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An Efficient Nash-Implementation Mechanism for Divisible Resource Allocation Rahul Jain

Jean Walrand

IBM T.J. Watson Research Center

EECS Department,

Hawthorne, NY 10532

University of California, Berkeley

[email protected]

[email protected]

Abstract We propose a mechanism for auctioning bundles of multiple divisible goods. Such a mechanism is very useful for allocation of bandwidth in a network where the buyers want the same amount of bandwidth on each link in their route. We allow for buyers to specify multiple routes (corresponding to a source-destination pair). The total flow can then be split among these multiple routes. We first propose a single-sided VCG-type mechanism. However, instead of reporting their valuation functions, the players only reveal a two-dimensional bid signal - the maximum quantity that they want and the per unit price they are willing to pay. The proposed mechanism is a weak Nash-implementation, i.e., it has a non-unique Nash equilibrium that implements the social-welfare maximizing allocation. We show the existence of an efficient Nash equilibrium in the corresponding auction game. We show through an example that there are other Nash equilibria that are not efficient. Further, we provide a sufficient characterization of all efficient Nash equilibria. We then generalize this to buyers getting arbitrary amounts of various goods. This require each buyer to submit a bid separately for each good but their utility function a general function of allocations of various divisible goods. Then, we present a doublesided auction mechanism for multiple divisible goods with buyers and sellers. We show that there exists a Nash equilibrium of this auction game which yields the efficient allocation.

I. I NTRODUCTION Many problems involve multiple divisible resources (i.e., those that can be divided infinitely, e.g., bandwidth when it is available in any real fraction of Mbps) which are to be shared among many entities. The allocation of resources is to be done to achieve a global objective (such as maximization of the sum of individual objective functions). There are information asymmetries: December 26, 2007

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each entity knows only its own objective function (henceforth, called a utility function) and the system administrator knows the class to which the utility functions belong but does not actually know the individual realized utility functions. The system administrator, given this limited information, is to design a system that determines an allocation to the various entities achieves that achieves a global objective. Any such design is possible only if some information indicative of the entities’ utility functions is elicited from them, and used to determine the allocation. However, each of the entities is an independent, self-interested and strategic player, and thus may attempt to manipulate the system to its advantage by misreporting information about its utility function. A basic question then is how can we design “rules of interaction” or a game that despite strategic behavior on the part of players, and without a priori knowledge of realized utility function by the system administrator, still achieves an allocation that maximizes the global objective function. In some sense, this is an “inverse game theory” problem. A theory that studies design of “strategy-proof” resource allocation mechanisms has been under development since the 1960s ([5], [20] are good references). In this paper, we are interested in solving network resource allocation and exchange problems in a particular environment. Our problem is motivated by the problem of resource allocation in communication networks where service providers want bandwidth on a whole route, hence same bandwidth on all links in the route. The first problem is allocating multiple divisible resources among strategic agents. Let there be L divisible goods available in quantities C1 , · · · , CL . Let r ⊆ [1 : L] denote a bundle of goods, such as those links that form a route. Let there be n agents and let Ri for agent i denote a set of bundles, such as set of routes between a source-destination pair. For each agent, his allocation might be split between r ∈ Ri (such as multiple routes) but within each r, the share of allocation on good l for route r has to same for all l ∈ r (such as requiring the same capacity on all links on a route). All the goods belong to the system administrator which must determine how the goods should be allocated among the agents. Each agent derives different satisfaction from owning a certain quantity of the various goods, i.e., they have different utility functions. The system administrator would like to allocate the various goods among the agents to maximize the sum of utility derived by all the agents. However, user utilities are unknown to the system administrator. Thus, it must elicit some information from the agents to determine the optimal allocation. This can be done through an auction mechanism wherein each agent is asked to reveal December 26, 2007

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a bid signal representative of its utility function. However, each agent is selfish, acts strategically and has an incentive to misrepresent its bid-signal. Thus, we must design an auction mechanism that is robust to such strategic manipulation by the agents. We then generalize this to the case where the allocated bundle to a buyer can be arbitrary. The second problem addresses a more general (multilateral trading) environment where there are many buyers and many sellers. We will assume that each buyer wants capacity between a source-destination pair (over multiple routes). Each seller sells goods individually (i.e., without forming bundles), and for simplicity we will assume that each seller sells only one type of good though there may be multiple sellers selling the same good. We will require each buyer and each seller to reveal a bid-signal representative of his utility or cost function. And our goal is to determine an allocation of resources that maximizes the social welfare (sum of utility derived by all buyers minus sum of cost incurred by all the sellers). Each of the agents has his own utility and cost function, and acts strategically. Thus, it might be difficult to obtain an optimal allocation. Our goal is to design an exchange mechanism which despite strategic behavior by the participants yields an allocation that maximizes the social welfare. Literature Review Resource allocation mechanism design in a general setting has been extensively studied by Economists and Operations Researchers. Unfortunately, most of the fundamental results are negative (such as the various impossibility theorems that specify economic environments for which it is impossible to design mechanisms with certain specified properties). The VickreyClarke-Groves mechanism [23], [3], [6] is the most prominent positive result. Attention was drawn to similar resource allocation problem in networks by the work of [11], [15], [16]. This work was largely motivated by the need to design and analyze distributed pricing signal-based network congestion control algorithms. In particular, [11], [12] showed that when agents in a network do not act strategically, the resource allocation problem can be solved efficiently in a distributed manner. In fact, it was suggested that the internet transport control protocol (TCP) can be understood to be doing exactly such a distributed optimization. This work inspired a mechanism design (the Kelly mechanism) for allocation of divisible goods (such as bandwidth in a network) [18]. This mechanism was analyzed for the case when users are strategic in [9]. It was discovered that with a single divisible good, the Kelly mechanism December 26, 2007

