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).