1. CPGomes – AEM 03 1 Electronic Markets Combinatorial Auctions Notes by Prof. Carla Gomes

2. CPGomes – AEM 03 2 Why Combinatorial Auctions? More expressive power to bidders In combinatorial auctions bidders have preferences not just for particular items but for sets or bundles of Items due because of complementarities or substitution effects. Example Bids: Airport time slots [ (take-off right in NYC @ time slot X ) AND (landing right in LAX @ time slot y)] for $9,750.00 Delivery routes (“lanes”) [(NYC - Miami ) AND [((Miami – Philadelphia) AND (Philadelphia – NYC)) OR ((Miami – Washington) AND (Washington – NYC))]] for $700.00

3. Managing over 100,000 trucks a day (June 2002), >$8 billion worth of transportation services. OPTIBID - software for combinatorial auctions Procurement Transportation Services on the web. FCC auctions spectrum licenses ( geographic regions and various frequency bands). Raised billions of dollars Currently licenses are sold in separate auctions USA Congress mandated that the next spectrum auction be made combinatorial.

4. FCC Auction #31 700 MHz Winner Determination Problem Choose among a set of bids such that: Revenue to the FCC is maximized Each license is awarded no more than once Bid Bid amt. 2 $12e6 3 $30e6$22e6 1 4 $16e6 5 $8e6 Package B ABCABDADC 6 $11e6 BC 7 $10e6 A 8 $7e6 D (source: Hoffman) Hard Computational Problem x 3 + x 5 + x 6 + x 3 x1x1 + x 4 + x 7 x1x1 + x 4 + x 8 B C A D <=1 1 1 1 + x 2 + x 3 x1x1 + x 6 Example: 4 licenses, 8 bids $30e6 $22e6 + $8e6 = $36e6 $12e6 + $16e6 +$8e6 =  $36e6 $28e6 $37e6 $27e6 $36e6

5. FCC Auction #31 700 MHz Winner Determination Problem Choose among a set of bids such that: Revenue to the FCC is maximized Each license is awarded no more than once Bid Bid amt. 2 $12e6 3 $30e6$22e6 1 4 $16e6 5 $8e6 Package B ABCABDADC 6 $11e6 BC 7 $10e6 A 8 $7e6 D (source: Hoffman) Hard Computational Problem x 3 + x 5 + x 6 + x 3 x1x1 + x 4 + x 7 x1x1 + x 4 + x 8 B C A D <=1 1 1 1 + x 2 + x 3 x1x1 + x 6 Example: 4 licenses, 8 bids $30e6 $22e6 + $8e6 = $36e6 $12e6 + $16e6 +$8e6 = 

6. CPGomes – AEM 03 6 Combinatorial Auctions cont. There exists a combinatorial auction mechanism (“Generalized Vickrey Auction”), which guarantees that the best each bidder can do is bid its true valuation for each bundle of items. (“Truth revealing”). However, finding the optimal allocation to the bids is a hard computational problem. No guarantees that an optimal solution can be found in reasonable time. What about a near-optimal solution? Does this matter? Yes! Problem: if the auctioneer cannot compute the optimal allocation, no guarantee for truthful bidding. So, computational issues have direct consequences for the feasibility and design of new electronic market mechanisms. A very active area in discrete optimization. (Bejar, Gomes 01)