost-sharing games, where the integer m > 0 is the number of distinct outputs and

Q1, ..., Qm are the aggregate amounts of these outputs. We shall first introduce these axioms for

general games, and then tailor them so as to fit our network flow problem. We define Pl(F,Q)

as the unit price for output l if the cost structure is C(·) and the output profile is Q, and define

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P(C,Q) ≡ (P1(C,Q), ..., Pm(C,Q)) as the price vector.

Axiom 1. (Rescaling). Let α1, ..., αm be m positive numbers. If C(Q) = F (α1Q1, ..., αmQm),

∀Q ∈ Rm+ , then Pl(C,Q) = αlPl(F, (α1Q1, ...αmQm)), ∀l ∈ {1, ..., m}.

The rescaling axiom requires that the price mechanism should be invariant to the change of

scale. Next, if the input is homogeneous, the price should be identical.

Axiom 2.(Weak consistency). If C(Q) = G(∑m

l=1 Ql), ∀Q ∈ Rm+ , then Pl(C,Q) = P(G,

∑ml=1 Ql),

∀l ∈ {1, ..., m}.

Moreover, if the cost can be decomposed into different components, then so should the price

scheme.

Axiom 3. (Additivity). For any pair of continuous cost functions (F,G) such that C(Q) =

F (Q) + G(Q), ∀Q ∈ Rm+ , the price mechanism satisfies P(C,Q) = P(F,Q) + P(G,Q).

Finally, if the aggregate cost increases as the output profile Q1, ..., Qm increases, then the price

mechanism should be nonnegative at that point Q1, ..., Qm. Two vectors Q′,Q are said to satisfy

Q′ ≤ Q if Q

′l ≤ Ql, ∀l ∈ {1, ..., m}.

Axiom 4. (Positivity). Let Q ∈ Rm+ . If C(Q

′) is nondecreasing for all Q

′ ≤ Q, then P(C,Q)

is nonnegative.

3.2 Price mechanisms and Nash equilibria in single-link problems

We now focus on the single-link problems and characterize the price mechanisms that satisfy these

axioms. In the following we define P (C, Q) as the unit price if the cost structure is C(·) and the

aggregate traffic flow is Q.

Note that in the single-link problem, the rescaling axiom in the single-link problems requires

only a single scaling factor α, and hence the rescaling is valid and we need not consider the flow con-

servation constraint. The weak consistency axiom degenerates in this special case. The additivity

and positivity axioms can be applied directly.

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In this paper we focus on the quadratic cost function C(Q), and hence the price mechanism

is defined only for the space of quadratic functions C(Q) = bQ + 12aQ2, ∀Q ≥ 0. According to the

following lemma, the price mechanism that satisfies the axioms turns out to be linear:

Lemma 1. In single-link problems, a price mechanism for C(Q) = bQ+ 12aQ2 satisfies the rescaling,

additivity, and positivity axioms if and only if P (C, Q) = λ(b(1 + β) + aQ), where λ ≥ 0, β ≥ −1.

Proof. Let us first show that the price scheme satisfies those axioms. Consider the additivity under

quadratic costs. Suppose C(Q) = F (Q) + G(Q), ∀Q ≥ 0, and C(·), F (·), G(·) are all quadratic.

We obtain the relation between their parameters: bC = bF + bG and aC = aF + aG, where the

superscripts denote the corresponding cost functions. Now our price scheme satisfies the following:

P (C,Q) = λ(bC(1 + β) + aCQ) = λ(bF (1 + β) + aF Q) + λ(bG(1 + β) + aGQ) = P (F, Q) + P (G,Q),

and therefore the additivity under quadratic costs holds.

Next, we verify the rescaling axiom. When C(Q) = F (αQ), ∀Q ≥ 0, it must be that bF =bC

α , aF = aC

α2 . Thus, ∀Q ≥ 0, we have

αP (F, αQ) = αλ(bF (1 + β) + aF αQ) = αλ(bC

α(1 + β) +

aC

α2αQ) = P (C, Q).

The price scheme is clearly nonnegative when λ ≥ 0, β ≥ −1, and therefore it satisfies the positivity

axiom as well. We conclude that the price scheme satisfies all those axioms.

Now we proceed to show that if a price mechanism satisfies all the axioms, it must take the

linear form as presented. Suppose P (C, Q) satisfies those axioms. Since C(Q) can be parametrized

by the coefficients a, b, we define without loss of generality that P (C, Q) ≡ P (c, Q), where the row

vector c ≡ (a, b).

From the additivity under quadratic costs, we have P (c, Q) = P (c1, Q) + P (c2, Q), ∀Q ≥ 0,

whenever c = c1 + c2. Equivalently, ∀c1, c2,

P (τc1 + (1− τ)c2, Q) = τP (c1, Q) + (1− τ)P (c2, Q), ∀τ ∈ [0, 1], ∀Q ≥ 0.

The above equality implies that P (c, Q) is both concave and convex in the vector c, ∀Q ≥ 0, where

the domain of c is (0,∞) × [0,∞), a convex set. Therefore, P (c, Q) is linear in c, and we can

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represent P (c, Q) as P ((a, b), Q) = af(Q) + bg(Q), with both f(Q), g(Q) being independent of the

cost parameters a, b.

Now apply the rescaling axiom. Consider two convex and quadratic cost functions C, F such

that C(Q) = F (αQ), ∀Q ≥ 0. We obtain bF = bC

α , aF = aC

α2 , and the rescaling axiom requires that

α(aC

α2f(αQ) +

bC

αg(αQ)) = aCf(Q) + bCg(Q), ∀Q ≥ 0, ∀α > 0, ∀aC > 0, ∀bC ≥ 0. (3)

Plugging in bC = 0 leads to f(Q) = f(αQ)α , ∀Q ≥ 0, ∀α > 0, which immediately implies that f(Q)

is linear in Q, i.e., f(Q) = CfQ, for some constant Cf . Moreover, g(αQ) = g(Q), ∀Q ≥ 0, ∀α > 0,

and therefore g(Q) = Cg is a constant function.

In sum, P (C,Q) = CgbC + CfaCQ, ∀Q ≥ 0. This can be rewritten as P (Q) = λ(bC(1 + β) +

aCQ), and λ ≥ 0 and β ≥ −1 are necessary so that P (Q) ≥ 0, ∀Q ≥ 0, ∀(aC , bC).

We could also enlarge the space of cost functions to include all continuous functions. In this

case, the price mechanism shall be defined for all continuous functions, and the unit price for our

flow allocation problem is considered only for quadratic cost structure as a special case of the

general price scheme. We now show that the unit price is still linear, but the parameter β can take

value from [0,∞). The feasible region for β becomes narrower since we include a larger space of

cost functions and hence the axioms impose a stronger constraint.

Lemma 2. Suppose the price mechanism is defined for all continuous cost functions. In the single-

link problem, the price mechanism satisfies the rescaling, additivity, and positivity axioms if and

only if there exist nonnegative constants λ and β such that the unit price for C(Q) = bQ+ 12aQ2 is

P (Q) = λ(b(1 + β) + aQ), ∀Q.

Proof. When the number of outputs m = 1, we can follow Samet and Tauman [30, Proposi-

tions 1 and 2] to show that for a general cost function C(Q), if a price mechanism satisfies the

rescaling, additivity, and positivity axioms, then there exists a nonnegative measure µ such that

P (C, Q) =∫ 10

∂C∂Q(tQ)dµ(t), ∀Q ≥ 0. Note that the weak consistency is not used for proving these

two propositions in their paper.

