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1 MANAGEMENT SCIENCE do /mnsc ec e-companon ONLY AVAILABLE IN ELECTRONIC FORM nforms 2009 INFORMS Electronc Companon Truthful Bundle/Multunt Double Auctons by Leon Yang Chu, Management Scence, do /mnsc

2 Proofs of Statements Onlne Appendx for Truthful Bundle/Mult-unt Double Auctons EC.1. Proofs for the IPB Paddng Mechansm In ths secton, we only focus on the case n whch Q c qj c = max{q j j J,c = c j } for all c C. Thus, Q c q j I(c j = c) for all j J, where I(c j = c) s the ndcator functon, whch equals 1 when c j = c and 0 otherwse. For commodty c, the total number of unts suppled s q j J,c j =c j. We ran all of these unts accordng to ther as prces from low to hgh; p c [h] s the as prce for the lowest hth unt of commodty c, where h s an nteger. Lemma EC.1. For Ĩ, xb = 1 n the optmal soluton to f Q c q c J c C. P(Ĩ,J) and P(Ĩ,J\{j}) for any j J Proof of Lemma EC.1. At the optmal soluton (x b,y j ) to P(I,J,Q), for each c C, a total of I,b B qc bx b + Q c = j J,c j =c y j unts s nvolved n the supply/demand balance constrant, and the hghest q as prces among these unts are p c [ P I,b B qc b xb +Qc ],pc [ P I,b B qc b xb +Qc ],,pc [ P I,b B qc b xb +Qc q+1]. f b If c C Ĩ, xb = 1. Comparng the soluton to P(I,J,Q) at xb qb c p c [ P. I,b B qc b xb +Qc h] = 1 and xb = 0, we conclude that At the optmal soluton (x b,y j ) to P(Ĩ,J), for each c C, a total of Ĩ qc b xb = j J,c j =c y j unts s nvolved n the supply/demand balance constrant. To show x b = 1 for buyer, notce that I,b B qc b xb + Q c Ĩ qc b x b = Ĩ qc b Ĩ\{} qc b x b + qb c ; therefore, f x b = 0, q c b p c [ P Ĩ qc b x b q c b +q b c h] p c [ P I qc x b b, and f b +Q c h] c C q c b p c [ P Ĩ qc b x b +q c b h]. x b must equal 1 because we can ncrease the objectve functon by ncreasng x b to 1 otherwse. At the optmal soluton (x b,y j ) to P(Ĩ,J\{j}), for each c C, a total of Ĩ qc x b = j J\{j},c j =c y j unts s nvolved n the supply-demand balance constrant. To show x b = 1 for buyer, notce that I,b B qc x b + Q c Ĩ qc b x b + q j I(c j = c) = Ĩ qc b + q j I(c j = c) ec1

3 ec2 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons Ĩ\{} qc b x b + q c + q j I(c j = c); therefore, f x b = 0, q c b p c [ P and f g Ib B qc x b b +Qc h] c C q c b q c b because we can ncrease the objectve functon by ncreasng x b p c [ P Ĩ qc b x b +q j I(c j =c)+q c b h] p c [ P. Ĩ qc x b b +q j I(c j =c)+q b c h] xb must equal 1 to 1 otherwse. Therefore, for Ĩ, xb = 1 n the optmal soluton to P(Ĩ,J) and P(I,J\{j}) for any j J. Theorem EC.1. Bddng truthfully s a domnant strategy for the buyers. Proof of Theorem EC.1. If buyer does not contrbute n the system P(I, J, Q), she does not trade n the allocaton; f buyer bds such that P(I,J,Q) > P(I\{},J,Q), she trades n the allocaton and acqures the bundle specfed by the optmal soluton to P(I, J, Q). Furthermore, her buyng prce s the VCG prce that s, her bd for the bundle mnus her contrbuton to the system. Bddng truthfully s a domnant strategy for the buyers. Lemma EC.2. When Q c > q c J 1, seller does not trade f y < q n the optmal soluton to P(I,J,Q). Proof of Lemma EC.2. A total of Ĩ qc b unts of commodty c are traded n the allocaton. At the optmal soluton (x b,y j ) to P(I,J,Q), a total of I qc b xb + Q c = j J,c j =c y j unts are nvolved n the supply-demand balance constrant for commodty c. Notce that Ĩ qc b I,b B qc b xb = j J,c j =c y j Q c j J\{},c j =c y j. If y < q, ether y = 0 or the as prce g by seller s the hghest among all of the sellers j wth y j > 0. In ether case, the supply of commodty c by sellers who as less than g (or as for g but have a hgher preference under the perturbaton) s no less than Ĩ qc b, whch means that seller does not trade. Lemma EC.3. If y = q n the optmal soluton to P(I,J,Q), the optmal soluton to P(I,J,Q) and Ĩ reman the same when seller lowers hs prces. Proof of Lemma EC.3. Suppose that (x b,y j ) s the optmal soluton to P(I,J,Q) and that y = q. If seller ass for g δ (δ > 0) nstead of g, (x b,y j ) remans optmal, wth the objectve functon value ncreased q δ. Ĩ = { xb = 1, b B} remans the same.

