Multi-Bid Auctions for Bandwidth Allocation in Communication Networks

Patrick Maille ENST Bretagne 2. rue de la Chataigneraie - CS 17607 35576 Cesson Sevigne Cedex - FRANCE Bruno Tuffin IRISA-INRIA Campus de Beaulieu 35042 Rennes Cedex - FRANCE Multi-Bid Auctions for Bandwidth Allocation in Communication Networks In Proc. of IEEE INFOCOM, Mar 2004 Presented by: Ming-Lung Lu

Outline • Introduction • Multi-Bid Auctions: Allocation and Pricing Rules • Properties of The Multi-Bid Mechanism • Incentive Compatibility • “Quantile Uniform” Choice of Bids • Determination of the Number of Bids Admitted by the Auctioneer • Conclusions and Perspectives • Comments

Introduction • The demand for bandwidth in communication networks has been growing exponentially. • The available capacities are often insufficient. • Congestion occurs frequently. • The pricing scheme base on a fixed charge does not take into account the negative externalities among users. • A user may block another user. • Designing new allocation and pricing schemes appears as a solution for solving congestion problems. • Pricing network resources can have 2 different goals: • Reaching a maximum revenue for the network • Allocating efficiently the resource • We concentrate on the latter objective in this paper.

Introduction (cont’d) • In [7], Lazar and Semret introduce the Progressive Second Price (PSP) Mechanism. • An iterative auction scheme that allocates bandwidth on a single communication link among users in a set I. • Players submit two-dimensional bids si = (qi, pi) • qi is the quantity of resource asked by user (player) i • pi is the unit price that player i is willing to pay for qi • Users can modify their bid, knowing the bid submitted by the others, until an equilibrium is reached. • Users’ preferences are modeled by the difference between the valuation of that player i for the quantity ai and the price ci that he is charged: • Ui(s) = θi(ai(s)) – ci(s) • Lazar and Semret prove that if players are informed of the other players’ bids when they submit their own bids, • the bid profile s converges after a finite time to a Nash equilibrium that corresponds to an efficient allocation of the resource. • The main drawback is that: • the convergence phase can be quite long • it corresponds to a signaling burst (to send the necessary information to players) • which may present a non-negligible part of the available bandwidth. [7] A. A. Lazar and N. Semret, “Design and analysis of the progressive second price auction for network bandwidth sharing,” Telecommunication Systems – Special issue on Network Economics, 1999

Introduction (cont’d) • The mechanism was modified by Delenda, who propsed in [12] a one-shot scheme: • players are asked to submit their demand function, • and the auctioneer directly computes the allocations and prices to pay without any convergence phase. • It is the continuous version of the Generalized Vickrey Auction. • It is a direct revelation auction mechanism, meaning that players have to give their whole valuation function in their bid. • However, communicating a general function is not feasible in practice. • Delenda suggests that • only a finite number of demand functions be proposed • players choose among them • Nevertheless, this scheme suppose that the auctioneer has a idea of what the demand functions of users could be. [12] A. Delenda, “Mecanismesd’encheres pour le partage de ressources telecom,” France Telecom R&D, Tech. Rep. 7831, 2002.

Introduction (cont’d) • In this paper, we suggest an intermediate mechanism, which is still one-shot, but which does not suppose any knowledge about the demand functions. • We allow players to submit several two-dimensional bids (qi, pi). • This mechanism will be called multi-bid auction scheme.

Multi-Bid Auctions: Allocation and Pricing Rules • Let us consider a communication link with capacity Q. • We assume that this resource is infinitely divisible. • When a player i enters the game, he submits a set of Mi two-dimensional bids si = {si1, … siMi} • where sim = (qim, pim) • We assume the bids are sorted: pi1 <= pi2 <= … <= piMi. • Let S denotes the set of multi-bids that a player can submit:

Reserve price p0 • Our model allows the auctioneer to fix a unit price p0 >= 0 under which she prefers not to sell the resource. • This is equivalent to considering that the auctioneer may use the resource if it is not sold, with a valuation function θ0(q) = p0q. • In the following, the auctioneer will be denoted player 0. • And p0 will be called the reserve price. • We suppose that this reserve price is known by all players. • We thus assume that a bid s0 = (q0, p0), with q0 > Q is introduced. • Therefore, the set of bids that the auctioneer may submit is:

Pseudo-demand function, pseudo-market clearing price • In this sub-section, we provide some definitions that will be helpful to understand the behavior of the mechanism. • Definition 1: • A player i∈ I is said to submit a truthful multi-bid si∈ S if si = φ, or if • We write SiT the set of truthful multi-bids that can be submit by player i. • We also denote • the set of truthful multi-bids for which all prices are above the reserve price.

