Download Multi-Bid Auctions for Bandwidth Allocation in

Patrick Maille
Bruno Tuffin
ENST Bretagne
2. rue de la Chataigneraie - CS 17607
35576 Cesson Sevigne Cedex - FRANCE
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
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Presented by: Ming-Lung Lu
Outline
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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.
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Congestion occurs frequently.
The pricing scheme base on a fixed charge does not take into
account the negative externalities among users.
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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:
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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.
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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)
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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:
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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,
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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:
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the convergence phase can be quite long
it corresponds to a signaling burst (to send the necessary information to players)
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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:
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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
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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.
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[12] A. Delenda, “Mecanismes d’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.
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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}
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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:
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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:
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A player i ∈ I is said to submit a truthful multi-bid si ∈ S if si =
φ, or if
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We write SiT the set of truthful multi-bids that can be submit
by player i.
We also denote
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the set of truthful multi-bids for which all prices are above the
reserve price.
Pseudo-demand function, pseudo-market
clearing price (cont’d)
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Definition 2:
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We define the demand function of player i as
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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)
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Definition 3:
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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)
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Definition 4:
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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)
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We now derive a property that the pseudo-demand and
pseudo-marginal valuation functions are smaller than their
“real” counterparts:
Lemma 1:
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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)
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Definition 5:
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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
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p0 otherwise
Pseudo-demand function, pseudo-market
clearing price (cont’d)
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Note that the efficiency measure corresponds to the
usual social welfare criterion
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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)
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Definition 6:
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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
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For every function f: R -> R and all x ∈ R, we define
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when this limit exists.
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If player i submits the multi-bid si then he receives a quantity
ai(si, s-i), with
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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
.
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This share is done with weights proportional.
Pricing rule
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Each player is charged a total price ci(s), where
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The intuition behind this pricing rule is an exclusion
compensation principle, which lies behind all second-price
mechanisms:
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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
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For a given bid profile, PSP allocations and prices can be
computed with complexity
For our model:
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The computation of the aggregated pseudo-demand function
needs the bids to be sorted, which can be done in time
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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)
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Consequently, the total complexity is
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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
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In this section, we establish some basic properties of the
multi-bid mechanism, showing its interest.
Property 3:
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All the resource is allocated.
Property 4:
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Player i‘s allocation is the difference between
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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)
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Property 5:
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Property 6:
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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:
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When a player i leaves the game, the allocations of all other players in
the game increase.
Property 7:
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A player increases his allocation by declaring a higher pseudo-demand
function.
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:
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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)
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A mechanism is said to be individually rational if
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no player can be worse off from participating in the auction
than if he had declined to participate.
Property 10:
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(individual rationality)
Formally,
Incentive Compatibility
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In this section, we prove that a player cannot do much
better than simply reveal his true valuation.
Proposition 1:
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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)
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Proposition 1 is then extended to Proposition 2
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Incentive Compatibility (cont’d)
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Thus the incentive compatibility in this model is in the
sense that:
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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
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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,
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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)
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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
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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
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In this section, we want to discuss the determination of
the number of bids admitted by the auctioneer.
Increasing the value of M increases
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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)
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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
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And the expected cost is
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Determination of the Number of Bids
Admitted by the Auctioneer (cont’d)
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Assumption 3:
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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:
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If the marginal valuation functions are uniformly bounded by a vale
pmax (that is,
),
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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,
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but not how to find that M
Conclusions and Perspectives
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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
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The incentive compatibility in this paper makes me
uncomfortable.
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Is Ci-best action really reasonable?
The determination of the number of bids seems to be
incomplete.
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The method to find the number is expected.