Determining Fairness by the Numbers
Mathematics now may be used to solve a problem that has tormented people since the dawn of humanity: how to divide things fairly.
Mathematician Alan Taylor and political scientist Steven Brams say they have devised a system based on “preference points” that can split just about anything--from the spoils of war to a child’s birthday cake--into “envy-free pieces.”
Not only do all parties get what they think is fair, they say, each thinks it got the better of the other guys.
Taylor and Brams’ work is only a small part of a rapidly surfacing trend. Mathematics is invading political science in attempts to find rational approaches to complex, often highly emotional questions.
Caltech, for example, recently received an $800,000 grant from the National Science Foundation to apply mathematics to finding fair ways to divide society’s scarce resources. It’s the biggest grant the foundation has given in the social sciences.
Among the questions Caltech is tackling: how to spread the cost of cleaning up Los Angeles’ polluted skies and how to divide listening time on NASA’s Deep Space Network among all the scientists that need its services. Caltech researchers are even investigating how human behavior--including good manners--can muck up some cherished mathematical models.
“Philosophers have argued about fairness for thousands of years,” said John Ledyard, who heads Caltech’s social sciences division. “What’s different now is we have a formal mathematical structure. That takes it out of ideological debate. There’s science here.”
Professional mediators are divided over whether logic can really help in cases where passions run deep. “[The mathematical approach to fairness] presupposes you have very rational people,” said Diane Blank, a New York attorney who specializes in family and employment law. “I don’t think people could relate to it unless they were engineers.”
But Montana mediator Norman Lavery has used the Brams / Taylor system in divorce cases with some success. “The couples I’ve dealt with have been very receptive,” he said. Lavery says he has even worked out a solution to property division in the former Yugoslavia, based on Brams’ and Taylor’s new book. “Our government should be on the forefront of this kind of creative thinking.”
At the very least, the new approaches are a useful tool for figuring out what’s fair in tricky situations.
At the outset, of course, one needs to define what “fairness” is. “When my kids say something’s unfair,” Ledyard said, “what they really mean is, they didn’t get what they want.”
One of the earliest notions of fairness is the biblical story of King Solomon, who had to decide which of two women claiming a baby was the mother. He ordered the child cut in half, which prompted the mother to give up her claim to save her infant’s life.
Solomon’s story illustrates the idea that fair division means more than cutting things into equal pieces. It involves the value that the parties--and society--place on what is to be divided.
Finding out how much people really value things is often difficult. Most people try to take whatever they can get rather than honestly state their choices.
Trying to encode the wisdom of Solomon into equations hasn’t been easy, but scientists are making progress.
Mathematicians do most of their research on fairness on a simple but versatile model called the cake-cutting problem.
Suppose two people want to share a small cake. The fairest way to divide it is to let one person cut the cake and let the other choose a piece first.
The cutter reveals her true preferences in the way she cuts the cake. For example, if she values icing, she might cut one piece smaller but with more icing, hoping her friend will go for the bigger slice.
Either way, both people can feel they are winners; one gets a bigger piece and the other gets more icing.
Perception is as important as mathematics in making the solution work. In this example, each person gets a role in choosing a slice.
The cake-cutting model works well for two. In more complicated situations, more players often are involved.
In 1992, in response to a challenge posed in the Sciences magazine, Brams found a solution that worked for three players. The first divides the cake into three; the second is allowed to trim one piece he thinks is bigger than the others. The third player gets to choose first. In the end, everyone has a say, and everyone gets to choose a piece. Therefore, Brams says, the solution is envy-free.
But when he tried to expand his system to four people, it didn’t work. So he called his friend Taylor, a mathematician at Union College in Schenectady, N.Y. Taylor, who had never worked on fair division, thinks that freshness helped him devise a radical approach.
Essentially, his new method involved cutting a cake into an extra piece--four for three players, and so forth. This allowed everyone to take a role in trimming and choosing.
Taylor’s breakthrough won praise in the mathematics community because it showed how to divide anything into any possible number of “envy-free” pieces.
In practice, though, the method is too unwieldy to use in everyday life because eventually the number of extra pieces required increases much faster than the number of players. Besides, there are things one can’t divide or trim, such as the family dog in a divorce.
So this past year, Brams and Taylor turned their attention to a more workable method. Instead of viewing the goods to be divided as a cake, in the new system--called Adjusted Winner--each player gets 100 points to distribute based on value preferences.
In a divorce, one spouse might assign 90 points to retaining custody of the children; the other spouse might care more about the house and put 70 points there.
In the first step of this process, each person wins whatever he placed more points on. In this case, the spouse with 70 might get, say, a life insurance policy rated at 30 points; then the other spouse might get the family computer, worth 10.
The rest is split according to a mathematical formula that Taylor and Brams guarantee is envy-free. They have even taken out a patent on the process. The key, they say, lies in the fact that different parties value things differently. Each perceives he is getting more than 50% because of the greater value he placed on things received.
“The big problem we can help solve is divorce,” said Brams, who teaches at New York University. He also sees immediate applications in issues ranging from labor disputes to the federal budget. Adjusted Winner, says Brams, would make both major political parties happy in budget fights because the Democrats could bid a lot for the things they value most--such as health care and education--while the Republicans could spend more points on tax relief or a balanced budget. Of course, both sides would also have to give up some things they cared less about. “It’s a very efficient way for them to put their money where there mouth is,” said Brams.
