Divorcing couples wrangling over possessions and executives embroiled in corporate disputes could benefit from a mathematical recipe that guarantees fair shares for both sides. Academics in the US have just been granted the first-ever patent for a method of resolving disputes called Adjusted Winner, which they have mathematically proved is the best way to divide up anything from homes to corporate responsibilities.
Steven Brams, a political scientist at New York University, and Alan Taylor, a mathematician at Union College in Schenectady, New York, devised the algorithm after working on ways of solving an apparently simple problem: how to divide up a cake fairly between a group of three or more people ("Fair shares for all", New Scientist, 17 June 1995, p 42).
One challenge in such decisions is to ensure the division is "envy-free", so everyone feels they have a slice at least as good as anyone else's. With a cake, where one slice is pretty much like any other, this isn't hard to achieve. But in many other situations, from divorce settlements to corporate mergers, avoiding envy can be very difficult. Even so, Adjusted Winner does it automatically.
To see how, suppose Adam and Barbara are in the throes of a divorce and can't agree on who gets the town house, the holiday cottage and the sports car. First, Adam and Barbara allot points to each item out of a total of 100, according to how much they value them. Adam is besotted by the car, giving it 60 points, and quite likes the town house (25 points). But he isn't particularly bothered about the cottage, which gets his remaining 15 points. Barbara, in contrast, wants the town house (65 points) and the car (25 points), but is also unimpressed by the cottage, which gets her remaining 10 points. Each item initially goes to whoever allotted them the most points. So Adam gets the car and cottage, and Barbara gets the town house. But this doesn't seem fair: Adam has two items, and 75 of his 100 points, while Barbara got just one, worth 65.
So now the "adjustment" part comes in. The item rated most similarly by both Adam and Barbara-the cottage, at 15 and 10 res-pectively-is taken from Adam's heap, so his total drops to 60. Simple algebra then shows that if the cottage is sold off with Adam keeping three-fifths of its selling price, worth 9 points to him, and Barbara getting the remaining two-fifths, worth 4 points to her, both end up with the same final total: 69 points.
So the final division is fair, as both parties get the same number of points. It is also envy-free, as both ended up with the bulk of what they wanted.
Brams and Taylor stress that the algorithm is no panacea. "But it takes much of the worry out of being an inept bargainer by providing a guarantee of fairness," says Brams. They expect software that runs the algorithm to become available soon.
Hilary Siddle, chairman of the Family Law Committee of Britain's Law Society, says the idea of a formula for solving divorce disputes is attractive. But she doubts it can deal with common complexities such as dividing up homes suffering from negative equity. "I'd be very interested to see it, though," she adds.
Author: Robert Matthews
New Scientist issue 17th July 1999
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