Buyer i wants a bundle of goods Ri â L and wants the same quantity x of all goods in his bundle. We will assume that each buyer has quasi-linear utility function ...
Mechanisms for Efficient Allocation in Divisible Capacity Networks Antonis Dimakis, Rahul Jain and Jean Walrand EECS Department University of California, Berkeley {dimakis,rjain,wlr}@eecs.berkeley.edu
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 first propose a single-sided VCG-type mechanism. However, instead of reporting types, 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. We show the existence of an efficient Nash equilibrium in the corresponding auction game of the mechanism. We show through an example that not all Nash equilibria are efficient but provide a distributed algorithm that yields the efficient one. Further, we provide a sufficient characterization of all efficient Nash equilibria. We then present a double-sided auction mechanism for multiple divisible goods, and show that there exists a Nash equilibrium of the auction game which yields the efficient allocation.
I. I NTRODUCTION We address two related problems of network resource allocation and exchange in this paper. The first problem is allocating multiple divisible resources among strategic agents. Let there be L divisible goods available in quantities C1 , · · · , CL . Let there be n agents, each of whom wants a bundle of goods, say Ri for agent i. The 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. All the goods belong to an agency 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. Thus, the agency would like to allocate the various goods among the agents to maximize the sum of utility derived by all the agents (there could be other objectives as well). However, user utilities are unknown to the agency. Thus, it must elicit some information from the agents so that it can determine the optimal allocation. This can be done through an auction mechanism wherein each agent is asked to reveal 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. The second problem is of designing a multilateral trading environment. Here, there are many buyers and many sellers. Each buyer wants the same capacity on a bundle of goods but each seller sells only one type of good. We 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. The first problem has received a lot of attention in the literature, [5], [9], [10] being some key papers. Kelly [5], [6] showed that when agents in a network act as price-takers (i.e., do not act strategically), the resource allocation problem can be solved efficiently in a distributed manner. Inspired by this work, a proportional allocation (PA) mechanism for divisible goods was proposed [12]. The case when users are strategic and price-anticipating was considered in [3]. In the case of a single divisible good, they showed that this can result in an efficiency loss of 25%. A similar result is obtained for the network case, when players desire multiple goods (such as capacity on links along a route) when an “itemizedbid” PA mechanism is considered. In the “itemized-bid” PA mechanism, the player specifies a bid on each link in its route. This however is not realistic for large networks. Thus a sum-bid PA mechanism was proposed in [2]. It was however shown that in this case, the efficiency loss of the proportional allocation mechanism can be 100%. A generalized class of proportional allocation (ESPA) mechanisms was first introduced in [11], and further analyzed in [12], [15]. These have been shown to be efficient for allocation of a single divisible good. Such ESPA mechanisms require one-dimensional bid-signals and have a unique Nash equilibrium at which the allocation is efficient. However, the mechanisms trade off dominant-strategy implementation for ease in implementation as compared to the VCG class of mechanisms. In [2], a similar generalization of the proportional allocation mechanism was proposed for a single divisible good. But its extension to multiple divisible goods has not been provided. This paper is more directly related to the work of Lazar and Semret [7], [8], [14]. They proposed a VCG-style auction mechanism for a single divisible good [8]. Attempts have been made to generalize this mechanism to multiple divisible goods so that it can be useful for network resource allocation problems [1], [7], [13]. The setting of [7] 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 [14], chapter 3, wherein sellers place ask bids to sell 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 [7], [14] 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 [13], 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 [4]. However, the results of [4] do not hold in the case where the routing matrix has full rank. The proposals in this paper are inspired by [8]. 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 utility function vˆ(x) = β ·max{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 other 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 dominantstrategy implementation of VCG mechanisms, i.e., truthful reporting of utility functions is not a dominant strategy equilibrium. Thus, each agent need not have knowledge of the utility functions of others, nor of the actions being taken by them. Each agent needs to know only his own utility function but the agents must in some way coordinate their actions so that they can reach a Nash equilibrium. Thus, they must have some information about the actions being taken by others. Without such information, it is difficult to envisage how a game with multiple Nash equilibria can be implemented in a real setting. Thus, as in [8], an iterative algorithm for computing a Nash equilibrium (preferably efficient) is highly desirable. We present such an algorithm and prove its convergence in a simple case. This can be generalized to the network case with multiple players.