This price mechanism is defined for all continuous cost functions C(·). In particular, when the

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cost is quadratic, i.e., C(Q) = bQ + 12aQ2, the corresponding price mechanism translates to

P (C, Q) =∫ 1

0

∂C

∂Q(tQ)dµ(t) = b

∫ 1

0dµ(t) + aQ

∫ 1

0tdµ(t), ∀Q. (4)

Thus, if we define λ =∫ 10 tdµ(t) and β =

R 10 dµ(t)R 10 tdµ(t)

− 1, the price mechanism for this particular cost

can be expressed as P (C,Q) = λ(b(1 + β) + aQ), ∀Q ≥ 0. The positivity of λ and β follows from

that µ is nonnegative and∫ 10 dµ(t) ≥ ∫ 1

0 tdµ(t).

On the other hand, suppose that there exist nonnegative constants λ and β such that P (C, Q) =

λ(b(1+β)+aQ), ∀Q ≥ 0. We shall extend this unit price to a price mechanism that is defined for all

continuous cost functions. Let us define the measure µ as such: µ(0) = λβ, µ(1) = λ, and µ(x) =

0, ∀x /∈ {0, 1}. With this choice, µ is nonnegative and P (C,Q) = b∫ 10 dµ(t)+aQ

∫ 10 tdµ(t). For gen-

eral continuous cost C, we propose to use the same measure and define P (C, Q) =∫ 10

∂C∂Q(tQ)dµ(t).

Since this price is a linear functional, the rescaling and additivity axioms hold, and the positivity

follows from the nonnegativity of measure µ.

Therefore, we conclude that the unit price for our single-link problems has to take the linear

form when the price mechanism is defined for either the quadratic functions or all continuous

functions. We in the sequel focuses on the case with merely quadratic cost structures. It can be

verified that our proposed price mechanism in Section 4 is still optimal in the single-link problems

even if we enlarge the domain of cost functions, and the optimality extends to the general network

structure.

Lemma 1 reduces the problem of finding the price mechanism that could potentially be very

complicated to an optimization problem over two parameters λ and β in single-link problems. Hence

from now on we will focus on the design of λ and β.

3.2.1 Existence and uniqueness of Nash equilibrium

Given the price mechanism, a Nash equilibrium {qi}’s requires that for all i ∈ I,

qi = argmaxqi{Ui(qi)− qiλ(b(1 + β) + a(qi +

∑

j 6=i

qj)) : s.t. qi ≥ 0}. (5)

We shall neglect the trivial case λ = 0, i.e., P (Q) = 0, ∀Q ≥ 0, since in this case all users would

13

like to send unbounded flows and no equilibrium exists. Except this, the existence and uniqueness

of Nash equilibrium is established for all cases.

Lemma 3. Suppose λ > 0. In a single-link problem with price P (Q) = λ(b(1 + β) + aQ), there

exists a unique Nash equilibrium, ∀ λ > 0, β ≥ −1.

Proof. This is a direct extension of Watts [39, Theorem 2]. We merely have to redefine the cost

function C(Q) = Qλ(b(1 + β) + aQ) = λb(1 + β)Q + λaQ2, which is strictly convex.

Remark 1. This argument also gives the uniqueness of Nash equilibrium in the single-link model

of Johari and Tsitsiklis [13].

The existence and uniqueness of Nash equilibrium given any price mechanism allows us to

predict users’ behavior in the decentralized system, and hence we shall move on to study the

efficiency loss, according to which we can design the optimal price mechanism. We separate the

design into two scenarios. In next section, we assume that the central planner has no access to

{Ui(·)}’s, and hence the price mechanisms shall depend merely on the cost parameters, i.e., be, ae’s,

∀e ∈ E. In Section 5, we allow the dependence of price mechanisms on {Ui(·)}’s, and focus on the

single-link problems.

4 Efficiency loss when the price mechanism is independent of

users’ preferences

In this section we assume that the information of users’ preferences is not available to the central

planner and hence the price mechanism is independent of Ui(·)’s. We will first focus on the single-

link problem with linear utilities and obtain the optimal price mechanism. We then propose to use

common parameters obtained in the special case for unit prices per link in the general problem,

and establish the efficiency lower bound.

14

4.1 Efficiency loss in single-link problems with linear utilities

We first consider the single-link problem and assume that utilities are all linear, i.e., C(Q) =

bQ + 12aQ, ∀Q ≥ 0, and Ui(qi) = uiqi, ∀qi ≥ 0, ∀i ∈ I.

4.1.1 Efficiency bounds

For ease of notation, we replace {qi(λ, β)}’s by {qi}’s and Π(λ, β) by Π while no confusion occurs.

Recall that the centralized solution is q∗1 = Q∗ = u1−ba , whose associated aggregate payoff Π∗ =

(u1−b)2

2a . The following lemma establishes a lower bound of aggregate payoff in the Nash equilibrium.

Lemma 4. ∀λ > 0, β ≥ −1, if q1 > 0, then

Π ≥ (λ(1 + β)− 1)b(1λ

Q∗ +b

a

1− λ(1 + β)λ

− q1) + (λ− 12)a(

1λ

Q∗ +b

a

1− λ(1 + β)λ

− q1)2 + λaq21

+λa

n− 1(1λ

Q∗ +b

a

1− λ(1 + β)λ

− 2q1)2.

Proof. The best response of user i requires that (ui−λb(1+β)−λaQ−λaqi)qi = 0. Summing over

i ∈ I, we have

Π ≡n∑

i=1

uiqi − (bQ +12aQ2) = (λ(1 + β)− 1)bQ + (λ− 1

2)aQ2 + λa

n∑

i=1

q2i . (6)

Since q1 > 0, by the first-order condition for user 1,

q1 + Q =u1/λ− b

a− b− βb

a=

1λ

Q∗ +b

a

1− λ(1 + β)λ

. (7)

We can rewrite

n∑

i=1

q2i = q2

1 +n∑

i≥2

q2i ≥ q2

1 +1

n− 1(

n∑

i≥2

qi)2 = q21 +

1n− 1

(Q− q1)2. (8)

Using Eq. (8) and replacing Q by 1λQ∗ + b

a1−λ(1+β)

λ − q1, we have

Π ≥(λ(1 + β)− 1)bQ + (λ− 12)aQ2 + λaq2

1 +λa

n− 1(Q− q1)2

=(λ(1 + β)− 1)b(1λ

Q∗ +b

a

1− λ(1 + β)λ

− q1) + (λ− 12)a(

1λ

Q∗ +b

a

1− λ(1 + β)λ

− q1)2 + λaq21

+λa

n− 1(1λ

Q∗ +b

a

1− λ(1 + β)λ

− 2q1)2.

15

We next derive a lower bound of aggregate payoff for some special cases.

Lemma 5. ∀λ > n−14(n+1) , β ≥ −1, if q1 > 0, then

Π ≥ 12a(Q∗)2

2(n(2λ− 1) + 2λ)λ(n(4λ− 1) + 4λ + 1)

+ bQ∗−2(n(λ− 1) + λ)(βλ + λ− 1)λ(n(4λ− 1) + 4λ + 1)

− b2

a

(n(λ + 2)− λ)(βλ + λ− 1)2

2λ(n(4λ− 1) + 4λ + 1).

Proof. We can rearrange the formula in Lemma 4 according to the descending order of q1. The

coefficient of q21 is a(2n+2

n−1 λ− 12), and the coefficient of q1 is

−a((2n + 2)λ− n + 1)λ(n− 1)

Q∗ + (λ(1 + β)− 1)b(−1 +1λ

(2n + 2)λ− n + 1n− 1

). (9)

Recall that Ay2 + By ≥ −B2

4A provided that A > 0, we have

Π ≥(λ(1 + β)− 1)b(

1λ

Q∗ +b

a

1− λ(1 + β)λ

)+ a(λ− 1

2+

λ

n− 1)(

1λ

Q∗ +b

a

1− λ(1 + β)λ

)2

−(−a((2n+2)λ−n+1)

λ(n−1) Q∗ + (λ(1 + β)− 1)b(−1 + 1λ

(2n+2)λ−n+1n−1 ))2

4a(2n+2n−1 λ− 1

2)

(10)

if 2n+2n−1 λ− 1

2 > 0. After some algebra, we can rearrange Eq. (10) by descending order of Q∗. Hence

if 2n+2n−1 λ− 1

2 > 0,

Π ≥ 12a(Q∗)2

2(n(2λ− 1) + 2λ)λ(n(4λ− 1) + 4λ + 1)

+ bQ∗−2(n(λ− 1) + λ)(βλ + λ− 1)λ(n(4λ− 1) + 4λ + 1

− b2

a

(n(λ + 2)− λ)(βλ + λ− 1)2

2λ(n(4λ− 1) + 4λ + 1.

