In Proc. 17th National Conference on Artificial Intelligence (AAAI-00), pp. 74-81. 1
Iterative Combinatorial Auctions: Theory and Practice
David C. Parkes and Lyle H. Ungar Computer and Information Science Department University of Pennsylvania 200 South 33rd Street, Philadelphia, PA 19104 [email protected]; [email protected]
Abstract route. Although combinatorial auctions can be approxi- mated by multiple auctions on single items, this often results Combinatorial auctions, which allow agents to bid directly for in inefficient outcomes (Bykowsky, Cull, & Ledyard 2000). bundles of resources, are necessary for optimal auction-based solutions to resource allocation problems with agents that iBundle is the first iterative combinatorial auction that is have non-additive values for resources, such as distributed optimal for a reasonable agent bidding strategy, in this case scheduling and task assignment problems. We introduce myopic utility-maximizing agents that place best-response iBundle, the first iterative combinatorial auction that is op- bids to prices. In this paper we prove the optimality of timal for a reasonable agent bidding strategy, in this case iBundle with a novel connection to primal-dual optimization myopic best-response bidding. Its optimality is proved with theory (Papadimitriou & Steiglitz 1982) that also suggests a a novel connection to primal-dual optimization theory. We useful methodology for the design and analysis of iterative demonstrate orders of magnitude performance improvements auctions for other problems. over the only other known optimal combinatorial auction, the Generalized Vickrey Auction. iBundle has many computational advantages over the only other known optimal combinatorial auction, the Generalized Vickrey Auction (GVA) (Varian & MacKie-Mason 1995). As Introduction an iterative auction, agents can incrementally compute val- ues for different bundles of items as prices change, and make Auctions provide useful mechanisms for resource alloca- new bids in response to bids from other agents. In compar- tion problems with autonomous and self-interested agents. ison, the GVA is a sealed-bid auction, in which agents first Typical applications include task assignment and distributed submit bids simultaneously, and then the auctioneer deter- scheduling problems, and are characterized with distributed mines an allocation and payments. In the GVA an agent’s information about agents’ local problems and multiple con- optimal strategy is to bid for, and compute the value of, all
flicting goals (Wellman 1993; Clearwater 1996). Auctions bundles for which it has positive value. This is often im-
¢£¡ ¤¦¥ §¨¥ can minimize communication within a system, and generate possible, since for ¡ items there are bundles to value, optimal (or near-optimal) solutions that maximize the sum each of which may require solving a difficult optimization value over all agents. problem (Parkes 1999a; Sandholm 1993). More recently, electronic commerce has generated new However, combinatorial auctions introduce new compu- interest in auction-based systems, both as dynamic mech- tational complexities in mechanism execution. In particu- anisms to sell items to individuals, and as systems for lar, the auctioneer’s winner-determination (WD) problem, business-to-business transactions. Many retailers have on- the problem of choosing bids to maximize revenue, is © - line consumer auctions, e.g. www.onsale.com, and there hard by reduction from the maximal weighted clique prob- are nascent auctions for procurement in the supply-chain, lem (Rothkopf et al. 1998). e.g. www.freemarkets.com. However, at present the In iBundle the auctioneer must solve a sequence of WD vast majority of online auctions are simple variations on the problems (one in each round) to maintain a provisional allo- traditional English auction, an ascending-price single-item cation as agents bid. In comparison, in the GVA the auction- auction. eer must solve one WD problem for each agent in the final We introduce iBundle, an iterative combinatorial auction allocation. Each WD problem in iBundle is smaller than that allows agents to bid for bundles of items while the auc- in the GVA, because agents bid for less bundles. In addi- tioneer increases prices and maintains a provisional alloca- tion, the auctioneer can increase the minimal bid increment tion. Bundles are important in many real-world problems: and reduce the number of rounds to termination, reducing consider a manufacturer that needs either components A and computation for some loss in economic efficiency. Further B, or just component C; consider a mobile agent that needs speed-ups are achieved through caching of solutions from an interval of compute time; consider a train that needs a previous rounds in the auction, and introducing approximate bundle of departure and arrival times on tracks across its WD algorithms that maintain the same incentives for myopic Copyright c 2000, American Association for Artificial Intelli- agents to bid truthfully in each round. gence (www.aaai.org). All rights reserved. We note that the GVA has stronger truth-revelation prop- erties than iBundle. Truthful bidding is optimal in the GVA Approximate Winner-determination. The auction-
whate ver the bids of other agents. In comparison, ratio- eer can also use an approximate algorithm for winner- nal agents with lookahead could manipulate the outcome of determination, and still maintain the same incentives for iBundle to their advantage, and lead to suboptimal alloca- myopic agents to follow the same bidding strategy. To tions. However, there is some evidence that myopic bid- achieve this an approximate algorithm must have the bid- ding may be a reasonable assumption in practice, perhaps monotonicity property: because of the computational complexity of strategic behav- Definition 1. Bid monotonicity. An algorithm for winner- ior. For example, in the FCC broadband spectrum auction, determination satisfies bid monotonicity if whenever an
conducted as a set of simultaneous ascending-price auctions
% 0 agent is allocated a bundle with bids , it is also allo-
on spectrum licenses, bids were rarely above minimum ask %132 cated a bundle with bids 0 that include a bid for an prices and jump bids were the exception (Cramton 1997). additional bundle 2 . In Parkes & Ungar (2000) we present a simple extension to iBundle that makes it robust to strategic manipulation in It is straightforward to prove that optimal winner- several interesting problems; we adjust the final prices in determination algorithms are bid-monotonic. iBundle towards Vickrey prices. Prices. The price-update rule generalizes the rule in the English auction, which is an ascending-price auction for a The Ascending-Price Bundle Auction single item. In the English auction the price is increased whenever two or more agents bid for the item at the current iBundle is an ascending-price auction that allows agents to price. In iBundle the price on a bundle is increased when bid on arbitrary combinations of items during the auction. one or more agents that do not receive a bundle in the current The auctioneer increases prices on bundles as bids are re- allocation bid at (or above) the current ask price for a bundle. ceived and maintains a set of winning bids that maximize The price is increased to + (the minimal bid increment) above
revenue. the greatest failed bid price. The initial ask prices are zero.