II. T HE N ETWORK S ECOND -P RICE M ECHANISM Consider L divisible goods, L = {1, · · · , L}, with Cl units of good l being available. Let there be n buyers. Buyer i wants a bundle of goods Ri ⊆ L and wants the same quantity x of all goods in his bundle. We will assume that each buyer has quasi-linear utility function ui (x) = vi (x) − ωi with a strictly increasing, strictly concave and differentiable valuation function vi (x), and ωi is the payment made by buyer i. The buyers specify bundles of goods R1 , · · · , Rn 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 . The auctioneer then determines an allocation x ˜ = (˜ x1 , · · · , x ˜n ) as a solution of the following optimization problem: max s.t.
P
i βi xi x ≤ Cl , ∀ l ∈ L, i:l∈Ri i xi ∈ [0, di ], ∀i = 1, · · · , n.
(1)
P
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
(2)
j6=i
Recall that this is the “externality” that the buyer i imposes on the other player based on his revealed utility function vˆi (x) = βi max(x, di ). Now, the payoff of buyer i is ui (bi , b−i ) = vi (˜ xi (b)) − Pi (b). We will say an allocation x∗∗ is efficient if it is a solution of the following X X max{ vi (xi ) : xi ≤ Cl , ∀l ∈ L}. i
(3)
i:l∈Ri
Note that such an allocation will be Pareto-efficient. The strategy space of the buyer i is Bi = [0, ∞) × [0, C i ] where C i = minl∈Ri Cl A Nash equilibrium is a bid profile b∗ = (b∗1 , · · · , b∗n ) such that ui (b∗i , b∗−i ) ≥ ui (bi , b∗−i ),
∀bi ∈ Bi .
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 i
vi (x∗i )/
X
vi (x∗∗ ).
i
Note that this will lie in [0,1], where 1 will mean that full efficiency is achieved.
III. P ROPERTIES OF THE NSP M ECHANISM
B. Inefficient Nash Equilibria and Reserve Prices
A. Existence of an Efficient Nash Equilibrium We first show existence of a Nash equilibrium in the corresponding auction game by construction. Theorem 1: There exists an efficient Nash Equilibrium in the NSP auction game. Proof: Let x∗∗ be an efficient allocation. Then, there exist P λ1 , · · · , λL ≥ 0 such that vi0 (x∗∗ i )= l∈Ri λl , ∀i. Consider 0 the strategy profile di = x∗∗ i and βi = vi (di ). Note that this implies X λl , ∀i. βi = i:l∈Ri
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, 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 (2), we get that the payment of player i does not change. However, since vi is strictly increasing ∗∗ and concave, his valuation reduces by vi (x∗∗ i )−vi (xi −∆). Thus, his payoff actually reduces. Now, suppose player i changes his bid to b0i such that he increases his allocation x∗∗ i by a small ∆ > 0. Let the 0 resulting allocation be x ∗ . Then, the change in his payment (later denoted ∆Pi ) is given by X 0 ∗ Pi (b0i , b−i ) − Pi (bi , b−i ) = βj (x∗∗ j − xj )
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, player 2 bids β2 = θ1 > β1 = θ2 if there is a d2 such that v2 (d2 ) − θ2 (d1 + d2 − 1) − p ≥ v2 (1 − d1 ) − p ≥ 0. Similarly, player 1 bids β1 < β2 and a d1 such that v1 (1 − d2 ) − p ≥ v1 (d1 ) − θ1 (d1 + d2 − 1) − p ≥ 0.
