Combinatorial auctions in theory and practice

1. Combinatorial auctions in theory and practice Graham Louth, Director of Spectrum Policy11 March 2010

2. Ofcom • The UK’s communications regulator with responsibilities across: • TV and radio broadcasting • fixed and mobile networks and services • management of the electro-magnetic spectrum (for wireless communications and other purposes)

3. The Radio Spectrum Electric Waves Radio Waves Visible Light Ultra Violet Gamma Rays Cosmic Rays Infra-red X-Rays Radio Spectrum “Sweetspot” 3G LMDS DECT WiFiBluetooth TETRA GSM FM Radio Medium Wave Radio Microwave Radio Links Long Wave Radio TV VLF LF MF HF VHF UHF SHF EHF 3 30 300 3 30 300 3 30 300 kHz MHz GHz Increasing RangeDecreasing Bandwidth Decreasing RangeIncreasing Bandwidth

4. Ofcom’s statutory duties • 3(1)(a) Further the interests of citizens in relation to communications matters • 3(1)(b) Further the interests of consumers in relevant markets, where appropriate by promoting competition • 3(2)(a) Secure the optimal use for wireless telegraphy of the electro-magnetic spectrum • 3(4)(d) Have regard for the desirability of encouraging investment and innovation in relevant markets • 3(4)(f) Have regard for the different needs and interests of all persons who may wish to make use of the spectrum

5. Why auctions? • Need to choose between conflicting demands for limited amount of spectrum • Want a mechanism that is fair, transparent and robust • Want to avoid pre-judging what constitutes ‘best use’ • Want to minimise constraints on future use • NOT to raise revenue – that is just a by-product of the process • Whilst not perfect, experience suggests that well designed auctions are generally better at achieving our objectives than alternative award approaches (e.g. beauty contests)

6. Why combinatorial auctions? • We want to facilitate competition, so want to divide up the available spectrum into a number of lots • We want to allow bidders the flexibility to win the amount and configuration of spectrum that best suits their needs, so we want the lots to be no larger than necessary and to allow bidders to bid for combinations of lots of their choosing (each bid winning or losing as a unit, with the bidder facing no risk of winning only part of what they need) • recognise possibility of (strong) complementarity between lots – lots worth (significantly) more in combination than the sum of their individual values • (Alternative is to allow bidders to simultaneously but separately bid on a number of different lots – puts risks on bidders to manage their exposure, which may lead them to bidding cautiously and consequently lead to an inefficient outcome)

7. Challenges of combinatorial auctions - 1 • If demand is ‘lumpy’ (strong complementarities) the combination of bidders who are individually willing to pay the highest prices may not constitute the optimal allocation of the spectrum since a different combination of bidders might, in aggregate, be willing to pay more (because they would take up more of the available spectrum): • Need to give bidders the opportunity and incentive to reveal their maximum willingness to pay across a wide range of packages of interest • Need to solve the “winner determination problem” – identify the package of bids which maximises total value subject to accepting at most one bid from each bidder and being able to fit all of the winning bids into the available spectrum (a potentially hard integer programming problem)

8. Challenges of combinatorial auctions - 2 • There may be a very large number of different combinations of lots of potential interest to bidders; How can they identify those that are most likely to win? • Use a multi-round process so that bidders can see how demand changes as prices increase, and can estimate what final prices are likely to be • Give bidders incentives to reflect their demand truthfully in their bids, so that other bidders can rely upon the demand information that is revealed through earlier rounds

9. An example – the 10-40GHz award

10. Combinatorial clock auction • Auctioneer announces prices • Bidders bid for the quantity they would most like to buy at those prices • Auctioneer looks at total demand and increases price where demand exceeds supply • Bidders respond with new bids, expressing the new quantity that they would most like to buy at those new prices – generally required to be no more in total than the quantity bid for in the previous round, but free to switch demand between different types of lot at will • Process continues until demand no longer exceeds supply

11. 10-40GHz auction – actual results

12. Overall price and demand

13. Evolution of demand

14. Evolution of demand

15. Evolution of prices

16. A real bidding example - Transfinite

17. Need for supplementary bids • If bidders interested in particular combinations of lots, rather than just more or less of the same thing, result at end of combinatorial clock stage may not be optimal – there may no longer be demand for some of the available spectrum • Need to give bidders the opportunity to express full range of their demand e.g. bid for combinations of lots that they didn’t bid on in the combinatorial clock stage – supplementary bids • Auctioneer then needs to work out which combination of bids generates greatest value – the winner determination problem – (a potentially hard integer programming problem)

18. Second price rule • In almost all simple multi-round ascending auctions, the auction stops as soon as the last losing bidder drops out; the winning bidder(s) then have to pay an amount equal to (or one bid increment greater than) the highest amount that any losing bidder was willing to pay: this is the key characteristic of a second price rule • The same idea can be applied in sealed bid auctions (e.g. this is how bidding works on e-Bay), and has the benefit of reducing the incentive for bidders to ‘shade’ their bid – bid less than their true value in order to avoid paying more than necessary – which can lead to inefficient outcomes • Second price rules can also be applied in combinatorial auctions, but require the auctioneer (but not bidders) to perform a complex set of calculations; objectives is to give bidders strong incentive to bid the true value to them of the lots they are bidding for: risk of paying more than true value if over-bid; risk of not winning when could have done if under-bid; little chance of reducing amount required to pay by reducing bid

19. Second price rule – further details • True second price rule in case of combinatorial auction is Vickrey-Clarke-Groves (opportunity cost) pricing • But there are a number of serious problems with this pricing rule, not least that VCG prices may not be in the core: • A different coalition of bidders may have offered to pay more in total for the same spectrum – liable to challenge outcome! • We currently use a pricing rule that uses the set of prices in the core that are closest to VCG prices – (an even harder non-linear optimisation problem) • No particularly strong theoretical justification for this rule – but seems to work • Means that in some circumstances winning bidders might be able to reduce their payments (individually, but not collectively) by under-bidding, but likely to be very difficult for a prospective winning bidder to predict this possibility, and doing so creates risk of losing when otherwise would have won

20. Final allocation

21. Final prices

22. Current challenges • Winner determination and pricing in a combined buy/sell auction with relative price constraints (!) • Incentive impacts of different variants of core pricing: VCG-closest vs Reference price rules • Auction designs for efficient allocation when bidders have budget constraints

23. Graham LouthDirector of Spectrum Policygraham.louth@ofcom.org.ukwww.ofcom.org.uk/radiocomms/spectrumawards