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can result in an efficiency loss of upto 25%, i.e., the value of the social welfare function at the equilibrium outcome allocation is 25% less than the one determined by a centralized mechanism with complete knowledge of all the players’ utility functions. It was however shown in [7] that in the network version of Kelly mechanism (a player submits a single bid for bandwidth on all links that constitute his route), the efficiency loss can be arbitrarily bad. Following up on this work, a generalized class of proportional allocation (ESPA) mechanisms was introduced in [18], and further analyzed in [17], [24]. It was shown that these are efficient for allocation of a single divisible good. Such ESPA mechanisms require one-dimensional bidsignals and have a unique Nash equilibrium at which the allocation is efficient. However, the mechanisms trade off dominant-strategy implementation, a very desirable property, for ease in implementation as compared to the VCG class of mechanisms. In [7], a similar generalization of the proportional allocation mechanism was proposed for a single divisible good. In [10], a general class of convex VCG-type mechanisms were introduced that required one dimensional bid signals. It was established that there exists one Nash equilibrium at which the corresponding allocation is efficient. Conditions were provided under which the Nash equilibrium is unique and the outcome is guaranteed to be efficient. A proposal, very similar in spirit, and really a sub-case of the above was presented in [25]. Both these mechanism require that the pseudo-utility functions that the players report be twice continuously differentiable. A mechanism for a single good, in the same spirit but with non-differentiable pseudo-utility functions was first reported in [14]. Note that all the mechanisms mentioned above are single-sided, i.e., they only involve the auctioneer and multiple buyers. Double-sided mechanisms with both buyers and sellers are of interest for actual bandwidth exchanges. Our Contribution This paper is directly related to the work of Lazar and Semret [13], [14], [21]. They proposed a VCG-style auction mechanism for a single divisible good [14]. Attempts have been made to generalize this mechanism to multiple divisible goods so that it can be useful for network resource allocation problems [1], [13], [19]. The setting of [13] addresses the case where agents want bundles of links (goods), and a different auction is held for every link. However, each agent’s utility only depends on the minimum allocation it obtains on any link in its route. A slightly different setting is provided in [21], chapter 3, wherein sellers place ask bids to sell December 26, 2007

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bandwidth on individual links. Moreover, a buyer has to effectively bid separately for bandwidth in each link in its route. Thus, there is a separate double auction for each link. Such auctions when agents have complementarities across goods can lead to outcomes where an agent does not get all goods in its bundle, and thus might end up with zero valuation for his allocation. In fact, [1] considers the networked PSP mechanism of [13], [21] and proposed strategies for the agents that will improve the efficiency of the outcome. But, in our view, what is required is a network auction mechanism wherein agents can express their bid for a whole path (such a setting is considered in the combinatorial auctions literature but for indivisible goods). In [19], a variation of the basic PSP mechanism is provided which uses a higher dimensional bid-signal space to yield the same efficiency results. This in our opinion is not necessary as was also shown in the recent paper [10]. However, the results of [10] do not hold in the case where the routing matrix has full rank. The proposals in this paper are inspired by [14]. We propose a VCG-style mechanism but instead of reporting their types (or complete utility functions), agents only report a two-dimensional bid: a per-unit price β and the maximum quantity d that the agent is willing to buy at that price. Note that this corresponds to a valuation function vˆ(x) = β · min{x, d} which are continuous, concave, non-decreasing but non-differentiable. The mechanism determines an allocation which maximizes the social welfare corresponding to the reported utility functions. The payment of each agent is exactly the externality it imposes on the others through its participation, just as in the VCG mechanism. What is remarkable here is that for divisible goods, when the utility functions are strictly increasing, strictly concave and differentiable, it suffices for agents to report only a quantity and their marginal valuation at that quantity (instead of the full valuation function) for the mechanism to yield the efficient outcome at a Nash equilibrium. What is lost is the dominant-strategy implementation of VCG mechanisms, i.e., truthful reporting of utility functions is not a dominant strategy equilibrium: Each agent need not have knowledge of the utility functions of others, nor of the actions being taken by them. II. P ROBLEM S TATEMENT Consider L divisible goods, L = {1, · · · , L}, with Cl units of good l being available. Let Γ be the power set of L. Let there be n buyers. Buyer i wants a bundle of goods r ∈ Ri ⊆ Γ and wants the same quantity xi of all goods in this bundle. For example, a buyer might desire December 26, 2007

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any route between a source-destination pair. Ri would then denote all the routes r between this source-destination pair. Moreover, we would allow that the buyer’s total flow is split between various routes in Ri , e.g., buyer i might receive zi1 = xi /2 on route r1 ∈ Ri and zi2 = xi /2 on route r2 ∈ Ri for a total of xi . We will assume that each buyer has a quasi-linear utility function ui (xi , ωi ) = vi (xi ) − ωi where ωi is the payment made by buyer i and vi (xi ) is a strictly increasing, strictly concave and twice differentiable valuation function. Denote x = (x1 , · · · , xn )0 and C = (C1 , · · · CL )0 . We will denote by zir the flow of i carried on route r ∈ Ri . We will use the notation Hir = I(r ∈ Ri ) and Alr = I(l ∈ r), where I is the indicator function. We will call S(x) =

n X

vi (xi )

i=1

as the social welfare function, which is a strictly increasing concave function. We will require capacity constraints X

Alr zir ≤ Cl , ∀l ∈ L,

(1)

i,r

xi =

X

Hir zir , ∀i,

(2)

r∈Ri

xi , zir ≥ 0, ∀i, r.