Let denote the set of items to be auctioned, denote
4'576 &(*) The auctioneer announces a new ask price, in
the set of agents, and denote a bundle of items. The round , for all bundles that increase in price. Other
auction proceeds in rounds, indexed . We describe the bundles are implicitly priced at least as high as the great- 48596
types of bids that agents can place, and the allocations and 48596
&:<;=)>? &(@) est price of any bundle they contain, i.e.
price updates computed by the auctioneer.1
for ;*A . These ask prices are anonymous, the same for
Bids. Agents can place exclusive-or bids for bundles, e.g. all agents. XOR , to indicate than an agent wants either all items Price discrimination. In some problems the auctioneer
2
in or all items in but not both and . introduces price discrimination based on agents’ bids, with
! "$# %'&(*)
Agent associates a bid price with a bid for bun- different ask prices to different agents, when this is neces- dle in round , non-negative by definition. The price must sary to achieve an optimal allocation. A simple rule dynam-
either be within + of, or greater than, the ask price announced ically introduces price discrimination on an agent-by-agent
by the auctioneer (see below). Parameter + ,.- defines basis, when an agent submits bids that are not safe: the minimal bid increment, the minimal price increase in the Definition 2. Safe bids. An agent’s bids are safe if the agent auction. Agents must repeat bids for bundles in the current is allocated a bundle in the current allocation, or it does allocation, but can bid at the same price if the ask price has not bid at or above the ask price for any pair of compatible
increased since the previous round.3 *CDEGFH bundles $B8 , such that . Winner-determination. The auctioneer solves a winner- Suppose agent bids unsuccessfully for compatible bun-
determination problem in each round, computing an alloca-
dles and in round . It is still possible that bids for
tion of bundles to agents that maximizes revenue. The auc- bundles and from two different agents can be suc- tioneer must respect agents’ XOR bid constraints, and cannot cessful at the prices. Remember that the XOR bid constraint allocate any item to more than one agent. The provisional prevents the auctioneer accepting both bids from agent . allocation becomes the final allocation when the auction ter- When an agent’s bids are not safe the agent receives
minates.
48596I# % &(*) individual ask prices, , in future rounds. Indi- vidual prices are initialized to the current general prices,
1The iBundle auction has three variations, that differ in their
48596I# % 48596
&:*)JFK &(@) + , and increased to above the agent’s price update rules (Parkes 1999b). In this paper, we use iBundle bids in future rounds that the agent receives no bundle in the both to refer to the family of auctions in general, and also to varia- tion iBundle(d), which we describe in detail. provisional allocation. 2Exclusive-or bids provide complete expressibility, but are not Termination. The auction terminates when: [T1] all necessarily computationally efficient for all problems. We can de- agents submit the same bids in two consecutive rounds, or rive price-update rules for other bid languages (Parkes 1999b). 3 [T2] all agents that bid receive a bundle.
An agent can also bid / below the ask price for any bundle in any round— but then it cannot bid a higher price for that bundle in the future. This allows an agent to bid for a bundle priced slightly A Myopic Best-Response Bidding Strategy above its value. iBundle computes an optimal allocation with myopically ra-
2 tional agents that play a best (utility-maximizing) response ficiency5 (Parkes 1999b) compared to 82% allocative effi-
to theL current ask prices and allocation in the auction. The ciency from non-combinatorial auctions in the same prob- agents are myopic in the sense that they only consider the lems. We found that price discrimination only had a no-
current round of the auction. ticeable effect on allocative efficiency with very small bid
S Q
Let MON8P(Q@R denote agent ’s value for bundle , and assume increments, and after many rounds of bidding.
