Double-Auction Mechanisms for Resource Trading Markets

Double-Auction Mechanisms for
Resource Trading Markets

K. P. Naveen and Rajesh Sundaresan K. P. Naveen is with the Department of Electrical Engineering, Indian Institute of Technology Tirupati, India (Email: naveenkp@iittp.ac.in). R. Sundaresan is with the Department of Electrical Communication Engineering and the Robert Bosch Centre for Cyber Physical Systems, Indian Institute of Science, Bangalore, India (Email: rajeshs@iisc.ernet.in).The work of the first author was supported by an INSPIRE Faculty Award of the Department of Science and Technology, Government of India. The second author was supported by the Robert Bosch Centre for Cyber-Physical Systems, Indian Institute of Science, Bangalore. This paper was presented in part at the 16th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt 2018) [1].
Abstract

We consider a double-auction mechanism, which was recently proposed in the context of a mobile data-offloading 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 (link-suppliers). Link-suppliers experience costs for the additional capacity that they provide. Users and link-suppliers (collectively referred to as agents) have their pay-offs and cost functions as private knowledge. A system-designer decomposes the problem into a network problem (with surrogate pay-offs and surrogate cost functions) and agent problems (one per agent). The surrogate pay-offs 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 (price-anticipating). 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 link-suppliers 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 pay-off functions are linear. Specifically, when the link-supplier’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.

Network utility maximization, double-auction, KKT conditions, Nash equilibrium, Stackelberg equilibrium.

I Introduction

We consider double auction mechanisms motivated by two examples – mobile data offloading and network slicing-based virtualization.

Mobile data offloading is an effective way to manage growth in mobile-data traffic. Traffic meant for the macrocellular network can be offloaded to already installed third-party Wi-Fi or femtocell networks. This provides an alternative means of network expansion. Wi-Fi access-point 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 high-quality 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 double-auction mechanism where a set of mobile network operators (buyers or users in this work) compete for resources from access-point operators (sellers or links in this work). The pay-offs 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 access-point 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 access-point 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 access-point operators, (b) each operator is aware only of his own needs or costs, and (c) all agents are price-taking (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 double-auctions 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 price-anticipating scenario open [7, Sec VII, p.1646].

Our contributions in this paper are as follows.

  1. We first re-derive the result on efficient allocation when the agents are price-taking, mainly to set up the notation for the next three results.

  2. We then analyze the price-anticipating scenario along the lines of Johari et al. [10]. When agents are price-anticipating, 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 access-point operators are not strategic. The efficiency loss due to price-anticipating mobile offloading agents is then at most 34%. However, when the access-point agents (suppliers) are also strategic and price-anticipating, 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 double-auction mechanism of [7] works when agents are price-taking, but fails in the more real situation when agents are price-anticipating. One must then look for alternative double-auction mechanisms.

  3. 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 pay-off functions are linear. For instance, for the quadratic link-cost function, the worst-case efficiency loss (with the worst-case taken over linear user pay-off functions) is at most 25%.

  4. 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 price-taking 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 price-anticipating 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 price-anticipating scenario with non-strategic 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 pay-offs. This of course raises the question of what is the minimal signalling dimension in the price-anticipating 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 link-supplier. Specifically, in Section II, we discuss the system model and problem definition. In Section III, we discuss the price-taking scenario for the single-link case. In Section IV, we analyze the price-anticipating scenario. As a positive result, in Section V, we discuss our proposed mechanism and characterize the worst-case efficiency loss for linear user pay-offs 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 link-suppliers. 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 link-supplier. In the context of mobile-data offloading [7], users and link-supplier correspond to mobile-network operators and the single access-point operator (e.g., Wi-Fi, femtocell), respectively. The mobile-network 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 link-supplier is the seller. We refer to the users and the link-supplier 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 network-manager,

Let denote the rate requested by user , and let be the rate the link-supplier is willing to allocate to user . Thus, and represent the rate-request and rate-allocation vectors, respectively. Let denote the aggregate-rate allocated by the link-supplier to all users. For user , the benefit of acquiring a rate of is represented by a pay-off function ; we assume that , , are concave, strictly increasing and continuously differentiable with finite . Similarly, the cost incurred by the link-supplier 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 rate-requests (demand) and the rate-allocations (supply) are matched although the capacity may not be fully utilized.

