0, then si = ∅ and d¯i (x) = max {q m : pm ≥ x} +
= =
i i 1≤m≤Mi m0 m0 qi with pi ≥ 0 di (pm i ) ≤ di (x)
0-7803-8356-7/04/$20.00 (C) 2004 IEEE
x
.
q q1i q2i
s2i p3i
di ( p)
d¯i ( p)
q3i 0
p
s1i
p1i
θ0i ( p) θ¯ 0i ( p)
p2i
s3i
p2i
s3i
Prices
m ∀m, 1 ≤ m ≤ Mi , pm i = θi (qi ).
.
Quantities
Definition 1: A player i ∈ I is said to submit a truthful multi-bid si ∈ S if si = ∅, or if
s1i
p1i
p3i
p
Prices
0
s2i
q3i
.
Quantities
q2i q1i
q .
Fig. 1. Demand and pseudo-demand functions (left), marginal valuation and pseudo-marginal valuation functions (right) for Mi = 3 and a truthful multi-bid.
where the non-increasingness of di is used. Relation (5) is then proved. Relation (6) is established exactly the same way by inverting the roles of prices and quantities. Fig. 1 illustrates this result. Definition 5: Consider a set of players i ∈ I, each submitting a multi-bid si ∈ S. We call agregated pseudo-demand function associated with the profile s = (si )i∈I∪{0} the function d¯ : R+ → R+ defined by ¯ = (7) d(p) d¯i (p). i∈I∪{0}
When the objective of the allocation problem is to maximize the efficiency i∈I∪{0} θi (ai ), it can be proved that, under Assumption 1, the optimal allocation is such that ∀i ∈ I, ai = is the market clearing di (u), where u price, i.e. the unique price such that i∈I di (u) = Q if i∈I di (p0 ) > Q and p0 otherwise [17]. Remark: Note that the efficiency measure that we use corresponds to the usual social welfare criterion i Ui (s) (see [4]): since the utility of the seller is U0 (s) = θ0 (a0 (s)) + i∈I ci (s), we have Ui (s) = U0 (s)+ Ui (s) i
i∈I
ci (s)+ (θi (ai (s))−ci (s)) = θ0 (a0 (s))+ =
i∈I
i∈I
θi (ai (s)).
i
Here the auctioneer cannot compute the market clearing price, for she does not know the agregated demand function. Nevertheless, she can estimate the clearing price thanks to the agregated pseudo-demand. Definition 6: Consider a multi-bid profile s = (si )i∈I∪{0} . Denoting by d¯ the agregated pseudo-demand function associated with this profile, we define the pseudo-market clearing price u ¯ by ¯ >Q . u ¯ = sup p : d(p) (8) ¯ Such a u ¯ always exists since d(0) ≥ d¯0 (0) = q0 > Q. i ¯ Moreover d(p) = 0 ∀p > maxi∈I∪0 (pM i ), which implies
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that u ¯ < +∞. Remark: As d¯ is a left-continuous stair-step function, the sup in (8) is actually a max. Thus we have ¯ u) > Q. d(¯
(9)
Fig. 2 shows an example of an agregated pseudo-demand function and a pseudo-market clearing price. C. Allocation rule Given all these definitions, we are now ready to describe the allocation rule. For every function f : R → R and all x ∈ R, we define f (x+ ) =
lim
z→x,z>x
f (z)
(10)
when this limit exists. We suggest that the resource be allocated the following way: if player i submits the multi-bid si (and thereby declares the associated functions d¯i and θ¯i ) then he receives a quantity ai (si , s−i ), with u) − d¯i (¯ u+ ) d¯i (¯ ¯ u+ )), (11) u+ ) + ¯ ai (si , s−i ) = d¯i (¯ ¯ u+ ) (Q − d(¯ d(¯ u) − d(¯ where d¯ is the agregated pseudo-demand function associated with the bid profile s. In other words: • each player receives the quantity he asks at the lowest price u ¯+ for which supply excesses pseudo-demand. • If all the resource is not allocated yet, the surplus Q − ¯ u+ ) is shared among players who submitted a bid with d(¯ price u ¯. This share is done with weights proportional u) − to the “hops” of the pseudo-demand functions d¯i (¯ u+ ). d¯i (¯ Remark: We know from (9) that the denominator in (11) ¯ u+ ) ≤ Q and d(¯ ¯ u) > Q. is always strictly positive, since d(¯ ¯ u+ ) Q−d(¯ Consequently 0 ≤ d(¯ ¯ u)−d(¯ ¯ u+ ) ≤ 1 and we have d¯i (¯ u+ ) ≤ ai (s) ≤ d¯i (¯ u).
(12)
Remark: As noticed by Tuffin in [8], the PSP allocation rule does not always allocate all the bandwith even if demand ¯i (¯ u)−d¯i (¯ u+ ) ¯ u+ )) exceeds supply. Here, the term dd(¯ ¯ u)−d(¯ ¯ u+ ) (Q − d(¯ ensures that all the resource is allocated (see Property 3). .
Quantities
q
d¯( p)
Q
0
u¯ Fig. 2.
Prices
The pseudo-market clearing price u ¯
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p .