M P(TUR*VXW M P(Q*YZR>[\M P:Q*R
N N
N and free disposal of items, so that
Q Q
for all Y^] . Consider a risk-neutral agent, with a quasi- Proof of Optimality Q linear utility function _`N'P(Q*RaVbMON8P(Q*RdcfegP:Q*R for bundle The proof of iBundle’s optimality is inspired by a proof at price egP(Q@R . Further, assume that agents are indifferent to due to Bertsekas (1990) for a simpler iterative auction, and within a utility of hji , the minimal bid increment. This is makes an interesting connection with primal-dual theory of reasonable as i¨klW . linear programming. It helps to motivate the price-update By definition, a myopic agent bids to maximize utility at rules, the safety condition for introducing price discrimina- the current ask prices (taking an i discount when repeating tion, and the conditions for termination. a bid for a bundle in the provisional allocation or bidding Primal-dual is an algorithm-design paradigm that is of- for a bundle priced just above its value). The myopic best- ten used to solve combinatorial optimization problems (Pa-
response strategy is to submit an XOR bid for all bundles padimitriou & Steiglitz 1982). A problem is first formulated
i _`N8P(Q*R Q that maximize (to within ) utility at the current both as a primal and a dual linear program (see the exam- prices. This maximizes the probability of a successful bid ples below for iBundle). A primal-dual algorithm searches for bid-monotonic WD algorithms. for feasible primal and dual solutions that satisfy comple- mentary slackness conditions, instead of searching for an Theoretical Results optimal primal (or dual) solution directly. Complementary- slackness (CS for short) expresses logical relationships be-
We are now ready to introduce our main theoretical results. tween the values of primal and dual solutions that are neces-
n£m m opm Recall that m is the number of items, is the number of sary and sufficient for optimality: agents, and i is the minimal bid increment. Complementary-Slackness Theorem. Feasible primal and
Theorem 1. iBundle terminates with an allocation that is dual solutions are optimal if and only if they satisfy comple-
n£mzyIm opm|{Ii within q Theorem 2. iBundle(2) terminates with an allocation that ple transformation between iBundle and iBundle(2). n£mym o`mz{i is within q*r}sZuw~m of the optimal solution when bids Figure 1 presents a standard integer program formula- are safe, for myopic best-response agent bidding strategies. tion of the combinatorial resource allocation problem. The objective is to maximize the total value of the allocation, As an example, bids are safe if each agent bids for a set of M P(Q@R Q S given value N for bundle to agent . Integer variables conflicting bundles in every round of the auction. iBundle(2) S N8P(Q*RdfwW y!O{ indicate whether or not agent receives bun- also provably solves the following problems without price dle Q . Constraints (IP-1) ensure that each agent receives discrimination: (1) every agent demands different bundles; at most a single bundle, constraints (IP-2) ensure that each (2) agents have additive or superadditive values, i.e. MpP(Q item is allocated to at most one agent. Q Ra[M`P:Q*R< MpP(Q R Q Q Y Y Y for non-conflicting bundles and ; Bikchandani & Ostroy (1998) formulate the combinato- (3) the bundles that receive bids throughout the auction are rial resource allocation problem as a linear program, see from a single partition of items, e.g. all bids are for pairs of P(Q*RwIW¦yO{ [LP ] in Figure 2. The integer constraints N matching shoes, or single items. 3 in [IP] are relaxed to N'P:Q*R[W , and new variables In experimental tests iBundle(2) performs well in many are introduced which correspond to a partition of items into hard problems, achieving an average of 99% allocative ef- bundles. is the set of all possible partitions. Constraints 4Label 2 refers to “second-degree” price discrimination, non- 5Allocative efficiency is a measure of optimality, computed as linear prices in bundles of items but identical prices across agents the ratio of the total value of the allocation across all agents to the (Bikchandani & Ostroy 1998). value of the optimal allocation. 3 ¡:¢*£¥¤O ¡:¢*£ U the dual problem computes competitive equilibrium bundle JO [IP] Z prices that minimize the sum of agent utility and auctioneer ¡:¢*£>©Xª ` revenue, see (1) and (2) below. ¨§ ¦§ We prove that iBundle(2) implements a primal-dual algo- ¬p « (IP-1) º º ¡:¢*£>©Xª LP DLP ` rithm for [ ] and [ ], and computes integral solutions º to [LP ] when agents follow myopic best-response bidding ¬¦± « (IP-2) strategies and bids are safe. First, we show that the alloca- ¡:¢*£>²f³´ ªOµ ¢ `*®°¯ tion and prices in each round of the auction correspond to feasible primal and dual solutions. Then, we show that the « « ¬`°« primal ¢Eand dual solutions satisfy CS when the auction ter- ¢ minates.¡(¢*£ Figure 1: Combinatorial resource allocation problem: Integer pro- Å Ë8Ì9ÍLet denote the provisional allocation to agent , and gram [IP] formulation. denote the ask price for bundle . '¡:¢(£dÎϪ 8¡(¢<ÐÑ£¨Î»´ ¢<Ð*ÒÎÓ¢ |¡:¢*£¥¤O ¡:¢*£ ¾ ÔÖÎ× ¢Ø ¢<Ù Ù Ú ¡(¾¦£*ÎX´ ¡:¾ ÔÕ£¨ÎªFeasible primal. To construct a feasible primal solution U tO assign and for all . Partition º [LP ] §§!