4 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons ec3 Let p (I,J,Q) be the supremum of as prce g for seller such that y = q n the optmal soluton to P(I,J,Q). Accordng to ths defnton, when seller bds above p (I,J,Q), y < q n the optmal soluton to P(I,J,Q). By Lemma EC.3, when seller bds below p (I,J,Q), y = q n the optmal soluton to P(I,J,Q). Let (x b,,yj ) be the optmal soluton to P(I,J,Q), wth the addtonal constrant y = q, and Ĩ = { x b, = 1, b B}, the correspondng tradng buyer set. For commodty c, the total unts of supply from sellers other than s j J\{},c j =c q j. We ran all of these unts accordng ther as prces from low to hgh. p,[h] s the as prce for the lowest hth unt. Lemma EC.4. Sellers face a non-unform prce scheme, and bddng truthfully s a domnant strategy for sellers f Q c > qj c 1 c C. Proof of Lemma EC.4. We frst show that seller does not trade when g > p,[ P Ĩ qc b ]. If the as prce g s such that y < q n the optmal soluton to P(I,J,Q), seller does not trade by Lemma EC.2; f hs as prce g s such that y = q n the optmal soluton to P(I,J,Q), the tradng buyer set Ĩ s Ĩ by Lemma EC.3, and the as prces for the lowest Ĩ q c b unts from sellers other than are p,[1],p,[2],,p P Ĩ,[ qc when g b ] > p P,[ Ĩ qc b ], whch means that seller does not trade. Therefore, seller does not trade when the as prce g s greater than mn{p,[ P Ĩ q c b ],p (I,J,Q)} by Lemma EC.2. If g < mn{p,[ P Ĩ q c b ],p (I,J,Q)}, the tradng buyer set Ĩ s Ĩ. By Lemma EC.1, x b = 1 for Ĩ, n the optmal solutons to both P(Ĩ,J) and P(Ĩ,J\{}). Thus, a total of Ĩ q c b unts of commodty c s traded n the optmal solutons to both P(Ĩ,J) and P(Ĩ,J\{}). Therefore, the dfference n these two optmal solutons results from the supples of commodty c. The allocaton s the effcent allocaton for P(Ĩ,J). If p,[h] < g < p,[h+1] for h Ĩ q c b q + 1, accordng to the effcent allocaton for P(Ĩ,J), seller trades Ĩ q c b h unts of commodty c, and the VCG compensaton awards hm p,[h+1] +p,[h+2] + +p,[ P Ĩ qc b ]. If g < p,[ P Ĩ qc b q +1], seller trades all of hs q unts wth a total compensaton q p,[ P Ĩ qc. If g b h] = p,[h] for q c Ĩ b q +1 h q c Ĩ b, seller may or may not trade ths hth unt.