D. Pricing rule Each player i ∈ I is charged a total price ci (s), where aj (s−i ) ci (si , s−i ) = (13) θ¯j (q)dq. j∈I∪{0},j=i
aj (s)
The intuition behind this pricing rule is an exclusioncompensation principle, which lies behind all second-price mechanisms [18]: player i pays so as to cover the “social opportunity cost”, that is to say the loss of utility he imposes on all other users by his presence.Actually, applying directly this principle would lead to ci = j∈I∪{0},j=i θj (aj (s−i )) −
a (s ) θj (aj (s)) = j∈I∪{0},j=i ajj(s)−i θj , as in [12]. However, in our case the auctioneer does not know the valuation functions θj nor the marginal valuation functions θj , j ∈ I. That is the reason why we use the pseudo-marginal valuation functions instead, that are computed thanks to submitted bids. Note here that we do not define c0 , since we consider that the auctioneer cannot buy resource to herself. We now give a lemma that will be used in the rest of the paper. Lemma 2: ∀i ∈ I ∪ {0}, ∀si ∈ S, ∀x, y ∈ R+ , i 0 if si = ∅ or pM ≤ x, i + m m d¯i (x ) = max {qi : pi > x} otherwise. 1 ≤ m ≤ Mi 0 if si = ∅ or qi1 ≤ y, m θ¯i (y + ) = : q > y} otherwise. max {pm i i 1 ≤ m ≤ Mi
Consequently we have ∀i ∈ I ∪ {0}, ∀si ∈ S, ∀x ∈ R+ , θ¯i (d¯i (x+ )+ ) ≤ x i si = ∅ ⇒ ∀x ∈ [0, pM θ¯i (d¯i (x)) ≥ x. i ], A proof of Lemma 2 is given in Appendix I.
(14) (15)
E. Computational considerations The change of PSP into a one-shot scheme has also consequences in terms of computational complexity: whereas for a given bid profile, PSP allocations and prices can be computed with complexity O(|I|2 ) where |I| is the number of players in I (see [4]), the multi-bid scheme may involve more calculations. We briefly study here the total complexity corresponding to a given multi-bid profile s. • The computation of the agregated pseudo-demand function which can be done in time to be sorted, needs thebids . Then the computation of the M log M i i i∈I i∈I pseudo-market clearing price u ¯ can be performed in time ¯ u+ ) Q−d(¯ ¯, the value of d(¯ ¯ u)−d(¯ ¯ u+ ) is computed i Mi . Given u only once, therefore all allocations can be calculated with a total complexity O(| i Mi |) (computing an allocation ai with (11) can be done with complexity O(Mi )). • To calculate charges, the computation of allocations must be done for all profiles s−i , i ∈ I, which gives a complexity O (|I| i Mi ). • Once all allocations ai (s−j ) are calculated, a price ci can be computed using (13) with a complexity less than i Mi .
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Consequently, the total complexity involved bythe multi-bid scheme for a given multi-bid profile is O(|I| i Mi ). If all players submit the same number of bids, (i.e. ∀i ∈ I, Mi = M ), then the total complexity is O(M |I|2 ). Notice that in the case of PSP, M = 1 and the same complexity is reached. Therefore, the computational time for both methods is of the order |I|2 , this being multiplied by the number of bids for the multi-bid algorithm. However, the PSP has to compute allocations and prices several times (until the equilibrium is reached), and we believe that even if the convergence of PSP is fast (less than M iterations), the gain in signalization overhead is worth the cost in computational time. III. P ROPERTIES OF THE MULTI - BID MECHANISM In this section, we establish some basic properties of the multi-bid mechanism, showing its interest, before dealing in next sections with the central notions of incentive compatibility and efficiency. Property 3: All the resource is allocated, i.e. for all multibid profile s = (s0 , (si )i∈I ) ∈ S0 × S |I| , ai (s) = Q, (16) i∈I∪{0}
where |I| is the number of players in the set I. Proof: u) − d¯i (¯ u+ ) d¯i (¯ ¯ u+ )) ai (s) = u+ ) + ¯ d¯i (¯ ¯ u+ ) (Q− d(¯ d(¯ u) − d(¯ i∈I∪{0}
i∈I∪{0}
¯ u) − d(¯ ¯ u+ ) ¯ u+ ) + d(¯ ¯ u+ )) = Q. = d(¯ ¯ u) − d(¯ ¯ u+ ) (Q− d(¯ d(¯
In the following, s denotes the multi-bid profile, i.e. s = (s0 , (si )i∈I ). For i ∈ I, we also write s−i = (s0 , (sj )j∈I,j=i ) the multi-bid profile without player i’s multi-bid. Therefore s = (si , s−i ). Property 4: ∀(si )i∈I , ∀i ∈ I, ai (si , s−i ) = [aj (∅, s−i ) − aj (si , s−i )] , j∈I∪{0},j=i
meaning that player i’s allocation is the difference between what other players would have obtained if player i was not part of the game and what they actually obtain. Proof: Apply (16) to the multi-bid profiles (si , s−i ) and (∅, s−i ), and remark that ai (∅, s−i ) = 0. Property 5: A player increases his allocation by declaring a higher pseudo-demand function. More precisely, consider a player i ∈ I and two multi-bids si , s˜i ∈ S (which may not have the same number of bids). We denote d¯i and d˜i the associated pseudo-demand functions. Then ∀s−i ∈ S0 × S |I|−1 , ∀j ∈ I ∪ {0} \ {i}, ai (si , s−i ) ≤ ai (˜ si , s−i ) d¯i ≤ d˜i ⇒ (17) si , s−i ). aj (si , s−i ) ≥ aj (˜ A proof of Property 5 is provided in Appendix II. Property 6: When a player i ∈ I leaves the game, the allocations of all other players in the game increase. Formally,
