DoubleAuction Mechanisms for
Resource Trading Markets
Abstract
We consider a doubleauction mechanism, which was recently proposed in the context of a mobile dataoffloading market. It is also applicable in a network slicing market. Network operators (users) derive benefit from offloading their traffic to third party WiFi or femtocell networks (linksuppliers). Linksuppliers experience costs for the additional capacity that they provide. Users and linksuppliers (collectively referred to as agents) have their payoffs and cost functions as private knowledge. A systemdesigner decomposes the problem into a network problem (with surrogate payoffs and surrogate cost functions) and agent problems (one per agent). The surrogate payoffs and cost functions are modulated by the agents’ bids. Agents’ payoffs and costs are then determined by the allocations and prices set by the system designer. Under this design, so long as the agents do not anticipate the effect of their actions, a competitive equilibrium exists as a solution to the network and agent problems, and this equilibrium optimizes the system utility. However, this design fails when the agents are strategic (priceanticipating). The presence of strategic supplying agents drives the system to an undesirable equilibrium with zero participation resulting in an efficiency loss of 100%. This is in stark contrast to the setting when linksuppliers are not strategic: the efficiency loss is at most 34% when the users alone are strategic. The paper then proposes a Stackelberg game modification with asymmetric information structures for suppliers and users in order to alleviate the efficiency loss problem. The system designer first announces the allocation and payment functions. He then invites the supplying agents to announce their bids, following which the users are invited to respond to the suppliers’ bids. The resulting Stackelberg games’ efficiency losses can be characterized in terms of the suppliers’ cost functions when the user payoff functions are linear. Specifically, when the linksupplier’s cost function is quadratic, the worst case efficiency loss is 25%. Further, the loss in efficiency improves for polynomial cost functions of higher degree.
I Introduction
We consider double auction mechanisms motivated by two examples – mobile data offloading and network slicingbased virtualization.
Mobile data offloading is an effective way to manage growth in mobiledata traffic. Traffic meant for the macrocellular network can be offloaded to already installed thirdparty WiFi or femtocell networks. This provides an alternative means of network expansion. WiFi accesspoint operators and femtocell network operators will however expect compensation for allowing macrocellular network traffic through their access points. Technological, security, and preliminary economic studies for secure and seamless offloading have been discussed in [2, 3, 4, 5].
Network slicing [6] is a virtualization technique that allows many logical networks to run atop shared physical networks. It allows physical mobile network operators to partition their network resources and offer them to different users or tenants (IoT streams, mobile broadband streams, etc.) in return for suitable compensation. It enables network operators to focus on their core strength of delivering highquality network experiences while the tenants or virtual network operators can focus more on business, billing, and branding relations.
In the rest of the paper, we discuss double auction mechanisms in the context of mobile data offloading. But the mapping to the context of network slicing will be obvious.
In recent work, Iosifidis et al. [7] proposed a doubleauction mechanism where a set of mobile network operators (buyers or users in this work) compete for resources from accesspoint operators (sellers or links in this work). The payoffs of the users and costs of the links are private information to the respective parties. The mechanism works as follows. A network manager collects how much each network operator is willing to pay each accesspoint operator, scalar signals on the costs at each access point, and then determines how much traffic should be offloaded to each access point and how much each agent will pay or get. The mobile network operators and the accesspoint operators then comply. This is a scenario with an asymmetric information structure where (a) the broker is not aware of the actual needs and costs of network and accesspoint operators, (b) each operator is aware only of his own needs or costs, and (c) all agents are pricetaking (made precise in the next section). Following Kelly et al. [8], Iosifidis et al. [7] showed that a tâtonnement procedure converges to the system optimal operating point.
Iosifidis et al. [7, p.1635] point out that designing incentive compatible mechanisms for doubleauctions which are weakly budget balanced (the broker should not end up subsidizing the mechanism) is ‘notoriously hard’ and has been done only in certain simplified settings (McAfee auction [9]) or can be computationally intensive. So [7] took a network utility maximization approach and left the analysis of the priceanticipating scenario open [7, Sec VII, p.1646].
Our contributions in this paper are as follows.

