Fair Division

Disputes involving division of property come up in divorces, settling estates, and dividing the assets of a business that is closing. Related cases are assigning household chores, negotiating salaries, sentencing criminals, setting income tax rates, and allocating resources (e.g., water among farmers, fish, homeowners, industry, and recreational users).

Some animals seem to have a sense of fairness. Capuchin monkeys prefer grapes to cucumbers. In a lab in which capuchins had learned to exchange tokens for food, capuchins who observed one animal receiving a grape for a token refused to give up a token just to get a cucumber, going so far as to throw their tokens away or even at the researcher who only offered cucumber slices.⁶ When two dogs sitting side-by-side got rewards for “shaking hands,” one with sausage and one with brown bread, both kept playing. But, when one got a reward and the other did not, the unrewarded animal stopped playing. Unlike primates, the quality of reward (as perceived by humans) didn't seem to matter to the dogs.⁷

The Ultimatum Game lets one person decide how to divide money between two people, and the other person to decide to accept the division or to reject it, in which case nobody gets anything. Deciders routinely reject lowball offers in much the same way monkeys refuse to pay the same price for cucumbers as for grapes. There appears to be a cultural dimension to this, as Americans usually accept greater inequality than Europeans (Hochschild 1981).

Many traditional stories raise the complex issue of what we mean by “fair.” One is the story of the prodigal son (Luke 15: 11-32). Another is the parable of the man who hired field workers at the first, third, sixth, and eleventh hours of the day but paid them all the same amount at the end of the day.

One possibility is to pay each dancer 12 rupees based on the contract. Another is to divide the 48 rupees among the dancers based on hours actually danced by each (3, 7, 17, and 25 rupees per dancer). As 48 rupees divided by four dancers for four hours implies an hourly rate of 3 rupees per person per hour danced, a third possibility is to pay the dancers 3, 6, 9 and 12 rupees, the remaining 18 rupees going unpaid as the contract was only partially completed. A fourth possibility is division based on ten hours of dancing (4.80, 9.60, 14.40, and 19.20 rupees per dancer). A fifth possibility is to refuse any payment because the dancers did not fulfill the contract for a group performance. A sixth is to pay only the dancer who fulfilled her contract. But, should she receive 12 rupees or the entire amount of 48 rupees?

Imagine a failed business in which each partner supplied a third of the capital. One partner provided the management expertise, the second the manufacturing expertise, and the third the marketing expertise. Should each get a third of the remaining assets? Should the partner who took the biggest risk by delaying payment of school loans get more than the others? Should the one who exaggerated his skills and caused the failure get less?

If you chose the first possibility, you equate fairness with equality. If you chose the second, you think that fairness requires consideration of need.

A driver in the US guilty of going 17 miles (25 km) per hour over the speed limit might pay a fine of $200. Fairness means treating everyone equally (the justification for a flat tax). In Finland, the fine depends on income. This defines fairness as equity (the justification for progressive taxation). It resulted in Anssi Vanjoki paying a $103,600 fine just after cashing in some stock options.⁸ Exempting someone from the fine because he was speeding to a hospital with a heart attack victim defines fairness as need (the justification for consumption or negative income taxes). Whenever you hear someone—especially a politician—speaking of “fairness,” make sure you understand which of the three possible definitions he means and whether or not you agree that it is appropriate to the circumstance.

Brams and Taylor (1966) identified four additional requirements for fairness in thought experiments dividing a piece of cake under various circumstances. Parents often divide a cake between two children by allowing one to cut it and the other to choose first. A procedure with this characteristic is said to be proportional, the first requirement for fair division. In that neither child will feel the other child got a larger piece (although the chooser may sometimes think he got the larger one), the solution also is envy-free, the second requirement.