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∀i ∈ I, ∀j ∈ I ∪ {0} \ {j}, ∀s ∈ S |I| , aj (∅, s−i ) ≥ aj (si , s−i ). Proof: Just apply (17) to pseudo-demand functions d˜i (associated with the multi-bid si ) and d¯i = 0 (that corresponds to the multi-bid ∅, following (3)). 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. Formally, if ∀i, j ∈ I ∪ {0}, ∀s ∈ S0 × S |I| and d¯i and d¯j are the pseudo-demand functions associated with multi-bids si and sj , we have d¯i ≤ d¯j ⇒ ai (s) ≤ aj (s). Proof: We write d¯ the agregated pseudo-demand function computed using the multi-bid profile s and u ¯ the corresponding pseudo-market clearing price. Thus ¯ u+ ) Q − d(¯ + + ¯ ¯ u ) − di (¯ u ) 1− ¯ aj (s)−ai (s) = dj (¯ ¯ u+ ) + d(¯ u) − d(¯ ¯ u+ ) Q − d(¯ u) − d¯i (¯ u) ¯ + d¯j (¯ ¯ u+ ) d(¯ u) − d(¯ ≥ 0, ¯
+
Q−d(¯ u ) ¯ ¯ where we have used 0 ≤ d(¯ ¯ u)−d(¯ ¯ u+ ) ≤ 1 and di ≤ dj . Property 8: (reserve price). 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 :
∀s ∈ S0 × S |I| , ci (s) ≥ p0 ai (s). (18) A proof of Property 8 is given in Appendix III. ¯ Remark: Notice that a bid (qim , pm i ) does not affect di (p) for m ¯ ¯ ≥ p0 since d(p0 ) ≥ q0 > p > pi , and that we always have u Q. For that reason, submitting a multi-bid si with p1i < p0 is useless for player i ∈ I: he would get exactly the same utility i as if he had submitted the multi-bid (s2i , ..., sM i ). Therefore we can consider that players do not submit such bids, i.e. ∀i ∈ I : si = ∅, p1i ≥ p0 . In the next property, we show that the arrival of a new player i with the multi-bid si corresponds to an increase of the network revenue. This result implies that the network will not deter any player from entering the auction game. Property 9: The seller’s revenue is always greater with all players than when a player is excluded from the game: ∀s ∈ S0 × S |I| , ∀i ∈ I, cj (s) ≥ cj (s−i ). (19) j∈I
j∈I\{i}
More precisely, if the seller has a marginal valuation p0 of the resource, (i.e. θ0 (q) = p0 q), then the seller’s net utility is larger when all players are in the game than in the case when a player is excluded: ∀s ∈ S0 × S |I| , ∀i ∈ I, cj (s) ≥ p0 a0 (s−i ) + cj (s−i ). p0 a0 (s) + j∈I
j∈I\{i}
We provide a proof of Property 9 in Appendix IV. Let us now introduce our last property. 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
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participate [13]. The following property states that this holds for truthful bidders, since the price a player is charged is lower than his declared willingness-to-pay. Property 10: (individual rationality) ai (s) |I| (20) θ¯i (q)dq. ∀i ∈ I, ∀s ∈ S0 × S , ci (s) ≤ 0
Moreover, if player i submits a truthful multi-bid (si ∈ SiT ), then ai (s) θi (q)dq = θi (ai (s)), (21) ci ≤ 0
is always less than max
0≤m≤Mi
di (pm i )
) di (pm+1 i
Ui (si , s−i ) ≥ Ui (˜ si , s−i ) − Ci ,
IV. I NCENTIVE COMPATIBILITY In this section, we focus on the problem of the multi-bid choice by a player. In particular, we prove that a player cannot do much better than simply reveal his true valuation, which in our context means bidding with prices equal to the marginal m valuations: ∀m, 1 ≤ m ≤ Mi , pm i = θi (qi ). Proposition 1: If a player i ∈ I submits a truthful multibid si = ∅, then every other multi-bid s˜i (truthful or not) necessarily to an increase of utility that is less
di (¯u) corresponds ¯)dq. than d¯i (¯u+ ) (θi (q) − u si ∈ S, ∀s−i ∈ S0 × S |I|−1 , Formally, ∀si ∈ SiT , ∀˜ di (¯u) si , s−i ) − (θi (q) − u ¯)dq. (22) Ui (si , s−i ) ≥ Ui (˜ d¯i (¯ u+ )
A proof of Proposition 1 is given in Appendix VI. This result is illustrated in Fig. 3 where the shaded area corresponds to the maximum utility gain player i could expect by submitting a different multi-bid. Since the pseudo-market clearing price is necessarily higher than p0 , we can therefore also note in Fig. 3 that, when p1i ≥ p0 , the quantity di (¯u) (θi (q) − u ¯)dq d¯i (¯ u+ )
p
0≤m≤Mi
with = 0 and qi0 = di (p0 ). Proposition 2 implies that a player i who knows the reserve price p0 can give a truthful multi-bid that brings him the best utility possible, up to a value Ci that can be controlled through the choice of the bids sm i on the demand curve. One important point is that this value does not depend on the number of other players, nor on the multi-bid they submit. Remark: Since submitting a truthful multi-bid is a Ci -best action for player i independently of the other players’ actions, we can say that submitting such a bid is an ex post Ci dominant strategy for i (see [19]). Therefore, the situation where all players submit truthful multi-bids is an ex post KNash equilibrium, with K = maxi∈I Ci , in the sense that no player could have improved his utility by more than K if he had submitted a different multi-bid. Remark: Note that if θi (Q) > p0 , player i can also ensure that u ¯ ≥ p1i ≥ p0 with p1i ≤ θi (Q) by submitting the truthful bid s1i = (q1 , θi (q1 ) = p1 ) with di (p0 ) ≥ q1 > Q. Choosing a bid sufficiently close to (Q, θi (Q)), player i may reduce his constant Ci . p
Ci
θ0i (q) p0
d¯i (u¯+ ) di (u¯)
Q
q.