We first rederive the result on efficient allocation when the agents are pricetaking, mainly to set up the notation for the next three results.

We then analyze the priceanticipating scenario along the lines of Johari et al. [10]. When agents are priceanticipating, they recognize the effect of their bids on the allocation. The appropriate equilibrium notion is a Nash equilibrium. The situation in Johari et al. [10], when mapped to the current offloading setting, would be one where the accesspoint operators are not strategic. The efficiency loss due to priceanticipating mobile offloading agents is then at most 34%. However, when the accesspoint agents (suppliers) are also strategic and priceanticipating, the equilibrium is one where the offloading agents prefer not to offload any traffic. The efficiency loss is then 100%. The main message is that the earlier proposed doubleauction mechanism of [7] works when agents are pricetaking, but fails in the more real situation when agents are priceanticipating. One must then look for alternative doubleauction mechanisms.

We then propose a modified mechanism where the supplying agent bids first and the users bid in response. To show that the situation is now improved, we characterize the new efficiency loss in terms of the supplier’s cost function, when the user payoff functions are linear. For instance, for the quadratic linkcost function, the worstcase efficiency loss (with the worstcase taken over linear user payoff functions) is at most 25%.

We extend all of the above results to the setting with multiple links.
From an implementation theory perspective, the Iosifidis et al. [7] mechanism in the pricetaking scenario implements the social welfare maximization rule under the competitive equilibrium solution concept with the minimal message dimension of 1 (scalar signals). The above implementation ignores strategic behavior of individual agents. It is not possible to enforce such mechanisms in general because individual preferences may diverge from social welfare maximization. This is the priceanticipating scenario. It is anticipated that if we do not enlarge the signal space dimension there may be no mechanism, let alone the Iosifidis et al. mechanism, that can implement the social welfare maximization rule, under now the Nash equilibrium solution concept. This is why the priceanticipating scenario with nonstrategic link suppliers suffered from an efficiency loss. What is surprising in our current setting is the dramatic increase in efficiency loss from at most 34% (Johari et al. [10]) to 100% (contribution (2) of this paper). What is promising from our study is that this efficiency loss can be mitigated by structuring the interaction, by making the link player lead the interaction (contribution (3) of this paper). The solution concept is that of a Stackelberg equilibrium. Efficiency loss drops down to a value that depends on the supplier’s cost function and is at most 25% for quadratic costs and linear user payoffs. This of course raises the question of what is the minimal signalling dimension in the priceanticipating scenario that implements the social welfare maximization rule in the Nash equilibrium solution concept. This a very interesting question that is beyond the scope of this work. Our proposed scheme, which structures the interactions by asking the supplier to lead, reduces efficiency loss. It would be of utmost interest if this structuring also reduces the minimum signalling dimension for social welfare maximization in the Stackelberg equilibrium solution concept. We refer the reader to [11] for an excellent discussion on the implementation theory perspective.
The paper is organized into two parts. In Part I we study a setting with a single linksupplier. Specifically, in Section II, we discuss the system model and problem definition. In Section III, we discuss the pricetaking scenario for the singlelink case. In Section IV, we analyze the priceanticipating scenario. As a positive result, in Section V, we discuss our proposed mechanism and characterize the worstcase efficiency loss for linear user payoffs in terms of the single supplying agent’s cost function. In Part II (Sections VI to IX) we generalize the above results to the setting with multiple linksuppliers. To focus on the flow of key ideas, we have moved all the proofs to the Appendix. The paper concludes with some remarks in Section X.
Part I: Single Link
Ii System Model and Problem Definition
