Patterns and proofs

Patterns and proofs

At auctions, does the house always ‘win’?

For crying out loud.

The Lichtenstein went for $56m and the Pollock fetched $58m. Someone bought Rothko’s piece for $27m, and the wining bid for the Basquiat was over $48m. In total, auction house Christie’s record-breaking art sale in New York last month raised almost half a billion dollars. But could they have taken even more?

Auctions are common in many industries. Literary agents collect bids when selling books to publishers. Car dealers use them to sell vehicles for prices that wouldn’t fit on a number plate. Estate agents employ them in the hectic housing market.

There are several different types of auction. The “open outcry” auctions is often used for art sales, where an auctioneer lists higher and higher prices until nobody is willing to bid anymore. There are also “silent auctions”, in which everyone writes down a secret bid beforehand and the winner is the person with the largest when the numbers are revealed. Then there are “Dutch auctions”, where the auctioneer opens with a ludicrously high price, gradually lowering it until someone jumps in to buy.

People who sell something at auction want to maximise the value of the winning bid, and hence their profit. At the same time, buyers are trying to bid enough to win, but no more. So which type of auction favours the seller, and which the buyer?

Game on

Auctions are essentially a game. And like most games, there is a science to winning.

Suppose you are bidding against a friend for a painting, which is being sold in a silent auction. You both have a fixed budget of, say, £100, and a view of how much the artwork is really worth.

According to economist William Vickrey, who pioneered “auction theory” in 1961, if you have no idea how much the other person values the item, you should write down half your valuation as your secret bid. So if you think the picture is worth £60, put down £30.

If there are three of you bidding instead, you should bid two-thirds your valuation. If there are four, you should bid three-quarters your valuation. And so on, for n potential buyers, your valuation needs to be multiplied by 1–(1/n).

Brad darling, this sculpture is a masterpiece but is it worth what you paid for it? PA/David Parry

Unfortunately, people tend to bid more than is optimal in silent auctions. Which means they can fall victim to the “winner’s curse”: if they win, they risk overpaying. This can occur in any auction that involves an item with a fixed “true” value, which none of the potential buyers knows.

For example, say you hold a silent auction for a jar full of pound coins. None of the bidders knows precisely how many are inside. But if you add all the guesses and divide them with the total number of guesses, the average will probably be close to the true value, a neat illustration of the wisdom of the crowd. However, unless everyone bids less than their valuation, as Vickrey suggested they should, the winner will end up paying more than the average, and therefore more than the jar is really worth.

From paying too much for shares in a newly public company to overestimating a house’s value, the winner’s curse frequently makes an appearance in auctions. There are exceptions, of course. If an item has personal, perhaps sentimental, value to one of the bidders, it’s unreasonable to expect them to pay the true value. Alternatively, one of the potential buyers might have information or expertise that others don’t, and use this advantage to outbid everyone else.

Be careful what you bid for. Kentucky Photo File

Money talks

Knowing the optimal bidding strategy helps buyers, but it doesn’t address the question of which auction type generates most money for sellers. Fortunately we have an answer thanks to the “revenue equivalence theorem”.

This theorem covers auctions involving an item that has a fixed true value, which none of the bidders know. According to the theorem, if all bidders employ optimal strategies, the seller should expect to make the same profit, regardless of which kind of auction is used. Silent, open outcry or Dutch; it shouldn’t matter if everyone is playing the game properly.

Like any model of human behaviour, there are inevitably things that the theorem does not account for. For instance, bidders might be influenced by a desire to avoid risks (or to take them). Or they might get emotional and make mistakes. Researchers are thus not sure how well the conclusions match real life.

Despite its simplifications, however, auction theory has some important uses. It shows that there are optimal strategies for bidding in auctions, which means that nobody should suffer the winners curse. It also shows that, in an ideal world, different auctions types should yield the same profit. Though with almost $500m in art sales, Christie’s probably won’t be changing their format anytime soon.