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Multi-Bid Auctions for Bandwidth Allocation in Communication Networks

In Proc. of IEEE INFOCOM, Mar 2004

Patrick Maille ENST Bretagne2. rue de la Chataigneraie - CS 1760735576 Cesson Sevigne Cedex - FRANCE

Bruno TuffinIRISA-INRIACampus de Beaulieu35042 Rennes Cedex - FRANCE

Presented by: Ming-Lung Lu

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Outline1. Introduction2. Multi-Bid Auctions: Allocation and Pricing

Rules3. Properties of The Multi-Bid Mechanism4. Incentive Compatibility5. “Quantile Uniform” Choice of Bids6. Determination of the Number of Bids

Admitted by the Auctioneer7. Conclusions and Perspectives8. Comments

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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.

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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

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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, “Mecanismes d’encheres pour le partage de ressources telecom,” France Telecom R&D, Tech. Rep. 7831, 2002.

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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.

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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 = (qi

m, pim)

We assume the bids are sorted: pi1 <= pi

2 <= … <= pi

Mi. Let S denotes the set of multi-bids that a

player can submit:

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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:

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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.

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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.

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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

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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

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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

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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

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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.

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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.

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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.

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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.

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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

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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.

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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,

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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.

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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,

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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,

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

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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.

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“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.

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“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.

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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

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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

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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

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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.

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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.