A network-manager, however, cannot solve the formulation in (1) without the knowledge of user pay-offs and the link-cost 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 link-supplier 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 link-supplier, respectively.

The network-manager 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 link-supplier is willing to share. For instance, a lower value of implies that the bandwidth shared by the link-supplier with user is low, and vice versa. In particular, implies that the link-supplier is unwilling to share any bandwidth with user . This will be useful while interpreting the Nash equilibrium bid-vectors (Theorem 2).

The above NETWORK problem is identical to the SYSTEM problem but with the true pay-off and cost functions replaced by surrogate pay-off and cost functions. In the following, we first review the case when the users and the link-supplier are price-taking. This means agents assume prices are given and do not anticipate the effect of their bids on the prices set by the network-manager. See Definition 1 below of a competitive equilibrium. We then proceed to study the more-involved price-anticipating scenario. Here the agents recognize that the effective price is based on their bids, anticipate the resulting allocation, payment, and therefore their pay-off, and act accordingly. The resulting pay-off 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 price-anticipating scenario is dramatically negative due to the presence of the strategic link-supplier, as we will soon see. We then propose a remedy via a Stackelberg framework where the link-supplier is a lead player and the users are followers.

Iii Price-taking scenario

The sequence of exchanges (between the network-manager and the agents) in the price-taking scenario is as shown in the box describing the price-taking mechanism (PTM) below.

{tcolorbox}

PRICE-TAKING MECHANISM (PTM)

  1. 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.

  2. The network-manager then initiates the bidding process by fixing the prices .

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

  4. The network-manager allocates a rate of to user and receives a payment of . Simultaneously, the link-supplier is asked to allocate a rate of to user ; the total payment made to the link-supplier is .

The prices set by the network-manager are . The pay-off to user , for bidding , is given by

(3)

Similarly, the pay-off to the link-supplier is given by

(4)

Using the above pay-off 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 link-supplier's prices are non-negative, i.e., ; also, we use 0 to denote the vector of all-zeros of appropriate length):

Definition 1 (Competitive Equilibrium [10, 12])

We say that constitutes a competitive equilibrium if the following conditions hold:



  1. 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 network-manager are fixed. Similarly, (C2) implies that the link-supplier has no benefit in deviating from the equilibrium bid-vector . Although (C1) and (C2) result in the optimality of the users' and the link-supplier's problem of maximizing their respective pay-offs, these conditions by themselves do not guarantee system-optimal performance. The conditions in (C3) (essentially derived from the optimality conditions for NETWORK) are crucial to guarantee that the prices set by the network-manager 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 link-supplier as a strategic agent.

Theorem 1

When the agents are price-taking, 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).

Proof:

The result can be gleaned from the results in [7] though it is not explicitly stated. Our proof of Theorem 1 is a direct one that does not rely on any learning dynamics. Instead, it is based on Lagrangian techniques. Details are available in Appendix A. \qed

Iv Price-anticipating scenario

In contrast to the price-taking scenario, agents initiate the bidding process in the price-anticipating scenario. Specifically, the sequence of exchanges is as given below.

{tcolorbox}

PRICE-ANTICIPATING MECHANISM (PAM)

  1. 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.

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

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

  4. The payments and the allocated rates are exactly as in the price-taking mechanism, but with replaced by .

In the following lemma we report the expression for the prices .