j6=i
=
XX
0
∗ λl (x∗∗ j − xj )
j6=i l∈Rj
=
X
X
0
∗ λl (x∗∗ j − xj )
l∈Rj :j6=i j6=i
≥
X
λl · ∆,
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. C. A Sufficient Characterization of Efficient Nash Equilibria
(4)
l∈Ri
where the last inequality follows for the following reason: First note that due to player i changing his bid, his allocation increases by ∆, but the allocation of no other player can increase since each player j, j 6= i already gets dj . Further, as player i’s allocation increases by ∆, there must be at least one player on each link l ∈ Ri whose allocation decreases by ∆. However, this can happen on otherP links l ∈ / Ri as well, hence the inequality. But vi0 (x∗∗ i ) = l∈Ri λl and vi is strictly concave. Thus, X ∗∗ vi (x∗∗ λl · ∆ ≤ ∆Pi , i + ∆) − vi (xi ) < l∈Ri
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. Note that the above result implies the existence of an εefficient ε-Nash equilibrium, a result obtained in [8] for the special case of a single good.
We will now provide a sufficient condition for a Nash equilibrium to be efficient. For a Nash equilibrium b, let λl be the Lagrange multiplier corresponding to the capacity constraint for good l and µi be the Lagrange multiplier corresponding to the xi ≤ di demand constraint in the auction optimization (1). Note that the dual of the linear program (1) is given by X X X min{ λl Cl + µi di : λl +µi ≥ θi , ∀i, λl , µi ≥ 0}. l
i
l∈Ri
(5) Let λ∗l , 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. Proof: It is sufficient toPshow that the λ∗1 , · · · , λ∗L ≥ 0 are such that x∗i · (vi0 (x∗i ) − l∈Ri λ∗l ) = 0. Thus, we only need to consider agents i with x∗i > 0. By assumption µ∗i > 0, which implies that x∗i = di . Suppose agent i changes his bid to d0i = di + ∆ to increase his allocation by (small enough) ∆ > 0. Let x0∗ i denote the new allocation. Then, by ∗ complementary slackness, x0∗ i > xi .
Now, from sensitivity analysis P of linear programs, we ∗ ∗ know that for small enough ∆ > 0, j θj (x0∗ j −xj ) = µi ∆. Thus, the change in payment of agent i is X X ∗ ∆Pi = θj (x∗j − x0∗ λ∗l . j ) = (θi − µi )∆ = ∆ j6=i
l∈Ri
TheP last inequality follows from complementary slackness: x∗i ( l∈Ri λl + µi − θi ) = 0. Since bi is a Nash equilibrium strategy, it must be that X λ∗l . vi (x∗i + ∆) − vi (x∗i ) < ∆ l∈Ri
Now, suppose buyer i wants to decrease his allocation by ∆. Suppose he changes his bid to d0i = di − ∆. Then, by ∗ complementary slackness x0∗ as i < xi . By a similar argument P above, we can see that the change in payment is l∈Ri λl ∆ and as ∆ → 0, we get that X vi (x∗i ) − vi (x∗i − ∆) < ∆ λ∗l l∈Ri
and we establish that vi0 (x∗i ) =
P
l∈Ri
λ∗l .
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. D. Distributed Computation of an Efficient Nash Equilibrium We now provide a distributed algorithm that computes an efficient Nash equilibrium. We will provide the algorithm for the simple setting of one good and two agents. This can be generalized to multiple users and multiple goods Let (θ10 , x01 ) = (θ20 , x02 ) = (0, 0) be the initial bids. The algorithm works as follows: At each step only one player makes a bid. If at the nth step, for n ≥ 1, it is player i’s turn, he checks if vi0 (xn−1 ) > θ¯ın−1 . (Here ¯ı denotes the i other player). If yes, then he computes the new bid (θin , xni ) and the resulting allocation x¯nı , of the other user, according to: vi0 (xni ) = θ¯ın−1 x¯ı = 1 − xni θin = θ¯ın−1 + ,
(6) (7) (8)
for some fixed > 0, common to all agents. In the case where vi0 (xn−1 ) ≤ θ¯ın−1 , i
(9)
player i stops. Since player ¯ı cannot further increase his surplus, both players do not have any incentive to move and the algorithm terminates. If at step n, it is player i’s turn, and (9) does not hold, since (8) implies θin > θin−2 , we have vi0 (xni ) > vi0 (xn−2 ). i Strict convexity of valuations implies xni < xn−2 . Thus, i xni , xn−2 , xn−4 , . . . is increasing and xn−1 , xn−3 , . . . is dei i i i creasing, for i = 1, 2. In particular, the monotonicity of the latter implies that the algorithm will terminate since the left
hand side of (9) decreases, while it’s right hand side increases by a constant amount > 0, each time. We now show that if at step n player i decides to stop, then the final allocation (x1n−1 , xn−1 ) is arbitrarily close to 2 ∗∗ the efficient allocation (x∗∗ 1 , x2 ). The idea is to show that ), v¯ı0 (x¯n−1 ) are close to each the marginal valuations, vi0 (xn−1 ı i other, in which case the allocations must be close to the efficient ones as well, as we will see below. By the mean value theorem, v¯ı0 (x¯ın−1 ) − v¯ı0 (x¯n−3 ) = v¯ı00 (ψ)(x¯ın−1 − x¯ın−3 ) ı
(10)
for some ψ ∈ (x¯ın−3 , x¯ın−1 ). Using (6) for player ¯ı, and steps n − 1, n − 2, (10) yields xn−3 − xn−1 = x¯ın−1 − x¯n−3 = ı i i
2 v¯ı00 (ψ)
.