Now we study the efficiency loss for given price mechanisms. We first show that, a tight lower

bound can be obtained in a special case.

Theorem 1. Suppose that price mechanisms depend merely on the cost structures. For a special

case when P (Q) = b + λaQ where λ ∈ [ n2n+2 , 1], the efficiency is at least 2(n(2λ−1)+2λ)

λ(n(4λ−1)+4λ+1) .

Proof. When P (Q) = b+λaQ, (qi = 0,∀i ∈ I) cannot be an equilibrium since if this were the case,

the Karush-Kuhn-Tucker (KKT) condition for user 1 would require u1 − b ≤ 0, a contradiction to

16

Assumption 1. Now assume that in the Nash equilibrium, q1 = 0 but ∃i 6= 1, s.t. qi > 0. Then

u1 − b− λaQ ≥ ui − b− λaQ > ui − b− λaQ− λaqi = 0, and hence q1 = 0 cannot be user 1’s best

response. Therefore, we conclude that q1 > 0 in equilibrium.

Recall the formula in Lemma 5. The price scheme P (Q) = b + λaQ implies that λ(1 + β) = 1,

in which case the last two terms vanish, and the coefficient of 12a(Q∗)2 = Π∗ becomes the lower

bound of the desired ratio. Note that within the range λ ∈ [ n2n+2 , 1], both λ and β are nonnegative

under condition λ(1 + β) = 1, and the efficiency bound 2(n(2λ−1)+2λ)λ(n(4λ−1)+4λ+1) is nonnegative.

We next show that these bounds are all tight.

Lemma 6. Suppose P (Q) = b+λaQ where λ ∈ [ n2n+2 , 1] is independent of ui’s, the price of anarchy

is at most 2(n(2λ−1)+2λ)λ(n(4λ−1)+4λ+1) .

Proof. Let ui−bu1−b = 2nλ+2λ+1

n(4λ−1)+4λ+1 ≡ θ, ∀i ≥ 2, where θ ∈ [12 , 1] if λ ∈ [ n2n+2 , 1]. Consider the following

allocations:

q1 =1λ

(1− (n− 1)θ + 1n + 1

)Q∗, qi =1λ

(θ − (n− 1)θ + 1n + 1

)Q∗, i ≥ 2. (11)

Summing over i ∈ I to obtain Q, we can find the following equalities for the flow quantities

q1 + Q =1λ

Q∗, qi + Q =1λ

θQ∗, i ≥ 2. (12)

It can then be verified that qi > 0, ∀i ∈ I and the KKT conditions are satisfied. Thus, {qi}’sconstitute a Nash equilibrium.

Since Π =∑

i∈I(ui − b)qi − 12aQ2, the ratio of aggregate payoff is

ΠΠ∗

=2λ

(1− (n− 1)θ + 1n + 1

) +2(n− 1)

λθ(θ − (n− 1)θ + 1

n + 1)− 1

λ2((n− 1)θ + 1

n + 1)2, (13)

where the right-hand side can be simplified to 2(n(2λ−1)+2λ)λ(n(4λ−1)+4λ+1) while plugging in θ = 2nλ+2λ+1

n(4λ−1)+4λ+1 .

Thus, the price of anarchy in this case is upper bounded as mentioned in the lemma.

When λ, β satisfy the conditions of Theorem 1, the efficiency bound is decreasing in the number

of users n, and hence the coordination problem does deteriorate as the system size becomes large.

However, when n(1+δ)2n+2 ≤ λ ≤ 1, where δ > 0 is a constant independent of n, a constant limit always

17

exists. This demonstrates that the efficiency loss cannot go arbitrarily large as the price mechanism

is carefully chosen.

The marginal cost pricing corresponds to λ = 1 and β = 0, in which case the bound2(n(2λ−1)+2λ)

λ(n(4λ−1)+4λ+1) becomes 2n+43n+5 , and it converges to 2

3 as obtained in Johari and Tsitsiklis [13].

However, our result refines that of Johari and Tsitsiklis [13] since we provide a tight bound for

any given number of users (we in Section 4.2 show that this bound works for the general net-

work problems with concave utilities). The worst case occurs when users have linear utilities withui−bu1−b = 2n+3

3n+5 , ∀i ≥ 2, with 23 being the limit.

Letting λ = 12 and β = 1, this gives us the popular average cost pricing, whose corresponding

efficiency guarantee is 4n+3 that vanishes as n → ∞. Note that the average cost pricing under

quadratic cost structures coincides with Aumann-Shapley price, Shapley-Shubik price and nucleolus

rule (Sudholter [36]). The efficient sharing rule proposed from the cooperative game theory performs

extremely poorly while considering the non-cooperative strategic interaction among users.

Now we characterize the optimal price mechanism and provide the efficiency bounds.

Theorem 2. Suppose that price mechanisms depend merely on the cost structures. Let λ∗ =2n+2n2+

√2√

n(1+n)3

4(1+n)2, and β∗ = 4(1+n)2

2n+2n2+√

2√

n(1+n)3− 1, the tight bound of efficiency loss is

4√

2(1+n)2√

n(1+n)3

(1+2n+n2+√

2√

n(1+n)3)(2n+2n2+√

2√

n(1+n)3), with 4(3 − 2

√2) ≈ 0.686 being its limit. More-

over, ∀λ > 0, β ≥ 0 that differ from the above pair, the price of anarchy is strictly less than4√

2(1+n)2√

n(1+n)3

(1+2n+n2+√

2√

n(1+n)3)(2n+2n2+√

2√

n(1+n)3).

Proof. We first consider the price parameters λ = 2n+2n2+√

2√

n(1+n)3

4(1+n)2, and β = 4(1+n)2

2n+2n2+√

2√

n(1+n)3−

1. Note that such a choice satisfies the conditions in Theorem 1, and hence we obtain a tight bound

given this price mechanism.

Now we shall prove that for any other choice, the efficiency bound cannot be further improved.

We first consider the region discussed in Theorem 1. The ratio 2(n(2λ−1)+2λ)λ(n(4λ−1)+4λ+1) does not depend on

β, is unimodal in λ, and achieves the maximum when λ = 2n+2n2+√

2√

n(1+n)3

4(1+n)2∈ [ n

2n+2 , 1]. Hence the

efficiency lower bound in this region is always less than 4√

2(1+n)2√

n(1+n)3

(1+2n+n2+√

2√

n(1+n)3)(2n+2n2+√

2√

n(1+n)3).

Now we consider the case λ(1+β)−1 = 0 but λ /∈ [ n2n+2 , 1], and construct examples to obtain

18

the upper bound of our efficiency measure. When λ ∈ (0, 14 ], we choose θ ≡ ui−b

u1−b ∈ (0, 12 ], ∀i ≥ 2.

It can be verified that qi = 0, ∀i ≥ 2 in equilibrium. Now by applying the first-order condition

on user 1’s quantity decision, in equilibrium q1 = u1−b2λa , and hence the aggregate payoff becomes

Π = (2λ − 12)a (u1−b)2

4λ2a2 = 1λ(1 − 1

4λ)Π∗, which gives us a non-positive efficiency as λ ∈ (0, 14 ].

