Maegan Ayers and her then-boyfriend, Nathan Socha, faced a dilemma in the fall of 2009. They had found the perfect little condo for sale in the Boston neighborhood of Jamaica Plain: on the ground floor, just a mile from the nearest “T” train station, and close by Boston’s Emerald Necklace, a seven-mile chain of parks and bike paths. Federal incentives, low prices, and high rents had made home-buying an unusually attractive proposition, and the pair was eager to snap up the condo.
But, as the couple’s parents gently pointed out, Ayers and Socha were not yet married, or even engaged. If their relationship were to sour, they would have none of the protections that married homebuyers enjoy. As “tenants in common,” one of them could legally rent out or even sell his or her share of the condo to a total stranger. In the event of a break-up, Ayers and Socha wondered, how could they avoid conflict over their jointly owned condo?
Many families are faced with some version of Ayers and Socha’s quandary—how to fairly divide coveted goods. Some achieve a satisfying division, and are even strengthened in the process; others are ripped apart. Despite these high stakes, the methods families use to divide their assets tend to be very ad hoc.
Ayers and Socha did something different: They turned to mathematics.
Perhaps the oldest fair division method on the books—one which has been used by children from time immemorial—is the “I cut, you choose” method for dividing up, say, a cake between two people. One person cuts the cake into two pieces, and the other person gets to choose which piece to take. Abraham and Lot used this method to split up the land in which they would settle: Abraham divided the land, and Lot chose Jordan, leaving Canaan for Abraham.
“I cut, you choose” has one very appealing property: It is envy-free, meaning that neither participant would willingly trade her share for the other share. The person who cuts the cake—or tract of land, or other divisible good—has an incentive to make the two shares as equal as possible from her perspective, since she doesn’t know which she’ll end up with. If she does a good cutting job, she will be content with either piece. The other participant gets to choose her favorite piece, so neither person will wish to trade.
But when the good being divided is not homogenous—when the cake has an assortment of different frostings, or the land has a mix of fertile valleys, mineral-rich mountains, and arid deserts—the “I cut, you choose” method falls short on other important measures of fairness and desirability. (...)
Mathematicians have proven that when two people are dividing a cake, there is always some division that is simultaneously envy-free, equitable, and efficient (to get a sense of why this is true, see Sidebar: Cakes Are Fair Game). But there’s no simple algorithm for identifying this ideal split. And, in some other division problems, mathematicians have shown that no ideal split even exists. In its stead, mathematicians have, over the past 20 years, developed a rigorous framework for exploring the trade-offs required by different kinds of divisions, helping to bring clarity to the fallout from divorce, death, and divestment.
One straightforward approach that Ayers and Socha might have taken is called “the shotgun clause,” a close analogue to “I cut, you choose” that is common in business contracts. This clause stipulates that if, for example, two owners of a business want to part ways, one of them will propose a buyout price, and the other will choose either to buy or be bought out at that price. Like “I cut, you choose,” this method is envy-free but not equitable: It’s better to be the chooser than the proposer. As a result, arguments about who should propose and who should choose sometimes lead to years of litigation, says James Ring, lawyer and CEO of Fair Outcomes, a Boston company that provides division algorithms.
Instead, Ayers and Socha committed that in the event of a break-up, they would use a relatively new algorithm called Fair Buy-Sell to determine which of them would buy out the other’s share, and at what price. Fair Buy-Sell was devised in 2007 by Ring and Steven Brams, a professor of politics at New York University, and requires each partner to simultaneously propose a buyout price. If John proposes $110,000 and Jane proposes $100,000 then John, the higher bidder, will buy out Jane for $105,000. Unlike the shotgun clause, this method is equitable: Each participant ends up with something—either money or the business—at a price that is better than his or her offer. “Both participants always get a solution that’s better than what they proposed,” Ring says. And the business always goes to the partner who values it more.
This algorithm joins a long list of others, with names like Adjusted Winner and Balanced Alternation. Just as important as prescriptions for fair division, though, is understanding when perfect fairness is impossible, or comes at the cost of social welfare (which measures the extent to which items are going to the people who value them most.). In the January 2013 issue of The American Mathematical Monthly, Brams—together with Christian Klamler of the University of Graz and Michael Jones of Montclair State University—showed that when three people are dividing a cake, it is sometimes impossible to find a division that is simultaneously envy-free, equitable, and efficient. Similarly, when three people have to divide a collection of indivisible items, it is sometimes necessary to choose between an envy-free solution and an efficient solution...