(11)
Now, the dual problem of the social welfare optimization problem gives the upper bound v1 (x∗1 ) + v2 (x∗2 ) ≤ v1 (x1 (λ)) + v2 (x2 (λ)) −λ(x1 (λ) + x2 (λ) − 1) , ∀λ ≥ 0, where (x1 (λ), x2 (λ)) are such that v10 (x1 (λ)) = v20 (x2 (λ)) = λ. (If this cannot hold for any λ, then x1 (λ) = x2 (λ) = ∞.) Now, set y¯ı = x¯n−1 and define yi such that vi0 (yi ) = ı n−1 θ¯ı . By the stopping condition (9), vi0 (yi ) ≥ vi0 (xin−1 ), so yi ≤ xin−2 . This and (7), yields yi + y¯ı ≤ 1. Hence, v1 (x∗1 ) + v2 (x∗2 ) − v¯ı (x¯n−1 ) − vi (xn−1 ) ı i ∗ ∗ ≤ v1 (x1 ) + v2 (x2 ) − v1 (y1 ) − v2 (y2 ) ≤ θ¯ın−1 (1 − y1 − y2 ) = θ¯ın−1 (xn−1 − yi ) i ≤ θ¯ın−1 (xn−1 − xn−3 ) i i 2 ≤ θ¯ın−1 00 , |u¯ı (ψ)| Where the last inequality follows from (11). this shows that provided maxx∈[0,1] vi00 (x) < 0, the inefficiency can be made arbitrarily small, provided one picks small enough . IV. T HE N ETWORK S ECOND -P RICE D OUBLE AUCTION M ECHANISM Consider L divisible goods, L = {1, · · · , L}. Let there be n buyers, buyer i wants a bundle of goods Ri and wants the same quantity x of all goods in his bundle. Let there be m sellers, seller j sells only one good Lj . 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. Buyers i specifies a bundle of goods Ri and corresponding bid bi = (βi , di ) which specifies the maximum per unit price βi that i is willing to pay, and demands up to di units of the bundle Ri . 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 . The auctioneer then determines an allocation (˜ x, y˜) as a solution of the following optimization problem: P P max αj yj (12) i βi xi − P P j s.t. i:l∈Ri xi ≤ j:l=Lj yj , ∀l,
his allocation x∗∗ by a small ∆ > 0, then note that the i allocation of all the other buyers does not change but some sellers on link l ∈ Ri sell less. From equation (14), we get the change in payment of buyer i (later denoted ∆T˜i ) is X X T˜i (b0i , b−i , a) − T˜i (b, a) = αj (yj0∗ − yj∗ )
xi ∈ [0, di ], ∀i, yj ∈ [0, sj ], ∀j.