§ Z¶® ·$Z¸¹ « for « , and otherwise. ¡:¢*£d©»ª ` ½ ½ ¦§ ¨§ ¡:¢*£dÎ ¡(¢@£ ¬p « (LP-1) Å Å Feasible dual.Ë8Ì9Í To construct a feasible dual solution as- ¡:¢*£d© ¡(¾¦£ ¢ ¦p sign . Constraints (DLP-1) and (DLP-2) are U U¼ ¬ « satisfied with assignments: £*Î ´ ³¤O ¡(¢*£^É ¡:¢*£°µÝÜ (LP-2) ¡ ¡:¾¦£d©»ª ¸ ® ¸p½ Å Å tOaÛ tO « U¼ (1) p ¡(¢*£ ÇÎ (LP-3) U¼ ¡(¢@£ ¡:¾¦£d¿À´ ¢ ¾ Å ¸ ½ tO (2) ¸ ¡ £ « « ¬p8« « ¸ Å Ç ½ ¡ £EÆÈÇ The value can be interpreted as agent ’s maximum 9UÅ tÁZ utility at the prices, and can be interpreted as the maximum [DLP º ] Ã$ ¶® Ã$Z¶® Ä ¡ £EÆ ¡:¢*£d¿\¤O ¡:¢*£ ¢ £ revenue that the auctioneer can achie¡ ve at the prices (irre- Å Å spective of the bids placed by agents).Å The auctioneer does ¦!§ ¨§ ÇDÉ ¡:¢*£d¿À´ ¾ « ¬`°« (DLP-1) not explicitly compute the value of , rather we prove that Å ¬ « (DLP-2) the allocation and prices in the auction satisfy CS with these ¡ £ ¡(¢*£ Ç3¿À´ ¢ ¸ assignments¤OÞ¡(¢@£ when the auction terminates, based on the bids Å Å placed by agents. This is just as well, because the values « « « ¬p8« remain private information to agents during the auc- tion. Figure 2: Combinatorial resource allocation problem: Primal lin- Complementary-slackness conditions. The first primal Ê Ê ¡(¢*£dßÖ´áà ¡ £Æ ¡(¢@£*Î ¤ ¡(¢*£ ¢ ear program [LP ] and dual linear program [DLP ] formulations. CS condition,6 (CS-1) is: Å Å « ¬`°« (CS-1) (LP-2) and (LP-3) replace constraints (IP-2), and ensure that Given (1) it states that all agents must only receive bun- a feasible solution does not allocate more than one of each dles that maximize utility at the current prices. (CS-1) item. is maintained throughout the auction because bundles¢ are In general, an optimal solution to the linear program [LP º ] ¡:¢*£© ¡(¢*£ only allocated according¡:¢*£âÉ㻩to bids from agents, and agents Å äå æ ÅÝË'Ì9Í can allocate fractional items to agents, and need not be a fea- ÅÝË'Ì9Í ¤ ¡(¢@£~É ¡(¢*£ÆDãÖ¿ ³¤ ¡:¢<Ð=£~É ¡(¢<ÐÑ£°µ place best-response bids. Formally, for any bundle bid ® IP LP º Å Å æ ä!å æ sible solution to [ ]. In fact, the optimal solution to [ ] äå by agent : (i) ã ; (ii) tO èç ® is integral and solves [IP] if and only if non-discriminatory ® ¡(¢*£j¿Ó´ ¤OÞ¡(¢@£*É because æ bundle prices exist that support the optimal allocation in ÅÝä!å agents bid for bundles that maximize'¡:¢¶£éÎêª utility within ; (iii) ® competitive equilibrium with best-response agent bidding ¢ , because agents only bid for bundles strategies (Bikchandani & Ostroy 1998). Competitive equi- with positive utility. Since implies agent bid ¡(¢*£gÆìëUã¿ librium implies that agents’ maximize utility and the auc- for bundle8¡(¢*£dß ´Xàê¤O'¡:¢*£^É, we have: Å Ë8Ì9Í tioneer maximizes’ revenue given the final prices and the Ð Ð ³¤O8¡(¢ £^É ¡(¢ £¹µ~î final allocation. ´ ¡ £ ¡:¢*£ Ç Å Ë8Ì9Í J$áí J$ « º º Å The dual problem,Å [DLP ], to primal [LP ] is shown in ç Figure 2. Variables , and correspond to con- LP LP LP 6 ¡(¢*£ ¡(¾¦£ straints ( -1), ( -2) and ( -3) respectively, and dual Complementary slackness states that if a primal variable is constraints (DLP-1) and (DLP-2) correspond to primal vari- non-zero then its corresponding dual inequality constraint is bind- ables and . When the primal solution is integral ing. Similarly for dual variables. ½ 4 ï ô8õ9öÞð:÷*ò*øïùð(÷*ò ú Substituting for ïgð7ñóò and , we prove -CS- the ask price and have values just below ask prices, other- ü 1: ûÝü wise prices would increase and their bids would change. ¡ ¡§¦ ïgð7ñóò ïùð(÷*ò£¢¥¤ ð:÷*ò ú©¨ pñ ¨8÷ ð(÷*òdý þXÿ Finally, the last pair of dual CS conditions, (CS-4) and ( ú -CS-1) (CS-5), are: ü û ü Z ð(÷*ò¨ø ð ¦ò\¨ ÷ ïùð(÷*òdý þáÿ (CS-4) B[ The second primal CS condition, (CS-2), is: ©BA ; ¦òdý þâÿ ïgð:÷*ò@ø þ¨ ð (CS-2) fý þáÿ] ð0 ¦ò*ø (CS-5) ©BA Given (2) it states that the allocation must maximize the /òêø /lø The assignment ð for the partition ^ auctioneer’s revenue at prices ïgð:÷*ò , over all possible allo- [ ÷+a a b ÷ trivially satisfies the RHS of both conditions. cations and irrespective of bids received from agents. We #`_6_6_ prove (CS-2) is maintained in all rounds because it is not Termination. By contradiction, assume the auction never binding that the auctioneer must allocate bundles according terminates. Informally, [T1] implies that agents must submit to agents’ bids. Through the price-update rules the auction- different bids in successive rounds, but with myopic best- eer is able to maximize revenue given prices in every round. response bidding this implies that prices must increase, and Formally: (i) Agent ñ with one of the highest losing bid agents must eventually bid above their values for bundles. ü ÷ for bundle ÷ in round will continue to bid for bundle in We prove a contradiction with myopic best-response bidding ¡ ü ü ð:÷*ò ñ rounds . Let denote agent ’s utility for bundle strategies. ü ü "!$# ÷ ð:÷*òáø% &\ú in round . Then, ð(÷*ò because the ask Putting it all together. Summing ú -CS-1 over all agents ÷ ú ð(÷*ò(') ð(÷+*Ñò ü ü ü price for increases by . Also, for all ü ïgð9ñóòø þ ü ü in the final allocation, and with for agents not 1 1 ÷,* bundles the agent did not bid in round . Hence, with ü ü B[ B[ c¢ ¤ ð:÷ òd CS ïùð9ñóò ü ü in the allocation by ( -3), "!