Caltech scientists are taking yet another approach to fairness, based on the age-old idea of auctions. Ledyard says competitive bidding counteracts the “natural tendency to ask for more than you really need.” Auctions generally prevent this because one may have to pay for and take whatever one bids on. In other words, choices carry consequences.
In working with NASA, Ledyard’s group is trying to find a better way to mete out listening time on the Deep Space Network--a constellation of ground-based telescopes designed to pick up remote signals. Scientists receiving data from spacecraft want as much time as they can get. And until now, the time slots were divvied up by inefficient haggling.
“They sit around a table and tell each other what they need and decide who gets what,” said Ledyard. The problem is everyone wants the biggest piece.
A better way, Ledyard says, might be to hold some kind of auction. In a model scenario, Caltech researchers awarded an equal number of tokens (or play money) to players posing as scientists in charge of spacecraft; the players bid against one another for time slots--for example, the first day of the month, or the first four hours of every day for a week.
The auction, said Ledyard, did significantly better than the old system. It was more efficient, and fairer in the eyes of the players. It was certainly better, he said, than “the committee process, which is deadlocked half the time arguing over what’s fair.”
However, for an auction to be fair, everyone has to have an equal start. But not everyone agrees on what “equal start” means. What if one person starts out with much more--or less--than the others? Should one first level the field? If so, what does “level” mean?
This problem emerged recently when Caltech helped design a system to divide responsibility for local air pollution. Say, Company A churns out 1,000 pounds of noxious gases a year and Company B releases 50. How much should each be forced to cut down?
The new method works like an auction, with players bidding for allowances to pollute a given amount.
In order to make the process fair, however, the total amount of pollution permitted had to be divided fairly at the start. “In this case, equal is not fair,” said Ledyard. “It’s not fair to give the same allowance to Joe’s Bar and Grill as to SoCal Edison.”
Instead, the allowances granted by the air quality board are based on levels of pollution during 1989 and 1990. Based on those figures, Company A might receive a permit to spew out 1,000 pounds of pollution per year; Company B might receive an allowance of 50 pounds. Any amount of pollution over that level would be fined. But Company A might find it less expensive to buy pollution rights from companies like B, which could use the money it earned from selling its rights to buy a new anti-pollution system. In theory at least, everybody wins.
Of course, as Caltech economist Colin Camerer pointed out, that system “rewards people for polluting in the past.”
The system, in other words, is not perfect. But at least it’s a step toward inventing a method that gives incentives to everyone to keep pollution down--and save money in the process. Even further complicating attempts to divide things fairly is the human factor. The science foundation grant to Caltech was mainly intended for experimental testing to determine how well people’s behavior matches mathematical models.
Researchers have assumed, for example, that people would always act in a way that would maximize their gain--in other words, logically. However, in a series of experiments that rewarded subjects with real cash Camerer found this wasn’t often the case. Instead, people gave up tangible rewards in order to avoid appearing greedy or selfish.
People also make choices in one-to-one situations that are quite different from the choices they might make as a member of a larger group: what if a community were trying to decide whether a tax break is fair compensation for having a chemical factory in its backyard?
Rakesh K. Sarin of UCLA studies how people weigh relative risk (say, of producing toxic fumes) against possible gain (more jobs). He’s found that what an individual might perceive as a good deal might not be perceived that way if that individual were part of a group.
The individuals might perceive the trade-off as fair. “But the group may have an outcry,” he said.
The main benefit of the new mathematical approach to fairness, says Ledyard, is the potential for creating new kinds of social systems, more or less from scratch. Until now, he said, economists and political scientists have studied existing systems. “We would look at auctions and elections and ask: Are they fair?”
Now they can turn the process on its head, he says, first getting agreement on what fairness is (for example, it is envy-free) then designing a process to deliver it.
One example of such an artificially designed system is the trading of pollution allowances. Similar approaches could be applied to everything from deciding who gets college dorm rooms to who gets called up for military service.
At the very least, the social scientists and mathematicians are learning how to step around the paradoxes inherent in most approaches to fairness.
“When someone says they want a situation to be fair,” said Ledyard, “we now understand that opens up a huge range of possibilities.”
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Test Case
New mathematical procedures are being used to decide issues of fairness more scientifically. They can be applied to a wide range of problems, even highly emotional issues such as divorce.
* A divorce
1. The two parties make a list of assets.
2. Each side is given a total of 100 points to spend on the assets. They bid independently on the items they desire, not knowing what the other side is offering.
3. Bids are compared; whoever bid the most on an item is the winner (W in the chart below) and takes it.
4. A tie is resolved by totaling points from each side’s winning bids, then giving the item to the lower score. Since husband has 65 to wife’s 55, she gets the “other furniture.”
5. The rest is divided into “envy-free” shares according to a mathematical formula.
****
*--*
Husband Item Wife 40 (W) House 30 20 Vacation home 25 (W) 10 Car 20 (W) 15 (W) Dog 10 0 Antique desk 10 (W) 10 (W) Big-screen TV 0 5 (tie) Other furniture 5 (tie)
*--*