5 ec4 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons For seller s payoff, we consder two scenaros: p (I,J,Q) p,[ P Ĩ qc ] and p (I,J,Q) < p,[ P Ĩ qc ]. p (I,J,Q) p P Ĩ,[ qc : when p ] p P,[ s supply s not among the Ĩ q c Ĩ qc, V (Ĩ,J g ] = p (I,J,Q)) = V (Ĩ,J\{}) as seller unts wth the lowest as prces. The revenue of the tradng seller s hs VCG compensaton. Seller can sell hs q unts unt-by-unt at non-ncreasng prces from p,[ P Ĩ qc b ], p,[ P Ĩ qc b ],, to p,[ P Ĩ qc b q +1]. p (I,J,Q) < p P Ĩ,[ qc : under ths scenaro, V (Ĩ,J g ] = p (I,J,Q)) can be greater than V (Ĩ,J\{}) as seller s supply may be among the Ĩ q c b unts wth the lowest as prces. The revenue of the tradng seller s then reduced by hs VCG payoff at prce p (I,J,Q). Ths reducton guarantees a zero payoff for seller wth valuaton p (I,J,Q) and put a cap p (I,J,Q) on sellng prces of the tems. In summary, seller can sell hs q unts unt-by-unt at non-ncreasng prces n both scenaros from mn{p (I,J,Q),p P Ĩ,[ qc }, mn{p P b ] (I,J,Q),p,[ Ĩ qc b ] },, to mn{p (I,J,Q),p P Ĩ,[ qc b q +1] }. Bddng truthfully s a domnant strategy for the sellers. Theorem EC.2. The nteger-program-based paddng mechansm s strategy-proof and ndvdually ratonal f Q c q c J for all commodtes c C. Proof of Theorem EC.2. By Lemmas EC.1 and EC.4, gven the bundle and quantty preferences, bddng truthfully s a domnant strategy for both buyers and sellers. By reportng the true valuatons, both buyers and sellers secure a non-negatve payoff, and the IPB paddng mechansm s ndvdually ratonal. Recall qi c = max{qc b b B}, the sze of the largest demand of commodty c from a sngle buyer. Theorem 5. The nteger-program-based paddng mechansm s strategy-proof, ndvdually ratonal, and (wealy) budget-balanced f Q c max{qj c,qc I } for all commodtes c C. Proof of Theorem 5. Gven Theorem EC.2, t suffces to show that the IPB paddng mechansm s (wealy) budget-balanced f Q c max{q c J,q c I} for all commodtes c C.

6 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons ec5 We frst consder the buyer sde and calculate a lower bound on the revenue from the buyers. Let (x b,y j) be the optmal soluton to P(I,J,Q) and buyer be a tradng buyer.e., x b = 1. Her buyng prce s f b (V (I,J,Q) V (I\{},J,Q)) f b I\{}). By the proof of Lemma EC.1, ths buyng prce s at least c C c C q c b p c [ P Ĩ qc b +Q c h] s no less than Ĩ q c b V (I,J,Q) + V (I\{},J,Q) xb = xb, q c b p c [ P I,b B qc b xb +Qc h]. The revenue based on the prces of commodty c from all of the buyers p c [ P Ĩ qc b +Q c h]. Notce that, as q b vares, we have more p c [ P Ĩ qc b +Q c h 1 ] than p c [ P Ĩ qc b +Q c h 2 ] n the summaton f h 1 < h 2 that s, we have more hgher prce terms than lower prce terms among the total Ĩ qc b unts. When Q c Q c I + 1, the lowest prce term s at least p c [ P Ĩ qc b +1]. In the allocaton, Ĩ qc b unts are traded for commodty c C. Let Ĩ qc b = aq c + b, where a and b are ntegers and b < Q c. In ths case, the revenue s at least (a+1) b pc [ P Ĩ qc +Q b c h] +a Q c h=b p c [ P Ĩ qc b +Q c h] by nducton on both Qc and aq c +b. Now, we consder the seller sde and calculate an upper bound on the payment to the sellers. The revenue of tradng seller s no more than hs VCG revenue n P(Ĩ,J). Because all buyers Ĩ trade n the optmal soluton to P(Ĩ,J) and P(Ĩ,J\{}) by Lemma EC.1, the VCG payment n P(Ĩ,J) s y h=1 pc [ P Ĩ qc b +h], f y = q.e., seller exhausts hs supply. For the commodty c = c, at most one seller j sells