$# 1 ð(÷+*Ñò ð:÷+*Ñò£'§ ÷+* because the price of can only increase ¡e¦ =fEgGOIKJ LMJ J N JhP B[ ¡- ò ¨ ú ïgð:÷ , because an allocation can "!$# "!$# ð(÷@ò£'§ ð(÷+*Ñò.}ú in round , we have and a bid for include no more bundles than there are items or agents. ÷+* ÷ ñ ü can never exclude a bid for from agent ’s best-response ¡¥ ð /!ò»ø Introducing ú -CS-2, because for the bundle- ü bids in round . A similar argument can be made for the ü i¢ set that corresponds to the final allocation ÷ , then 1 ¡ =FEHGIKJ LMJ J NOJQP ü utility of bundles that the agent did bid in round ; (ii) No ü ü ü B[ ò ¨ ú ïgð:÷ . Finally, adding these two 1 1 ¡ single agent causes the price to increase to its current level ¡ B[ B[ ïùð9ñóò¢ ¤ ð(÷ ò equations, we have on a pair of compatible bundles. This follows because price j =FEHGOIkJ LMJ J N JhP ú updates are due to safe bids from agents. ¨ . The LHS is the value of the final dual ü ü ü solution, lmVn7o , and the first-term on the RHS is the value ð0 /ò ø Therefore, for partition / such that , 1 1 ü ü ü ln7o lp/ ¢qlmVn7o 72465 nBo 32465 of the final primal solution, . We know , ï ð(÷ ò ' ï869 :3; ð(÷ ò ô8õ9ö , because 1 / l ü ü where n7o is the value of the optimal primal solution by 5 32< 869 :3; 869 :3; ' ï ð(÷*ò ï ð(÷ ò ' ï ô8õ9öóð:÷*ò , and 1 =?>3@ the weak duality property of linear programs. Thus, be- ©BA 2 j ¡ 869 :3; ï ð(÷ ò LMJ J NOJQP because of (i) and (ii), i.e. =FEHGIKJ lUmVn7o]¢rlUnBo ¨ ú ü ü cause , it follows that j LMJ J NOJQP the constraints to allocate to agents’ bids are not binding. =FEgGOIKJ 1 / lUn7o('¥l ¨ ú n7o =?>7@ ü . Finally, because the primal ©BA 324 8C9 :7; ð:÷ òD' Finally, with (2) we have ï solution is integral (by construction during iBundle), it is a =FEHGIKJ LMJ J NOJQP 869 :3; ¨ ú ï ð(÷*òR' ï ô8õ9öóð:÷*òS»ú ¥ because and feasible and optimal solution to the combinatorial resource an allocation can include no more bundles than there are allocation problem [IP]. agents or items. We prove ú -CS-2: In addition, it follows immediately that iBundle(2) ter- =FEHGIKJ LMJ J NOJQP minates in competitive equilibrium when agents are myopi- ð0 ¦òý\þéÿDT§ ïùð(÷*ò£¢ ¨ ú©¨ V U cally rational and place safe bids. ( ú -CS-2) Proof of iBundle A simple transformation of agents’ bids allows iBundle to be The first dual CS condition, (CS-3), is: ü û implemented within iBundle(2) and ensures that agents’ bids remain safe throughout the auction. Whenever bids from W ð(÷@ò*ø ¨ pñ ïùð9ñóòdýÀþáÿ (CS-3) UXOY agent ñ are not safe in iBundle(2) we can simulate the price- update rule in iBundle by introducingü a new dummy item Given (1) it states that every agent with positive utility for that is specific to that agent, call it s . This item is concate- some bundle at the current prices must receive a bundle in nated by the auctioneer to all bids from agent ñ in this round the allocation. (CS-3) is only satisfied during the auction and all future rounds. It has the following effects: for agents that receive bundles in the provisional allocation, 1. The outcome of winner-determination, or the allocative but we prove (CS-3) for all agents when iBundle(2) termi- ü nates. In termination case [T2] every agent that bids receives efficiency of the auction, is unchanged because no other agent bids for item s . a bundle, so we immediately have (CS-3) with myopic best- ü response agents. In case [T1] some agents may bid and re- 2. Agent ñ ’s bids are always safe because every bid includes s ceive no bundles. However, these agents must bid at ú below item , and no pair of bids is compatible. 5 3. The price increases due to bids from agent t are isolated to items, with XOR valuation functions, such that agents want thatu agent in all future rounds because all price increases at most one bundle. In our main experiments the number of M`7 are for bundles that include item v?w . items, , and we scale the problems by increasing The optimality of iBundle follows immediately from the the number of agents from 5 to 40, with values for 10 bun- . optimality of iBundle(2). dles per agent. We set Sandholm’s parameter q in Decay, and select bundles of size 10 in Uniform. Computational Analysis Results are presented for iBundle(2), the auction variation without price discrimination. A variation on Sandholm’s As an iterative auction, iBundle has many computational ad- depth-first branch-and-bound search algorithm (Sandholm vantages for agents over the sealed-bid GVA, as we discussed 1999) solves winner-determination (WD) in each round, and in the introduction. In Parkes (1999b) we present results that computes the allocation and prices in the GVA. We introduce demonstrate savings in agent valuation work in iBundle. a new heuristic to make search more efficient for XOR bids. However, the winner-determination (WD) problem that The heuristic computes an overestimate of the possible value the auctioneer solves in each round of iBundle to compute of a partial allocation based on allocating at most one bundle the provisional allocation is xzy -hard, just as in the GVA. to each remaining agent without a bundle. The auctioneer must solve one WD problem in each round, In addition, we measure the performance of iBundle with and a naive worst-case analysis gives {F|0}p~+.7\ rounds a greedy approximate winner-determination algorithm due to converge, for a total of } bundles with positive value over to Lehmann et al. (1999) that satisfies the bid-monotonicity ~ all agents, maximum value + for any bundle, and mini- property (Definition 1). mal bid increment . In the worst-case the price of a single bundle must increase by at least in each round the auc- 6 tion remains open, and prices are bounded by the maximum 10 value over all agents. The number of rounds to termina- 5 GVA tion is inversely proportional to the minimal bid increment. 