(3)

The first constraint simply says that it is not possible to allocate more than the available quantity of any good, the second constraint says that total flow allocation to a buyer equals the sum of flow allocations to him on various routes r ∈ Ri , and the third constraint says that only non-negative allocations are allowed. The three constraints together determine a convex domain. Let λl and νi be the Lagrange multipliers corresponding to constraints (1) and (2). System Objective: To determine an allocation x∗ that satisfies max

S(x)

(4)

Az ≤ C, Hz = x, x, z ≥ 0. We will call such an allocation efficient. December 26, 2007

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Observe that it is a convex optimization problem. Thus, a solution exists and moreover it is unique. It is characterized by the following set of conditions (vi0 (x∗i ) − νi∗ ) x∗i = 0,

∀i

(vi0 (x∗i ) − νi∗ ) ≤ 0, ! X ∗ Cl − Alr zir λ∗l = 0,

∀i

(5)

∀l

i,r

! Cl −

X

∗ Alr zir

≥ 0,

∀l

i,r

νi∗ −

X

λ∗l = 0,

l∈r ∗ ∗ ∗ λl , νi , x∗i , zir

∀r ∈ Ri , ∀i

≥ 0, ∀l, ∀r ∈ Ri , ∀i.

The above conditions are derived from the KKT necessary and sufficient conditions for optimality in convex programs [2]. Note that it is possible for a system administrator to achieve this objective only if he knows the valuation functions of all the agents exactly. This however may not be true in distributed systems with selfish agents who may not reveal their actual valuation functions. In that case, we need an incentivized mechanism ((˜ x1 , P1 )), · · · , (˜ xn , Pn ) which asks agent i to report a signal bi indicative of its valuation function vi , and determines an allocation x˜i and a payment Pi to be made by it. Agent’s Objective: To pick a bi to maximize its net utility ui (bi ; b−i ) = vi (xi (bi , b−i )) − Pi (bi , b−i ) where b−i are the bid signals of all the other agents. This gives rise to a strategic game between the agents. The allocation and payment rule is to be designed in such a way that each agent reports a signal that enables the system administrator to determine the allocation x∗ even without knowing the actual valuation functions. III. T HE N ETWORK S ECOND -P RICE M ECHANISM

WITH

M ULTIPLE ROUTES

We now propose a mechanism to be used by the system administrator (also called the auctioneer) to allocate multiple divisible goods available in certain quantities among many buyers. The buyers specify R1 , · · · , Rn ⊆ Γ (with r ∈ Ri , a subset of L) and corresponding bids b1 , · · · , bn . The bid bi = (βi , di ) specifies the maximum per unit price βi that i is willing to pay and demands up to di units of Ri . Denote d = (d1 , · · · , dn )0 . December 26, 2007

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Note that any buyer i derives zero utility if he gets flow on a route r ∈ / Ri . Thus, he has no incentive to not truthfully report Ri . The auctioneer then determines an allocation x˜ = (˜ x1 , · · · , x˜n ) as a solution of the following optimization problem: max

P

i

βi xi

(6)

s.t. Az ≤ C, Hz = x, x ≤ d, x, z ≥ 0. Let x˜−i denote the solution of the above with di = 0. Then, the payment to be made by buyer i is Pi (bi , b−i ) =

X

βj (˜ x−i ˜j ). j −x

(7)

j6=i

The above defines the Network Second Price (NSP) mechanism. This is a VCG-style mechanism [23] where the players instead of reporting their type or a full valuation function, only report the parameters (βi , di ) of the revealed valuation function vˆi (x) = βi min(x, di ). The payment of i is the “externality” or the decrease in social welfare that the buyer imposes on all the other players by his participation based on this revealed valuation function. The payoff of buyer i is ui (bi , b−i ) = vi (˜ xi (b)) − Pi (b). Recall that an allocation x∗∗ is efficient if it is a solution of the optimization (3). Note that such an allocation cannot be changed to improve any player’s payoff without decreasing some other player’s payoff and hence is Pareto-efficient. The strategy space of buyer i is Bi = [0, ∞) × [0, C i ] where C i = maxr∈Ri minl∈r Cl .

1

A Nash equilibrium is a bid profile b∗ = (b∗1 , · · · , b∗n ) such that ui (b∗i , b∗−i ) ≥ ui (bi , b∗−i ), 1

∀bi ∈ Bi .

Note that we are after “allocative” efficiency (social welfare maximization) here, which also happens to be a Pareto-efficient

allocation. However, there would be other Pareto-efficient allocations as well.

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Nash equilibria which yield efficient allocation will be said to be efficient. For any Nash equilibrium allocation x∗ , we will say that its relative efficiency is X X η := vi (x∗i )/ vi (x∗∗ ). i

i

Note that this will lie in [0,1], where η = 1 will mean that full efficiency is achieved. A. Properties of the NSP Mechanism We first note the KKT conditions for the auction optimization problem. Let λl be the Lagrange multiplier corresponding to the capacity constraint 1 for good l, νi the Lagrange multiplier corresponding to the flow balance constraint 2 and µi be the Lagrange multiplier corresponding to the demand constraint 3 in the auction optimization (6). (βi − νi∗ − µ∗i ) x∗i = 0,

∀i

(βi − νi∗ − µ∗i ) ≤ 0, ! X ∗ Cl − Alr zir λ∗l = 0,

∀i

(8)

∀l

i,r

! Cl −

X

∗ Alr zir

≥ 0,

∀l

(di − x∗i ) µ∗i = 0, X λ∗l = 0, νi∗ −

∀i

i,r

l∈r ∗ ∗ λl , νi , µi , x∗i , zir

∀r ∈ Ri , ∀i

≥ 0, ∀l, ∀r ∈ Ri , ∀i.

1) Existence of an Efficient Nash Equilibrium: We first show existence of a Nash equilibrium in the corresponding resource allocation game by construction. Theorem 1: There exists a Nash Equilibrium b∗ of the NSP mechanism whose corresponding allocation x∗ is efficient (i.e., η(x∗ ) = 1). The proof is by construction. We relegate it to the appendix. Note that the above result implies the existence of an ε-efficient ε-Nash equilibrium, a result obtained in [14] for the special case of a single good.