On the other hand, if λ ∈ (14 , n

2n+2 ], we set ui = u1, ∀i ≥ 2. By symmetry the equilibrium

quantity qi = qj , ∀i, j ∈ I, and hence the first-order condition yields qi = 1λ(n+1)Q

∗, ∀i ∈ I. The

corresponding ratio of aggregate payoff is n(2λ+n(2λ−1))λ2(n+1)2

≡ A(λ). Differentiating A(λ), we obtain

A′(λ) =

2n(n− λ(n + 1))λ3(n + 1)2

≥ 2n

λ3(n + 1)2(n− n

2n + 2(n + 1)) =

n2

λ3(n + 1)2> 0,

where the first inequality follows from λ ≤ n2n+2 . Thus, A(λ) is increasing in λ. If we fix ui =

u1, ∀i ≥ 2, the upper bound of efficiency when λ ∈ (14 , n

2n+2 ] is less than A( n2n+2) = 0, i.e., no

positive efficiency bound can be obtained in this case as well.

If λ(1 + β) − 1 ≡ γ > 0, we can let u1 ∈ (b, λ(1 + β)b). KKT conditions suggest that

qi = 0, ∀i ∈ I, which makes the aggregate payoff 0 and hence no strictly positive bound can be

obtained.

Finally, suppose that λ(1 + β) − 1 ≡ γ < 0. Define D = bu1−b and let ui = u1, ∀i ≥ 2. From

the first-order condition and symmetry among users, qi = 1n+1( 1

λ − γD)Q∗, ∀i ∈ I. Note that1λ − γD > 0, ∀D > 0 since γ < 0. The aggregate payoff becomes

Π =∑

i∈I

(ui − b)qi − 12aQ2 = Π∗

(2n

n + 1(1λ− γD)− (

n

n + 1(1λ− γD))2

). (14)

The ratio can be expressed as a quadratic function of D where the coefficient of D2 is − n2γ2

(n+1)2< 0,

and hence it approaches −∞ while letting D →∞. Thus for any fixed constant M , there exists D

s.t. ΠΠ∗ < −M if γ < 0.3

Note that under this price mechanism β ≥ 0, and hence it remains feasible even if the price

mechanism is defined for all continuous cost functions, and the optimality follows immediately since

we restrict the domain of β while considering all continuous cost functions.

The above theorem shows that our proposed price mechanism indeed achieves the smallest price

of anarchy if the central planner has no access to users’ preferences. The optimal pricing scheme3Following a similar argument, we can show that when γ > 0 and 0 < λ < 1

2, as n → ∞ and D → 0, the ratio

will be in the neighborhood of 2λ−1λ

< 0.

19

is P (Q) = b + 2n+2n2+√

2√

n(1+n)3

4(1+n)2aQ, which converges to P (Q) = b + 2+

√2

4 aQ as n → ∞. This

price scheme has never been proposed elsewhere. The worst case occurs when users possess linear

utilities, and except the highest one they are identical. While using the optimal price mechanism,

it is achieved when ui−bu1−b →

√2

2 as n →∞. Such relation among users’ preferences is nontrivial as

well.

Recall that the marginal cost pricing yields a constant lower bound 23 ≈ 66.7%, which is fairly

close to the best price of anarchy one can come up with. The marginal cost pricing has been shown

to achieve social optimality with price takers, and our result further proves that it is nearly optimal,

although suboptimal, when users are price-anticipating.

Finally, since we change the price users are facing, in some cases (λ(1 + β) 6= 1) the cost

in equilibrium actually exceeds the aggregate utility of users. Although a user can always ensure

herself a zero payoff by setting qi = 0, in the modified game the price may induce users to request

too much. Thus, the ratio could be negative and is not bounded below. Inappropriate selections

of price mechanisms may cause disasters.

4.2 Extension to general network problems with concave utilities

We now consider the general network structure when users possess concave utilities. We propose

to adopt linear unit prices for each link in the network structure, and use the common parameters

λ∗, β∗ for all unit prices. That is, we choose Pe(Q) = λ∗(be(1 + β∗) + aeQe), ∀e ∈ E, where

λ∗ = 2n+2n2+√

2√

n(1+n)3

4(1+n)2, and β∗ = 4(1+n)2

2n+2n2+√

2√

n(1+n)3− 1 (and the corresponding measure µ∗ for

decomposed costs in the additivity axiom). We will show that the efficiency bound obtained in

Theorem 2 continues to be valid in the general network problems with concave utilities. Note that

a slight generalization of Lemma 1 shows that this price mechanism satisfies the four axioms.

We use {qip(λ, β)}’s and Π(λ, β) to denote the equilibrium flow profile and the aggregate payoff

for given constants λ and β. From the definition of Nash equilibrium, the equilibrium flow profile

20

of user i must be the solution of the following problem:

max Ui(∑

p∈Γi

qip)−∑

p∈Γi

qip

∑e∈p

λ

be(1 + β) + ae(

∑p:e∈p

qip +∑

l 6=i

qle(λ, β))

s.t. qip ≥ 0 ∀i ∈ I, p ∈ Γi

(15)

The existence of Nash equilibrium follows from an argument similar to Johari and Tsitsiklis

[13]. We replace {qip(λ, β)}’s by {qip}’s and Π(λ, β) by Π while no confusion occurs, and regard∑

i∈I Ui(∑

p∈Γiqip) ≡

∑i∈I Ui(qi) as a function of (q1, ..., qn) ≡ q.

We first show that to obtain the price of anarchy when Pe(Q) = λ∗(be(1+β∗)+aeQe), ∀e ∈ E,

it suffices to consider the case when users possess linear utilities. The following lemma proves a

more general result, with our choice being a special case. Note that this result has been obtained

by Johari and Tsitsiklis [13] for marginal cost pricing.

Lemma 7. Suppose Pe(Q) = λ(be(1 + β) + aeQe), ∀e ∈ E. Let λ ≥ 12 and λ(1 + β) = 1. Suppose

{qip} is a Nash equilibrium. For each i ∈ I, let ξi be a subgradient of Ui(qi) at qi =∑

p∈Γiqip.

Then

ΠΠ∗

≥∑

i∈I ξi∑

p∈Γiqip −

∑e∈E [beQe + 1

2aeQ2e]

maxq≥0∑

e∈E [(

max i: ∃p∈Γi,s.t.e∈p

ξi

)Qe − (beQe + 1

2aeQ2e)]

. (16)

Proof. We first show that∑

i∈I ξiqi −∑

e∈E [beQe + 12aeQ

2e] ≥ 0. From the KKT condition on qip

where p ∈ Γi, i ∈ I, we have(

ξi −∑e∈p

be −∑e∈p

λaeQe −∑e∈p

λaeyie

)qip = 0.

Summing over all available paths p ∈ Γi and all users i ∈ I, we obtain

∑

i∈I

∑

p∈Γi

ξiqip =∑

i∈I

∑

p∈Γi

∑e∈p

beqip +∑

i∈I

∑

p∈Γi

∑e∈p

λaeQeqip +∑

i∈I

∑

p∈Γi

∑e∈p

λaeyieqip

≥∑

i∈I

∑

p∈Γi

∑e∈p

beqip +∑

i∈I

∑

p∈Γi

∑e∈p

λaeQeqip,(17)

where the inequality follows from the nonnegativity of traffic flows.

21

Observe that∑

i∈I

∑p∈Γi

∑e∈p beqip =

∑e∈E beQe and

∑p∈Γi

∑e∈p aeQeqip =

∑e∈E aeQ

2e, we

obtain that∑

i∈I

∑

p∈Γi

ξiqip ≥∑

e∈E

beQe + λ∑

e∈E

aeQ2e ≥

∑

e∈E

beQe +12

∑

e∈E

aeQ2e,

where we have used λ ≥ 12 . Thus,

∑i∈I ξiqi −

∑e∈E [beQe + 1

2aeQ2e] ≥ 0 in equilibrium.