10 The auctioneer can solve less WD problems by increasing the minimal bid increment, for some loss in economic effi- 4 95% ciency. 10 99% A number of optimizations are possible within iBundle 3 Truthful to speed-up computation on winner-determination in each 10 round. First, the provisional allocation from the previous 2 round provides a good initial solution to the WD problem, 10 because agents must re-bid bundles received in the previ- 85% Auctioneer CPU Time (s) ous round. This allows pruning of the search for a revenue- 1 80% maximizing allocation. An additional saving in computa- 10 tion time is achieved by limiting search to an allocation at 0 least better than the value of the allocation in the previous 10 5 10 15 20 25 30 35 40 round. Moreover, although each intermediate WD problem Number of Agents in iBundle may be intrinsically more difficult than each WD problem in GVA because all agents bid at similar prices for bundles (Andersson et al. 2000), the problems are typically Figure 3: Total computation time in iBundle(2), the GVA, and a sealed-bid auction with truthful agents, in problem set Decay. The much smaller than in the GVA. performance of iBundle is plotted with different bid increments , The auctioneer only announces price increases in each selected to give allocative efficiency of 80%, 85%, 95% and 99%. round, and need not maintain explicit prices for all possible bundles. Bid prices are verified dynamically in each round, Figure 3 plots the total auctioneer winner-determination to check that bids are at least as large as the ask price of 7 all contained bundles. With a simple sorted-list implemen- and price-update time in iBundle in the Decay problem set. tation, the total work in checking each bid is linear in the Performance is measured for different bid increments, with the bid increment selected to give allocative efficiency of number y of bundles that have explicit ask prices. Simi- 80%, 85%, 95% and 99% ( p ). Figure 3 also plots per- larly, prices can be maintained in linear-time in y for each formance for the GVA, and for a sealed-bid auction in which new price increase. In addition, yZ} , with agents that agents are assumed to bid truthfully.8 Results are averaged have values for } bundles, because only bundles that receive bids can receive explicit ask prices. over 10 trials. First, note that the curves are sublinear on the logarithmic value axis as the number of agents increases, Experimental Results 7Time is measured as user time in seconds on a 450 MHz Pen- We compare the computation and communication cost of tium Pro with 1024 MRAM, with iBundle coded in C++. iBundle with the Generalized Vickrey Auction (GVA). 8The GVA proved intractable for 30 and 40 agents. In those We consider problems Decay, Weighted-random (WR), problems the run time is estimated as the time to compute the opti- Random and Uniform from Sandholm (1999). Each prob- mal solution in a single WD problem multiplied by the number of lem defines a distribution over agents’ values for bundles of agents in the optimal allocation. 6 indicating polynomial computation time in the number of change in problems with agents that have values for many agents. bundles because all values must be reported in the GVA, or The performance improvement of iBundle over GVA is in easier problems because iBundle will terminate quickly striking, achieving at least one order of magnitude improve- with less bids. ment with 99% allocative efficiency and three orders of mag- The performance of iBundle with the greedy WD algo- nitude with 85% allocative efficiency. For up to 95% effi- rithm is noteworthy: iBundle performs well in the hard De- ciency we essentially get the myopic truth-revelation prop- cay problem set, with allocative efficiency 85.1%, giving at erties of iBundle for free, because iBundle’s run-time is least a 1000-fold reduction in WD time. We believe that approximately the same as for the sealed-bid auction with other, slightly less greedy, approximate algorithms will give truthful agents. even further performance improvements. Problem GVA iBundle Approx- Speeding up iBundle In addition to using the allocation ©©C© Bundle from the previous round to prune search, it is also useful to Decay Eff (%) 100 91.5 94.9 98.3 85.1 cache all previous provisional allocations and select the best 67.3% WD-time (s) 41700 831 2400 5650 0 cached allocation as an initial solution for WD. A simple ¡ 13.4 Pr-time (s) – 26 34.5 44 39.2 linear program is used to select the best allocation from the Comm ¢ (kBit) 18.8 221 306 394 377 cache, and requires negligible computation. In our main tri- WR Eff (%) 100 90.7 94.9 99.2 79.4 als we use a cache size of 1, i.e. take the solution from the 71.5% WD-time (s) 3 0.6 1.7 6 0 previous round as an initial solution to the WD problem. 