Brams and Taylor extended the process to three people, but could not solve it for four. They turned to Alan Taylor, who in a “Eureka!” moment saw that dividing a cake among “n” people requires cutting it into 2n-2 + 1 pieces at the start. That comes to five pieces in a four party dispute, reminiscent of the problem Britain, France, the US, and USSR faced in dividing Germany into occupation zones until they treated Berlin as a fifth “piece.” After the first person cuts the cake, the next person has the option of setting one piece aside then trimming any of the remaining pieces so that he perceives all of them to be equal.

Suppose that half the cake is vanilla and the other half is chocolate. Alita likes only vanilla, but Mike, who likes only chocolate and is the cutter does not know this. The only way Mike is sure to have some of his favorite flavor is to cut the cake so each piece has equal amounts of chocolate. In this case, if Mike and Alita announced their preferences, Mike could cut the cake so that he got all the chocolate and Alita got all the vanilla. The solution would remain proportional and envy-free but also would be efficient, a third desirable characteristic of fair division. But, if Alita was being deceptive about her preference, which really is for chocolate, she will end up with all the chocolate and Mike will end up with the vanilla he dislikes. Efficiency requires truthfulness, rare among people in conflict.

“Making the cake larger” is a second aspect of efficiency. Consider a sister and brother deciding who gets a diamond and who gets a rifle, both appraised at $2000. The sister personally values the rifle at $2500 and the diamond at the $2000 for which she could sell it. The brother values the diamond at $2500 but the rifle at the $2000 for which he could sell it. In this artificially simple situation, giving the diamond to the sister and the rifle to the brother would give $2000 to each for a joint gain of $4000. But, if the diamond went to the brother and the rifle to the sister, each would feel they had received $2500, a joint gain of $5000. The second solution is more efficient. It also is likely to be stable, the characteristic originally identified by Nash and sought by Fraser and Hipel.

All this speculation clarifies “fairness.” Sometimes we mean equality, sometimes equity, and sometimes need. However defined, the problem remains of finding a practical method to achieve it that is efficient, envy-free, proportional, stable, and encourages truthfulness. Nobody has devised a perfect method to do all this. Divide and choose is a simple technique going back at least to the Roman army, where pairs of soldiers each received a single large loaf of bread each day. Another classic, used by survivors of shipwrecks, is to have one person divide a captured bird or fish into as many parts as there are survivors. The divider holds up a piece for all but one person to see, and that person calls out the name of the person to receive that piece. But, like Taylor’s solution to dividing a cake among “n” people, it is not useful in many disputes.

Four mathematical methods come close to the ideal and are adaptable to various circumstances The first technique, Adjusted Winner, begins by having each claimant distribute 100 (or 1000 if a lot of items are involved) points to the items to indicate their relative importance. Let us assume two claimants assign their points among five items as shown in Figure 2.6:

Tentatively assign each item to the individual who values it the most, as shown by underlining. The items are re-ordered from left to right based on the ratios of the points (always dividing the largest by the smallest number). Add the underlined points to determine the perceived value received by each, as shown in Figure 2.7:

We now must determine what and how much Mike must transfer to Alita to equalize the perceived result. Work from smallest to largest ratio, which is to say left to right, through as many items as necessary to achieve the goal. In this case, transferring all of C to Alita gives her 71 points but reduces Mike to 40 points.

This system and this order ensure efficiency, envy-freeness, and proportionality, so is likely to prove stable. It also makes it possible to divide goods in any proportion. For example, if Alita is entitled to two-thirds of the property, merely rewrite the equation so that twice Mike’s shares equals Alita’s share:

The method is difficult to understand, works only with items that are divisible into small units, only with two claimants, and only if both honestly represent their interests. It may work sometimes but is not a useful general solution.