Fig. 3. The multi-bid si = (s1i , s2i , s3i , s4i ) is optimal for player i up to the
d (¯ u) value d¯ i(¯ (θ (q) − u ¯)dq of shaded surface u+ ) i
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s3i s2i
s1i
i
qim+1
qiMi +1
s2i u¯
(24)
Note that Ci can also be written m qi m Ci = max (θi (q) − θi (qi ))dq
θ0i (0)
s3i
di (pm+1 ) i
(23)
i +1 with pM = θi (0) and p0i = p0 . i
. s4i
,
i +1 where pM = θi (0) and p0i = p0 . i This last quantity is the largest shaded area in Fig. 4. The following proposition is then straightforward: Proposition 2: Under Assumption 1 we have ∀i ∈ I, ∀si ∈ si ∈ S, ∀s−i ∈ S0 × S |I|−1 , SiT (p0 ) \ ∅, ∀˜
0≤m≤Mi
θ0i (0)
−
pm i )dq
where SiT (p0 ) is defined in (2), and where di (pm i ) m Ci = max (θi (q) − pi )dq
which means that Ui (s) ≥ 0. Appendix V gives a proof of Property 10.
.
(θi (q)
s1i θ0i (q) q.
Fig. 4. The multi-bid si = (s1i , s2i , s3i ) is optimal for player i up to a constant Ci , whatever the multi-bids submitted by others s−i be. Ci is the surface of the darkest shaded area.
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V. “Q UANTILE 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. To do so, it would be of interest to consider that players have beliefs (a priori probability distributions, as in [20]) on the number of users in the game and on their preferences, in order to deduce a probability distribution of the pseudo-market price u ¯. (In this sense, the auction game can be seen as a game with population uncertainty [21].) Given this distribution, player i may choose his bids so as to use Proposition 1 to di (¯ u) ¯)dq . However, estimating such minimize E d¯i (¯u+ ) (θi (q) − u a distribution on the pseudo-market price may imply high cost for information-gathering and market appraisal [18]. For this reason, and 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 . With this in mind, 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 in his multi-bid, he will give a number Mi of bids as large as possible, in order to make Ci tend to zero. We therefore focus here on the choice of the multi-bid by player i, after the number of bids Mi is determined, for instance in the case where it is fixed by the auctioneer (see also Section VII). It is then clear that i for a fixed Mi , the multi-bid (s1i , ..., sM i ) that minimizes Ci is such that ∀m, n, 0 ≤ m, n ≤ Mi ,
di (pm
di (pni ) n i ) (θ (q) − pm i )dq = d (pn+1 ) (θi (q) − pi )dq d (pm+1 ) i i
i
i
i
i +1 with pM = θi (0) and p0i = p0 , i
i.e. all the shaded areas are equal. In the following, we will call quantile uniform this bid repartition. An example of quantile uniform repartition of bids is presented in Fig. 5. Example: For parabolic valuation functions, i.e. of the form θi (q) = αi −(q ∧ q¯i )2 /2 + q¯i (q ∧ q¯i ) with parameters αi and q¯i , the marginal valuation function is linear: + qi − q] . θi (q) = αi [¯
When θi (0) > p0 , the quantile uniform repartition of bids is then easy to compute: prices pm i , 1 ≤ m ≤ Mi are such that
θi (0) − p0 αi q¯i − p0 = p0 + m . Mi + 1 Mi + 1 VI. E FFICIENCY One goal of the auction is to make sure that the capacity goes to users who value it most. In this section, we prove that the multi-bid second-price mechanism verifies this property. To obtain this result, we make another regularity assumption on the valuation functions: Assumption 2: ∃κ > 0, ∀i ∈ I, • θi (0) < +∞ • ∀z, z , z > z ≥ 0, θi (z) − θi (z ) > −κ(z − z ). Notice that this assumption is also needed to prove the efficiency of the PSP (see [7]). Proposition 3: If Assumptions 1 and 2 hold, then ∀s ∈ S0 × S |I| , θi (˜ ai ) − θi (ai (s)) ≤ Q κ max Ci , max pm i = p0 + m
A
i
i∈I
i
where A = a ˜ ∈ [0, Q]|I|+1 : i a ˜i ≤ Q . We provide a proof of Proposition 3 in Appendix VII. Remark: This proposition states that the allocation a(s) is optimal up to a certain value, in terms of efficiency as well as social welfare (see the remark in sub-section II-B).
