How to Divide Something Fairly: Cut and Choose, Worked Out

A candy bar, a pair of children and an argument over who got the bigger half: parents have settled that fight for generations with a rule that is also a piece of game theory. You cut, I choose. This page shows why the rule works, runs the numbers on a textbook example, and then covers what to do when a third person arrives, because the rule breaks at that point.

How do you divide something fairly between two people?

Use divide and choose, also called “I cut, you choose”². One person is the divider and the other is the chooser, perhaps picked by a coin toss¹. The steps run in this order¹:

• First, the divider cuts the item into two pieces that are, in the divider’s eyes, equal in value¹.
• Second, the chooser selects either piece¹.
• Third, the divider receives the piece that is left¹.

The trick is the divider’s position. The divider does not know which piece will be theirs, so there is no incentive to cheat¹. A divider who cuts one piece larger risks being left with the smaller one². The chooser can pick either piece, so the chooser is guaranteed at least 50% of the whole in their own eyes¹. The divider is guaranteed a piece worth 50%¹.

That outcome has a name. It is an envy-free allocation, meaning neither person can claim they would rather have the other’s share⁹. To an outside observer the split may look uneven, but to the two people involved no one has cause to envy the other².

What is fair division in math?

Fair division is the mathematical study of splitting resources among parties who each have an entitlement, so that each party gets its due share³. A central rule of the field is that the players divide the goods themselves, because only they know how they value them³. Real settings include inheritance, partnership breakups and divorce settlements³.

Fairness has to be defined before it can be measured, and the field uses several definitions³. There is no objective measure of what an item is worth, so these definitions rest on each person’s own valuation³.

For two people with additive values, these first two criteria coincide: every envy-free division is also proportional². Cut and choose delivers both. The equitable criterion is the hard one, since players need not be truthful when asked their valuation³.

Worked example: splitting a year at a beach house

Two retirees, Fred and Martha, buy a vacation house in Florida and agree to split the year into 2 parts¹. Fred is the divider and cuts the year into November to February and March to October¹. The first piece is 4 months and the second is 8 months, yet Fred values them equally because he likes Florida in the winter¹. Martha values every month the same¹.

Worked example (our arithmetic): the cut and the two valuations

Martha picks March to October, a piece she values at 8 ÷ 12 = 66.7% of the whole (our arithmetic)¹. Fred keeps November to February, which he values at 50%¹. Both are at or above the 50% guarantee, and neither would swap. The unequal result is not a flaw. Martha gets more only because she values the months differently from Fred, and the procedure respects that difference.

The method suits people who distrust each other’s motives or cannot talk it through, such as two countries drawing a border or two children splitting a candy bar¹. Fred and Martha could simply discuss it, but the formal rule works even when they cannot¹.

How do you split 12 equally?

When the thing is uniform, such as 12 identical cookies or a $12 bill, plain division is enough: 12 ÷ 2 = 6 each for two people, 12 ÷ 3 = 4 each for three, and 12 ÷ 4 = 3 each for four (our arithmetic)⁶. Cut and choose is for the other case, where the parts are not interchangeable.

Here is a pack where equal counts are not equal values. A pizza has 4 slices of cheese and 4 slices of veggie with a total value of $12⁶. Marcus and Ben like both flavors equally, so each values every slice at $1.50⁶. Abby only likes cheese and values each cheese slice at $3 and veggie at $0⁶. A fair share for any of the 4 players is $3⁶.

Worked example (our arithmetic): why counting slices fails

Give Abby 2 slices of veggie and she holds $0 of value, even though she received 2 of the 8 slices, a fourth of them (our arithmetic)⁶. Splitting by count is even. Splitting by value is fair. They differ whenever tastes differ.

How do you split 100% three ways evenly?

Each share is 1/3 of the whole, or about 33.3%⁴. The harder part is making cuts that all 3 people accept as that much, and the obvious first attempt fails⁴.

The tempting approach picks a divider, lets the first chooser take any piece and the second chooser take whatever remains of the 3 pieces, and gives the divider the last one⁴. The divider is fine and so is the first chooser. The second chooser is the problem⁴.

Worked example (our arithmetic): the three-person failure

Chooser 1 takes Piece 1, which leaves Chooser 2 to choose between Piece 2 and Piece 3, and neither is worth a fair share of 1/3 or about 33.3% to her⁴. At 30% and 25% she is below the line. The method does not guarantee a fair division⁴. Divide and choose works only for two parties².

How do 3 or more people divide fairly?

Two methods work for any number of people. Both give every person a piece they consider at least a fair share, which with N people means at least 1/N of the total⁸.

