Stromquist moving-knives procedure

Last updated October 28, 2024

The Stromquist moving-knives procedure is a procedure for envy-free cake-cutting among three players. It is named after Walter Stromquist who presented it in 1980.^[1]

This procedure was the first envy-free moving knife procedure devised for three players. It requires four knives but only two cuts, so each player receives a single connected piece. There is no natural generalization to more than three players which divides the cake without extra cuts. The resulting partition is not necessarily efficient.^[2]^{: 120–121}

Procedure

A referee moves a sword from left to right over the cake, hypothetically dividing it into small left piece and a large right piece. Each player moves a knife over the right piece, always keeping it parallel to the sword. The players must move their knives in a continuous manner, without making any "jumps".^[3] When any player shouts "cut", the cake is cut by the sword and by whichever of the players' knives happens to be the central one of the three (that is, the second in order from the sword). Then the cake is divided in the following way:

The piece to the left of the sword, which we denote Left, is given to the player who first shouted "cut". We call this player the "shouter" and the other two players the "quieters".
The piece between the sword and the central knife, which we denote Middle, is given to the remaining player whose knife is closest to the sword.
The remaining piece, Right, is given to the third player.

Strategy

Each player can act in a way that guarantees that—according to their own measure—no other player receives more than them:

Always hold your knife such that it divides the part to the right of the sword to two pieces that are equal in your eyes (hence, your knife initially divides the entire cake to two equal parts and then moves rightwards as the sword moves rightwards).
Shout 'cut' when Left becomes equal to the piece you are about to receive if you remain quiet (i.e. if your knife is leftmost, shout 'cut' if Left=Middle; if your knife is rightmost, shout if Left=Right; if your knife is central, shout 'cut' if Left=Middle=Right).

Analysis

We now prove that any player using the above strategy receives an envy-free share.

First, consider the two quieters. Each of them receives a piece that contains their own knife, so they do not envy each other. Additionally, because they remained quiet, the piece they receive is larger in their eyes than Left, so they also don't envy the shouter.

The shouter receives Left, which is equal to the piece they could receive by remaining silent and larger than the third piece, hence the shouter does not envy any of the quieters.

Following this strategy each person gets the largest or one of the largest pieces by their own valuation and therefore the division is envy-free.

The same analysis shows that the division is envy-free even in the somewhat degenerate case when there are two shouters, and the leftmost piece is given to any of them.

Dividing a 'bad' cake

The moving-knives procedure can be adapted for chore division - dividing a cake with a negative value.^[4]^{: exercise 5.11}

Related Research Articles

Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inheritance, partnership dissolutions, divorce settlements, electronic frequency allocation, airport traffic management, and exploitation of Earth observation satellites. It is an active research area in mathematics, economics, and dispute resolution. The central tenet of fair division is that such a division should be performed by the players themselves, without the need for external arbitration, as only the players themselves really know how they value the goods.

In the mathematics of social science, and especially game theory, a moving-knife procedure is a type of solution to the fair division problem. The canonical example is the division of a cake using a knife.

An envy-free cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the envy-free criterion, namely, that every partner feels that their allocated share is at least as good as any other share, according to their own subjective valuation.

Divide and choose is a procedure for fair division of a continuous resource, such as a cake, between two parties. It involves a heterogeneous good or resource and two partners who have different preferences over parts of the cake. The protocol proceeds as follows: one person cuts the cake into two pieces; the other person selects one of the pieces; the cutter receives the remaining piece.

The Austin moving-knife procedures are procedures for equitable division of a cake. To each of n partners, they allocate a piece of the cake which this partner values as exactly $of the cake. This is in contrast to proportional division procedures, which give each partner at least of the cake, but may give more to some of the partners.$

Chore division is a fair division problem in which the divided resource is undesirable, so that each participant wants to get as little as possible. It is the mirror-image of the fair cake-cutting problem, in which the divided resource is desirable so that each participant wants to get as much as possible. Both problems have heterogeneous resources, meaning that the resources are nonuniform. In cake division, cakes can have edge, corner, and middle pieces along with different amounts of frosting. Whereas in chore division, there are different chore types and different amounts of time needed to finish each chore. Similarly, both problems assume that the resources are divisible. Chores can be infinitely divisible, because the finite set of chores can be partitioned by chore or by time. For example, a load of laundry could be partitioned by the number of articles of clothing and/or by the amount of time spent loading the machine. The problems differ, however, in the desirability of the resources. The chore division problem was introduced by Martin Gardner in 1978.