part of hs supply, and hs payment s y j h=1 pc [ P Ĩ qc +q b j y j +h]. The pay- ment based on commodty c to all of the sellers s no greater than J,c =c, j y h=1 pc [ P Ĩ qc b +h] + yj h=1 pc [ P Ĩ qc b +q j y j +h]. Notce that, as y vares, we have more lower prce terms than hgher prce terms n the summaton y J,c =c,,j h=1 pc [ P. When Ĩ qc +h] Qc Q c J, the hghest prce term s at b most p c [ P Ĩ qc b +Q c ]. Let J,c =c, j y = cq c + d, where c and d are ntegers and d < Q c. The frst term s at most c Q c d p c [ P Ĩ qc +Q b c h] +(c+1) Q c h=q c d pc [ P Ĩ qc +Q b c h] by nducton on both Qc and cq c +d, and the second term y j h=1 pc [ P Ĩ qc b +q j y j +h] s at most y j pc [ P Ĩ qc b +Q c h]. Because Ĩ qc b unts are traded for commodty c C, aq c + b = cq c + d + y, by nducton on both Q c and aq c +b, t can be shown by that (a+1) b pc [ P Ĩ qc +Q b c h] +a Q c h=b p c [ P Ĩ qc b +Q c h] c Q c d p c [ P Ĩ qc +Q b c h] +(c+1) Q c h=q c d pc [ P Ĩ qc +Q b c h] + y j pc [ P Ĩ qc +Q b c h]. Therefore, when Q c max{q c J,q c I} for all c C, the dfference between the revenue from the buyers and the payment to the sellers s non-negatve. The IPB paddng mechansm s (wealy) budget-balanced.

7 ec6 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons EC.2. Proofs for the Asymptotcal Property of the LPB Paddng Mechansm Recall that there are a fnte number of possble commodtes c C and that there exsts a number M such that q j M for every seller j and q c M for every buyer and every commodty c. That s, M s the lmt to how many unts of commodty c a buyer can acqure or a seller can supply. Let valuatons of the bundles and goods be drawn accordng to some dstrbutons wth support contaned n [0,a] for some constant a. When the paddng functon Q c s solely based on M, the IPB paddng mechansm s asymptotcally effcent. To smplfy the presentaton and elmnate the techncal dscusson of the feasblty of ˆP(I,J,Q) gven a large Q, we assume that we have enough sellers to provde all types of commodtes at prce a so that ˆP(I,J,Q) s always feasble. These fcttous sellers do not trade n the allocaton accordng to the optmal soluton to ˆP(Ĩ,J) because no buyer has valuaton greater than a. Lemma EC.5. Let Q 1 and Q 2 be two vectors such that Q c 2 = Qc 1 + and Qc 2 = Qc 1 c c and c C. Then ˆV (I,J,Q 2 ) ˆV (I,J,Q 1 ) a. for > 0, Proof of Lemma EC.5. Let (x b,y j ) denote the optmal soluton to ˆP(I,J,Q 1 ). Consder the formulaton ˆP(I,J,Q 2 ) wth one more seller j provdng unts of commodty c at prce a. ˆV (I,J,Q 2 ) = ˆV (I,J {j },Q 2 ) because no buyer has valuaton greater than a and wants to trade wth seller j. Now, (x b,y j,y j = ) s a feasble soluton to ˆP(I,J {j },Q 2 ) wth objectve value ˆV (I,J,Q 1 ) a. Therefore, ˆV (I,J,Q 2 ) ˆV (I,J,Q 1 ) a. Lemma EC.6. The effcency loss s bounded for the nteger-program-based paddng mechansm gven a fxed paddng Q (defned by M). Proof of Lemma EC.6. By Lemma EC.5, ˆV (I,J,Q) ˆV (I,J) a c C qc by reducng Q to zero for all commodtes c C. Notce that for each possble bundle, at most one buyer acqures a partal bundle n the optmal soluton to ˆP(I,J,Q). Gven M, the lmt on the acqurng quantty, only a fnte number of bundles