When truthfulness seems unlikely, Proportional Allocation gives up efficiency to insure equality, proportionality, and envy-freeness. It is much simpler to carry out. Start with the same situation as before (Figure 2.6). Assign Mike 15/24 and Alita 9/24 of item A; Mike 25/45 of item B, and so on. Determine the value each receives by squaring the points assigned to each item by the individual and dividing by the column total. For example, for item A, Mike’s value received is 15*15/24 = 9.37 and Alita’s is 9*9/24 = 3.37. Thus, Mike would receive 9.37 + 13.89 + 18.85 + 3.33 + 6.25 = 51.69 points. Alita would receive 3.37 + 8.89 + 13.85 + 13.33 + 12.25 = 51.69 points. Alita and Mike each receive equal shares but 8.88% less value than the 56.69 points they would receive using Adjusted Winner. The method remains envy free and proportional but surrenders efficiency to ensure truthfulness. As in the case of Adjusted Winner, the process is limited to two claimants and to items that are easily subdivided.

What if there are many items and claimants? Taylor did develop a technique (described above) based on cake cutting, but it quickly becomes unmanageable. Imagine three siblings trying to divide an inheritance consisting of $20,000 and fifty items that include such typical family possessions as a piano and a car, items that are hard to “trim.” Of course, the siblings could simply agree to sell everything and divide the resulting cash. The siblings could even buy some items themselves, knowing they would get back one third of the price paid for anything as part of their equal share of the proceeds from the sale when the cash was divided. However, two siblings might bid against one another for the same item, reducing efficiency. Furthermore, one sibling might try to gain an advantage by bidding on an unwanted item to force one who wants it to pay more.

There are two alternatives to selling. The first is the Steinhaus fair-division procedure (Raiffa 1982). It assumes that each claimant has cash to give to other claimants if necessary to equalize the distribution. Each claimant submits a valuation for each item to a mediator. The mediator divides each claimant’s total valuation by the number of claimants, designating this amount as the “initial fair share.” First, assign each item to the claimant who values it most highly. Second, subtract the initial fair share from the total value each claimant perceives himself to be receiving. Third, divide the total excess (it will be positive so long as the initial valuations differ) equally among all claimants. Figure 2.12 provides a specific example. The result is efficient, envy-free, and proportional but complex, so disputants may not trust it.

The final possibility is an “imaginary auction.” Adopting this method, the negotiation will not be over who gets what, but over the rules for the auction. The most important of these rules are the minimum raise, whether to use the last bid or a professionally appraised value in balancing the books, and agreeing to use cash to balance the books at the end. Balancing based on the last bid is best except perhaps when it is worth paying for a professional appraisal. With the rules set, claimants make and raise bids on each item in turn until someone wins it. Instead of actually paying for items, the winning bid is credited as value received. After auctioning off all items, the total received by each person is determined and cash is used to balance the books. The cash might be part of the property to be divided (the reason for exempting it from the bidding), some of which might come from sale of items on which no claimant bids. If there is no cash, or not enough, the winner of the greatest value has to make the payments from his own funds. If there is more cash remaining than necessary to balance the books, it can be divided among the claimants. Figure 2.9 shows what might have happened in such an auction in which fifty items and $15,000 were divided equally among four siblings. In this case, Sibling C had to contribute personal cash to balance the books.

A claimant could try to gain an advantage if he knows how much someone else values an item. By overbidding, he can force the other person to pay too much, which will increase the value he receives in the end. However, it is risky. The person seeking the advantage may overestimate the amount the other will bid and end up winning the item for more than he values it, reducing rather than increasing the total value he receives. Making sure everyone understands the trick and its implications before the bidding starts tends to increase truthfulness and reduce even if it does not eliminate strategic bidding.

The imaginary auction is likely to assign each item to the person who wants it most, so the result is envy-free and efficient. Proportionality is impossible to calculate because we do not know how much more each winning bidder would have paid—something the bidders might not even know themselves.

The method has much to commend it. It is envy-free and efficient, and can handle almost any number of claimants and items. Nobody has to decide in advance how much to bid on an item. Disproportionate entitlements remain possible through cash adjustments. It minimizes bickering and bitterness, reduces the time wasted in trying to reach a solution, and is easy to understand.