1. Multi-Bid Auctions for Bandwidth Allocation in Communication Networks In Proc. of IEEE INFOCOM, Mar 2004 Patrick Maille ENST Bretagne 2. rue de la Chataigneraie - CS 17607 35576 Cesson Sevigne Cedex - FRANCE 1 Bruno Tuffin IRISA-INRIA Campus de Beaulieu 35042 Rennes Cedex - FRANCE Presented by: Ming-Lung Lu

2. Outline 2 1. Introduction 2. Multi-Bid Auctions: Allocation and Pricing Rules 3. Properties of The Multi-Bid Mechanism 4. Incentive Compatibility 5. “Quantile Uniform” Choice of Bids 6. Determination of the Number of Bids Admitted by the Auctioneer 7. Conclusions and Perspectives 8. Comments

3. Introduction 3  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.

4. Introduction (cont’d) 4  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 s i = (q i, p i )  q i is the quantity of resource asked by user (player) i  p i is the unit price that player i is willing to pay for q i  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 a i and the price ci that he is charged:  U i (s) = θ i (a i (s)) – c i (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

5. Introduction (cont’d) 5  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.

6. Introduction (cont’d) 6  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 (q i, p i ).  This mechanism will be called multi-bid auction scheme.

7. Multi-Bid Auctions: Allocation and Pricing Rules 7  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 M i two-dimensional bids s i = {s i 1, … s i Mi }  where s i m = (q i m, p i m )  We assume the bids are sorted: p i 1 <= p i 2 <= … <= p i Mi.  Let S denotes the set of multi-bids that a player can submit:

8. Reserve price p 0 8  Our model allows the auctioneer to fix a unit price p 0 >= 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 θ 0 (q) = p 0 q.  In the following, the auctioneer will be denoted player 0.  And p 0 will be called the reserve price.  We suppose that this reserve price is known by all players.  We thus assume that a bid s 0 = (q 0, p 0 ), with q 0 > Q is introduced.  Therefore, the set of bids that the auctioneer may submit is:

9. Pseudo-demand function, pseudo-market clearing price 9  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 s i ∈ S if s i = φ, or if  We write S i T 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.

10. Pseudo-demand function, pseudo-market clearing price (cont’d) 10  Definition 2:  We define the demand function of player i as  the function d i (p) = ( θ ’ i ) -1 (p) if 0 < p <= θ ’ i (0)  And 0 otherwise.  d i (p) is the quantity player i would buy if the resource were sold at the unit price p, in order to maximize his utility.

11. Pseudo-demand function, pseudo-market clearing price (cont’d) 11  Definition 3:  Consider a player i ∈ I U {0} having submitted a multi-bid s i ∈ S.  We call pseudo-demand function of i associated with si the function, defined by

12. Pseudo-demand function, pseudo-market clearing price (cont’d) 12  Definition 4:  Consider a player i ∈ I U {0} and s i ∈ S a multi-bid submitted by i.  We call pseudo-marginal valuation function of i, associated with si, the function, defined by

13. Pseudo-demand function, pseudo-market clearing price (cont’d) 13  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 s i, then  Fig. 1 illustrate this result

14. Pseudo-demand function, pseudo-market clearing price (cont’d) 14  Definition 5:  Consider a set of players i ∈ I, each submitting a multi-bid s i ∈ 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, a i = d i (u), where u is the market clearing price, i.e., the unique price such that  if  p 0 otherwise

15. Pseudo-demand function, pseudo-market clearing price (cont’d) 15  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.

16. Pseudo-demand function, pseudo-market clearing price (cont’d) 16  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.

17. Allocation rule 17  For every function f: R -> R and all x ∈ R, we define  when this limit exists.  If player i submits the multi-bid s i then he receives a quantity a i (s i, 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.

18. Pricing rule 18  Each player is charged a total price c i (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.

19. Computational considerations 19  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 a i (s -j ) are calculated, a price c i can be computed using (13) with a complexity less than

20. Computational considerations (cont’d) 20  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.

21. Properties of the Multi-Bid Mechanism 21  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,

22. Properties of the Multi-Bid Mechanism (cont’d) 22  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 p 0 that the auctioneer declares in her bid ensures her that the resource is sold at a unit price higher than p 0.  Property 9:  The seller’s revenue is always greater with all players than when a player is excluded from the game.

23. Properties of the Multi-Bid Mechanism (cont’d) 23  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,

24. Incentive Compatibility 24  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 s i ≠ φ, then every other multi-bid (truthful or not) necessarily corresponds to an increase of utility that is less than  Formally,

25. Incentive Compatibility (cont’d) 25  Proposition 1 is then extended to Proposition 2

26. Incentive Compatibility (cont’d) 26  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 C i.  Thus submitting a truthful multi-bid is called a C i -best action.

27. “Quantile Uniform” Choice of Bids 27  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 p 0.  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 M i of bids as large as possible, in order to make C i tend to zero.

28. “Quantile Uniform” Choice of Bids (cont’d) 28  We therefore focus on the situation where the number of bids M i is determined.  Then for a fixed M i, the multi-bid minimizing C i is such that  i.e., all the shaded areas are equal.  We call this quantile uniform.

29. Determination of the Number of Bids Admitted by the Auctioneer 29  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

30. Determination of the Number of Bids Admitted by the Auctioneer (cont’d) 30  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 P T on N T.  Then the expected revenue is given by  And the expected cost is

31. Determination of the Number of Bids Admitted by the Auctioneer (cont’d) 31  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 p max (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

32. Conclusions and Perspectives 32  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. (C i -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.

33. Comments 33  The incentive compatibility in this paper makes me uncomfortable.  Is C i -best action really reasonable?  The determination of the number of bids seems to be incomplete.  The method to find the number is expected.