8 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons ec7 can be acqured by the buyers. Let the number of possble bundles be. Then, V (Ĩ,J) V (I,J,Q) a because no buyer has valuaton greater than a. Furthermore, notce that V (Ĩ,J) s the socal welfare realzed and that V (I,J) ˆV (I,J). Therefore, the effcency loss V (I,J) V (Ĩ,J) s bounded by a( c C qc + ), whch s a constant for a fxed paddng Q (defned by M). Theorem EC.3. Wth bounded valuaton dstrbutons, the lnear-program-based paddng mechansm s asymptotcally effcent gven a fxed paddng Q (defned by M). Proof of Theorem EC.3. By Lemma EC.6, the effcency loss s bounded. As the aucton becomes large and the maxmal socal welfare goes to nfnty, the rato between realzed socal welfare and maxmal socal welfare goes to one. The lnear-program-based paddng mechansm s asymptotcally effcent. EC.3. Proofs for the Effcency Comparson on the MBC Mechansm In ths secton, we ntroduce the modfed lmt LPB paddng mechansm and show that ths mechansm s strategy-proof, ndvdually ratonal, and (wealy) budget-balanced. Furthermore, the modfed lmt LPB paddng mechansm domnates the modfed buyer competton (MBC) mechansm. The MBC mechansm s suted for an exchange envronment n whch each buyer wants to acqure a bundle of commodtes and each seller supples one sngle unt of a commodty. The MBC mechansm can be vewed as a BC-LP mechansm wth an addtonal screenng stage accordng to the optmal soluton to P(I,J). When no transacton costs exst, the MBC mechansm can be mplemented by the followng procedure: Collect the bds from the agents. Calculate the optmal soluton to P(I,J) and the VCG payment p (I,J) for each agent. Let Ī denote the set of tradng buyers n the optmal soluton. Solve ˆP(Ī,J). Calculate p (Ī,J), the mnmum shadow prce of the constrant x 1 for buyer such that x = 1 n the optmal soluton to ˆP(Ī,J). Let Ĩ denote the set of buyers wth p (I,J) > 0, and let the tradng prce for buyer be max{p (I,J),f p (I,J)}.

9 ec8 Solve e-companon to Author: Truthful Bundle/Mult-unt Double Auctons ˆP(Ĩ,J) and trade accordng to ts optmal soluton. The revenue for seller j s unt s mn{p j (I,J),g j + ˆV (Ĩ,J) ˆV (Ĩ,J\{j})}. We can defne the modfed lmt LPB paddng mechansm by ntroducng the same addtonal screen stage. The detaled procedure s as follows: Pc the postve lmt vector q = (q c ) c C, where q c > 0 for all c C. Collect the bds from the agents. Calculate the optmal soluton to P(I,J), the VCG payment p (I,J) for each buyer, and p j (I,J), the supremum of as prce g j for seller j such that y = q n the optmal soluton to P(I,J). (When seller j only supples a sngle unt, p j (I,J) s also hs VCG prce.) Let Ī denote the set of tradng buyers n the optmal soluton. Calculate q c J = max{q j j J,c = c j }, the sze of the largest supply of commodty c from a sngle seller. Solve the followng parametrc lnear program wth Q =q J +λq for small λ > 0: ˆP(Ī,J,Q) : Maxmze Ī f x j J g jy j Subject to: Ī qc x +q c J 1 = j J,c j =c y j λq c for each c C 0 x 1 for each Ī 0 y j q j for each j J. There exsts λ such that the optmal soluton s (x +λx,y j +λy j ) for 0 < λ < λ. Let Ĩ denote the set of buyers wth x = 1 and x = 0 n the optmal soluton. In other words, Ĩ s the set of buyers wth x = 1 n the optmal soluton to ˆP(Ī,J,q J + λq) when 0 < λ < λ. The tradng prce for buyer s max{p (I,J), ˆp (Ī,J,q J +λq)}. Solve the followng lnear program: ˆP(Ĩ,J) : Maxmze f Ĩ x g j J jy j Subject to: Ĩ qc x = j J,c j =c y j for each c C 0 x 1 for each Ĩ 0 y j q j for each j J. Trade accordng to the optmal soluton to ˆP(Ĩ,J). The revenue for seller j s g jy j + ˆV (Ĩ,J) ˆV (Ĩ,J g j = p j (I,J),q j = y j ). (When seller j only supples a sngle unt, hs sellng prce can be wrtten as mn{p j (I,J),g j + ˆV (Ĩ,J) ˆV (Ĩ,J\{j})}.)