Consider a scenario where users intend to share the bandwidth of a (single) link of capacity owned by a linksupplier. In the context of mobiledata offloading [7], users and linksupplier correspond to mobilenetwork operators and the single accesspoint operator (e.g., WiFi, femtocell), respectively. The mobilenetwork operators want to buy a share of the limited bandwidth resource available at the access point to offload their macrocellular traffic, while the access point operator is interested in maximizing his profit. In the double auction terminology [9], users are synonymous to buyers bidding for a share of a resource while the linksupplier is the seller. We refer to the users and the linksupplier collectively as agents. The social planner, the entity that designs the mechanism (i.e., sets up the rules for information transfer, allocation, and payments) is referred to as the networkmanager,
Let denote the rate requested by user , and let be the rate the linksupplier is willing to allocate to user . Thus, and represent the raterequest and rateallocation vectors, respectively. Let denote the aggregaterate allocated by the linksupplier to all users. For user , the benefit of acquiring a rate of is represented by a payoff function ; we assume that , , are concave, strictly increasing and continuously differentiable with finite . Similarly, the cost incurred by the linksupplier for accepting to serve an aggregate rate of is given by , where is strictly convex, strictly increasing and continuously differentiable. Thus, the system optimal solution is the solution to the optimization problem:
SYSTEM
Maximize:  (1a)  
Subject to:  (1b)  
(1c)  
(1d) 
Continuity of the objective function and compactness of the constraint set imply that an optimal solution and exists. Further, if are strictly concave then (since is strictly convex) the solution is unique. Since are strictly increasing in , an optimal solution must satisfy . Thus, at optimality, the raterequests (demand) and the rateallocations (supply) are matched although the capacity may not be fully utilized.
A networkmanager, however, cannot solve the formulation in (1) without the knowledge of user payoffs and the linkcost function. Hence, consider the following mechanism proposed by Iosifidis et al. in [7] for rate allocation. Each user submits a bid that denotes the amount he is willing to pay, while the linksupplier communicates signals that implicitly indicate the amounts of bandwidth that he is willing to provide; we refer to and as the bids submitted by the users and the linksupplier, respectively.
The networkmanager is responsible for fixing the prices and that determines the rate allocation. The prices and are supposed to be the optimal dual variables of the following network problem proposed by Iosifidis et al. in [7]:
NETWORK
Maximize:  (2a)  
Subject to:  (2b)  
(2c)  
(2d) 
In the NETWORK problem above we choose to use instead of a related that was used in the original formulation by Iosifidis et al. in [7]; the quantities and are related by . Then each is valued, with values on the positive real line, while each is in general valued. Moreover, the signals in are directly proportional to the amount of bandwidth the linksupplier is willing to share. For instance, a lower value of implies that the bandwidth shared by the linksupplier with user is low, and vice versa. In particular, implies that the linksupplier is unwilling to share any bandwidth with user . This will be useful while interpreting the Nash equilibrium bidvectors (Theorem 2).
The above NETWORK problem is identical to the SYSTEM problem but with the true payoff and cost functions replaced by surrogate payoff and cost functions. In the following, we first review the case when the users and the linksupplier are pricetaking. This means agents assume prices are given and do not anticipate the effect of their bids on the prices set by the networkmanager. See Definition 1 below of a competitive equilibrium. We then proceed to study the moreinvolved priceanticipating scenario. Here the agents recognize that the effective price is based on their bids, anticipate the resulting allocation, payment, and therefore their payoff, and act accordingly. The resulting payoff functions are new functions of the bids; see Definition 2. Our methodology in Sections III and IV is similar to Johari et al. [10], but the outcome in the priceanticipating scenario is dramatically negative due to the presence of the strategic linksupplier, as we will soon see. We then propose a remedy via a Stackelberg framework where the linksupplier is a lead player and the users are followers.
Iii Pricetaking scenario
The sequence of exchanges (between the networkmanager and the agents) in the pricetaking scenario is as shown in the box describing the pricetaking mechanism (PTM) below.
PRICETAKING MECHANISM (PTM)

The network manager announces to the agents how the allocation will be done and the payments will be fixed, as a function of prices and agents’ bids.

The networkmanager then initiates the bidding process by fixing the prices .

The agents accept the prices and respond by announcing their respective bids, p and .