VII. D ETERMINATION OF THE NUMBER OF BIDS ADMITTED BY THE AUCTIONEER
In this section, we assume that the auctioneer imposes the number of bids for all players i ∈ I to be M . We further suppose that players, knowing the reserve price p0 and the number M of bids allowed, choose their multi-bid according to the quantile uniform distribution, as mentionned in Section IV. Since increasing the value of M increases the signaling overhead, the memory storage and the complexity of all underlying allocation and price computations, we introduce a cost function C(M, I) that models these negative effects, which the auctioneer will have to take into account when calculating her benefit B(M, I): ci − C(M, I), B(M, I) = p0 a0 + i∈I
.
p θ0i (0)
s3i s2i s1i
p0
θ0i (q)
.q Fig. 5. Quantile uniform repartition of bids for Mi = 3: the four shaded zones have the same surface.
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where allocations and prices correspond to the situation when each user submits exactly M bids. We denote T the set of possible player types, characterizing the valuation function (in other words, a type-t player has valuation function θ(t) ). We model the auctioneer’s beliefs about the number of players of each type by an a priori law PT on NT . Therefore, the auctioneer can estimate her expected revenue EI [RM ] when players submit M bids. We have EI [RM ] = EI p0 a0 + ci M = I∈NT
i∈I
p0 a0 + ci M
dPT (I).
i∈I
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We now make an assumption about the cost function C(M, I): =
Assumption 3: The expected cost EI [C](M ) C(M, I)dP (I) is non-decreasing, and tends to T T I∈N infinity when M tends to infinity: lim EI C(M ) = +∞.
M →+∞
This assumption seems intuitive. Actually, if we deal with memory costs, we have C(M, I) = M |I|, so Assumption 3 is verified as soon as EI [|I|] > 0. Considering computation costs as in Subsection II-E or signaling costs would also lead to a cost function that would verify Assumption 3. The following result gives an idea on how the auctioneer may choose M : 4: If the marginal valuation functions Proposition θ(t) are uniformly bounded by a value pmax (that is t∈T
∀t ∈ T , θ(t) (0) ≤ pmax ), then under Assumption 3 there exists a finite M that maximizes the expected net benefit of the seller, i.e. that maximizes EI p0 a0 + ci M − EI [C(M, I)] . i∈I
Proof: We assume without loss of generality that pmax ≥ p0 . Applying Property 10, we have ∀I, ∀M, ci ≤ θi (ai ) ≤ ai θi (0) p0 a0 + i∈I
i∈I∪{0}
≤ pmax
i∈I∪{0}
ai = pmax Q.
i∈I∪{0}
Consequently EI p0 a0 + i∈I ci M ≤ pmax Q for all M ∈ N. Therefore lim EI p0 a0 + ci − C(M, I)M = −∞, M →+∞
i∈I
which ensures us that there exists a finite M that maxi mizes the expected net benefit EI p0 a0 + i∈I ci M − EI [C(M, I)]. Remark: It is possible that the expected net benefit be nonpositive for all M ≥ 1. This means that organizing the auction is too expensive for the seller of the resource. In that case the owner will choose M = 0, i.e. she prefers not to sell the resource. Remark: We considered in this section the case when the auctioneer chooses the number M of bids allowed. However, one can imagine a high authority which may want to ensure a certain efficiency of the allocation. This authority could then fix M so as to reach a good trade-off between efficiency and costs, and subsidize the auctioneer in order to compensate her net benefit loss. VIII. C ONCLUSIONS AND PERSPECTIVES In this paper, we have designed and studied a one-shot auction-based mechanism for sharing an arbitrarily divisible resource, like the capacity of a communication link. This mechanism requires each user i to submit Mi two-dimensional
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bids when entering the auction. With respect to the progressive second price (PSP) auction of the literature, our mechanism saves a lot of signaling overhead. Considering an elastic-demand model of user preferences, we have proved that our rule incites players to submit truthful bids. We have highlighted a reasonable behavior for users, assuming that they know their own preferences, but do not have beliefs about the other players (neither about their numbers nor about their valuation functions). We have shown that the rule leads to an efficient allocation of resource, and have given some hints to understand how the number of bids can be chosen. This mechanism is particularly well-suited for pricing bandwidth on a communication link, for it implies relatively low computational costs and signaling traffic, and adapts instantly to a change in the user set (start or end of a connection). Future work in that context will essentially focus on the application of our mechanism to the network case, i.e. to the situation where users want to buy bandwidth on several links from one point to another. The PSP mechanism was studied in the network case in [4], [10], with the same signaling overhead than in the single link case. We are currently working on a scheme based on multi-bids that would still be efficient in a network context. A PPENDIX I P ROOF OF L EMMA 2 Proof: The equations giving d¯i (.+ ) and θ¯i (.+ ) are straightforward. We focus here on (14): • if si = ∅ then θ¯i (.) = 0 ≤ x, and (14) is verified. i i • If si = ∅ and pM ≤ x, (14) comes from θ¯i (.) ≤ pM i i . Mi • If si = ∅ and pi > x then d¯i (x+ ) =
max {qim : pm i > x}.