Pseudo-demand function, pseudo-market clearing price (cont’d) • Definition 2: • We define the demand function of player i as • the function di(p) = (θ’i)-1(p) if 0 < p <= θ’i(0) • And 0 otherwise. • di(p) is the quantity player i would buy if the resource were sold at the unit price p, in order to maximize his utility.

Pseudo-demand function, pseudo-market clearing price (cont’d) • Definition 3: • Consider a player i∈ I U {0} having submitted a multi-bid si∈ S. • We call pseudo-demand function of i associated with si the function , defined by

Pseudo-demand function, pseudo-market clearing price (cont’d) • Definition 4: • Consider a player i∈ I U {0} and si∈ S a multi-bid submitted by i. • We call pseudo-marginal valuation function of i, associated with si, the function , defined by

Pseudo-demand function, pseudo-market clearing price (cont’d) • We now derive a property that the pseudo-demand and pseudo-marginal valuation functions are smaller than their “real” counterparts: • Lemma 1: • If player i∈ I submits a truthful multi-bid si, then • Fig. 1 illustrate this result

Pseudo-demand function, pseudo-market clearing price (cont’d) • Definition 5: • Consider a set of players i∈ I, each submitting a multi-bid si∈ S. • We call aggregated pseudo-demand function associated with the profile the function defined by • When the objective of the allocation problem is to maximized the efficiency , it can be proved that the optimal allocation is such that , ai = di(u), where u is the market clearing price, i.e., the unique price such that • if • p0 otherwise

Pseudo-demand function, pseudo-market clearing price (cont’d) • Note that the efficiency measure corresponds to the usual social welfare criterion : • Here the auctioneer cannot compute the market clearing price, for she does not know the aggregated demand function. • Nevertheless, she can estimate the clearing price thanks to the aggregated pseudo-demand.

Pseudo-demand function, pseudo-market clearing price (cont’d) • Definition 6: • Consider a multi-bid profile • Denoting by the aggregated pseudo-demand function associated with this profile, we define the pseudo-market clearing price by • Such a always exists since • Moreover , which implies that • Fig. 2 shows an example of an aggregated pseudo-demand function and a pseudo-market clearing price.

Allocation rule • For every function f: R -> R and all x ∈ R, we define • when this limit exists. • If player i submits the multi-bid si then he receives a quantity ai(si, s-i), with • Each player receives the quantity he asks at the lowest price for which supply excesses pseudo-demand. • If all the resource is not allocated yet, the surplus is shared among players who submitted a bid with price . • This share is done with weights proportional.

Pricing rule • Each player is charged a total price ci(s), where • The intuition behind this pricing rule is an exclusion compensation principle, which lies behind all second-price mechanisms: • player i pays so as to cover the “social opportunity cost”, the loss of utility he imposes on all other users by his presence.

Computational considerations • For a given bid profile, PSP allocations and prices can be computed with complexity • For our model: • The computation of the aggregated pseudo-demand function needs the bids to be sorted, which can be done in time • The computation of the pseudo-market clearing price can be performed in time • All allocations can be calculated with total complexity • To calculate charges, the computation of allocations must be done for all profiles s-i, which gives a complexity • Once all allocations ai(s-j) are calculated, a price ci can be computed using (13) with a complexity less than

Computational considerations (cont’d) • Consequently, the total complexity is • Question: (sorting) O( ) <= ? • Answer: possibly because the bids are assumed to be given in the correct order. • If all players submit the same number of bids, then the total complexity is • Thus both methods have the same order • However, PSP has to compute allocations and prices several times. • We believe that the gain in signalization overhead is worth the cost in computational time.

