Game theory is being tested out in telecom and TV rights auctions as bidders try to beat the system, says Faisal Islam

Sotherby's must be casting envious eyes over two lots currently up for auction. Neither lot number one, licences for third-generation mobile phone services, nor lot number two, the rights to televise Premiership football, could be described as antique, fine art, or rock star memorabilia.

The commissions from these multi-billion pound deals could, however, warrant the two lots a place under the hammer at the New Bond Street auction house.

But auctions like these, rather than lining the pockets of traditional auction houses, are providing work for an army of mathematical economists and game theorists.

Auction theory, as this complex amalgam of microeconomics is known, has a wide application. It is relevant to the £22 billion battle for control of mobile phones. BSkyB's bid to take control of Manchester United was blocked, in part, because of the insights of auction theory. Sales of gold and electricity, and the increasing use of the Internet for procurement of business supplies, have also brought the theory back to the fore.

Essentially auctions are an efficient method of distributing a scarce resource: by collecting all the possible bidders, an auction should get the maximum price from the most eager bidder. In a commercial auction, bids indicate the amount of profit each bidder expects to make.

So, although much is said about the £22bn mobile phone auction amounting to a tax on technology, access to this portion of the airwaves offers a licence to print money. Auction theory says it would be 'economically inefficient' to hand these licences - or, more specifically, these tranches of the electromagnetic spectrum - over to selected companies for a low price in a beauty contest.

'Well-run auctions assign rights to the bidders who are willing to pay most - not the party with the most political influence,' says Professor Paul Milgrom of Stanford University. He designed the mechanism used by the Federal Communications Commission for the first 'spectrum auction' in America in 1994. The auction of next-generation mobile phone licences in Britain is a modified version of those auctions.

The essentials remain the same regardless of the auction: bids are the lowest a firm believes will secure the lot. A bidder has to make a careful analysis of what the other bidders may do, which in turn will be dependent to some degree on what they think he may bid. This is the realm of game theory.

It has identified two main problems with the auction process. When all the bidders for a licence can only secure the same profits, any victor in an auction faces the 'winner's curse'. The winner is likely to be the bidder most optimistic about potential profits. In many circumstances it will have paid over the odds and is really a loser. At worst this can lead to bankruptcy.

In an ascending bid auction this problem is lessened. Throughout the auction, bidders reveal more and more information about how they value a contract or licence, and learn from each other's reactions. Game theorists employed by telecoms companies will have played out this high-stakes poker game already. Using advanced computer models they will have worked through every possible permutation of response from each bidder.

However, these auctions leave the door open to a second problem - collusion between bidders.

Professor Paul Klemperer of Oxford University, the economic theorist behind the current next-generation mobile auctions, says auctions in Germany last year offering extra capacity for conventional mobile phones illustrate this point.

Mannesmann bid a low price for half the licences and an even lower price for the other half. Though no discussions had taken place, the structure of their bid meant the other bidder, T-mobil, only needed to raise the bid on half the licences slightly. The result was that after two rounds of bidding each of the bidders had half the licences for the same low price.

A 'second-price' auction in New Zealand in 1990 (see box) enabled one bidder to pick up the rights to a tranche of the mobile phone spectrum for just NZ$6.

In America there have been instances of bidders using the final four numbers of their multi-million dollar bids to signal to each other which telephone code area they want to buy.

In a separate case, the US Justice Department was brought in to investigate possible collusion in a 1997 auction expected to raise almost $2bn that raised less than $14 million.

The use of these auctions has clearly not been an unmitigated success. 'It's a case of horses for courses,' says Klemperer. 'There are many different methods of auctioning and which is best is sensitive to the specific details of the market.'

But his theories have also been applied to other, seemingly unrelated areas. In Britain a report by the Competition Commission recommending that BSkyB's takeover of Man Utd should be blocked was rooted in Klem perer's analysis of 'toehold effects' in bidding situations.

Put simply, if BSkyB bought Man Utd, any TV rights deal would offer the same potential profit to both companies. Imagine that BSkyB owned United and was bidding against Ondigital for Premiership rights. If Ondigital were to win, 7 per cent of its bid would go to BSkyB. Thus, whether it wins the rights or not, BSkyB has an incentive to push up the bidding that much more aggressively. This exaggerates the winner's curse for OnDigital; in effect, it has to back off the bidding more than BSkyB.

The 'toehold' affects both sides of the bidding equation, setting off a vicious circle for OnDigital and a virtuous circle for BSkyB. Ondigital, knowing that BSkyB can bid more aggressively, will scale back its bidding. BSkyB will always win the rights in an auction like this.

While BSkyB was not able to take over Man Utd, it has taken strategic 9.9 per cent stakes in four Premiership clubs. It still has a toehold, but this time one shared across several different clubs rather than concentrated on one.

The last negotiation for Premiership rights had a gaping flaw. Awarding BSkyB the right to match any rival offer may have been seen as a clever way to top any final bid for rights, but Klemperer says: 'Giving a bidder the right to match is not a costless exchange, since there is no reason to bid the price up in the first instance.' So knowing that it could always match any offer, BSkyB never pushed up the bid price.

Even the forthcoming Pre miership auction seems questionable, given the insights of auction theory. The parties have been invited to lodge a sealed bid before 10 May, after which there will be an auction of the rights by July. Given that the first sealed bid appears not to be binding, it is difficult to see why potential bidders would want to give too much information away before the serious business begins.

Auction theory also shows up glaring inconsistencies in government policy.

Toehold effects, identified as a primary reason for blocking BSkyB's bid for Man Utd in the Competition Commission's report, have not been ended. BSkyB, and to a smaller extent other broadcasters, continue to collect small stakes in Premier League clubs.

Similarly, why in a climate of converging technologies are auctions designed to extract the maximum fees from bidders for mobile phone operators, when television operators need only have their licences rubber-stamped by the ITC for comparatively pitiful sums of money?

As consumers begin to get used to auctions through Internet sites such as QXL and eBay, analysis of auction behaviour will become an increasingly fertile area for research.

And given that, the Premier League might consider employing a leading game theorist to design an auction before they are going, going, gone.

Under the hammer

1. Types of auction

Ascending price (English) auction

Example: mobile phone licences. Bidders offer sequentially higher prices until there is one left. Such auctions can lead to collusion.

Descending price (Dutch) auction Example: tulip, tobacco. Prices start high and descend until a buyer is found.

Sealed-bid auction

Example: tenders for government contracts and, to some degree, TV rights for Premiership football. Each bidder makes one bid, without knowing the others.

Second-price auctions

Example: New Zealand's spectrum auction in 1990. The licence is awarded to the highest bidder but for the second highest price, so avoiding 'winner's curse'.

2. Problems with auctions

Winner's curse

In a sealed bid auction, the winner is the optimistic bidder who most overvalues the 'prize'. Ascending price auctions get round this problem by gradually revealing information from the whole marketplace about the value of the auction.

The theory assumes 'common value' - that the value of whatever is bid for is approximately the same to each bidder.

Collusion

This can happen when two or more bidders make illegal arrangements not to bid up the auction price. But tacit collusion - where bidders signal their intentions or make 'offers' to other bidders through the bid itself which can also depress prices - is not illegal.