10 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons ec9 Theorem EC.4. The modfed lmt lnear-program-based paddng mechansm s strategy-proof, ndvdually ratonal, and (wealy) budget-balanced. Proof of Theorem EC.4. We frst show that the modfed lmt LPB paddng mechansm s budget-balanced. Comparng the payment and revenue of the modfed lmt LPB paddng mechansm wth buyer set I and seller set J and those of the lmt LPB paddng mechansm wth buyer set Ī and seller set J, the revenue from the buyers under the modfed lmt LPB paddng mechansm s no less than that under the lmt LPB paddng mechansm, and the payment of the sellers under the modfed lmt LPB paddng mechansm s no more than that under the lmt LPB paddng mechansm. By Theorem 10, the lmt LPB paddng mechansm s (wealy) budget-balanced for any buyer set and seller set, ncludng buyer set Ī and seller set J. Therefore, the modfed lmt LPB paddng mechansm s (wealy) budget-balanced. To show that the modfed lmt LPB paddng mechansm s strategy-proof for the buyers, notce that p (I,J) s ndependent of buyer s bd f accordng to ts defnton. Once buyer bds more than or equal to p (I,J) and acqures her bundle n the optmal soluton to P(I,J), Ī and ˆp (Ī,J,Q) do not vary wth f. When buyer bds lower than max{p (I,J), ˆp (Ī,J,Q)}, / Ĩ, and she does not trade n the allocaton; when she bds hgher than max{p (I,J), ˆp (Ī,J,Q)}, Ĩ, and she trades n the allocaton at prce max{p (I,J), ˆp (Ī,J,Q)}. If she bds max{p (I,J), ˆp (Ī,J,Q)}, she may or may not trade at prce max{p (I,J), ˆp (Ī,J,Q)}. Therefore, when buyer s valuaton s hgher than max{p (I,J), ˆp (Ī,J,Q)}, she prefers to trade, whch can be acheved by bddng truthfully; when her valuaton s lower than max{p (I,J), ˆp (Ī,J,Q)}, she prefers not to trade, whch can also be acheved by bddng truthfully; and when her valuaton s equal to max{p (I,J), ˆp (Ī,J,Q)}, she s ndfferent to tradng. Bddng truthfully s a domnant strategy for the buyers. Now we show that the modfed lmt LPB paddng mechansm s strategy-proof for the sellers. Notce that p j (I,J) s ndependent of seller j s bd g j accordng to ts defnton. Once seller j bds less than or equal to p j (I,J) and y j = q j n the optmal soluton to P(I,J), Ī and ˆp j(ī,j,q) do not vary wth g j. When seller j bds hgher than mn{p j (I,J), ˆp j (Ī,J,Q)}, he does not trade n the

11 ec10 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons allocaton. When he bds less than or equal to mn{p j (I,J), ˆp j (Ī,J,Q)}, Ĩ = Ĩj does not vary wth g j, and buyer Ĩj acqures her bundle n both ˆP(Ĩ,J) and ˆP(Ĩ,J\{}). When seller s as prce s no more than p P Ĩ,[ qc ], the payment he receves s g j y j + ˆV (Ĩ,J) ˆV (Ĩ,J g j = p j (I,J),q j = y j ) = g j y j +(ˆV (Ĩ,J) ˆV (Ĩ,J\{j})) (ˆV (Ĩ,J g j = p j (I,J),q j = y j ) ˆV (I,J\{j}). The VCG compensaton g j y j +(ˆV (Ĩ,J) ˆV (Ĩ,J\{j})), awards seller j a total of y j p j,[ P Ĩj q c j, h] whle the VCG compensaton at prce p