Multiagent Coordination Using a Distributed Combinatorial Auction

C AUCTIONS are a great way to represent and solve distributed allocation problems. Unfortunately, most of the winner determination solutions that exists are centralized. They require all agents to send their bids to a centralized auctioneer who then determines the winners. The PAUSE auction, in contrast, is an increasing-price combinatorial auction in which the problem of winner determination is naturally distributed amongst the bidders. We present PAUSEBID, a bidding algorithm for agents in a PAUSE auction which always returns the bid that maximizes the bidder’s utility. Our test results show that a system where all agents use PAUSEBID finds the revenue-maximizing solution at least 95% of the time. Run time, as expected since this is an NP-complete problem, remains exponential on the number of items. 1. Bidding in the PAUSE Auction A PAUSE AUCTION for m items has m stages. Stage 1 consists of having simultaneous ascending price open-cry auctions for each individual item. During this stage the bidders can only place individual bids on items. At the end of this stage we know what is the highest bid for each individual item and who placed that bid. In each successive stage k = 2,3, . . . ,m we hold an ascending price auction where the bidders must submit sets of bids that cover all goods but each one of the bids must be for k goods or less. The bidders are allowed to use bids that other agents have placed in previous rounds when placing their bid, thus allowing them to find better solutions. Also, any new bidset has to have a sum of bid prices which is bigger than the currently winning bidset. That is, revenue must increase monotonically. Formally, let each bid b be composed of bitems which is the set of items the bid is over, bvalue the value or price of the bid, and bagent the agent that placed the bid. The agents maintain a set B of the current best bids, one for each set of items of size≤ k. At any point in the auction, after the first round, there will also be a set W ⊆ B of currently winning bids. This is the set of bids that currently maximizes the revenue, where the revenue of W is given by r(W ) = ∑ b∈W bvalue. (1) Agent i’s value function is given by vi : S→R where S is a subset of the items. Given an agent’s value function and the current set of winning bids W we can calculate the agent’s utility from W as ui(W ) = ∑ b∈W |bagent=i vi(b)−b. (2) Given that W is the current set of winning bids, agent i must find a g∗ such that r(g∗)≥ r(W )+ ε and g∗ = argmax g⊆2B ui(g), (3) where each g is a set of bids all taken from B and g covers all items. The goal of the PAUSEBID algorithm is to find this g∗.