The networkmanager allocates a rate of to user and receives a payment of . Simultaneously, the linksupplier is asked to allocate a rate of to user ; the total payment made to the linksupplier is .
The prices set by the networkmanager are . The payoff to user , for bidding , is given by
(3) 
Similarly, the payoff to the linksupplier is given by
(4)  
Using the above payoff functions we characterize the solution as a competitive equilibrium which is defined as follows (unless mentioned otherwise, we assume that the agents' bids and the linksupplier's prices are nonnegative, i.e., ; also, we use 0 to denote the vector of allzeros of appropriate length):
Definition 1 (Competitive Equilibrium [10, 12])
We say that constitutes a competitive equilibrium if the following conditions hold:



Define and
(5) Then, the following should hold:

For all ,
(6) 
For all , the equality holds, where
(7) 
Furthermore,
(8)

In the above definition, condition (C1) implies that the users do not benefit by deviating from their equilibrium bids , when the prices set by the networkmanager are fixed. Similarly, (C2) implies that the linksupplier has no benefit in deviating from the equilibrium bidvector . Although (C1) and (C2) result in the optimality of the users' and the linksupplier's problem of maximizing their respective payoffs, these conditions by themselves do not guarantee systemoptimal performance. The conditions in (C3) (essentially derived from the optimality conditions for NETWORK) are crucial to guarantee that the prices set by the networkmanager are dual optimal for SYSTEM. Condition (C3) along with (C1) and (C2) can then be used to show the optimality of a competitive equilibrium. We summarize this result in the following theorem; in particular, we first prove the existence of a competitive equilibrium, and then derive its optimality property. This theorem is essentially an extension of the result due to Kelly [13] and Kelly et al. [8] (see also [10] and [12]). The main difference that warrants an extension is the presence of the linksupplier as a strategic agent.
Theorem 1
When the agents are pricetaking, there exists a competitive equilibrium, i.e., there exist vectors satisfying (C1), (C2) and (C3). Moreover, given a competitive equilibrium , the rate vectors x and y defined as and () are optimal for the problem SYSTEM in (1).
Iv Priceanticipating scenario
In contrast to the pricetaking scenario, agents initiate the bidding process in the priceanticipating scenario. Specifically, the sequence of exchanges is as given below.
PRICEANTICIPATING MECHANISM (PAM)

The network manager first announces to the agents how the allocation will be done and the payments will be fixed, as a function of prices and agents’ bids.

Agents then initiate the bidding process by simultaneously announcing their bids, denoted p and .

The networkmanager sets prices where we have set . Note that the above prices are dual optimal for the NETWORK problem in (2).

The payments and the allocated rates are exactly as in the pricetaking mechanism, but with replaced by .
In the following lemma we report the expression for the prices .
Lemma 1
Given any vector of users' and linksupplier's bids, the prices set by the networkmanager are given by
(9) 
where is the inverse of defined as
(10) 
and for
(11) 
Proof:
See Appendix BA. \qed
Continuing with the discussion, using the above prices in (3), the payoffs to the users in the priceanticipating scenario can be expressed as follows for (for simplicity, we use ):
(12)  
where denotes the bids of all users other than , while is the bid submitted by the linksupplier. Similarly, for the linksupplier we have
(13)  
The quantity in the above expression is due to complementary slackness conditions which imply
The users and the linksupplier recognize that their bids affect the prices and the allocation. Acting as rational and strategic agents, they now anticipate these prices. The appropriate notion of an equilibrium in this context is the following.
Definition 2 (Nash Equilibrium)
A bid vector is a Nash equilibrium if, for all , we have
When , the link is not fully utilized. In this case the Lagrange multiplier . Examination of (12) and (13) indicates that the payments made by the users are all passed on to the linksupplier. This may be interpreted as follows: for a given set of payments, the linksupplier bids are such that the link is viewed as a costly resource and the networkmanager passes on all his revenue to the linksupplier. The linksupplier is thus assured of this revenue even if his link is not fully utilized. If, on the other hand, the linksupplier's bids are such that , then , and it is clear from (13) that not all the collected revenue is passed on to the linksupplier. Indeed, since , we have
where the righthand side is obtained when . The actions of the linksupplier as a strategic agent creates a situation of conflict and results in the following undesirable equilibrium.
Theorem 2
When the users and the linksupplier are priceanticipating, the only Nash equilibrium is where and for all .
Proof:
See Appendix BB. \qed
Thus, in the priceanticipating setting, efficiency loss is 100%, which we interpret as a market breakdown. Indeed, at , the linksupplier is assured an income of . Given this guaranteed income, he minimizes his cost by supplying zero capacity. The resulting equilibrium is one with the lowest efficiency, and the situation is vastly different from the setting when the linksupplier is not viewed as an agent [10].
V PriceAnticipation with link as lead player
In view of the breakdown of the market when both the users and the linksupplier are simultaneously price anticipating, we design an alternative scheme that involves an additional stage. The sequence of exchanges is as follows.
PRICEANTICIPATION WITH LINK AS LEADER (PALL)

