Proportional Division

For proportional division (proportionality), a resource is divided among n people with subjective evaluations, giving each person ate least 1/n of the resource by his/her own subjective valuation. It has been showed that for n people with additive valuation, a proportional division always exists and there are several known protocols for achieving such allocations, including:

The Banach-Knaster “Last Diminisher” Procedure (Steinhaus, 1948)

Inspired by the divide and choose protocol for dividing a cake between two partners, Hugo Steinhaus in 1947 challenged two of his doctoral students, Stefan Banach and Bronisław Knaster to find a procedure that could work for any number of people. Their solution was published in the paper The Problem of Fair Division in Econometrica 16 (1) in 1948. In the words of the authors, the protocol works as follows (Steinhaus, 1948 p. 102):

The partners being ranged A, B, C,.. N, A cuts from the cake an arbitrary part. B has now the right, but is not obliged, to diminish the slice cut off. Whatever he does, C has the right (without obligation) to diminish still the already diminished (or not diminished) slice, and so on up to N. The rule obliges the "last diminisher" to take as his part the slice he was the last to touch. This partner being thus disposed of, the remaining n−1 persons start the same game with the remainder of the cake. After the number of participants has been reduced to two, they apply the classical rule for halving the remainder. - Excerpt, The Problem of Fair Division (Steinhaus, 1948)

Explained somewhat more neatly:

The Banach-Knaster "Last Diminisher" Procedure

For three people Alice, Bob, Carol: 1. Alice begins by cutting from the cake, a part of an arbitrary size according to her individual utility function 2. Bob next has the right, but is not obliged to, cut a part from Alice's slice. Carol then next also has the right, but is not obliged to, cut another part from Alice's slice 3. Each person has to take the slice they were the last to cut from, and thus exit the game 4. The procedure repeats with the remainder of the cake among the people who have yet to receive a piece

The procedure is equitable but not envy-free because, although all players can ensure themselves to at least 1/n pieces by not diminishing a piece, all except the last two players may envy those who receive pieces later — i.e., although players exist the game satisfied with their piece, they cannot guarantee that they would not have preferred a piece later obtained by someone else (Brams & Taylor, 1996).

The Dubin-Spanier “Moving Knife” Method

A continuous-time version of the “Last Diminisher” method is the Dubin-Spanier “moving knife” method, proposed by Lester E. Dubins and Edwin H. Spanier (1961). It is the first moving-knife procedure described in the literature (although, as pointed out by Brams & Taylor, an earlier “pouring procedure” was published in Davis, 1955).

The Dubin-Spanier Moving-Knife Method 1. A referee holds the knife on the left side of the cake, and then moves the knife towards the right side at a continuous pace 2. At any point, any of the three players can call out "cut", and receive the piece to the left of the knife, existing the game 3. The procedure repeats with the remainder of the cake. If the remaining two players yell cut simultaneously, the cut piece is given to one of the players at random

The Dubin-Spanier procedure is equitable (each player values his/her allocation to be the same as the others, according to his/her own utility function) but not envy-free. A key difference between the procedure and the last diminisher is that each player still in the game has to make continuous choices, rather than by turns.

The Steinhaus-Kuhn “Lone Divider” Procedure

The so-called “lone divider procedure” is a fair division procedure which, like the last diminisher and the moving-knife method above, also results in a proportional division of the resource among an arbitrary number of people. It was proposed by game theorist Harold Kuhn in 1967 and works as follows (Brams & Taylor, 1996), in the case of n=3 for simplicity’s sake:

The Steinhaus-Kuhn "Lone Divider" Procedure

For three people Alice, Bob and Carol: 1. Let Alice cute the cake into three pieces of equal value based on her utility function 2. Now let Bob and Carol indicate which pieces are acceptable to them (at least of value 1/3 based on their own utility functions) This leads two two mutually exclusive cases: 1. Either Bob or Carol finds two or more pieces are acceptable. If Bob finds more than one piece acceptable, then choosing in the order: Carol, Bob and Alice will result in a proportional allocation 2. Or, both Bob and Carol indicate that at most one piece is acceptable. We then give the piece that both Bob and Carol find unacceptable to Alice, then do a divide and choose procedure between Bob and Carol. This will also result in a proportional allocation

Because it is impossible for either Carol or Bob to think that all three pieces cut by Alice are unacceptable (of sizes smaller than 1/3), case two must prevail if case one does not, as they are the only two possible situations.

The “lone divider” procedure is not envy-free either because in case one, although Alice and Bob will not envy anyone, Alice will envy Bob if he took the larger of the two pieces she considered acceptable. In case two, if the redivision by Bob and Carol is not exactly 50/50 according to Bob, then he will be envious of one of the other two (Brams & Taylor, 1996).

Kuhn published the result in a book chapter entitled On games of fair division in the 1967 book Essays on Mathematical Economics in Honor of Oskar Morgenstern edited by Martin Shubik for Princeton University Press. In the book, Kuhn presents the case of an arbitrary number of players, and proves fair proportional division by making use of the Frobenius-König theorem.