1≤m≤Mi
(25)
Now let us assume that θ¯i (d¯i (x+ )+ ) > x and see that we arrive at a contradiction. The fact that θ¯i (d¯i (x+ )+ ) > 0 implies that we are in the case when θ¯i (d¯i (x+ )+ ) =
m ¯ + max {pm i : qi > di (x )}.
1≤m≤Mi
Since θ¯i (d¯i (x+ )+ ) > x,
m1 m m si = (qi 1 , pi 1 ) ∈ si m1 ¯ ¯ pi = θi (di (x+ )+ ) > x ∃m1 , 1 ≤ m1 ≤ Mi : m qi 1 > d¯i (x+ ).
We now remark that this contradicts the definition of d¯i (x+ ) (Eq. (25)). Therefore (14) is verified. Now we establish (15). By definition, d¯i (x) = max1≤m≤Mi {qim : pm i ≥ x}. This means that there exists 0 m0 ≤ Mi such that d¯i (x) = qim0 and pm ≥ x. i Consequently we have θ¯i (d¯i (x)) =
m0 m0 m max {pm ≥ x, i : q i ≥ q i } ≥ pi
1≤m≤Mi
which gives (15).
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A PPENDIX II P ROOF OF P ROPERTY 5 Proof: We just need to prove the result for players j = i, si , s−i ) coming directly from the inequality ai (si , s−i ) ≤ ai (˜ (16). To simplify the notations we will write I0 = I ∪ {0}. So consider j = i. We write u ¯ (respectively u ˜) the pseudomarket clearing price when the multi-bid profile is s = (si , s−i ) (respectively (˜ si , s−i )). We also denote the respective ˜ and we define agregated pseudo-demand functions by d¯ and d, ¯ ¯ d−i = k∈I0 \{i} dk . We then have si , s−i ) aj (s) − aj (˜
u) − d¯j (¯ u+ ) d¯j (¯ ¯ u+ )) − = d¯j (¯ (Q − d(¯ u+ ) − d¯j (˜ u+ ) + ¯ ¯ d(¯ u) − d(¯ u+ ) u) − d¯j (˜ u+ ) d¯j (˜ ˜ u+ )). (Q − d(˜ (26) − ˜ u) − d(˜ ˜ u+ ) d(˜
Since by hypothesis d¯i ≤ d˜i , then d¯ = d¯i + d¯−i ≤ d˜ = ˜ ¯≤u ˜. di + d¯−i , and necessarily u We distinguish two cases: u)−d¯j (¯ u+ ) d¯j (¯ ¯ u+ )) ≥ 0 and • if u ¯ < u ˜, then since d(¯ ¯ u)−d(¯ ¯ u+ ) (Q − d(¯ + ¯ ¯ u)−dj (˜ u ) dj (˜ ˜ u+ )) ≤ d¯j (˜ (Q − d(˜ u) − d¯j (˜ u+ ), (26) implies
where u ¯ is the pseudo-market clearing price corresponding to the multi-bid profile s.
a (s) • If y ≤ aj (s), then y j θ¯j ≥ θ¯j (aj (s))(aj (s) − y). u), the nonSince we know from (12) that aj (s) ≤ d¯j (¯ u)) ≥ u ¯, increasingness of θ¯j implies that θ¯j (aj (s)) ≥ θ¯j (d¯j (¯ (s) − y ≥ 0 gives (27). by applying (15). Finally, a j
y • If y > aj (s), then aj (s) θ¯j ≤ θ¯j (aj (s)+ )(y − aj (s)). u+ ) to obtain θ¯j (aj (s)+ ) ≤ We now use (14) and aj (s) ≥ d¯j (¯ + + u ) )≤u ¯, which leads to (27). θ¯j (d¯j (¯ To prove Property 8, we apply (27) to the bid profile s−i and get aj (s−i ) ci (s) = θ¯j
≥
aj (s) − aj (˜ si , s−i ) ≥ d¯j (¯ u ) − d¯j (˜ u ) − d¯j (˜ u) + d¯j (˜ u ) + ¯ ¯ ≥ dj (¯ u ) − dj (˜ u) ≥ 0, +
where u ¯−i denotes the pseudo-market clearing price corresponding to the multi-bid profile s−i , and where Property 6 ¯−i ≥ p0 , has been used. Since d¯0 (p0 ) = q0 > Q, then u establishing Property 8. A PPENDIX IV P ROOF OF P ROPERTY 9
+
where the last inequality stems from u ¯ < u ˜ and from the non-increasingness of d¯j . • if u ¯=u ˜, then (26) becomes aj (s) − aj (˜ si , s−i ) =
˜ u+ ) ¯ u+ ) Q− d(¯ Q− d(¯ + ¯ ¯ (dj (¯ . u))− dj (¯ u )) ¯ ¯ u+ ) − d(¯ ˜ u)− d(¯ ˜ u+ ) d(¯ u)− d(¯