The network manager first announces to the agents how the allocation will be done and the payments will be fixed, as a function of prices and agents’ bids.

The linksupplier then announces his bidvector . This information is made available to all users.

The users then send their bids (). Let .

The networkmanager then computes the prices by solving the NETWORK problem in (2).

The payments and the ratesallocated are exactly as in the pricetaking mechanism, but with replaced by .
The analysis of this mechanism proceeds as follows. Given a , the expression for the prices set by the networkmanager are as in Lemma 1. As a result, the expressions for the users' and the linksupplier's payoff functions are exactly as in (12) and (13), respectively, but with p replaced by . Using these payoff functions, we characterize the solution in the form of Stackelberg equilibrium defined next.
Definition 3 (Stackelberg Equilibrium)
A bid vector is a Stackelberg equilibrium if, for all , we have
Observe that the bidvector announced by the linksupplier in step2 anticipates the user bids of step3. For a given , the bids submitted by the users is in anticipation of the prices the networkmanager announces in step4.
For the ease of exposition, we assume that so that the capacity constraint is not binding (the case where is finite can be similarly handled). Thus, recalling (9) and (11), we have and . As a result the payoff functions can be simply expressed as
(14)  
(15) 
This simplification will enable us to focus on the key ideas rather than dwell on the technicalities arising from a finite (which can be handled but is cumbersome and not enlightening).
From (14) we see that the user payoffs are independent of the bids submitted by the other users. As a result, for a given , the unique equilibrium strategy for user is given by
(16) 
In Lemma 2 we report the expression for that is obtained by solving (16).
Lemma 2
For a given we have
(17) 
where is the fixed point of .
Proof:
Since the objective function in (16) is continuously differentiable and strictly concave (both are easy to check), it suffices to show that of (17) solves the following optimality equation:
Indeed, with of (17) plugged into the above expression we have
and so satisfies . The case when is straightforward. \qed
We extend the definition of in the above lemma by defining if . It is then easy to see that is the allocation to user . Plugging the above result into (15), we compute the optimal that the linksupplier should announce in step1 as
(18) 
where means componentwise inequality.
For any it is clear that constitutes a Stackelberg equilibrium, where the rate allocated to user is given by . However, we first need to assert the existence of a solution , i.e., that the set is nonempty.
Theorem 3
Suppose and satisfy the following: and as . Then the set is nonempty. Hence, under the above assumptions on the payoffs and cost function, a Stackelberg equilibrium exists.
Proof:
See Appendix C. \qed
Remark: The above assumption excludes cost functions that are asymptotically linear, and payoffs such as . However, we note that these assumptions are not too restrictive. Also, note that it is not possible to assert the uniqueness of as it is not clear how varies as a function of (although it can be shown that increases with ).
In the remainder of this section, we restrict attention to linear user payoffs.
Va Stackelberg Equilibrium for Linear User Payoffs
An explicit expression for the Stackelberg equilibrium can be derived when the user payoffs are linear. Suppose that the user payoffs are of the form where (). Without loss of generality, assume that . The Stackelberg equilibrium can then be computed as follows.
First, fix a . Recalling Lemma 2, we have
so that the equilibrium bid of user can be written as
(19) 
Substituting for in (18), the optimal can be computed by solving
The solution to the above problem is given by
(20) 
where . The equilibrium bids of users in response to this optimized is then given by
(21) 
Thus, when the user payoffs are linear, the linksupplier allocates all the bandwidth to the “best” user (i.e., the one with the maximum slope ); in return, the best user alone makes a positive payment to the linksupplier.
The rate allocated to user at equilibrium is
(22)  
The total rate served by the linksupplier at equilibrium is given by .
VB Lower Bound on Efficiency for Linear User Payoffs
Given a Stackelberg equilibrium the efficiency is defined as the ratio of the utility at equilibrium (Stackelberg utility) to the system optimum (social utility):
(23) 
where denotes the social optimum allocation to user (obtained by solving SYSTEM in (1)). Note that we have emphasized the dependency of efficiency on by incorporating these into the notation for efficiency.
When the linksupplier is nonstrategic, from Johari et al. [10] it is known that the bound on efficiency is , i.e., for any general collection of user payoff functions (the loss in efficiency is thus no more than ). The above bound is obtained in [10] by doing the following.