Envy-Free Division

The added criterion of envy-freeness requires that in addition to proportional division (equity), the allocation of the resource has the property of envy-freeness, namely that

Envy-freeness: No player is willing to give up its allocation in exchange for the other players' allocations

Selfridge-Conway Discrete Procedure

The first envy-free discrete cake-cutting procedure for three players was discovered by John Selfridge in 1960, without publishing its proof. It was later re-discovered by John Horton Conway independently, although he also chose not to publish it (Brams & Taylor, 1996). The procedure may be described in the following way for players Alice, Bob and Carol:

The Selfridge-Conway Envy-Free Cake-Cutting Procedure

For players Alice, Bob and Carol: 1. Alice divides the cake into three pieces A,B,C she considers to be of equal size. 2. Bob thinks piece A is the largest piece 3. Bob now trims a bit off piece A to make it the same size as the second largest piece. Now, piece A is divided into: 1. The trimmed piece A₁ and the trimmings A₂ 4. Carol next chooses a piece among A₁ and the two other pieces B, C 5. Bob next chooses a piece with the limitation that if Carol didn't choose A₁, he has to 6. Alice finally chooses the last piece leaving only the trimmings A₂ to be divided among the three The trimmed piece A₁ was chosen by either Carol or Bob. Call the player that chose it Player A and the other Player B. To divide the trimmings A₂ among them: 1. Player B first cuts the trimmings A₂ into three equal pieces A₂₁, A₂₂ and A₂₃ 2. Player A then chooses a piece of A₂, call it A₂₁ 3. Alice next chooses a piece of A₂, call it A₂₂ 4. Player B finally chooses the last remaining piece of A₂, call it A₂₃

The outcome of the procedure is:

Player A received piece A₁ + A₂₁

Player B received piece B + A₂₃

Alice received piece C + A₂₂

The Selfridge-Conway procedure ensures envy-freeness because if Carol chooses first, she does so knowing what she has to choose from, and so will not envy anyone who chooses either of the other two pieces. Because Bob creates a two-way tie for the largest piece by trimming piece A, and at least one of these two is still available after Carol chooses, he will not either envy anyone. Finally, because if Carol didn’t choose A₁, Alice has to, the trimmed piece cannot be the one left over. Hence, Alice chooses among two untrimmed pieces which she herself cut, and so will not envy anyone.

The key observation is hence that Alice will not envy the player who receives the trimmed piece. Alice created a three-way tie among the pieces and received an untrimmed piece, while the trimmed piece, even with the trimmings added to it, would yield Carol only a piece that Alice considers to be exactly the same size as the one she received (Brams & Taylor, 1996)..

Stromquist “Moving-Knives” Procedure

For connected pieces, the so-called Stromquist moving-knives procedure first introduced by Walter Stromquist in 1980 ensures both proportionality and envy-freeness (). Similar to the Dubin-Spanier moving knife procedure, it requires that the players make continuous choices as to whether or not the slice to the left of the knife ensures them a piece which is proportional.

The Stromquist "Moving-Knives" Envy-Free Cake Cutting Procedure 1. A referee moves a knife from left to right over a one-dimensional cake, hypothetically dividing it into a small left piece and a large right piece 2. Each player has their own knife, kept parallel and to the right of the referee's knife, always positioning it so that it divides the remainder of the cake in half 3. At any time, any player can call out "cut", receiving the piece to the left of the referee's knife. Simultaneously, a cut is made by whichever of the three players' knives is in the middle of the three players' knives 4. The player that called cut receives the piece to the left of the referee's knife. The player whose knife was closest to the referee's knife receives the middle piece. The last player receives the right piece.

The Stromquist moving-knives procedure ensures envy-freeness because whoever receives the middle piece (of the two players who did not yell cut, had their knife closest to the referee’s knife) held either the second or third knife from the left. Hence, that player that received the middle piece thinks that the middle piece is either larger than the right piece (if his/her knife is closest to the referee’s knife) or tied with the right piece for largest (if his/her knife is the knife that divides the remainder of the cake). Similarly, the other player who did not yell cut held either the knife who cut the remainder of the cake, or the knife on the far right.

Players’ and referee’s knives in the Stromquist moving-knives cake cutting procedure (Photo: Brams & Taylor, 1996)

Epilogue

Hugo Steinhaus’ initial publication The problem of fair division set the scene for a century of ever-more complicated and complete cake-cutting procedures. A sub-branch of fair division, which itself is a sub-branch of game theory, cake-cutting today stands as an archetypal game which represents the incentives and limitations of certain kinds of social scientific conflicts of competition and cooperation in non-cooperative games.

Those interested in studying cake-cutting procedures further are encouraged to check out the book Fair Division by Steven J. Brams and Alan D. Taylor (1996). In addition, the YouTuber Numberphile, Brady Haran, has a video about cake-cutting procedures for n=3 which is quite good, available here.