1 Pr-time (s) – 5.4 11.5 40.9 12.2 Comm (kBit) 18.1 20.5 52.1 144 53.1 Problem WD Time % Cache Rand Eff (%) 100 89.3 97 99 95.8 ¯® 0 1 Correct 37.8% WD-time (s) 68 4.4 7.4 11 0 Decay 50/15/150 415 371 355 291 0 28 47 59 11.2 Pr-time (s) – 6.5 9.7 12.1 12.9 WR 50/50/1000 253 243 231 163 0 11 57 57 Comm (kBit) 18.7 49.5 66.4 82.6 85.6 Rand 50/30/600 1823 1616 1491 864 ° 0 6 30 78 Unif Eff (%) 100 – 95.6 99.1 76.2 Unif 50/40/800 343 337 336 110 0 14 29 49 58% WD-time (s) 25 – 6.6 18.7 0 3 Pr-time (s) – – 14.7 42.0 46 Comm (kBit) 18.2 – 56.5 120 124 Table 2: Winner-determination time with caches of size 0, 1 (last ¯® round), and (all previous rounds). In cache revenue maximiz- ing cached solutions from previous rounds are assumed optimal. ©³B´ µ© Table 1: Performance in the Decay, WR, random, and uniform C ± ² £ ¤¥£ Eff in all problems except ° , where Eff . / ¢ problems. Auctioneer WD time. ¡ Price-update time. Commu- ¶U£ £ / # bundle values. nication cost. Alloc. eff. of a sealed-bid auction with a greedy WD algorithm and truthful agents. Average number of agents in the optimal solution. Table 2 compares the WD time in each problem with and without caching of previous allocations. Although a Table 1 compares iBundle with the GVA for all Sand- full cache can provide an additional speed-up over using holm’s problems, for problems with 30 agents. With our no cached solutions, or just the allocation from the previ- parameters the WR and Uniform problems are quite easy ous round, the effect is not very dramatic. The reason is that because the optimal allocation sells large bundles to a few it remains expensive to verify that a cached solution is opti- agents, which allows considerable pruning during search. mal. For example, although an extended cache in the Decay The Random and, in particular, Decay problems tend to problem provides the correct allocation in 47% of problems, be harder because the optimal allocation requires coordina- the speed-up is limited to around 14%. tion across a number of agents, see also Sandholm (1999) In an attempt to leverage the correct solutions from the and Andersson et al. (2000). In all problems iBundle has cache, we tested the performance of iBundle under an ad- less WD time at 95% allocative efficiency than the GVA. ditional assumption that if a cached solution from before Note that price-update is relatively expensive in the oth- round ·,¸º¹ generates more revenue than the solution from erwise easy weighted-random (WR) problem, because bid round ·£¸»¹ , this is adopted as the new provisional alloca- prices for large bundles must be checked for price consis- tion without further computation. The rule is designed to tency against the price of all included bundles. capture “flip-flop” competition between a number of good There is a communication cost9 penalty in using iBundle allocations during an auction. compared with the GVA in these problems (Table 1) because Labeled ¼¾½ , the rule proves useful in Decay, WR and Uni- of repeated bids across a number of rounds. This would form, reducing computation by 30%, 36% and 68% from the 9 time with no cache for a negligible drop in allocative effi- We assume that bids and price information in iBundle must ciency. However, one must be careful: although we also see only specify a bundle, because bids are usually at the current ask a speed-up in Random, the allocative efficiency falls from ¿BÁU Ã.À price, and ask prices only increase by the minimal bid increment. ¿B¿.À We also assume a broadcast network infrastructure for price up- to . Further analysis shows that cached solutions prove optimal in 54%, 97% and 49% of rounds in Decay, ¤¥£ dates. A bundle is specified with £ bits. In the GVA a bid specifies both a bundle and a value. We assume that values re- WR and Uniform, but only optimal 34% of rounds in Ran- quire 10 bits, enough to specify a value to 3 significant figures dom. ©©©¬$ « ( ¦h§©¨B©©ª4« .) Further optimizations should be possible, for example 7 using cached solutions once a large enough cache is con- References structed,Ä and solving WD when an auction is about to ter- Andersson, A.; Tenhunen, M.; and Ygge, F. 2000. Integer pro- minate with cached solutions. Another useful approach is gramming for auctions with bids for combinations. Forthcoming, Å -scaling, that adjusts the bid increment during an auction Proc. ICMAS’00. (Bertsekas 1990). Banks, J. S.; Ledyard, J. O.; and Porter, D. 1989. Allocating uncertain and unresponsive resources: An experimental approach. Related Work The Rand Journal of Economics 20:1–25. Rassenti et al. (1982) describe an early single-round combi- Bertsekas, D. P. 1990. The auction algorithm for assignment and natorial mechanism for airport slot allocation, while Banks other network flow problems: A tutorial. Interfaces 