l
= −∆
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. Then, the money transfer (the payment) to be made by buyer i is X X T˜i (bi , b−i , a) = βk (˜ x−i ˜k ) − αj (˜ yj−i − y˜j ). (13) k −x 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 ). (14) 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 X X X X max{ vi (xi ) − cj (yj ) : xi ≤ yj , ∀l}. i
j
i:l∈Ri
j:l=Lj
j:l=Lj
(15) 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 now show existence of a Nash equilibrium in the double auction game by construction. Theorem 3: There exists an efficient Nash Equilibrium in the NSP double auction game. Proof: Let (x∗∗ , y ∗∗ ) be an efficient allocation. Then, there P exist λ1 , · · · , λL ≥ 0 such that vi0 (x∗∗ ) = i l∈Ri λl , ∀i and c0j (yj∗∗ ) = λLj , ∀j. Consider the strategy profile di = x∗∗ i , βi = vi0 (di ), sj = yj∗∗ and αj = c0j (sj ). Note that this implies X βi = λl , ∀i and αj = λLj , ∀j. (16) i:l∈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
X
λl .
l∈Ri
The first equality is obtained just by taking differences of the two payments, and the second equality is obtained noting that the allocations of sellers change only for l ∈ Ri and the total change in allocation of all sellers of a good l is ∆. Since vi is strictly increasing and concave, we get that X ∗∗ vi (x∗∗ λl = −∆T˜i , (17) i ) − vi (xi − ∆) < ∆ l∈Ri
i.e., the net change in his payoff is negative. Now, suppose player i with x∗∗ i ≥ 0, changes his bid to 0 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, that of some buyers decreases. Let the resulting 0 allocation of buyers be x ∗ . Then, X 0 ∗ T¯i (b0i , b−i , a) − T¯i (b, a) = βk (x∗∗ k − xk ) k6=i
=
X
λl
l∈∪k6=i Rk
X
≥ ∆·
X
0
∗ (x∗∗ j − xj )
k6=i
λl ,
(18)
l∈Ri
where the last inequality follows because of the increase in the allocation to buyer i on goods Ri and no change in allocation to sellers (their supply constraints are already active) which must result in a ∆ decrease in allocation to at least one buyer on each l ∈ Ri . Further, vi is strictly increasing and concave. Thus, X ∗∗ vi (x∗∗ λl · ∆ ≤ ∆T¯i , (19) i + ∆) − vi (xi ) < l∈Ri
From (17) and (19), 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 transfer of the seller is ∆T¯j = −∆·λl and since cj is strictly increasing and convex, we get that cj (yj∗∗ + ∆) − cj (yj∗∗ ) ≥ λl · ∆ = −∆T¯j . And if any seller j 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’s transfer is XX ∗ λl (x0∗ ∆T¯j = − i − xi ) i
=
l∈Ri
X l∈∪i Ri
X
λl
(x∗i − x0∗ i )
i:l∈Ri
≥ λl · ∆.
(20)
And again by strict convexity of cj (yj∗∗ − ∆) − cj (yj∗∗ ) ≥ −λl ∆ ≥ −∆T¯j , which implies 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.
V. C ONCLUSIONS AND F URTHER W ORK We have proposed an auction mechanism for allocating 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 [8] 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 which is ε-efficient. However, not all Nash equilibria are efficient as we show through an example. But we present a distributed algorithm that yields an ε-efficient ε-Nash equilibrium. This algorithm is different from that presented in [14] which is difficult to generalize to the network case. We also present a doublesided mechanism which has a Nash equilibrium with efficient allocation. Our work is also related to [4]. 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. As part of further work, we would like to obtain a distributed algorithm for computing the efficient Nash equilibrium in the double-sided case. 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] B.H AJEK AND S.YANG, “Strategic buyers in a sum-bid game for flat networks”, manuscript, 2004. [3] R.J OHARI AND J.T SITSIKLIS, “Efficiency loss in a network resource allocation game”, Mathematics of Operations Research, 2004. [4] R.J OHARI AND J.T SITSIKLIS, “Communication requirements of VCGlike mechanisms in convex environments”, manuscript, 2005. [5] F. K ELLY, “Charging and rate control for elastic traffic”, Euro. Trans. on Telecommunications, 8(1):33-37, 1997.
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