j (I,J) wth quantty y s y j max{0,p j,[ P Ĩj q c j p j (I,J)}. The revenue, whch s the dfference, can be wrtten as y j mn{p j(i,j),p P Ĩj j,[ q c j In summary, seller j can sell hs q j mn{p j (I,J),p P Ĩj j,[ q c j },mn{p ] j(i,j),p P j,[ h] h] }. unts unt-by-unt at non-ncreasng prces from Ĩj q c j ] },, to mn{p j(i,j),p j,[ P Ĩj q c j. q j +1] If seller j s unt valuaton s hgher than mn{p j (I,J),p P Ĩj j,[ q c j }, he prefers not to trade, whch ] can be acheved by bddng hs true valuaton. If hs unt valuaton s between mn{p j (I,J),p j,[h] } and mn{p j (I,J),p j,[h+1] } for h Ĩj q c j q j + 1, he prefers to trade Ĩj q c j h unts, whch can also be acheved by bddng hs true valuaton. If hs unt valuaton s lower than mn{p j (I,J),p P Ĩj j,[ q c j }, he prefers to trade all q q j +1] j unts, whch can be secured by bddng hs true valuaton. Bddng truthfully s a domnant strategy for the sellers. Theorem EC.5. For arbtrary postve lmt vector q = (q c ) c C, the modfed lmt LPB paddng mechansm s at least as effcent as the MBC mechansm. Furthermore, each agent s payoff under the modfed lmt LPB paddng mechansm s at least as hgh as that under the MBC mechansm. Proof of Theorem EC.5. Both mechansms go through the same screen stage and have the dentcal Ī. The effcences of the mechansms are equal to those of the BC-LP mechansm and the lmt LPB paddng mechansm startng wth buyer set Ī and seller set J. By Theorem 7, the lmt LPB paddng mechansm s at least as effcent as the BC-LP mechansm for any buyer set and seller set. Therefore, the modfed lmt LPB paddng mechansm s at least as effcent as the MBC mechansm. By Theorem 7, the buyng prce of buyer under the BC-LP mechansm s no less than her buyng prce under the lmt LPB paddng mechansm. The buyng prce of buyer under the MBC

12 e-companon to Author: Truthful Bundle/Mult-unt Double Auctons ec11 mechansm s the maxmum of her VCG prce p (I,J) and her buyng prce under the BC-LP mechansm wth buyer set Ī and seller set J. The buyng prce of buyer under the modfed lmt LPB paddng mechansm s the maxmum of her VCG prce p (I,J) and her buyng prce under the lmt LPB paddng mechansm wth buyer set Ī and seller set J. Therefore, the buyng prce of buyer under the MBC mechansm s no less than her buyng prce under the modfed lmt LPB paddng mechansm. By Theorem 7, the sellng prce of seller j under the BC-LP mechansm s no more than hs sellng prce under the lmt LPB paddng mechansm. The sellng prce of seller j under the MBC mechansm s the mnmum of hs VCG prce p j (I,J) and hs sellng prce under the BC-LP mechansm wth buyer set Ī and seller set J. The sellng prce of seller j under the modfed lmt LPB paddng mechansm s the mnmum of hs VCG prce p j (I,J) and hs sellng prce under the lmt LPB paddng mechansm wth buyer set Ī and seller set J. Therefore, the sellng prce of seller j under the MBC mechansm s no more than hs sellng prce under the modfed lmt LPB paddng mechansm.