Now we establish the lower bound of the ratio ΠΠ∗ . Let us first consider the denominator Π∗.

Π∗ =∑

i∈I

Ui(q∗i )−∑

e∈E

[beQ∗e +

12ae(Q∗

e)2]

≤∑

i∈I

Ui(qi) + ξ(q∗ − q)T −∑

e∈E

[beQ∗e +

12ae(Q∗

e)2]

=∑

i∈I

Ui(qi)− ξqT + ξq∗T −∑

e∈E

[beQ∗e +

12ae(Q∗

e)2]

≤∑

i∈I

Ui(qi)− ξqT + maxq≥0

∑

e∈E

[( maxi: ∃p∈Γi,

s.t.e∈p

ξi)Qe − (beQe +12aeQ

2e)].

(18)

In the above, the first inequality is by the subgradient inequality. To obtain the second inequality, we

shall consider the modified problem with linear utilities: Ui(qi) = ξiqi, ∀i ∈ I. Now in this modified

problem, q∗ is a feasible flow allocation, and hence it generates a weakly lower aggregate net utility

compared to the centralized solution, which gives us maxq≥0∑

e∈E [(max i: ∃p∈Γi,s.t.e∈p

ξi)Qe − (beQe +12aeQ

2e)]. Note that for link e ∈ E, max i: ∃p∈Γi,

s.t.e∈pξi corresponds to the maximum utility coefficient

among users who are allowed to send flows over this link. Thus under the centralized control, we

should allocate the entire link flow to this specific user in order to maximize the aggregate net

utility. Also note that maxq≥0∑

e∈E [(max i: ∃p∈Γi,s.t.e∈p

ξi)Qe − (beQe + 12aeQ

2e)] ≥ 0 because q = 0 is a

feasible flow allocation.

The efficiency ratio can be represented as follows:

ΠΠ∗

=∑

i∈I Ui(qi)−∑

e∈E [beQe + 12aeQ

2e]∑

i∈I Ui(q∗i )−∑

e∈E [beQ∗e + 1

2ae(Q∗e)2]

≥∑

i∈I Ui(qi)−∑

e∈E [beQe + 12aeQ

2e] + ξ(q− q)T

∑i∈I Ui(qi)− ξqT + maxq≥0

∑e∈E [

(max i: ∃p∈Γi,

s.t.e∈pξi

)Qe − (beQe + 1

2aeQ2e)]

=∑

i∈I Ui(qi)− ξqT + ξqT −∑e∈E [beQe + 1

2aeQ2e]

∑i∈I Ui(qi)− ξqT + maxq≥0

∑e∈E [

(max i: ∃p∈Γi,

s.t.e∈pξi

)Qe − (beQe + 1

2aeQ2e)]

.

(19)

22

By concavity and nonnegativity of utilities {Ui}’s, we know that∑

i∈I Ui(qi) − ξqT ≥ 0. Since

both the numerator and denominator in Eq. (19) are positive, removing the common positive term

will only make the ratio smaller. Note also that ξqT can be rewritten explicitly as∑

i∈I ξiqi ≡∑

i∈I ξi∑

p∈Γiqip, ∀q ∈ Rn

+. Therefore, we obtain the inequality mentioned in the lemma.

Now if we regard Ui(∑

p∈Γiqip) ≡ ξi

∑p∈Γi

qip as the linear utility function of user i, then

for each Nash equilibrium of the original problem, it is also an equilibrium of the modified game

with these linear utilities. The denominator of the lower bound in Lemma 7 corresponds to the

centralized solution of this modified game, and hence it suffices to consider the case with linear

utility functions.

We next show that, when users possess linear utilities, the price mechanism design problem

can be simplified as a single-link problem. This lemma again works for mechanisms that are more

general than our proposed price scheme.

Lemma 8. Suppose Pe(Q) = λ(be(1 + β) + aeQe), ∀e ∈ E. In the case with linear utilities,

∀λ > 0, β ≥ −1, if the price of anarchy is η for single-link problems, then the price of anarchy is η

for the problems with network structure as well.

Proof. Let us fix λ > 0, β ≥ 0, and consider the user equilibrium. In a Nash equilibrium, the

Karush-Kuhn-Tucker (KKT) condition on qip where p ∈ Γi, i ∈ I is(

ui −∑e∈p

λbe(1 + β)−∑e∈p

λaeQe −∑e∈p

λaeyie

)qip = 0,

where we recall that yie is the total flow user i sends via link e.

Define αie =(λbe(1 + β) + λaeQe + λaeyie

)/ui, ∀i ∈ I, ∀e ∈ E, then from this definition we

have

uiαie − λbe(1 + β)− λaeQe − λaeyie = 0, ∀i ∈ I, ∀e ∈ E. (20)

From KKT conditions, we have∑

e∈p αie ≥ 1, ∀p ∈ Γi, ∀i ∈ I. Moreover,∑

e∈p αie = 1 if qip > 0,

∀p ∈ Γi, ∀i ∈ I. Multiplying Eq. (20) by qip and summing over p while p ∈ Γi and e ∈ p, we obtain

that ∀i ∈ I, ∀e ∈ E,

uiαie

∑

p:p∈Γi,e∈p

qip =∑

p:p∈Γi,e∈p

qipλbe(1 + β) +∑

p:p∈Γi,e∈p

qipλaeQe +∑

p:p∈Γi,e∈p

yieqipλae. (21)

23

We can then replace∑

p:p∈Γi,e∈p qip by yie and get

uiαie

∑

p:p∈Γi,e∈p

qip = yieλbe(1 + β) + yieλaeQe + y2ieλae, ∀i ∈ I, ∀e ∈ E. (22)

Now summing over all users and all links,

∑

e∈E

n∑

i=1

uiαie

∑

p:p∈Γi,e∈p

qip =∑

e∈E

{(n∑

i=1

yie)λbe(1 + β) + (n∑

i=1

yie)λaeQe +n∑

i=1

y2ieλae}. (23)

The left-hand side can be rearranged as

∑

e∈E

n∑

i=1

uiαie

∑

p:p∈Γi,e∈p

qip =n∑

i=1

ui

∑

p∈Γi

∑e∈p

αieqip =n∑

i=1

ui

∑

p∈Γi

qip

∑e∈p

αie =n∑

i=1

ui

∑

p∈Γi

qip, (24)

where the last equality follows from the fact that∑

e∈p αie = 1 if qip > 0, ∀p ∈ Γi, ∀i ∈ I. Using

the equalities in Eqs. (23) and (24), the aggregate payoff Π in equilibrium becomes

Π =n∑

i=1

ui

∑

p∈Γi

qip −∑

e∈E

beQe − 12

∑

e∈E

aeQ2e

=∑

e∈E

n∑

i=1

uiαie

∑

p:p∈Γi,e∈p

qip −∑

e∈E

beQe − 12

∑

e∈E

aeQ2e

=∑

e∈E

{(λbe(1 + β)− be)Qe + (λ− 1

2)aeQ

2e + λae

n∑

i=1

y2ie

},

(25)

which can be decomposed link by link. Notice also that, Eq. (20) implies that (yie, ∀i ∈ I) is a Nash

Equilibrium for the game if we restrict ourselves to only link e and assume that the coefficient of

user i’s utility becomes uiαie, ∀i ∈ I. Thus for each decentralized solution of the original problem,

there is an equivalent Nash equilibrium where users compete on the per-link basis in terms of both

the equilibrium flow profiles and the aggregate payoff.