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2) Inefficient Nash Equilibria and Reserve Prices: However, not all Nash equilibria of the NSP mechanism are efficient. We show existence of an inefficient one through an example. Example 1: Consider two players with linear valuation functions, vi (x) = θi x for one good with C = 1, and with θ1 > θ2 . Thus, the efficient allocation is (1, 0). Let player 2 bid β2 = (θ1 , 1−) and player 1 bid β1 = (θ2 , ). The allocation is (, 1−) and the payments are (0, 0). It is easy to check that it is a Nash equilibrium. Further, the relative efficiency is (θ2 (1−)−θ1 )/θ1 . For  and θ2 arbitrarily small, this can be made arbitrarily close to zero. Note that in the example above, we assumed that the valuation functions are linear. Theorem 1 assumes that the utility functions are strictly concave. However, one can imagine strictly concave valuation functions arbitrarily close to being linear. Thus, for any 0 <  < 1, there exist valuation functions and a Nash equilibria in the two player auction game above which have relative efficiency smaller than . But note that this arbitrarily large efficiency loss can be mitigated by introducing reserve prices and eliminating some of the inefficient Nash equilibria. Example 2: Let p be a reserve price, the price that any participant has to pay. Then, in the example above, the players bid β1 = (θ1 , d1 ) and β2 = (θ2 , d2 ) with θ1 > θ2 if there is a d2 such that v2 (d2 ) − θ2 (d2 − (1 − d1 )) − p ≥ v2 (1 − d1 ) − p ≥ 0. The inequality follows because with such bids, player 2 prefers to be the ”winner” and get d2 and pay p + θ2 (d2 + d1 − 1). Similarly, player 1 bids β1 < β2 and a d1 such that v1 (1 − d2 ) − p ≥ v1 (d1 ) − θ1 (d1 + d2 − 1) − p ≥ 0. And again this inequality follows because player 1 prefers to ”lose” and get 1 − d2 and pay only the reserve price. The two above yield that d1 ≤ 1 − p/θ2 and d2 ≤ 1 − p/θ1 . Thus, d2 cannot be arbitrarily close to 1 and clearly, the worst relative efficiency of any Nash equilibria has now improved. This idea extends to general networks. However, unless the auctioneer has some a priory information about user valuation functions (such as a distribution on user types), it cannot be guaranteed that reserve pricing will not eliminate the efficient Nash equilibrium as well.

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3) A Sufficient Characterization of Efficient Nash Equilibria: We will now provide a sufficient condition for a Nash equilibrium to be efficient. Note that the dual of the linear program (6) is given by min{

X

λl Cl +

X

µi di : νi + µi ≥ βi ,

i

l

X

λl ≥ νi ,

l∈r

∀r ∈ Ri , ∀i, λl , νi , µi ≥ 0}.

(9)

Let λ∗l , νi∗ , µ∗i denote a solution of the above with Nash equilibrium b. Theorem 2: Consider a Nash equilibrium b of the NSP game with µ∗i > 0 for all i, then the corresponding allocation is efficient. The above result implies that if at any Nash equilibrium, the allocation is such that x∗i = di for all i, then it must be efficient. IV. T HE NSP M ECHANISM

FOR

A RBITRARY B UNDLES

We now consider a slightly different setting. There are still L divisible goods, L = {1, · · · , L}, with Cl units of good l being available. And there are n buyers. But now each buyer wants an arbitrary bundle, i.e., buyer i wants xi = (xi1 , · · · , xiL ). Its valuation function now is vi (xi1 , · · · , xiL ) which depends on amounts of various goods obtained. We still assume that these functions are nice, in the sense that they are strictly increasing, strictly concave and twice differentiable in each argument. We will call S(x1 , · · · , xn ) =

n X

vi (xi )

i=1

as the social welfare function, which is a strictly increasing concave function. Our system objective then is to determine an allocation x∗ that satisfies max

S(x1 , · · · , xn ) P ∀l, i xil ≤ Cl , xil ≥ 0,

(10)

∀i, l.

We will call such an allocation efficient.

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As before, it is a convex optimization problem and thus, a solution exists and is unique. Let λl be the Lagrange multipliers corresponding to the capacity constraint. Then, the optimal solution is characterized by the following set of conditions   ∂vi ∗ − λl x∗il = 0, ∂xil   ∂vi ∗ ≤ 0, − λl ∂xil ! X Cl − x∗il λ∗l = 0,

∀i, l

(11)

∀i, l ∀l

i

! Cl −

X

x∗il

≥ 0,

∀l

i

λ∗l , x∗il ≥ 0, ∀i, l. The buyers specify bids b1 , · · · , bn where bi = (βi , di ), where βi = (βi1 , · · · , βiL ), di = (di1 , · · · , diL ) which specifies the maximum per unit price βil that i is willing to pay for good l and demands up to dil units of it. The auctioneer then determines an allocation x˜ = (˜ x1 , · · · , x˜n ) as a solution of the following optimization problem: P

max s.t.

i,l

P

il

βil xil

xil ≤ Cl ,

0 ≤ xil ≤ dil ,

(12) ∀i, ∀i, l.

Let x˜−i denote the solution of the above with di = 0. Then, the payment to be made by buyer i is Pi (bi , b−i ) =

X

βj (˜ x−i ˜j ). j −x

(13)

j6=i

This defines the NSP mechanism for arbitrary bundles. As before, the payoff of buyer i is ui (bi , b−i ) = vi (˜ xi (b)) − Pi (b). The strategy space of buyer i is Bi = [0, ∞)L × ×l [0, Cl ]. The Nash equilibrium is then defined as before.

