1CPGomes – AEM 03

Electronic MarketsElectronic Markets

Combinatorial Auctions

Notes by Prof. Carla Gomes

2CPGomes – AEM 03

Why Combinatorial Auctions?Why Combinatorial Auctions?

More expressive power to biddersIn 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

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.

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 ABCABD AD C

6

$11e6

BC

7

$10e6

A

8

$7e6

D

(source: Hoffman)

Hard Computational

Problem

bidsallforxb 1,0

x3 + x5 + x6

+ x3x1 + x4 + x7

x1 + x4 + x8

B

C

A

D

<= 1

<= 1

<= 1

<= 1

+ x2 + x3x1 + x6

8

1bbb

xxBidAmtMax

Example: 4 licenses, 8 bids

$30e6$22e6 + $8e6 =

$36e6

$12e6 + $16e6 +$8e6 =

$36e6

$28e6

$37e6

$37e6$27e6

$36e6

6CPGomes – AEM 03

Combinatorial Auctions cont.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)