Since 2007, together with colleagues at the Smith Institute, I have spent a good deal of time working with clients around the globe on auctions for radio spectrum licences. Bearing in mind that auction outcomes can shape a country’s mobile telecommunications industry for a decade or more, the tasks of getting the auction designs right and running auctions without a hitch leave no room for mistakes.

There is always extensive discussion and consultation over proposed auction rules, and of course one would expect major industry players to speak up for rules they consider favourable to their own interests. But what are the real underlying issues?

Imagine there is to be an auction of fine furniture at your local saleroom, and there is a nice table and also a set of chairs that have caught your eye. Just the thing for that important anniversary coming up. You have £1,000 to spend (from matching 5 balls in last Saturday’s lottery draw). The only problem is that the table and chairs are being sold as separate lots. You go along to the sale, and the table comes under the hammer first. There are a few other bidders interested and the bids start to rise … £500, £550, £600, … The chairs have a reserve price of £250 and you think they might sell for about £350, but that’s only a guess – and the excitement seems high in the room today. Do you keep bidding on the table or not?

You might stop bidding on the table, but then how would you feel if it finally sold at £750 and then you find out a few minutes later that no-one else is interested in the chairs, when they come up for sale? Or you might carry on and win the table for £700, but then find that bidding on the chairs is just as fierce. In that case, you have to go home with just a table and carry on playing the lottery, hoping to come back to another sale soon.

What you would really like to do is to communicate to the auctioneer that you wish to bid up to £1,000 for both table and chairs, but the auction is not designed that way. This is an example of the exposure problem. Combinatorial auctions remove the exposure risk. This means that single bids can be placed on packages of items, in this case a package consisting of the table and chairs. Another bidder might be interested in the same set of chairs and the fine bookcase (which caught your eye too, but for the fact you already have one). In general, different bidders will be interested in different, but overlapping packages, and an individual bidder will generally bid on multiple packages at different amounts.

The construction of combinatorial auctions is now well established and there are numerous examples of their successful implementation. In addition to avoiding the exposure problem, they have high economic efficiency, meaning that in applications like radio spectrum their outcomes tend to be ultimately good for consumers. They also create strong incentives for bidders to be truthful and consistent in the bids that they place. For bidders, the main task is to decide which packages are of interest and at what valuations, and these decisions depend on being clear about their individual business plans.

One of the challenges is in managing the potential combinatorial ‘explosion’. The UK auction for 4G spectrum in February 2013 sold licences for 28 individual ‘lots’ of spectrum, which could be assembled into about 15,000 possible packages. On a different scale was the Canadian auction in the 2500MHz band, for which the results were announced just yesterday. This auction had 318 lots and in theory more than 1039 possible packages.

Canada’s auction had a more detailed lot structure to reflect the regional structure of its telecommunications market. Spectrum licences relate to a particular ‘lot’ of frequencies but also to a particular geographical region – 61 regions in Canada’s case. Regional service providers and local service providers are able to participate alongside national operators, in the same auction process. Each participant is able to concentrate on those packages that are compatible with its own business model, which will be some tiny fraction of the full 1039. This auction also broke new ground in the way bidders express their preferences, to handle such a large number of possibilities.

Spectrum auctions are conducted using secure electronic systems and may easily last several weeks, with multiple rounds of online bidding. Depending on the number of lots and how they can be combined into packages, we use different mathematical approaches to verify the results. The main tasks are identifying winners and prices from what might easily be several thousand bids in total.

If the number of possible packages is relatively low, say up to a few million, then the mathematical approach of ‘dynamic programming’ is a good choice, and works by building up the overall solution one bidder at a time. For larger auctions, we use ‘integer programming’, which is a more general-purpose approach to solving combinatorial problems (for example it is widely used to solve scheduling problems, including the famous travelling salesman problem). We have also developed a set of automated and manual checks on the results, which are founded on the underlying mathematical theory of optimisation problems, and allow us to be completely certain that the declared results are correct.

Listening to the press and other industry commentators, it’s easy to pick up the idea that auction implementation has somehow got out of hand compared with the local saleroom. The Guardian newspaper reported during the UK 4G auction in 2013 that ‘The outcome … is being determined by secret mathematical algorithms worked out using massive computer processing power’. This wasn’t true for the UK auction and it isn’t true for subsequent auctions. Chi Onwurah, currently the MP for Newcastle Central and formerly senior technologist at Ofcom was mischievously quoted as saying ‘[The auction rules] are something that only 10 people in the world understand and half of them are mad.’ As far as I know, that’s not true either!

What is clear, however, is that the latest developments in auction design and implementation are helping to ensure that telecommunication markets can keep pace with the seemingly insatiable demand for mobile services.