20(4):133– 149. et al. (1989) describe AUSM, an early iterative bundle auc- tion. AUSM has no explicit price-update rules, and agents Bikchandani, S., and Ostroy, J. M. 1998. The package assignment must solve hard problems to bid effectively. DeMartini et model. Technical report, Anderson School of Management and Department of Economics, UCLA. al. (1998) describe, RAD, an iterative extension of Rassenti et al., also with linear prices. No optimal properties have Bykowsky, M. M.; Cull, R. J.; and Ledyard, J. O. 2000. Mutually been proved for any of these auctions in general problems. destructive bidding: The FCC auction design problem. Journal of Regulatory Economics. The AkBA (Wurman 1999, chapter 5) auctions are con- ceptually similar to iBundle, but have different price-update Clearwater, S. H., ed. 1996. Market-Based Control. World Sci- rules and no price discrimination. AkBA shares many of entific. iBundle’s computational properties, but is not known to be Cramton, P. 1997. The FCC spectrum auctions: An early assess- optimal for any reasonable bidding strategy. ment. J. Economics and Management Strategy 6:431–495. There have been a number of proposals to reduce the com- DeMartini, C.; Kwasnica, A. M.; Ledyard, J. O.; and Porter, D. putational costs of combinatorial auctions while maintain- 1998. A new and improved design for multi-object iterative auc- ing incentives for truth-revelation; e.g. limit the types of tions. Technical Report SSWP 1054, California Institute of Tech- nology. Revised March 1999. bundles that agents can bid for (Rothkopf et al. 1998); or introduce an approximate solution for winner-determination Lehmann, D.; O’Callaghan, L.; and Shoham, Y. 1999. Truth rev- elation in rapid, approximately efficient combinatorial auctions. (Lehmann et al. 1999), but little success in designing good In Proc. ACM Conf. on Electronic Commerce (EC-99). auctions for general bundle problems. Moreover, most pre- vious work focuses on sealed-bid auctions. Papadimitriou, C. H., and Steiglitz, K. 1982. Combinatorial Op- timization: Algorithms and Complexity. Prentice-Hall. Parkes, D. C., and Ungar, L. H. 2000. Preventing strate- Conclusions gic manipulation in iterative auctions: Proxy agents and price- iBundle is a new iterative combinatorial auction that is opti- adjustment. In Proc. 17th National Conference on Artificial Intel- mal for myopically-rational agents. As an iterative auction, ligence (AAAI-00). To appear. iBundle is particularly useful when agents have hard local Parkes, D. C. 1999a. Optimal auction design for agents with valuation problems because it allows agents to compute es- hard valuation problems. In Proc. IJCAI-99 Workshop on Agent timates of the value of different outcomes incrementally, in Mediated Electronic Commerce. Stockholm. response to bids from other agents. We proved iBundle’s Parkes, D. C. 1999b. iBundle: An efficient ascending price bundle optimality within a primal-dual framework, which we be- auction. In Proc. ACM Conf. on Electronic Commerce (EC-99), lieve will provide a useful conceptual basis for the design 148–157. and analysis of iterative auctions for other problems. Rassenti, S. J.; Smith, V. L.; and Bulfin, R. L. 1982. A combina- It remains expensive to compute optimal solutions with torial mechanism for airport time slot allocation. Bell Journal of iBundle in many problems, because the auctioneer’s winner- Economics 13:402–417. determination (WD) problem is ÆzÇ -hard. We suggested a Rothkopf, M. H.; Pekec,ˇ A.; and Harstad, R. M. 1998. Compu- number of techniques to reduce computation, possibly for tationally manageable combinatorial auctions. Management Sci- some loss in allocative efficiency, for example: increase the ence 44(8):1131–1147. bid increment, use cached allocations, and introduce approx- Sandholm, T. 1993. An implementation of the Contract Net Pro- imate winner-determination algorithms. We demonstrated tocol based on marginal-cost calculations. In Proc. 11th National orders of magnitude performance improvements over the Conference on Artificial Intelligence (AAAI-93), 256–262. GVA, the only other known optimal combinatorial auction, Sandholm, T. 1999. An algorithm for optimal winner determina- in some hard problems. tion in combinatorial auctions. In Proc. 16th International Joint In future work we plan to test iBundle in some real prob- Conference on Artificial Intelligence (IJCAI-99), 542–547. lems, and experiment with additional bid restrictions and al- Varian, H., and MacKie-Mason, J. K. 1995. Generalized Vickrey ternative approximate WD algorithms. An interesting open auctions. Technical report, University of Michigan. problem is to adapt iBundle for two-sided markets, with Wellman, M. P. 1993. A market-oriented programming envi- multiple buyers and sellers. ronment and its application to distributed multicommodity flow problems. Journal of Artificial Intelligence Research 1:1–23. Acknowledgments Wurman, P. R. 1999. Market Structure and Multidimensional Auction Design for Computational Economies. Ph.D. Disserta- This research was funded in part by National Science Foun- tion, University of Michigan. dation Grant SBR 97-08965. 8