Lemma 1

Given any vector of users' and link-supplier's bids, the prices set by the network-manager are given by

(9)

where is the inverse of defined as

(10)

and for

(11)
Proof:

See Appendix B-A. \qed

Continuing with the discussion, using the above prices in (3), the pay-offs to the users in the price-anticipating 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 link-supplier. Similarly, for the link-supplier we have

(13)

The quantity in the above expression is due to complementary slackness conditions which imply

The users and the link-supplier 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 link-supplier. This may be interpreted as follows: for a given set of payments, the link-supplier bids are such that the link is viewed as a costly resource and the network-manager passes on all his revenue to the link-supplier. The link-supplier is thus assured of this revenue even if his link is not fully utilized. If, on the other hand, the link-supplier's bids are such that , then , and it is clear from (13) that not all the collected revenue is passed on to the link-supplier. Indeed, since , we have

where the right-hand side is obtained when . The actions of the link-supplier as a strategic agent creates a situation of conflict and results in the following undesirable equilibrium.

Theorem 2

When the users and the link-supplier are price-anticipating, the only Nash equilibrium is where and for all .

Proof:

See Appendix B-B. \qed

Thus, in the price-anticipating setting, efficiency loss is 100%, which we interpret as a market break-down. Indeed, at , the link-supplier 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 link-supplier is not viewed as an agent [10].

V Price-Anticipation with link as lead player

In view of the break-down of the market when both the users and the link-supplier are simultaneously price anticipating, we design an alternative scheme that involves an additional stage. The sequence of exchanges is as follows.

{tcolorbox}

PRICE-ANTICIPATION WITH LINK AS LEADER (PALL)

  1. 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.

  2. The link-supplier then announces his bid-vector . This information is made available to all users.

  3. The users then send their bids (). Let .

  4. The network-manager then computes the prices by solving the NETWORK problem in (2).

  5. The payments and the rates-allocated are exactly as in the price-taking mechanism, but with replaced by .

The analysis of this mechanism proceeds as follows. Given a , the expression for the prices set by the network-manager are as in Lemma 1. As a result, the expressions for the users' and the link-supplier's pay-off functions are exactly as in (12) and (13), respectively, but with p replaced by . Using these pay-off 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 bid-vector announced by the link-supplier in step-2 anticipates the user bids of step-3. For a given , the bids submitted by the users is in anticipation of the prices the network-manager announces in step-4.

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 pay-off 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 pay-offs 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 link-supplier should announce in step-1 as

(18)

where means component-wise 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 pay-offs and cost function, a Stackelberg equilibrium exists.

Proof:

See Appendix C. \qed

Remark: The above assumption excludes cost functions that are asymptotically linear, and pay-offs 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 pay-offs.

V-a Stackelberg Equilibrium for Linear User Pay-offs

An explicit expression for the Stackelberg equilibrium can be derived when the user pay-offs are linear. Suppose that the user pay-offs 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 pay-offs are linear, the link-supplier 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 link-supplier.

The rate allocated to user at equilibrium is

(22)

The total rate served by the link-supplier at equilibrium is given by .

V-B Lower Bound on Efficiency for Linear User Pay-offs

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 link-supplier is non-strategic, from Johari et al. [10] it is known that the bound on efficiency is , i.e., for any general collection of user pay-off 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 pay-off functions) constitutes an equilibrium in an alternate game with appropriately chosen linear pay-off 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 pay-offs; this can be explicitly computed and is .

In our case, although (a) holds111Formally, we can show that for any given , the equilibrium strategy for the users in the original game with pay-off functions is also an equilibrium strategy for the users in an alternate game with linear pay-offs , where with . for any given , there is a subtle issue222Our 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 pay-offs.. Since the link-supplier is also strategic, the original game and the alternate game (with linear user pay-offs) 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 pay-offs: with . Thus, (a) and (b) may not hold for general user pay-offs. However, an analog of (c) continues to hold if we restrict our attention to the ensemble of all linear user pay-offs. The lower bound on efficiency will however depend on the link-suppliers cost function . This result is detailed in the following theorem.

Fig. 1: Geometric interpretation for the efficiency bound.
Theorem 4

Fix a link-cost function . For any set of linear user pay-offs , we have

(25)

where .