Proof: To simplify the notations, we omit to precise the multi-bid profile when it is s, i.e. we denote aj (s) = aj for all j ∈ I0 . between the two terms of (19) is The difference c − j j∈I j∈I\{i} cj (s−i ) = ci + [cj − cj (s−i )] j∈I\{i}
=
˜ u) −d(¯ ˜ u+ ))(Q − d(¯ ¯ u + )) − ( d(¯ ! ! ¯ u) ≥d(¯
≥0
˜ u+ )) ¯ u) − d(¯ ¯ u+ ))(Q − d(¯ −(d(¯ ¯ u+ )) ≥ 0 ¯ u) − Q)(d(¯ ˜ u+ ) − d(¯ ≥ (d(¯ where we have used (9) and the assumption d˜i ≥ d¯i , which ¯ gives d˜ ≥ d. Finally we have proved that in all cases, aj (s) ≥ si , s−i ), which establishes the property. aj (˜ A PPENDIX III P ROOF OF P ROPERTY 8 Proof: We first show that ∀I, ∀s ∈ S0 × S |I| , ∀j ∈ I0 , ∀y ∈ R+ , aj (s) ¯(aj (s) − y) (27) θ¯j ≥ u y
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=
Since d¯j (¯ u) − d¯j (¯ u+ ) ≥ 0, we only need to show that A = + ˜ ˜ ¯ u+ )) − (d(¯ ¯ u) − d(¯ ¯ u+ ))(Q − d(¯ ˜ u+ )) is (d(¯ u) − d(¯ u ))(Q − d(¯ non-negative. We have A
u ¯−i (aj (s−i ) − aj (s)) = u ¯−i ai (s)
j∈I0 ,j=i
˜ u)−d(˜ ˜ u+ ) d(˜
+
aj (s)
j∈I0 ,j=i
aj (s−i )
aj
j∈(I0 )\{i}
θ¯j +
[cj − cj (s−i )]
j∈I\{i}
= p0 (a0 (s−i ) − a0 ) +
Aj
(28)
j∈I\{i}
with
Aj =
aj (s−i )
aj aj (s−i )
=
aj
=
θ¯j + cj − cj (s−i )
θ¯j +
+
ak (s−j )
θ¯k −
θ¯j +
k∈I0 \{i,j}
ai (s−j )
ai
ak
ak (s−i,j )
θ¯k
θ¯i +
θ¯k −
ak (s−j )
k∈I0 \{i,j} ak (s−i )
k∈I0 \{j} ak
aj (s−i )
aj
¯ θk .
ak (s−i,j )
ak (s−i )
(29)
To show the proposition, we just need to prove that Aj ≥ 0 for all j ∈ I \ {i}. In order to do that, we denote u ¯−i (respectively u ¯−j ) the pseudo-market clearing price associated with the multi-bid profile s−i (respectively s−j ), and consider two cases:
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• if u ¯−j ≥ u ¯−i then (27) applied to the profiles s−i and s−j gives
A PPENDIX V P ROOF OF P ROPERTY 10 Proof: Applying (27) we get
≥ u ¯−i [aj (s−i ) − aj ] + u ¯−j [ai (s−j ) − ai ] + + u ¯−j (ak (s−j ) − ak ) +
Aj
ci (s)
k∈I0 \{i,j}
+
u ¯−i (ak (s−i ) − ak (s−i,j ))
≥ u ¯−i
ak (s−i ) −
k∈I0 \{i}
ak (s−i,j ) +
k∈I0 \{i,j}
+¯ u−j
ak (s−j ) −
k∈I0 \{j}
≤
ak +
k∈I0
+(¯ u−j − u ¯−i )aj ≥ 0
(Property 3);
ak (s−j )
ak
θ¯k −
ak (s−i,j ) ak (s−i ) ak (s−j )
≥
ak ak (s−j )
≥
where the last line comes from Property 4. Now we consider two cases: • if ai (s) = 0 then ci (s) ≤ 0 and (20) is established. • if ai (s) > 0 then necessarily d¯i (¯ u) > 0, thus si = ∅ and i ¯ (d¯i (¯ ≥ u ¯ . Lemma 2 then implies θ u)) ≥ u ¯. Moreover, pM i i u), since θ¯i is non-increasing and ai (s) ≤ d¯i (¯ ai (s)
0
u)) ≥ ai (s)¯ u, θ¯i ≥ ai (s)θ¯i (ai (s)) ≥ ai (s)θ¯i (d¯i (¯
which gives (20). Proving (21) is then straightforward by applying (20) and Lemma 1. A PPENDIX VI P ROOF OF P ROPOSITION 1
θ¯k
θ¯k −
u ¯(aj (∅, s−i ) − aj (s))
u, ≤ ai (s)¯
• if u ¯−j < u ¯−i then we use the non-increasingness of the pseudo-marginal valuation functions θ¯k to modify the integration bounds in (29):
θ¯j
j∈I0 \{i}
aj (∅,s−i )
aj (s)
j∈I0 \{i}
k∈I0 \{i,j}
=
ak (s−i,j )+ak −ak (s−i )
ak
ak (s−i,j )+ak −ak (s−i )
θ¯k
θ¯k .