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We can now show existence of a Nash equilibrium in the corresponding resource allocation game by construction. Theorem 3: There exists a Nash Equilibrium b∗ of the NSP mechanism whose corresponding allocation x∗ is efficient (i.e., η(x∗ ) = 1). The proof is in the appendix. V. T HE N ETWORK S ECOND -P RICE D OUBLE -S IDED M ECHANISM Consider L divisible goods, L = {1, · · · , L}, with Cl units of good l being available. Let Γ be the power set of L. Let there be n buyers. Buyer i wants a bundle of goods r ∈ Ri ⊆ Γ and wants the same quantity (or flow) xi of all goods in this bundle r. Let there be m ≥ L sellers, seller j sells only one good lj and there can be more than one seller selling the same good. We will assume that each buyer has valuation function vi (x) which is strictly increasing, strictly concave and differentiable. And each seller has cost cj (y) which is strictly increasing, convex and differentiable. Note that this also includes the case where the costs are linear. The buyers specify R1 , · · · , Rn ⊆ Γ (with r ∈ Ri a subset of L) and corresponding bids b1 , · · · , bn . The bid bi = (βi , di ) specifies the maximum per unit price βi that i is willing to pay and demands up to di units of the bundle Ri . Denote d = (d1 , · · · , dn )0 . Seller j specifies the good lj , an ask-bid aj = (αj , sj ) where αj is the minimum per unit price that j is willing to accept and can supply up to sj units of the good lj . Denote s = (s1 , · · · , sm )0 . The auctioneer then determines an allocation (˜ x, y˜) as a solution of the following optimization problem: max

X i

βi xi −

X

αj yj

(14)

j

Az ≤ y, Hz = x, x ≤ d, y ≤ s, x, y, z ≥ 0. Let (˜ x−i , y˜−i ) denote the solution to the above with di = 0 and (¯ x−j , y¯−j ) denote the solution to the above with sj = 0. December 26, 2007

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Then, the money transfer (the payment) to be made by buyer i is X X T˜i (bi , b−i , a) = βk (˜ x−i − x ˜ ) − αj (˜ yj−i − y˜j ). k k

14

(15)

j

k6=i

and the money transfer to be made by seller j (negative would means transfer to the seller) X X T¯j (b, aj , a−j ) = βi (¯ x−j ˜i ) − αk (¯ yk−j − y˜k ). (16) i −x i

k6=j

Recall that these transfer are the “externality” that the agents impose on the others through their participation. The payoff of buyer i is u˜i (bi , b−i , a) = vi (˜ xi (b, a)) − T˜i (b, a), and the payoff of seller j is u¯j (b, aj , a−j ) = −T¯j (b, a) − cj (˜ yj (b, a)). We will say an allocation (x∗∗ , y ∗∗ ) is efficient if it is a solution of the following optimization problem max

X i

vi (xi ) −

X

cj (yj )

(17)

j

Az ≤ y,

(18)

Hz = x,

(19)

x, y, z ≥ 0.

(20)

Such an allocation is necessarily Pareto-efficient since no player can unilaterally improve his payoff without making another player worse off. The strategy space of the buyer i is Bi = [0, ∞) × [0, ∞). The strategy space of seller j is Aj = [0, ∞) × [0, ∞). A Nash equilibrium for this game is defined as before, and we say it is efficient if the corresponding allocation is efficient. We can show the existence of a Nash equilibrium in the double-sided mechanism by construction. Theorem 4: There exists an efficient Nash Equilibrium (x∗ , y ∗ ) in the NSP double-sided mechanism.

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VI. C ONCLUSIONS AND F URTHER W ORK We have proposed a mechanism for allocation of multiple divisible goods such as bandwidth in a communication network. The mechanism is VCG-like and the players are only asked to report two numbers: a price per unit, and the maximum quantity demanded, as opposed to the VCG mechanism which requires the full valuation function. Our mechanism is a generalization of that presented in [14] to the network case. We show the existence of a Nash equilibrium where the allocation is efficient. This immediately implies the existence of an ε-Nash equilibrium (where each player, given strategies of all other players, chooses a response which is within ε of the best response) which is ε-efficient (i.e., an allocation which is within κε of the social welfare maximizing allocation, where κ is a constant). However, not all Nash equilibria are efficient as we show through an example. A distributed, computationally efficient algorithm that yields an ε-efficient ε-Nash equilibrium can be obtained as a generalization of the algorithm presented in [21]. We also present a double-sided mechanism which has a Nash equilibrium with efficient allocation. Our work is also related to [10]. They present a limited communication VCG-like mechanism that yields an efficient Nash equilibrium and gives conditions under which all equilibria are efficient, some of which are restrictive. Further, while they require the revealed utility functions to be differentiable for every parameter, our revealed utility functions are not differentiable and hence is not a particular case of their mechanism. Further, experimental work in electricity markets have shown that mechanisms which express both quantity and per-unit price, such as in our mechanism, work better than one-dimensional bid mechanisms as proposed in [10] (see [4] for a discussion).

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A PPENDIX P ROOFS OF T HEOREMS Proof: (Theorem 1) Let x∗∗ be an efficient allocation. Then, there exist ν1 , · · · , νn ≥ 0, λ1 , · · · , λL ≥ P ∗∗ 0 such that vi0 (x∗∗ i ) ≤ νi = l∈r:r∈Ri λl , ∀r ∈ Ri , ∀i with equality if xi > 0 (a strict inequality ∗∗ 0 ∗∗ is is possible with x∗∗ i = 0). Consider the strategy profile di = xi and βi = vi (di ). Note that x

an auction outcome with λ’s and ν’s as above, and µi = 0, ∀i, i.e., these solve (8). This implies that ∀i such that x∗∗ i > 0 X

βi =

λl , ∀r ∈ Ri .