24

Finally, we consider the aggregate payoff in the centralized solution. Notice that

maxq∗ip≥0

n∑

i=1

ui

∑

p∈Γi

q∗ip −|E|∑

e=1

(beQ

∗e +

12ae(Q∗

e)2

)

≤maxq∗ip≥0

n∑

i=1

ui

∑

p∈Γi

q∗ip∑e∈p

αie −|E|∑

e=1

(beQ

∗e +

12ae(Q∗

e)2

)

= maxq∗ip≥0

∑

e∈E

n∑

i=1

uiαie

∑

p:p∈Γi,e∈p

q∗ip −|E|∑

e=1

(beQ

∗e +

12ae(Q∗

e)2

)

≤∑

e∈E

max

q∗ip≥0

n∑

i=1

uiαie

∑

p:p∈Γi,e∈p

q∗ip −(

beQ∗e +

12ae(Q∗

e)2

)

=∑

e∈E

{maxy∗ie≥0

n∑

i=1

uiαiey∗ie −

(be

n∑

i=1

y∗ie +12ae(

n∑

i=1

y∗ie)2

)},

(26)

where the first inequality follows from KKT condition. In the last equality of Eq. (26), each

term inside the brackets can be regarded as a centralized solution for a link e ∈ E, where the

coefficient of user i’s utility becomes uiαie and the central planner is optimizing over users’ choices

yie, ∀i ∈ I. Therefore, the efficiency of the network problem where users choose traffic flows via

their available paths shall be lower bounded by that on per link basis with such transformation

of utility coefficients. The price of anarchy while considering general network structure shall be

equivalent to that in the single-link problems.

Combining Theorem 1 and Lemmas 7 and 8, an efficiency bound 2(n(2λ−1)+2λ)λ(n(4λ−1)+4λ+1) is guar-

anteed for general network problems with concave utilities when Pe(Q) = be + λaeQe, ∀e ∈ E

is adopted and λ ∈ [12 , 1]. In particular, under the parameters λ∗ and β∗, the efficiency bound4√

2(1+n)2√

n(1+n)3

(1+2n+n2+√

2√

n(1+n)3)(2n+2n2+√

2√

n(1+n)3)goes through to this general problem. Moreover, we in

Section 4.1 prove that the proposed price scheme is the unique mechanism that provides the maxi-

mally achievable efficiency bound for the single-link problems, and hence it is suboptimal to adopt

any mechanism that is properly different from our proposed price scheme in the single-link problem.

Note that we need not characterize the set of price mechanisms that satisfy the four axioms in

the general network structure, nor do we modify the rescaling axiom to make it compatible to the

flow conservation. This simple price scheme outperforms all other mechanisms in terms of price of

anarchy.

25

5 Efficiency loss when the information of users’ preferences is

available: the single-link case

In this section we allow the price mechanism to depend on users’ preferences and focus on the

optimal price design problem for the single-link case. We again adopt P (Q) = λ(b(1+β)+aQ), ∀Q,

and use qi(λ, β) to denote the equilibrium flows. From Lemma 3, qi(λ, β) is uniquely determined

by λ > 0, β ≥ 0, ∀i ∈ I. We in the following separate our analysis into two cases, depending on

whether the utilities are linear.

5.1 Efficiency with linear utilities

We first consider the case with linear utilities, i.e., Ui(qi) = uiqi, ∀i ∈ I, and show that the efficiency

is achieved for any instance {ui}’s.

Theorem 3.Consider the single-link problem and suppose the information of the users’ preferences

is available. With linear utilities, there exist a continuum of pairs (λ, β) such that Π(λ, β) = Π∗.

Proof. Let Πi(qi, q−i(λ, β)) = uiqi−λ(b(1+β)+a(qi+∑

j 6=i qj(λ, β)) be user i’s payoff while sending

qi, assuming that other users follow the equilibrium strategies. The derivative of user i’s payoff

w.r.t. qi is ∂Πi(qi,q−i(λ,β))∂qi

= ui−λb(1+β)−λa(qi +∑

j 6=i qj(λ, β))−λaqi. This can be rewritten as

∂Πi(qi, q−i(λ, β))∂qi

= a[ui − b

a− b

a(λ(1 + β)− 1)]− λa(

∑

j 6=i

qj(λ, β) + 2qi)

= aQ∗[ui − b

u1 − b− b

aQ∗ (λ(1 + β)− 1)]− λa(∑

j 6=i

qj(λ, β) + 2qi).(27)

Without loss of generality assume that u1 > u2 ≥ ... ≥ un.4 Define θ = u2−bu1−b ∈ [0, 1). Since in the

centralized solution q∗i = 0, ∀i ≥ 2, our goal is to show that there exists a pair (λ, β) such that

qi(λ, β) = 0, ∀i ≥ 2 and q1(λ, β) = Q∗. From the derivatives ∂Πi(qi,q−i(λ,β))∂qi

, ∀i ∈ I, it suffices to

4If the first k users have the same utilities, then any redistribution within them will not change the aggregate

payoff.

26

check the feasibility of

1− (λ(1 + β)− 1) bu1−b

2λ= 1,

θ < (λ(1 + β)− 1)b

u1 − b,

(28)

plus the nonnegativity constraints on λ, β, where the second condition in Eq. (28) guarantees

that qi(λ, β) = 0, ∀i ≥ 2 and the first condition results in q1(λ, β) = Q∗. If we choose arbitrary

λ ∈ (0, 1−θ2 ) and β =

1+ bu1−b

−2λ−λ bu1−b

λb/u1−b accordingly, this combination is feasible to the above system

and λ, β are both positive.

In the case with linear utility functions, the efficiency can be achieved by a simple uniform

price that depends merely on the cost functions and the ratio between the (distinct) marginal

utilities of the highest two users. As opposed to the VCG mechanisms (Golany and Rothblum [10]

and Yang and Hajek [41]) where knowing the users’ identities is essential for achieving the system

optimality, the information requirement here is significantly weaker.

Due to the multiplicity of Nash equilibria, we cannot follow the approach in Section 4 to

generalize the results to the network case. Recall the proof of Lemma 8 where we redefine uiαie as

the coefficient of user i’s utility on every link e ∈ E. In order to implement the efficient mechanism

on each link, we have to set λ, β in Eq. (28) accordingly. Since the uniqueness of Nash equilibrium

is not guaranteed in the network problem, {αie}’s may not be known.

5.2 Efficiency bound with concave utility functions

Now we switch our attention to the case with concave utilities, and show that the efficiency can

be further improved from 0.686 if utility functions satisfy the regularity conditions in the following

assumption:

Assumption 2. The set of utility functions {Ui(·)}’s satisfy :

1. U′i exists, ∀i ∈ I, and maxi∈I U

′i (0) < ∞.

2. U′i (q) ≥ U

′j(q), ∀q ∈ R+, whenever i ≤ j.

27

Condition 1 in Assumption 2 guarantees the existence of utilities’ derivatives and ensures that

the marginal utility for any user at any point is bounded. It does rule out utility functions that

could potentially go unboundedly such as Ui(qi) = log(qi).5 The second condition is usually referred

to as the single-crossing (sorting) condition, or the Spence-Mirrlees condition. It implies that two

distinct utility functions intersect at most once. This condition is commonly adopted in a variety

of literature, including the nonlinear pricing (Laffont and Martimort [20]), Nash implementation

(Maskin [23]), auctions (Krishna [17]), and operations management (Cachon and Lariviere [3]).6

Note that if we require Ui(0) = 0, ∀i ∈ I, i.e., the status quo is the same for every user, then

single-crossing condition guarantees that Ui(q) ≥ Uj(q), ∀q ≥ 0, ∀i ≤ j. From Assumption 1, we

also have U′1(0) > b.

With Assumption 2, we are able to provide structural properties of users’ behavior in equilib-

rium. We first show that the equilibrium flows can be ordered.

Lemma 9. ∀λ > 0, β ≥ 0, we have qi(λ, β) ≥ qj(λ, β), ∀i ≥ j.