The lone divider method

One person is picked at random as the divider, and the rest are choosers⁵. The steps⁵:

• The divider cuts the item into N pieces⁵.
• Each chooser separately lists the pieces they consider a fair share⁵. This list is the chooser’s declaration, or bid⁵.
• If every party can be given a piece they declared, do so, and the divider takes the piece that remains⁵.
• If two or more parties want the same pieces and no others, give a piece nobody contested to the divider, recombine the rest and repeat with the remaining parties⁵.

Take the failed example above. Both choosers declare Piece 1 only, so the pieces cannot be handed out at once⁵. Pieces 2 and 3 are uncontested, so the divider receives one of them, say Piece 2⁵. Pieces 1 and 3 are recombined into one piece worth 70% to each chooser⁵. The two choosers then use divider-chooser on it, and each is guaranteed at least 35%, which is a fair share of the original⁵.

Worked example (our arithmetic): the recombined piece

Steinhaus developed the procedure for 3 people, and Harold W. Kuhn later extended it to more⁸.

The last diminisher method

This method also suits 3 or more people⁶. The players are assigned an order at random, perhaps by pulling names from a hat⁶. The first player cuts a slice they value as a fair share⁶. Each later player either passes, or trims the slice if they think it is worth more than a fair share, and claims it⁶. After the last player decides, the last person to trim receives the slice, and if nobody trimmed it the cutter keeps it⁶. Repeat with the rest until 2 people remain, who finish with divider-chooser⁶.

The pizza above shows it. Marcus cuts 2 slices of cheese, which he values at $3⁶. Abby values the same piece at $6, trims it down to 1 slice of cheese worth her fair share of $3, and takes it, since Ben and Julian pass⁶. The order matters. Being last to decide is an advantage, because the last player can trim a sliver off and still claim the piece⁶.

Worked example (our arithmetic): Abby’s trim

A simpler version exists for three players too: Ali cuts three pieces equal in her eyes, and Blake and Chris each say which pieces they would accept⁹. If both reject Ali’s cut, there is a piece neither wants, which goes to Ali, and the other two pieces are pushed together for Blake and Chris to split by cut and choose⁹. It is long-winded but guarantees a fair split⁹.

Where does cut and choose fall short?

Three limits are worth knowing.

It can waste value. Suppose Bob likes only chocolate and Carol likes only vanilla². If Bob cuts without knowing Carol’s taste, his safe move is to put equal chocolate in both halves². Then Bob gets half the chocolate whichever half Carol takes, when a smarter split would have given each person everything they want². Fair is not the same as efficient.

It does not give an exact half. The cutter may judge parts of the cake differently from the chooser, and the chooser simply takes what they think is better².

It fixes the roles. Arguments about who cuts are common, so a symmetric version has both people mark their midpoint, cuts between the two marks, and gives each person the piece holding their own mark².

For two parties and indivisible goods, one procedure goes further. Adjusted winner is envy-free, equitable and Pareto optimal, and it splits at most one good¹¹.

Where is it used?

The idea is old. In Genesis, Abraham offers Lot the choice of land after dividing it into a western part and an eastern part, and Lot chooses the east². A similar rule is in the law of the sea: a state applying to mine the ocean floor must offer 2 areas of similar value and the UN authority picks one².

Modern uses are less dramatic. A shared three-bedroom apartment with rooms of different sizes and one rent to split is what the field calls rental harmony¹⁰³. It means dividing indivisible rooms and a divisible rent at once³. A division by square feet cannot take every individual preference into account¹⁰. The solution described in a New York Times lesson plan uses an algorithm by the mathematician Francis Su, built on Sperner’s Lemma¹⁰.

Where does the idea come from?

Fair division as a field dates back to the end of the second world war³. It was devised by Polish mathematicians Hugo Steinhaus, Bronislaw Knaster and Stefan Banach, who met in the Scottish Cafe in Lvov³. Steinhaus, hiding from the Nazis during World War II, asked how to divide resources fairly⁷. Inspired by divide and choose, he asked his students Banach and Knaster for a procedure that works for any number of people⁷. Martin Gardner brought the problem of fair division among larger groups to a wide audience in his May 1959 column in Scientific American².

Which method should you use?

Cut and choose fits two people, lone divider fits 3 or more, and last diminisher is the easier of those two to run at a table²⁵⁶. In each, every person ends up with a piece they value at 1/N or more⁸. For deeper reading, the free divider-chooser chapter, the lone divider chapter and the last diminisher chapter of one open textbook work through more cases. Fair division is one of the quieter places where math shows up in everyday life, and the Math in daily life section collects more, such as the hidden math in a normal day and what calculus is used for.