Properties of the Multi-Bid Mechanism • In this section, we establish some basic properties of the multi-bid mechanism, showing its interest. • Property 3: • All the resource is allocated. • Property 4: • Player i‘s allocation is the difference between • what other players would have obtained if player i was notpart of the game and • what they actually obtain. • Formally,

Properties of the Multi-Bid Mechanism (cont’d) • Property 5: • A player increases his allocation by declaring a higher pseudo-demand function. • Property 6: • When a player i leaves the game, the allocations of all other players in the game increase. • Property 7: • If a player declares a pseudo-demand function that is higher than the pseudo-demand function of another player, then he obtains more bandwidth. • Property 8: • The reserve price p0 that the auctioneer declares in her bid ensures her that the resource is sold at a unit price higher than p0. • Property 9: • The seller’s revenue is always greater with all players than when a player is excluded from the game.

Properties of the Multi-Bid Mechanism (cont’d) • A mechanism is said to be individually rational if • no player can be worse off from participating in the auction than if he had declined to participate. • Property 10: • (individual rationality) • Formally,

Incentive Compatibility • In this section, we prove that a player cannot do much better than simply reveal his true valuation. • Proposition 1: • If a player i submits a truthful multi-bid si≠ φ, then every other multi-bid (truthful or not) necessarily corresponds to an increase of utility that is less than • Formally,

Incentive Compatibility (cont’d) • Proposition 1 is then extended to Proposition 2

Incentive Compatibility (cont’d) • Thus the incentive compatibility in this model is in the sense that: • The utility from multi-bid other than the truthful multi-bid is upper bounded by Ci. • Thus submitting a truthful multi-bid is called a Ci-best action.

“Quantile Uniform” Choice of Bids • It is reasonable to assume that each user i intends to ensure a utility that is as close as possible to the maximum. • For sake of simplicity, we assume that players have no idea of what the pseudo-market price will be, except that it will not be below p0. • The simplest way to choose a multi-bid that would be almost optimum, whatever the multi-bid profile is, is to minimize the quantity Ci of Proposition 2. • Nevertheless, if player i is allowed to submit as many bids as he wants, • he will give a number Mi of bids as large as possible, in order to make Ci tend to zero.

“Quantile Uniform” Choice of Bids (cont’d) • We therefore focus on the situation where the number of bids Mi is determined. • Then for a fixed Mi, the multi-bid minimizing Ci is such that • i.e., all the shaded areas are equal. • We call this quantile uniform.

Determination of the Number of Bids Admitted by the Auctioneer • In this section, we want to discuss the determination of the number of bids admitted by the auctioneer. • Increasing the value of M increases • the signaling over head • the memory storage • the complexity of all underlying allocation and price computations. • We introduce a cost function C(M, I) that models these negative effects. • Auctioneer’s benefit is then

Determination of the Number of Bids Admitted by the Auctioneer (cont’d) • We denote T the set of possible player types, characterizing the valuation function. • We model the auctioneer’s beliefs about the number of players of each type by PT on NT. • Then the expected revenue is given by • And the expected cost is

Determination of the Number of Bids Admitted by the Auctioneer (cont’d) • Assumption 3: • The expected cost is non-decreasing and tends to infinity when M tends to infinity. • The following result gives an idea on how the auctioneer may choose M: • Proposition 4: • If the marginal valuation functions are uniformly bounded by a vale pmax (that is, ), • then under Assumption 3 there exists a finite M that maximizes the expected net benefit of the seller, i.e. that maximizes • This section only show the existence of the finite M that maximizes expected net benefit, • but not how to find that M

Conclusions and Perspectives • We have designed and studied a one-shot auction-based mechanism for sharing and arbitrarily divisible resource. • With respective to the progressive second price (PSP) auction, our mechanism saves a lot of signaling overhead. • We have proved that our rule incites players to submit truthful bids. (Ci-best action) • “Quantile uniform” shows how does a bidder choose his multi-bid give the number of bids allowed. • Finally, we have given some hints to understand how the number of bids can be chosen.

Comments • The incentive compatibility in this paper makes me uncomfortable. • Is Ci-best action really reasonable? • The determination of the number of bids seems to be incomplete. • The method to find the number is expected.

Multi-Bid Auctions for Bandwidth Allocation in Communication Networks