Proof:

See Appendix D. \qed

V-C Efficiency Bound for Linear User Pay-offs and Polynomial Link-Costs

We apply the above theorem to derive explicit expressions for the lower bound on the efficiency when the link-cost function is the polynomial . We start with the simplest case of quadratic link-cost, i.e., where . We then have so that . Thus, using (25), we obtain

Thus, when the link-cost is quadratic, the worst-case efficiency loss for any linear user pay-off 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 worst-case efficiency loss improves to when the link-cost is cubic. In general, suppose the link-cost 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 link-cost 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 price-anticipating mechanism of Section IV whose efficiency loss (for any including linear user pay-offs and any ) is .

V-D Worst-Case Bound on Efficiency for Linear User Pay-offs

Although the class of polynomial link-cost functions yield favorable lower bounds on efficiency, we now show that there exists a family of link-cost functions , , such that the corresponding sequence of efficiency-bound converges to as . Thus, the worst-case 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 link-supplier , 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 link-cost functions whose efficiency-bounds are arbitrarily close to . Therefore, it is not possible to guarantee a less-than- efficiency loss (i.e., a positive efficiency) when the class of all possible link-cost functions are considered. Nevertheless, bounding the efficiency for a fixed link-cost 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 off-load different amounts of rates on different links. Simultaneously, the respective link-managers have to be competitive in terms of their bids in order to maximize their respective pay-offs333Extension 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 link-managers, in the upcoming Theorem 6 we show the contrary. We will see that, when the users and the link-managers are strategic, the market collapses due to zero participation from both types of agents. This outcome is similar to the single-link case. This also establishes that the break-down in the single-link case is not due to the monopolistic nature of the supplier in the single-link 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 link-manager is willing to allocate to user . Thus, is the rate-request vector of user , and is the rate-allocation vector of link . Let and denote the rate-request matrix and rate-allocation matrix, respectively. The user pay-off and the link-cost 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 Multi-Link):

ML-SYSTEM

Maximize: (27a)
Subject to: (27b)
(27c)

Similarly, denoting the users' and the link-managers' bid-vectors as

respectively, the analog of problem NETWORK in (2) is:

ML-NETWORK

Maximize: (28a)
Subject to: (28b)
(28c)

We introduce some more notation. Let denote the users' bid matrix. Similarly, the link-managers' bid matrix is denoted by . The network-manager sets prices and where . The prices and M are essentially the Lagrange multipliers associated with the constraints (28b) and (28c), respectively.

We investigate the price-taking and the price-anticipating scenarios separately, as was done in the single-link setting.

Vii Price-Taking Scenario

The mechanism under the price-taking scenario is exactly as in Section III (see PTM in Section III), except that now there are multiple link-managers who submit their respective bids () simultaneously. In this setting, given the prices set by the network-manager, the pay-off to user can be written as

(29)

Similarly, the pay-off to the link-manager 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:



  1. For each define and . Then,

    • ;

    • where

Theorem 5

When the users and the link-managers are price-taking, there exists a competitive equilibrium. Moreover, given a competitive equilibrium , the rate matrices X and Y, defined as and , are optimal for the problem ML-SYSTEM in (27).

Proof:

The proof is omitted since it is a straightforward extension of the proof of Theorem 1. \qed

Viii Price-Anticipating Scenario

Recall that when the users and the link-managers are price-anticipating they expect that the bids submitted by them affect the prices set by the network-manager. In particular, the users and the link-managers are aware that the prices set by the network-manager, in response to the bids (P,B) submitted by the agents, are dual-optimal for the problem ML-NETWORK in (28). The details of the mechanism under price-anticipating scenario is similar to PAM in Section IV, except that the setting now consists of multiple link-managers who submit their respective bids simultaneously.

Now, the expressions for the prices set by the network-manager is as reported in the following lemma (which is in-line with the result in Lemma 1).

Lemma 3

Given any matrix of users' and link-managers' bids, the prices set by the network-manager are given by,

where denotes the inverse of the function which is defined as