For many of us, an early lesson in fair division happens in elementary school with the choosing of sides for a spelling bee or when picking teams on the playground. In terms of importance, these pale in comparison with the issue of property settlement in a divorce. Remarkably, however, the same fair-division procedure—taking turns—is often used in both.

Taking turns is fairly self-explanatory. With two parties (and that’s all we’ll consider here), one party selects an object, the other party then selects one, the first party then selects again, and so on. But in this context, there are several interesting questions to consider.

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The answer to Question 1 is often “toss a coin,” but there are other possibilities— for example, the two parties could “bid” for the right to go first, as in an auction. The answer to Question 2 is less clear, but in Writing Project 2 (on page 570), we outline a discussion of the issue it raises.

Question 3, on the other hand, is remarkably interesting, and it is this one that we want to pursue. Let’s look at an easy example. Suppose that Bob and Carol are getting a divorce, and their four main possessions, ranked from best to worst by each, are as follows:

If Carol knows nothing of Bob’s preferences, then we can assume that she will choose sincerely—selecting at her turn whichever item she most prefers from those not yet chosen. Now, if Bob is also sincere, and if he chooses first, the items will be allocated as follows:

Hence, Bob gets his first and third favorites (the pension and the investments). However, if Bob opens by choosing the house—and bypassing the pension for the moment—then the allocation will be as follows:

Thus, by being insincere, Bob does better—getting his first and second favorites (the pension and the house).

In general, then, what is the optimal strategy for rational players to use, assuming that both know the preferences of the other? The answer is something called the bottom-up strategy, discovered by the mathematicians D. A. Kohler and R. Chandrasekaran in 1969. We will illustrate it with an example.

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Suppose that we have five objects—, , , , —and Bob is choosing first. Suppose that Bob and Carol have the following rankings of the objects (called preference lists in what follows):

It will turn out that Bob should open with (his third choice), followed by Carol’s choice of (skipping over , for the moment). Bob will then take , Carol will follow with , and finally Bob will get . Bob gets his first, second, and third choices without selecting his first choice first! Where does this strategy come from?

The intuition here is quite easy. Let’s make two assumptions about rational players: A rational player will never willingly choose his or her least-preferred alternative, and a rational player will avoid wasting a choice on an object that he or she knows will remain available and thus can be chosen later.

With these assumptions as motivation, let’s return to the preceding example and think about the mental calculation that Bob will go through in deciding what his first choice will be. Bob knows the eventual sequence of choices will fill in all the following blanks:

Now, working mentally from right to left, Bob knows that Carol will not choose because it is at the bottom of her list. Thus, he will get stuck with , and so he will avoid wasting anything but his last choice on alternative . Thus, Bob can pencil in alternative as his last choice.

Bob, placing himself momentarily in Carol’s shoes, knows she will reason the same way, and thus he pencils Carol in for the bottom alternative, , on his list.

Mentally now, Bob reasons as if alternatives and never existed (and the choice sequence had been Bob-Carol-Bob) and continues to pencil in alternatives from right to left, with Bob working from bottom to top on Carol’s preference list and Carol working from bottom to top on Bob’s preference list. This yields the following sequence of choices mentally penciled in by Bob:

Remember, this is just a mental calculation that Bob went through to decide upon the actual choice—in this case, —with which he will open. Bob has no guarantee that Carol will, in fact, respond with , so the use of this strategy involves some risk on Bob’s part.

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This bottom-up strategy can also be viewed as a procedure that a mediator could use to specify a division of several objects between two parties. Given the preference lists of both parties, the mediator could construct a list—exactly as we did for Bob and Carol above—and then offer this to the parties as the suggested allocation. In effect, the mediator is simultaneously playing the role of two rational parties who choose to employ optimal strategies.