Proof: Consider the difference of the prices charged to player i depending on his multi-bid: si , s−i ) − ci (s) ci (˜
=
j∈I0 \{i}
≥ u ¯
By applying (27) we obtain
aj (s)
si ,s−i ) aj (˜
θ¯j
(aj (s) − aj (˜ si , s−i ))
j∈I0 \{i}
si , s−i ) − ai (s)), ≥ u ¯(ai (˜ Aj
≥ u ¯−i (aj (s−i ) − aj + u ¯−j (ai (s−j ) − ai ) + +¯ u−j [ak (s−j ) + ak (s−i ) − ak − ak (s−i,j )] k∈I0 \{i,j}
≥ u ¯−j
ak (s−i ) −
k∈I0 \{i}
+¯ u−j
ak (s−i,j ) +
k∈I0 \{i,j}
ak (s−j ) −
k∈I0 \{j}
ak +
k∈I0
where the first inequality comes from (27) and the last one from Property 4. On the other hand, consider the difference of valuations si , s−i )). Dθi := θi (ai (s)) − θi (ai (˜ We distinguish several cases: • if ai (s) > ai (˜ si , s−i ), then Dθi =
Finally we always have ∀j ∈ I \ {i}, Aj ≥ 0, and (28) then gives the second part of the proposition. The first part immediately follows, using the inequality a0 ≤ a0 (s−i ) (Property 6).
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ai (s)
ai (˜ si ,s−i )
+(¯ u−i − u ¯−j )(aj (s−i ) − aj ) ≥ 0.
(30)
θ
≥
ai (s)
ai (˜ si ,s−i )
θ¯i
≥ u ¯(ai (s) − ai (˜ si , s−i )) from inequalities (5) and (27). si , s−i ) and u ¯ ≥ θi (0), then • If ai (s) ≤ ai (˜ Dθi =
ai (s)
ai (˜ si ,s−i )
θ
≥ θi (0)(ai (s) − ai (˜ si , s−i )) ≥ u ¯(ai (s) − ai (˜ si , s−i )).
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• If ai (s) ≤ ai (˜ si , s−i ) and u ¯ < θi (0), then θi (di (¯ u)) = u ¯ and Dθi
= θi (ai (s)) − θi (di (¯ u)) + θi (di (¯ u)) − θi (ai (˜ si , s−i )) ai (s) (θi (q) − u ¯)dq + u ¯(ai (s) − di (¯ u)) + ≥ di (¯ u)
u) − ai (˜ si , s−i )) +¯ u(di (¯ d¯i (¯u+ ) (θi (q) − u ¯)dq + u ¯(ai (s) − ai (˜ si , s−i )) ≥ di (¯ u)
where the last line comes from (12) and from the fact that ¯ ≥ 0 for all q ≤ di (¯ u). θi (q) − u Finally we always have di (¯u) Dθi ≥ u ¯(ai (s) − ai (˜ si , s−i )) − (θi (q) − u ¯)dq. (31) d¯i (¯ u+ )
Ui (s) − Ui (˜ si , s−i ) = Dθi + ci (˜ si , s−i ) − ci (s) di (¯u) (θi (q) − u ¯)dq. ≥ −
(32)
Consider a player i ∈ I such that ai (s) > 0. Since ai (s) ≤ u), we have di (¯ u) ≥ d¯i (¯ u) > 0. Thus θi (di (¯ u)) = u ¯, and d¯i (¯ u)) ≥ θi (di (¯ u)) = u ¯. θi (ai (s)) ≥ θi (d¯i (¯ On the other hand, we have (33)
• If θi (0) ≤ u ¯ then θi (ai ) ≤ u ¯. • If θi (0) > u ¯ then =
min
1≤m≤Mi +1
≤ u ¯+
m {pm ¯} i : pi > u
− pm max (pm+1 i ). i
0≤m≤Mi
(34)
i +1 with pM = θi (0) and p0i = p0 . i Now we remark that for all m, 0 ≤ m ≤ Mi pm+1 di (pm i ) i m (θi (q) − pi )dq = (di (p) − di (pm+1 ))dp i
di (pm+1 ) i
pm i
≥
pm+1 i
pm i (pm+1 i
pm+1 − pm i i dp κ
2 − pm i ) κ where the second line comes from (32).
≥
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≤ Q κ max Ci ,
I−
κCi (˜ ai − ai (s))
I+
i∈I
R EFERENCES
Proof: First notice that if Assumptions 1 and 2 hold, then ∀i ∈ I, ∀e, f : 0 < e ≤ f ≤ θi (0),
u+ )) θi (d¯i (¯
(˜ ai − ai (s)) +
i
&
This work has been partially supported by the INRIA project PRIXNET (see http://www.irisa.fr/armor/ Armor-Ext/RA/prixnet/ARC.htm)
A PPENDIX VII P ROOF OF P ROPOSITION 3
θi (ai (s)) ≤ θi (d¯i (¯ u+ )).
≤u ¯
ACKNOWLEDGMENT
d¯i (¯ u+ )
f −e . κ
I+
which establishes the proposition.
To conclude the proof, from (30) and (31), we get
di (e) − di (f ) ≥
m+1 − pm i ≤ √ Finally, (24) implies that ∀m, 0 ≤ m ≤ Mi , pi κCi . Therefore (34) gives & ¯ + κCi . θi (ai (s)) ≤ u ˜i ≤ Q , and take any Define A = a ˜ ∈ [0, Q]|I|+1 : i a ˜k ≥ ak (s)} and I− = {k : a ˜k < a ˜ ∈ A. Let I+ = {k : a ˜k ≥ 0, and therefore ak (s)}. For i ∈ I− , we have ak (s) > a (33) implies θi (ai (s)) ≥ u ¯. Applying (34), we then have ai ) − θi (ai (s)) i θi (˜ ≤ θi (ai (s))(˜ ai − ai (s)) − θi (ai (s))(ai (s) − a ˜i )
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