Show that the users’ equilibrium bids in the original game (with general user payoff functions) constitutes an equilibrium in an alternate game with appropriately chosen linear payoff functions.

Use this to show that the efficiency in the original game is bounded below by the efficiency achieved in the alternate game.

Finally, minimize the efficiency over the set of all linear payoffs; this can be explicitly computed and is .
In our case, although (a) holds^{1}^{1}1Formally, we can show that for any given , the equilibrium strategy for the users in the original game with payoff functions is also an equilibrium strategy for the users in an alternate game with linear payoffs , where with . for any given , there is a subtle issue^{2}^{2}2Our conference version [1] missed this subtle point and incorrectly made a more general claim that the lower bound held for a larger class of user payoffs.. Since the linksupplier is also strategic, the original game and the alternate game (with linear user payoffs) may not have identical Stackelberg equilibria. In particular, the that optimizes the objective in (18) may not necessarily optimize
(24) 
which is the objective corresponding to the game with linear payoffs: with . Thus, (a) and (b) may not hold for general user payoffs. However, an analog of (c) continues to hold if we restrict our attention to the ensemble of all linear user payoffs. The lower bound on efficiency will however depend on the linksuppliers cost function . This result is detailed in the following theorem.
Theorem 4
Fix a linkcost function . For any set of linear user payoffs , we have
(25) 
where .
Proof:
See Appendix D. \qed
VC Efficiency Bound for Linear User Payoffs and Polynomial LinkCosts
We apply the above theorem to derive explicit expressions for the lower bound on the efficiency when the linkcost function is the polynomial . We start with the simplest case of quadratic linkcost, i.e., where . We then have so that . Thus, using (25), we obtain
Thus, when the linkcost is quadratic, the worstcase efficiency loss for any linear user payoff is no more that .
Similarly, suppose for , with . (This is increasing and convex for .) Then, using the bound (25) and a similar calculation, we obtain
Thus, the worstcase efficiency loss improves to when the linkcost is cubic. In general, suppose the linkcost is polynomial of degree , i.e., , , then the bound on efficiency is given by
(26) 
The aforementioned lower bound is increasing as a function of and converges to as . Thus, if the linkcost can be modeled as , the efficiency loss reduces with increase in .
The above observation provides strong support for our proposed PALL mechanism when compared with the priceanticipating mechanism of Section IV whose efficiency loss (for any including linear user payoffs and any ) is .
VD WorstCase Bound on Efficiency for Linear User Payoffs
Although the class of polynomial linkcost functions yield favorable lower bounds on efficiency, we now show that there exists a family of linkcost functions , , such that the corresponding sequence of efficiencybound converges to as . Thus, the worstcase efficiency bound, over all possible linear and over all possible , is .
To see this, let us first rewrite (25) by expressing in the integral form to get
For a given and a marginal cost function for the linksupplier , can be geometrically interpreted with the aid of the illustration in Fig.1 as follows: the numerator in the formula for efficiency is the area of the region (light shaded region) while the denominator is total area of and (shaded dark). We then have
where denotes the area of region (). In Fig. 1 we have used to denote ; also, where is arbitrarily chosen in . Since is strictly convex and increasing, it follows that is strictly increasing.
Now, it is possible to construct a sequence of functions, say , such that , while ; an illustration of such a construction is depicted in Fig. 1. Observe that along such a sequence we have and . As a result we have as . Thus, for any given it is possible to produce pathological linkcost functions whose efficiencybounds are arbitrarily close to . Therefore, it is not possible to guarantee a lessthan efficiency loss (i.e., a positive efficiency) when the class of all possible linkcost functions are considered. Nevertheless, bounding the efficiency for a fixed linkcost function is reassuring.
Part II: Multiple Links
Vi System Model and Problem Definition