Proof. From the first-order condition for user i, we have qi(λ, β)[U′i (qi(λ, β))−λ(1+β)−λaQ(λ, β)−

λaqi(λ, β)] = 0. Suppose that qj(λ, β) > qi(λ, β), i ≤ j. From the concavity and single-crossing

condition, we obtain U′i (qi(λ, β)) ≥ U

′i (qj(λ, β)) > U

′j(qj(λ, β)). Hence

U′i (qi(λ, β))−λ(1+β)b−λaQ(λ, β)−λaqi(λ, β) > U

′j(qj(λ, β))−λ(1+β)b−λaQ(λ, β)−λaqj(λ, β) = 0,

where the last equality follows from qj(λ, β) > qi(λ, β) ≥ 0. But then qi(λ, β) cannot be a best

response.

We now establish the monotonicity of U′i (·)’s in equilibrium.

Lemma 10. ∀λ > 0, β ≥ 0, U′i (qi(λ, β)) ≥ U

′j(qj(λ, β)), ∀i ≤ j.

Proof. Suppose that user i is active, i.e., qi(λ, β) > 0. From the first-order conditions, in equilibrium5Unless the negative infinite utility brings intriguing scenarios, this assumption is mild and reasonable.6Examples that conform the single-crossing condition include the popular utility functions Ui(q) = log(q + Ci) for

empirical work, where constants Ci’s satisfy Ci ≤ Cj whenever i ≤ j. Another class is Ui(q) = uiq + G(q), with G(q)

being concave in q. Note that the linear utility Ui(q) = uiq is a special case of this class.

28

we have

U′j(qj(λ, β)) ≤ λ(1+β)b+λaQ(λ, β)+λaqj(λ, β) ≤ λ(1+β)b+λaQ(λ, β)+λaqi(λ, β) = U

′i (qi(λ, β)).

If qi(λ, β) = 0, then qj(λ, β) = 0 as well by Lemma 9, and it follows from the single-crossing

condition that U′j(0) ≤ U

′i (0).

Now we are ready to state our main result in this section. We provide a lower bound of

efficiency under the optimal price mechanism, which is bounded away from 4(3 − 2√

2). Instead

of characterizing the optimal price mechanism, for any instance {Ui}’s, we characterize a feasible

price scheme that yields such an efficiency bound.

Theorem 4.Consider the single-link problem and suppose the information of the users’ preferences

is available. If the utility functions satisfy Assumption 2, then we can choose λ = n−14(n+1) and a

corresponding β ≥ 0 such that the efficiency is at least 8n(n+1)(3n+1)2

.

Proof. We first assume that the following equation

λ(1 + β)− 1 =2(n + 1)3n + 1

maxi∈I U′i (qi(λ, β))− b

b(29)

has a solution β ≥ 0 when λ = n−14(n+1) . If we use linear approximation for utility functions, i.e.,

ui = U′i (qi(λ, β)), ∀i ∈ I, the formula of Lemma 4 leads to

Π(λ, β) ≥ (λ(1 + β)− 1)b(1λ

Q∗ − b(λ(1 + β)− 1)aλ

) + a(λ− 12

+λ

n− 1)(

1λ

Q∗ − b(λ(1 + β)− 1)aλ

)2

+ (2n + 2n− 1

λ− 12)aq2

1 + [−(2n + 2)λ− n + 1λ(n− 1)

aQ∗ + (λ(1 + β)− 1)b(−1 +1λ

(2n + 2)λ− n + 1n− 1

)]q1.

(30)

As we choose λ = n−14(n+1) and correspondingly β = 2(n+1)

3n+1aQ∗λb + 1

λ −1, both the coefficients of q1 and

q21 become zero. Note that with the chosen λ and β, λ(1 + β) − 1 = 2(n+1)

3n+1aQ∗

b , and therefore the

left-over terms on the right-hand side of Eq. (30) can be combined as a single term 8n(n+1)(3n+1)2

12aQ∗,

where 12aQ∗ = Π∗. Thus, we obtain a lower bound of the aggregate payoff that is independent of

q1: Π(λ, β) ≥ 8n(n+1)(3n+1)2

Π∗.

Now we prove that Eq. (29) indeed has a fixed point as λ = n−14(n+1) . We in the following

provide more general results that work for any λ ∈ (0, 1), and decompose the proof into a sequence

29

of lemmas. Note that Lemma 10 allows us to consider merely U′1(q1(λ, β)) = maxi∈I U

′i (qi(λ, β))

in Eq. (29) for any Nash equilibrium. The first step is to establish the continuity of qi(λ, β) while

varying β.

Lemma 11. Fixed any λ > 0, the equilibrium flows qi(λ, β) is continuous in β, ∀i ∈ I; so is the

the aggregate output Q(λ, β).

Proof. We first show that Q(λ, β) is decreasing in β. Suppose the claim is not true, then there

must exist (β, β) s.t. β > β but Q(λ, β) > Q(λ, β). Consider a user i with qi(λ, β) > 0. The KKT

condition for such user i becomes U′i (qi(λ, β))−λ(1 + β)b−λaQ(λ, β)−λaqi(λ, β) = 0. We obtain

that ∀i, s.t. qi(λ, β) > 0,

U′i (0)−[λ(1+β)b+λaQ(λ, β)] > U

′i (0)−[λ(1+β)b+λaQ(λ, β)] > U

′i (0)−U

′i (qi(λ, β))+λaqi(λ, β) > 0,

where the last inequality results from the concavity of Ui(·). Hence 0 is not user i’s best response,

i.e., qi(λ, β) > 0 as well.

Since qi(λ, β) > 0, the associated first-order condition must be binding at β. Hence ∀i, s.t.

qi(λ, β) > 0,

U′i (qi(λ, β))− λaqi(λ, β) = λ(1 + β)b + λaQ(λ, β)

< λ(1 + β)b + λaQ(λ, β)

= U′i (qi(λ, β))− λaqi(λ, β).

(31)

Thus by concavity of Ui(·) we have qi(λ, β) > qi(λ, β), ∀i ∈ I such that qi(λ, β) > 0. Summing

over i s.t. qi(λ, β) > 0, we have

Q(λ, β) =n∑

j=1

qj(λ, β) =∑

i:qi(λ,β)>0

qi(λ, β) <∑

i:qi(λ,β)>0

qi(λ, β) ≤n∑

j=1

qj(λ, β) = Q(λ, β),

which contradicts the assumption that Q(λ, β) > Q(λ, β).

Now we are ready to prove that for fixed λ, qi(λ, β) is continuous in β ≥ 0, ∀i ∈ I. This is

done by contradiction as well. Suppose that the flow of user i is not continuous at some β ≥ 0, i.e.,

∃ε > 0, such that ∀δ > 0, we can find β ∈ [β, β + δ) and |qi(λ, β) − qi(λ, β)| > ε, where we have

applied the same notation as above. Choose δ < aεnb . Since Q is decreasing, there must exist a user

30

j, such that qj(λ, β) < qj(λ, β) − εn (If qi(λ, β) < qi(λ, β) − ε, then choose j = i; otherwise, there

exists another user j that satisfies it). Now consider the incentive of user j:

U′j(qj(λ, β))− λaqj(λ, β)− λ(1 + β)b− λaQ(λ, β)

≥ U′j(qj(λ, β))− λaqj(λ, β)− λ(1 + β)b− λaQ(λ, β) + λa

ε

n

≥ U′j(qj(λ, β))− λaqj(λ, β)− λ(1 + β)b− λaQ(λ, β) + λa

ε

n− λbδ

≥ U′j(qj(λ, β))− λaqj(λ, β)− λ(1 + β)b− λaQ(λ, β) + λ(a

ε

n− bδ)

≥ λ(aε

n− bδ)

> 0,

(32)

where the first inequality is by concavity of U′j , the third inequality follows from monotonicity of

Q(λ, β), and the fourth inequality is because qj(λ, β) > qj(λ, β) + εn > 0. Eq. (32) contradicts the

fact that qi(λ, β) is user i’s best response.