(21)

l∈r:r∈Ri

Consider a buyer i with x∗∗ i > 0. Given the bids b−i of the others as fixed, if buyer i changes ∗∗ his bid bi to decrease his allocation x∗∗ i by a small ∆ > 0 (a buyer i with xi = 0 cannot

decrease his allocation), then the allocation of all the other players does not change since all of them already receive the maximum quantity they ask for. From equation (7), we get that the payment of player i does not change. However, since vi is strictly increasing and concave, his ∗∗ utility reduces by vi (x∗∗ i ) − vi (xi − ∆). Thus, his net payoff actually reduces. 0 Now, consider a buyer i (with x∗∗ i ≥ 0) changes his bid to bi such that he increases his P allocation x∗∗ i by a small ∆ > 0. Denote the change on route r by ∆r so that ∆ = r∈Ri ∆r . 0

Let the resulting allocation be x ∗ . Denote Li = {l ∈ r : r ∈ Ri }, the set of links on which buyer i’s traffic can flow for some route r ∈ Ri . Let L−i = {l ∈ s : s ∈ Rj , j 6= i}, the set of links on which any other buyer j’s traffic can flow for some route s ∈ Rj . We first note that for l ∈ Li ∩ L−i , X X 0 0 ∗∗ ∗∗ λl · (zjs − zjs∗ ) = λl · (zir∗ − zir ). ∗∗ >0,l∈s (j,s):j6=i,s∈Rj ,zjs

(22)

r∈Ri :l∈r

This is because if λl > 0, then the capacity constraint on l is tight, and any increase in i’s share of bandwidth on l is at the expense of other buyers i, who also want l. If the capacity constraint on l is not tight, then λl = 0, and the above still holds. We further note that for l ∈ Li \L−i , i.e., the set of links on which no other buyer’s traffic can flow, if the capacity constraint is tight, then an increase in i’s flow cannot come from increases for r ∈ Ri such that l ∈ r. If the capacity constraint is not tight, then λl = 0. In other words, X 0 ∗∗ λl · (zir∗ − zir ) = 0, ∀l ∈ Li \L−i . (23) r∈Ri :l∈r December 26, 2007

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The change in buyer i’s payment (later denoted ∆Pi ) as he changes his bid to b0i to increase his allocation by ∆ is given by Pi (b0i , b−i ) − Pi (bi , b−i ) X 0∗ = (x∗∗ j − xj )βj j6=i:x∗∗ j >0

=

X

0

X

∗∗ − zjs∗ ) (zjs

∗∗ >0 j6=i s∈Rj :zjs

λl

l∈s

X

=

X

X

0

∗∗ (zjs − zjs∗ )λl

∗∗ >0 l∈s (j,s):j6=i,s∈Rj ,zjs

X

X

∗∗ >0 (j,s):j6=i,s∈Rj ,zjs

l∈s∩Li

≥ =

X

(25)

∗∗ ) (zir∗ − zir

(26)

r∈Ri :l∈r

X

λl ·

0

∗∗ ) (zir∗ − zir

(27)

r∈Ri :l∈r

l∈Li

=

0

∗∗ (zjs − zjs∗ )

0

X

λl ·

l∈Li ∩L−i

=

(24)

∗∗ >0,l∈s (j,s):j6=i,s∈Rj ,zjs

X X

X

λl ·

l∈Li ∩L−i

=

0

∗∗ (zjs − zjs∗ )λl

XX

0

∗∗ (zir∗ − zir ) · λl

r∈Ri l∈r

=

X

X 0 ∗∗ λl ) )·( (zir∗ − zir

r∈Ri

=

X

l∈r 0

∗∗ ) · νi (zir∗ − zir

r∈Ri

= ∆ · νi . The first two equalities follow by definition and by equation (21). The third follows merely by compactifying notation and writing a double sum over j and s as a single sum over (j, s). Inequality (24) follows because the sum over l has less terms than before. Equality (25) is arrived at by rearrangement of terms. Equality (26) follows from equality (22). Equality (27) follows from equality (23). The rest of equalities are simple. Now, since vi is strictly concave, ∗∗ vi (x∗∗ i + ∆) − vi (xi ) < νi · ∆ ≤ ∆Pi .

This holds even if x∗∗ i = 0. December 26, 2007

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Thus, given the bids b−i of all the other players, the best response of player i is to bid bi so that he obtains x∗∗ i . This implies that b = (b1 , · · · , bn ) is a Nash equilibrium and the corresponding allocation is efficient. Proof: (Theorem 2) It is sufficient to show that the ν1∗ , · · · , νn∗ ≥ 0 are such that x∗i ·(vi0 (x∗i )−νi∗ ) = 0. Thus, we only need to consider agents i with x∗i > 0. By assumption µ∗i > 0, we have x∗i = di . Suppose agent i changes his bid to d0i = di + ∆ to increase his allocation by (small enough) ∗ 0∗ ∆ > 0. Let x0∗ i denote the new allocation. Then, by complementary slackness, xi ≥ xi .

Now, from sensitivity analysis of linear programs, we know that for small enough ∆ > 0, P 0∗ ∗ ∗ j βj (xj − xj ) = µi ∆. Thus, the change in payment of agent i is X ∗ ∗ ∆Pi = βj (x∗j − x0∗ j ) = (βi − µi )∆ = ∆ · νi . j6=i

The last equality follows from complementary slackness: x∗i (νi + µi − βi ) = 0. Since bi is a Nash equilibrium strategy, it must be that vi (x∗i + ∆) − vi (x∗i ) ≤ ∆ · νi∗ . Now, suppose buyer i wants to decrease his allocation by ∆. Suppose he changes his bid to ∗ d0i = di − ∆. Then, by complementary slackness x0∗ i < xi . By a similar argument as above, we

can see that the change in payment is νi ∆ and as ∆ → 0, we get that vi (x∗i ) − vi (x∗i − ∆) ≤ ∆ · νi∗ and we establish that vi0 (x∗i ) = νi∗ . Proof: (Theorem 3) Let x∗∗ be an efficient allocation. Then, there exist λ1 , · · · , λL ≥ 0 such ∂vi (x∗∗ i ) ≤ λl , ∂xil i (dil ) βil = ∂v∂x . Note il

that

∗∗ ∀i, l (with equality if x∗∗ il > 0). Consider the strategy profile dil = xil and

that this implies βil = λl , ∀i, l : x∗∗ il > 0.

