Sequential Bundle-Bid Single-Sale Auction Algorithms for Decentralized Control Sven Koenig USC Computer Science skoenig@usc.edu Craig Tovey Georgia Institute of Technology Industrial and Systems Engineering ctovey@isye.gatech.edu Xiaoming Zheng USC Computer Science xiaominz@usc.edu Ilgaz Sungur USC Industrial and Systems Engineering sungur@usc.edu Abstract We study auction-like algorithms for the distributed al- location of tasks to cooperating agents. To reduce the team cost of sequential single-item auction algorithms, we generalize them to assign more than one additional task during each round, which increases their similarity to combinatorial auction algorithms. We show that, for a given number of additional tasks to be assigned dur- ing each round, every agent needs to submit only a con- stant number of bids per round and the runtime of win- ner determination is linear in the number of agents. The communication and winner determination costs do not depend on the number of tasks and thus scale to a large number of tasks for small bundle sizes. We then demon- strate empirically that the team cost of sequential bundle- bid single-sale (= single-item) auction algorithms can be substantially smaller than that without bundles for multi- agent routing problems with capacity constraints. 1 Introduction We study the distributed allocation of tasks to cooperating agents, where each task has to be assigned to exactly one agent so that the team cost is small (= team performance is high). Auction algorithms promise to solve these combina- torial task-assignment problems with small communication and computation costs since the agents compress information into a small number of bids, which they compute in parallel and then exchange [Dias et al., 2005]. Ideal combinatorial auctions consist of a single round, after which all tasks have been assigned to agents. All agents bid on all bundles (= sets) of tasks and the auctioneer then assigns all tasks to agents, which allows the agents to take synergies among tasks into account in their bids and results in a minimal team cost but incurs prohibitively large communication and winner deter- mination costs. The communication and winner determina- tion costs remain large even if the agents bid on selected bun- dles only [Berhault et al., 2003]. Researchers have therefore recently advocated sequential single-item auctions (SSI auc- tions) instead [Lagoudakis et al., 2005]. SSI auctions consist of several rounds, until all tasks have been assigned to agents [Boutilier et al., 1999; Fatima, 2006]. During each round, all agents bid on all unallocated tasks and the auctioneer then as- signs one additional task to some agent, which incurs small communication and winner determination costs. However, the team cost of SSI auctions tends to be larger than that of combinatorial auctions since the agents cannot take as many synergies among tasks into account in their bids. To reduce the team cost of SSI auctions, we extend them to assign k> 1 additional tasks among the agents during each round. These sequential bundle-bid single-sale (= single-item) auction al- gorithms (short: SSI auctions with bundles) still consist of several rounds, until all tasks have been assigned to agents. During each round, all agents can now bid on bundles of at most k tasks and the auctioneer then assigns k additional tasks to agents, making SSI auctions with bundles similar to com- binatorial auctions. We expect the team cost of SSI auctions with bundles to be smaller than the one of standard SSI auc- tions since the agents can take more synergies among tasks into account in their bids. In this paper, we develop a gen- eral theory for such SSI auctions with bundles, proving that they can be implemented for small bundle sizes k without greatly increasing the communication and winner determina- tion costs of standard SSI auctions. 2 Task-Allocation Problem We now formalize the task-allocation problems. A task- allocation problem consists of a set of agents A = {a 1 ...a m } and a set of tasks T = {t 1 ...t n }. Any tuple (T a1 ...T am ) of pairwise disjoint bundles T ai ⊆ T , for all i =1 ...m, (= no task is assigned to more than one agent) is a partial solution of the task-allocation problem, with the meaning that agent a i performs the tasks T ai . Let c agent a (T ′ ) be the cost needed by agent a ∈ A to perform the tasks T ′ ⊆ T , called agent cost. There can be synergies among tasks, that is, c agent a (T ′ )+ c agent a (T ′′ ) does not necessarily equal c agent a (T ′ ∪ T ′′ ) even if T ′ ∩ T ′′ = ∅. The cost of the partial solution, called team cost, depends on the team objec- tive. In this paper, we consider two different team objectives. The team cost of the partial solution is ∑ a∈A c agent a (T a ) for the team objective MiniSum and max a∈A c agent a (T a ) for the team objective MiniMax. We use c team as a special opera- tor (shorthand) for either the sum and max operator, depend- ing on the team objective, and write c team a∈A c agent a (T a ) to make our notation independent of the team objective. Any partial solution (T a1 ...T am ) with ∪ a∈A T a = T (= each task is as- signed to exactly one agent) is a complete solution of the task- allocation problem. We want to find a complete solution of the task-allocation problem with a small team cost. IJCAI-07 1359