Finally, since Q(λ, β) =∑

i∈I qi(λ, β) is the sum of n continuous functions, it is continuous as

well.

Recall that our goal is to show the existence of fixed point for Eq. (29). Since maxi∈I U′i (qi(λ, β)) =

U′1(q1(λ, β)), ∀λ > 0, β ≥ 0, we can rearrange Eq. (29) as follows:

β =2(n + 1)3n + 1

U′1(q1(λ, β))

λb+

1λ

[1− 2(n + 1)3n + 1

]− 1 (33)

Define f(β) = 2(n+1)3n+1

U′1(q1(λ,β))

λb + 1λ [1 − 2(n+1)

3n+1 ] − 1. Then it suffices to show that f(β) = β has

a solution. Recall that U′1(0) > b, and denote ∆ = U

′1(0) − b > 0. It is easy to versify that

U′1(0)λb − 1− ∆

λb(1− 2(n+1)3n+1 ) ≥ 0. Now we are ready to prove the following lemmas.

Lemma 12. If λ ∈ (0, 1), then for any β ∈ [0, U′1(0)λb − 1− ∆

λb(1− 2(n+1)3n+1 )],

f(β) =2(n + 1)3n + 1

λ(1 + β)b + λaQ(λ, β) + λaq1(λ, β)λb

+1λ

[1− 2(n + 1)3n + 1

]− 1,

and f(β) is continuous in β, ∀β ∈ [0, U′1(0)λb − 1− ∆

λb(1− 2(n+1)3n+1 )].

Proof. We first show that if β ∈ [0, U′1(0)λb − 1 − ∆

λb(1 − 2(n+1)3n+1 )], then q1(λ, β) must be positive.

Otherwise, assume q1(λ, β) = 0, then by Lemma 9, qi(λ, β) = 0 for all i ∈ I and thus Q(λ, β) = 0.

31

However, on the other hand, we know that

U′1(q1(λ, β))− λ(1 + β)b− λaQ(λ, β)− λaq1(λ, β) ≤ 0.

It follows that U′1(0) − λ(1 + β)b ≤ 0, which contradicts to the assumption that β ≤ U

′1(0)λb − 1 −

∆λb(1− 2(n+1)

3n+1 ). Therefore, it must be the case that q1(λ, β) > 0.

Then from the first order condition we have U′1(q1(λ, β))−λ(1+β)b−λaQ(λ, β)−λaq1(λ, β) = 0.

Rearrange the terms, we get U′1(q1(λ, β)) = λ(1 + β)b + λaQ(λ, β) + λaq1(λ, β). Therefore,

f(β) =2(n + 1)3n + 1

λ(1 + β)b + λaQ(λ, β) + λaq1(λ, β)λb

+1λ

[1− 2(n + 1)3n + 1

]− 1.

By Lemma 11, both q1(λ, β) and Q(λ, β) are continuous in β. This completes the proof.

Finally, we prove that Eq. (33) has a solution.

Lemma 13. If λ ∈ (0, 1), f(β) has a fixed point β ≥ 0.

Proof. For any β ∈ [0,U′1(0)λb − 1− ∆

λb(1− 2(n+1)3n+1 )], from Lemma 12,

f(β) =2(n + 1)3n + 1

λ(1 + β)b + λaQ(λ, β) + λaq1(λ, β)λb

+1λ

[1− 2(n + 1)3n + 1

]− 1

≥ 2(n + 1)3n + 1

λ(1 + β)bλb

+1λ

[1− 2(n + 1)3n + 1

]− 1

≥ 2(n + 1)3n + 1

λb

λb+

1λ

[1− 2(n + 1)3n + 1

]− 1

= (1− 1λ

)[1− 2(n + 1)3n + 1

]

≥ 0

On the other hand

f(β) =2(n + 1)3n + 1

U′1(q1(λ, β))

λb+

1λ

[1− 2(n + 1)3n + 1

]− 1

≤ 2(n + 1)3n + 1

U′1(0)λb

+1λ

[1− 2(n + 1)3n + 1

]− 1

≤ 2(n + 1)3n + 1

U′1(0)λb

+U′1(0)−∆

λb[1− 2(n + 1)

3n + 1]− 1

≤ U′1(0)λb

− 1− ∆λb

(1− 2(n + 1)3n + 1

).

32

Therefore, we have shown that for any β ∈ [0, U′1(0)λb − 1 − ∆

λb(1 − 2(n+1)3n+1 )], f(β) is continuous

and f(β) ∈ [0, U′1(0)λb − 1 − ∆

λb(1 − 2(n+1)3n+1 )]. Thus, from fixed-point theorem, there exists β ∈

[0, U′1(0)λb − 1− ∆

λb(1− 2(n+1)3n+1 )], such that f(β) = β.

Since f(β) = β has a solution, for any instance Ui(·)’s, we can find a price mechanism that

provides an efficiency bound that is at least 8n(n+1)(3n+1)2

. The proof of Theorem 4 is now complete.

Note that the optimal price mechanism is at least as good as the proposed scheme, and hence

the efficiency loss under the optimal mechanism shall be between 0 and n2−2n−1(3n+1)2

, which converges

to 19 ≈ 11.1% in the limit. Thus, the efficiency loss is further suppressed from 31.4% to 11.1% while

allowing the central planner to use the information of users’ preferences.

The single-crossing condition is adopted in the economics literature to ensure the possibility of

truth-telling mechanism (Fudenberg and Tirole [9]), and in optimization literature to guarantee the

existence of optimal solution. Here the single-crossing condition helps us to show that the same user

has the highest marginal utility in any Nash equilibrium, which enables us to design a corresponding

price parameter based on this term. Without this assumption, the highest marginal utility in

equilibrium can switch among users while varying the price scheme, and hence the continuity

becomes problematic.

A counterexample shows that the efficiency cannot be achieved by all single-link problems with

single-crossing condition. Assume that n = 2, U1(q1) = 3q1− 12q2

1 and U2(q2) = 2q2− 12q2

2, and cost

functions C(Q) = Q+Q2. The centralized solution is (q∗1, q∗2) = (2

3 , 0). Suppose there exist a pair of

constants (A,B) such that (23 , 0) is a Nash equilibrium with P (Q) = B + AQ. Then the first-order

conditions yield 3 − 23 − B − 2

3A − 23A = 0 and 2 − B − 2

3A ≤ 0. The only nonnegative pair that

satisfies these two equations is (B, A) = (73 , 0). But this would imply λ = 0 and λ(1 + β) = 7

3 , a

contradiction.

We conclude that the case with linear utility functions, although achieves the lower bound of

efficiency when price mechanisms are restricted to depend only on the cost structures, becomes the

best scenario if users’ preferences are available to the central planner. The value of information

while designing the price mechanism achieves the maximum in this special case.

33

6 Conclusion

In this paper we consider the optimal congestion pricing design for a network model with quadratic

cost structures, where the criterion is the price of anarchy and the search is confined within the

mechanisms that satisfy four desirable axioms in cost allocation. If users’ preferences are unknown,

the optimal price mechanism guarantees a constant bound 4(3 − 2√

2) ≈ 0.686 of efficiency. The

marginal cost pricing is nearly optimal, whereas the average cost pricing performs extremely poorly.

We then consider the single-link problems while allowing the price mechanism to depend on users’

preferences. With linear utility functions, the system optimality can be implemented in the de-

centralized scenario. Note that the efficiency is achieved by a uniform price scheme that does not

hinge on users’ identities, as opposed to the VCG mechanism. If utility functions satisfy the single-

crossing condition, the efficiency loss is at most 19 ≈ 11.1%. The approach developed here can

also be applied to other cost-sharing games such as telephone companies (Tauman [37]), pollution

cleanup issues (Watts [39]), airline authorities, regulatory commissions, and water supply systems

(Young [42]).

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