Given the bids b−i of the others as fixed, suppose that buyer i changes his bid bi to change 0

0

∗ ∗ ∗∗ his allocation x∗∗ i to some other xi by some small ∆l = xil − xil . Without loss of generality,

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assume that there is some ¯l such that ∆l < 0 for l ≤ ¯l and ∆l ≥ 0 for l > ¯l. Define ¯ x˜il = x∗∗ il − |∆l |, for l ≤ l,

(28)

¯ = x∗∗ il , for l > l. ˜i , We consider two cases: (i) buyer i changes his bid to ˜bi to change his allocation from x∗∗ i to x 0

and (ii) buyer i changes his bid to b0i to change his allocation from x˜i to xi∗ . ˜i . Then, So, first consider that the buyer changes his bid to change his allocation from x∗∗ i to x buyer i now gets |∆l | less for goods l ≤ ¯l and same as before for the other goods. The allocation of all the other players does not change since all of them already receive the maximum quantity they ask for. From equation (13), we get then that the payment of player i does not change. However, since vi is strictly increasing and concave in each argument, his valuation strictly reduces by vi (x∗∗ xi ). Thus, i ) − vi (˜ ˜ vi (˜ xi ) − vi (x∗∗ i ) < Pi (bi , b−i ) − Pi (bi , b−i ) = 0.

(29)

Now, suppose buyer i changes his bid from ˜bi to b0i such that he changes his allocation from 0 x˜i to x ∗ . Note that now his allocation changes by ∆l ≥ 0 for l > ¯l. Then, the change in his i

payment is given by Pi (b0i , b−i ) − Pi (˜bi , b−i ) =

X X

0

βjl (˜ xjl − xjl∗ )

j6=i l>¯ l:˜ xjl >0

=

X

=

X

l>¯ l

λl

X

0

(˜ xjl − xjl∗ )

j6=i:˜ xjl >0

λl · ∆l ,

l>¯ l

where the last equality follows since the total change in allocation of all other players on an item l > ¯l is ∆l , which is how much more buyer i gets of l. Now, vi is strictly concave in each argument. Thus, 0

vi (xi∗ ) − vi (˜ xi ) <

X

λl · ∆l = Pi (b0i , b−i ) − Pi (˜bi , b−i ).

(30)

l>¯ l

From (29) and (30), we get that 0

0 vi (xi∗ ) − vi (x∗∗ i ) < Pi (bi , b−i ) − Pi (bi , b−i ),

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which implies that given the bids b−i of all the other players, the best response of player i is to bid bi so that he obtains x∗∗ i . Thus, b = (b1 , · · · , bn ) is a Nash equilibrium and the corresponding allocation is efficient. Proof: (Theorem 4) Let (x∗∗ , y ∗∗ ) be an efficient allocation. Then, there exist ν1 , · · · , νn ≥ 0 P ∗∗ 0 and λ1 , · · · , λL ≥ 0 such that vi0 (x∗∗ i ) ≤ νi = l∈r:r∈Ri λl , ∀i and cj (yj ) ≥ λlj , ∀j. Consider 0 ∗∗ 0 the strategy profile di = x∗∗ i , βi = vi (di ), sj = yj and αj = cj (sj ). Note that this implies X ∗∗ βi = (31) λl , ∀i : x∗∗ i > 0 and αj = λlj , ∀j : yj > 0. l∈r:r∈Ri

Consider a buyer i with x∗∗ i > 0. Given the bids (b−i , a) of the others as fixed, if buyer i changes his bid bi to decrease his allocation x∗∗ i by a small ∆ > 0, then note that the allocation of all the other buyers does not change but some sellers on links Li = {l ∈ r, r ∈ Ri } sell less. From equation (15), we get the change in payment of buyer i (later denoted ∆T˜i ) is T˜i (b0i , b−i , a) − T˜i (b, a) =

X

X

αj (yj0∗ − yj∗∗ )

l∈Li j:lj =l,yj∗∗ >0

=

X

X

λlj (yj0∗ − yj∗∗ )

l∈Li j:lj =l,yj∗∗ >0

X

=

X

0

λl (xir∗ − x∗∗ ir )

r∈Ri :x∗∗ ir >0 l∈r

= −

X

X

λl ∆r

r∈Ri :x∗∗ ir >0 l∈r

= −

X

∆r · νi = −∆ · νi .

r∈Ri :x∗∗ ir >0

The first equality is obtained just by taking differences of the two payments, the second equality by (31), and the third follows by an argument similar to (22). The last two are obvious. Since vi is strictly increasing and concave, we get that ∗∗ ˜ vi (x∗∗ i − ∆) − vi (xi ) < −∆ · νi = ∆Ti ,

(32)

i.e., his net payoff decreases. 0 Now, suppose buyer i with x∗∗ i ≥ 0, changes his bid to bi such that it increases his allocation

x∗∗ i by a small ∆ > 0 then note that while the allocation of all the sellers remains unchanged,

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0

that of some buyers decreases. Let the resulting allocation of buyers be x ∗ . Then, as in the proof of Theorem 1, T˜i (b0i , b−i , a) − T˜i (b, a) X X 0 ∗∗ = (zjs − zjs∗ )λl

(33)

∗∗ >0 l∈s (j,s):j6=i,s∈Rj ,zjs

X

≥

0

X

∗∗ (zjs − zjs∗ )λl

∗∗ >0 l∈s∩L (j,s):j6=i,s∈Rj ,zjs i

=

X l∈Li ∩L−i

=

X

=

0

∗∗ (zir∗ − zir )

r∈Ri :l∈r 0

X

λl ·

l∈Li

X

λl ·

∗∗ (zir∗ − zir )

r∈Ri :l∈r

XX

0

∗∗ (zir∗ − zir ) · λl

r∈Ri l∈r

= ∆ · νi . The reasoning is same as before. Further, since vi is strictly increasing and concave, we have ∗∗ ˜ vi (x∗∗ i + ∆) − vi (xi ) < νi · ∆ ≤ ∆Ti ,