But, as the couple’s parents gently pointed out, Ayers and Socha were not yet married, or even engaged. If their relationship were to sour, they would have none of the protections that married homebuyers enjoy. As “tenants in common,” one of them could legally rent out or even sell his or her share of the condo to a total stranger. In the event of a break-up, Ayers and Socha wondered, how could they avoid conflict over their jointly owned condo?
Many families are faced with some version of Ayers and Socha’s quandary—how to fairly divide coveted goods. Some achieve a satisfying division, and are even strengthened in the process; others are ripped apart. Despite these high stakes, the methods families use to divide their assets tend to be very ad hoc.
Ayers and Socha did something different: They turned to mathematics.Perhaps the oldest fair division method on the books—one which has been used by children from time immemorial—is the “I cut, you choose” method for dividing up, say, a cake between two people. One person cuts the cake into two pieces, and the other person gets to choose which piece to take. Abraham and Lot used this method to split up the land in which they would settle: Abraham divided the land, and Lot chose Jordan, leaving Canaan for Abraham.
“I cut, you choose” has one very appealing property: It is envy-free, meaning that neither participant would willingly trade her share for the other share. The person who cuts the cake—or tract of land, or other divisible good—has an incentive to make the two shares as equal as possible from her perspective, since she doesn’t know which she’ll end up with. If she does a good cutting job, she will be content with either piece. The other participant gets to choose her favorite piece, so neither person will wish to trade.
But when the good being divided is not homogenous—when the cake has an assortment of different frostings, or the land has a mix of fertile valleys, mineral-rich mountains, and arid deserts—the “I cut, you choose” method falls short on other important measures of fairness and desirability. (...)
Mathematicians have proven that when two people are dividing a cake, there is always some division that is simultaneously envy-free, equitable, and efficient (to get a sense of why this is true, see Sidebar: Cakes Are Fair Game). But there’s no simple algorithm for identifying this ideal split. And, in some other division problems, mathematicians have shown that no ideal split even exists. In its stead, mathematicians have, over the past 20 years, developed a rigorous framework for exploring the trade-offs required by different kinds of divisions, helping to bring clarity to the fallout from divorce, death, and divestment.
One straightforward approach that Ayers and Socha might have taken is called “the shotgun clause,” a close analogue to “I cut, you choose” that is common in business contracts. This clause stipulates that if, for example, two owners of a business want to part ways, one of them will propose a buyout price, and the other will choose either to buy or be bought out at that price. Like “I cut, you choose,” this method is envy-free but not equitable: It’s better to be the chooser than the proposer. As a result, arguments about who should propose and who should choose sometimes lead to years of litigation, says James Ring, lawyer and CEO of Fair Outcomes, a Boston company that provides division algorithms.
Instead, Ayers and Socha committed that in the event of a break-up, they would use a relatively new algorithm called Fair Buy-Sell to determine which of them would buy out the other’s share, and at what price. Fair Buy-Sell was devised in 2007 by Ring and Steven Brams, a professor of politics at New York University, and requires each partner to simultaneously propose a buyout price. If John proposes $110,000 and Jane proposes $100,000 then John, the higher bidder, will buy out Jane for $105,000. Unlike the shotgun clause, this method is equitable: Each participant ends up with something—either money or the business—at a price that is better than his or her offer. “Both participants always get a solution that’s better than what they proposed,” Ring says. And the business always goes to the partner who values it more.
This algorithm joins a long list of others, with names like Adjusted Winner and Balanced Alternation. Just as important as prescriptions for fair division, though, is understanding when perfect fairness is impossible, or comes at the cost of social welfare (which measures the extent to which items are going to the people who value them most.). In the January 2013 issue of The American Mathematical Monthly, Brams—together with Christian Klamler of the University of Graz and Michael Jones of Montclair State University—showed that when three people are dividing a cake, it is sometimes impossible to find a division that is simultaneously envy-free, equitable, and efficient. Similarly, when three people have to divide a collection of indivisible items, it is sometimes necessary to choose between an envy-free solution and an efficient solution...
by Erica Klarreich, Nautilus | Read more:
Image: Emmanuel Polanco