In this section we extend our results to the more general setting with multiple links. We assume an example scenario with parallel links so that the users have the flexibility to offload different amounts of rates on different links. Simultaneously, the respective linkmanagers have to be competitive in terms of their bids in order to maximize their respective payoffs^{3}^{3}3Extension to a general network as in Kelly [13] is straightforward and does not bring out any new phenomenon.. Although it is natural to expect active participation from both users and linkmanagers, in the upcoming Theorem 6 we show the contrary. We will see that, when the users and the linkmanagers are strategic, the market collapses due to zero participation from both types of agents. This outcome is similar to the singlelink case. This also establishes that the breakdown in the singlelink case is not due to the monopolistic nature of the supplier in the singlelink setting. In Theorem 6, alternative routes exist, and yet, the undesirable equilibrium ensues.
We begin by generalizing our notation from Section II. As before we assume that there are users in the system. However, we now generalize our earlier model by introducing parallel links. The capacity of link is given by . Let denote the rate requested by user on link , and let be the rate the linkmanager is willing to allocate to user . Thus, is the raterequest vector of user , and is the rateallocation vector of link . Let and denote the raterequest matrix and rateallocation matrix, respectively. The user payoff and the linkcost functions are given by and . As before, we assume that and are concave and strictly convex, respectively. In addition, both and are strictly increasing and continuously differentiable with finite.
The analog of the problem SYSTEM in (1) is given by (in the sequel, the acronym ML stands for MultiLink):
MLSYSTEM
Maximize:  (27a)  
Subject to:  (27b)  
(27c) 
Similarly, denoting the users' and the linkmanagers' bidvectors as
respectively, the analog of problem NETWORK in (2) is:
MLNETWORK
Maximize:  (28a)  
Subject to:  (28b)  
(28c) 
We introduce some more notation. Let denote the users' bid matrix. Similarly, the linkmanagers' bid matrix is denoted by . The networkmanager sets prices and where . The prices and M are essentially the Lagrange multipliers associated with the constraints (28b) and (28c), respectively.
We investigate the pricetaking and the priceanticipating scenarios separately, as was done in the singlelink setting.
Vii PriceTaking Scenario
The mechanism under the pricetaking scenario is exactly as in Section III (see PTM in Section III), except that now there are multiple linkmanagers who submit their respective bids () simultaneously. In this setting, given the prices set by the networkmanager, the payoff to user can be written as
(29) 
Similarly, the payoff to the linkmanager is given by
(30)  
where . The following are the generalizations of Definition 1 and Theorem 1, respectively.
Definition 4 (Competitive Equilibrium)
A vector of bids and prices is said to constitute a competitive equilibrium if the following conditions hold:



For each define and . Then,

;

where


Theorem 5
When the users and the linkmanagers are pricetaking, there exists a competitive equilibrium. Moreover, given a competitive equilibrium , the rate matrices X and Y, defined as and , are optimal for the problem MLSYSTEM in (27).
Proof:
The proof is omitted since it is a straightforward extension of the proof of Theorem 1. \qed
Viii PriceAnticipating Scenario
Recall that when the users and the linkmanagers are priceanticipating they expect that the bids submitted by them affect the prices set by the networkmanager. In particular, the users and the linkmanagers are aware that the prices set by the networkmanager, in response to the bids (P,B) submitted by the agents, are dualoptimal for the problem MLNETWORK in (28). The details of the mechanism under priceanticipating scenario is similar to PAM in Section IV, except that the setting now consists of multiple linkmanagers who submit their respective bids simultaneously.
Now, the expressions for the prices set by the networkmanager is as reported in the following lemma (which is inline with the result in Lemma 1).
Lemma 3
Given any matrix of users' and linkmanagers' bids, the prices set by the networkmanager are given by,
where denotes the inverse of the function which is defined as