(34)

From (32) and (34), we get that given the bids (b−i , a) of all the other players, the best response of a buyer i is to bid bi so that he obtains x∗∗ i . Now consider a seller j with yj∗∗ ≥ 0. Suppose a seller j changes his bid to increase yj∗∗ by a small ∆ > 0. This will not affect the allocation of the buyers but some sellers selling good l might get affected. Clearly, the net change in payment to the seller is −∆T¯j = ∆ · λl (follows easily from (31)) and since cj is strictly increasing and convex, we get that cj (yj∗∗ + ∆) − cj (yj∗∗ ) > λl · ∆ = −∆T¯j .

(35)

And if any seller j (selling l) with yj∗∗ > 0, were to change his bid to decrease his allocation by ∆ > 0 then the allocation to other sellers does not change but some buyers get ∆ less. Thus, the net change in seller j’s transfer is given by X X ∆T¯j =

0

∗∗ βi (zir − zir∗ )

(36)

∗∗ >0 i:l∈Li r∈Ri :l∈r,zir

≥

X

X

0

∗∗ λl (zir − zir∗ )

∗∗ >0 i:l∈Li r∈Ri :l∈r,zir

= λl · ∆. December 26, 2007

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And again by strict convexity of cj , cj (yj∗∗ − ∆) − cj (yj∗∗ ) > −λl ∆ ≥ −∆T¯j .

(37)

From, (35) and (37), we get that aj is a best response of seller j to bids of other players (b, a−j ). Thus, (b, a) is a Nash equilibrium. Moreover, the corresponding allocation is efficient.

R EFERENCES [1] M. B ITSAKI , G. S TAMOULIS AND C. C OURCOUBETIS, “A new strategy for bidding in the network-wide progressive second price auction for bandwidth”, Proc. CoNEXT, 2005. [2] S. B OYD AND L. VANDENBERGHE, Convex Optimization, Cambridge University Press,2004 [3] E. C LARKE “Multipart pricing of public goods”, Public Choice, 2:19-33, 1971. [4] W. E LMAGHRABY AND S. O REN, “The efficiency of multi-unit electricity auctions”, The Energy Journal, 20(4):89-116, 1999. [5] D. F UDENBERG AND J. T IROLE, Chapter 8, Game Theory, MIT Press, 1991. [6] T. G ROVES, “Incentives in teams”, Econometrica, 41:617-631, 1973. [7] B.H AJEK AND S.YANG, “Strategic buyers in a sum-bid game for flat networks”, manuscript, 2004. [8] R. JAIN , A. D IMAKIS AND J. WALRAND, ”Mechanisms for efficient allocation in divisible capacity networks”, Proc. Control and Decision Conference (CDC), December 2006. [9] R.J OHARI AND J.T SITSIKLIS, “Efficiency loss in a network resource allocation game”, Mathematics of Operations Research, 2004. [10] R.J OHARI AND J.T SITSIKLIS, “Efficiency of scaler parameterized mechanisms”, submitted to Operations Research, 2007. [11] F. K ELLY, “Charging and rate control for elastic traffic”, Euro. Trans. on Telecommunications, 8(1):33-37, 1997. [12] F. K ELLY, A. M AULLO AND D. TAN, “Rate control in communication networks: Shadow prices, proportional fairness and stability, J. Operational Research Soc., 49:237-252, 1998. [13] A. L AZAR AND N. S EMRET, “The progressive second price auction mechanism for network resource sharing”, Proc. Int. Symp. on Dynamic Games and Applications, 1997. [14] A. L AZAR AND N. S EMRET, “Design and analysis of the progressive second price auction for network bandwidth sharing”, Telecommunication Systems - Special issue on Network Economics, 1999. [15] S. L OW AND P. VARAIYA, “A new approach to service provisioning in ATM networks”, IEEE/ACM Trans. on Networking, 1(5):547-553, 1993. [16] J. M ACKIE -M ASON AND H. VARIAN, “Pricing congestible network resources”, IEEE J. Selected Areas in Comm., 13(7):1141-149, 1995. [17] R. M AHESWARAN AND T.BASAR, “Social welfare of selfish agents: Motivating efficiency for divisible resources”, Proc. CDC, 2004. [18] R. M AHESWARAN AND T.BASAR, “Nash equilibrium and decentralized negotiation in auctioning divisible resources”, J. Group Decision and Negotiation, 13(2), 2003.

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[19] P. M AILLE AND B. T UFFIN, “Multi-bid auctions for bandwidth allocation in communication networks”, Proc. IEEE INFOCOM 2004. [20] A. M AS -C OLELL , M. W HINSTON AND J. G REEN, Chapter 23, Microeconomic Theory, Oxford University Press, 1995. [21] N. S EMRET, Market Mechanisms for Network Resource Sharing, PhD Dissertation, Columbia University, 1999. [22] T. S TOENESCU AND J. L EDYARD, “A pricing mechanism which implements a network rate allocation problem in Nash equilibria”, unpublished, 2006 [23] W. V ICKREY, “Counterspeculation, auctions, and sealed tenders”, J. Finance, 16:8-37, 1961. [24] S.YANG AND B.H AJEK, “Revenue and stability of a mechanism for efficient allocation of a divisible good”, manuscript, 2005. [25] S. YANG AND AND B. H AJEK, “VCG-Kelly Mechanisms for allocation of divisible goods: Adapting VCG mechanisms to one-dimensional signals”, IEEE J. Selected Areas of Communications 25:1237-1243, 2007.

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