Consensus splitting, also called exact division, is a partition of a continuous resource ("cake") into some k pieces, such that each of n people with different tastes agree on the value of each of the pieces. For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided into three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is an exact division (with k = 3 and n = 2), as both Alice and George value the three pieces as 20%, 50% and 30% respectively. Several common variants and special cases are known by different terms:

The Selfridge–Conway procedure is a discrete procedure that produces an envy-free cake-cutting for three partners. It is named after John Selfridge and John Horton Conway. Selfridge discovered it in 1960, and told it to Richard Guy, who told many people, but Selfridge did not publish it. John Conway later discovered it independently, and also never published it. This procedure was the first envy-free discrete procedure devised for three partners, and it paved the way for more advanced procedures for n partners.

The Brams–Taylor procedure (BTP) is a procedure for envy-free cake-cutting. It explicated the first finite procedure to produce an envy-free division of a cake among any positive integer number of players.

Fair cake-cutting is a kind of fair division problem. The problem involves a heterogeneous resource, such as a cake with different toppings, that is assumed to be divisible – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be unanimously fair – each person should receive a piece believed to be a fair share.

The last diminisher procedure is a procedure for fair cake-cutting. It involves a certain heterogenous and divisible resource, such as a birthday cake, and n partners with different preferences over different parts of the cake. It allows the n people to achieve a proportional division, i.e., divide the cake among them such that each person receives a piece with a value of at least 1/n of the total value according to his own subjective valuation. For example, if Alice values the entire cake as $100 and there are 5 partners then Alice can receive a piece that she values as at least $20, regardless of what the other partners think or do.

The Simmons–Su protocols are several protocols for envy-free division. They are based on Sperner's lemma. The merits of these protocols is that they put few restrictions on the preferences of the partners, and ask the partners only simple queries such as "which piece do you prefer?".

The fair pie-cutting problem is a variation of the fair cake-cutting problem, in which the resource to be divided is circular.

The Brams–Taylor–Zwicker procedure is a protocol for envy-free cake-cutting among 4 partners.

Equitable (EQ) cake-cutting is a kind of a fair cake-cutting problem, in which the fairness criterion is equitability. It is a cake-allocation in which the subjective value of all partners is the same, i.e., each partner is equally happy with his/her share. Mathematically, that means that for all partners $i$ and $j$ :

The Levmore–Cook moving-knives procedure is a procedure for envy-free cake-cutting among three partners. It is named after Saul X. Levmore and Elizabeth Early Cook who presented it in 1981. It assumes that the cake is two-dimensional. It requires a referee, two knives and four cuts, so some partners may receive disconnected pieces.

The Robertson–Webb rotating-knife procedure is a procedure for envy-free cake-cutting of a two-dimensional cake among three partners. It makes only two cuts, so each partner receives a single connected piece.

The Barbanel–Brams rotating-knife procedure is a procedure for envy-free cake-cutting of a cake among three partners. It makes only two cuts, so each partner receives a single connected piece.

Symmetric fair cake-cutting is a variant of the fair cake-cutting problem, in which fairness is applied not only to the final outcome, but also to the assignment of roles in the division procedure.

In computer science, the Robertson–Webb (RW) query model is a model of computation used by algorithms for the problem of fair cake-cutting. In this problem, there is a resource called a "cake", and several agents with different value measures on the cake. The goal is to divide the cake among the agents such that each agent will consider his/her piece as "fair" by his/her personal value measure. Since the agents' valuations can be very complex, they cannot - in general - be given as inputs to a fair division algorithm. The RW model specifies two kinds of queries that a fair division algorithm may ask the agents: Eval and Cut. Informally, an Eval query asks an agent to specify his/her value to a given piece of the cake, and a Cut query asks an agent to specify a piece of cake with a given value.

References

↑ Stromquist, Walter (1980). "How to Cut a Cake Fairly". The American Mathematical Monthly. 87 (8): 640–644. doi:10.2307/2320951. JSTOR 2320951.
↑ Brams, Steven J.; Taylor, Alan D. (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. ISBN 0-521-55644-9.
↑ The importance of this continuity is explained here: "Stromquist's 3 knives procedure". Math Overflow. Retrieved 14 September 2014.
↑ Robertson, Jack; Webb, William (1998). Cake-Cutting Algorithms: Be Fair If You Can. Natick, Massachusetts: A. K. Peters. ISBN 978-1-56881-076-8. LCCN 97041258. OL 2730675W.

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[1] Stromquist, Walter (1980). "How to Cut a Cake Fairly". The American Mathematical Monthly. 87 (8): 640–644. doi:10.2307/2320951. JSTOR 2320951.

[2] Brams, Steven J.; Taylor, Alan D. (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. ISBN 0-521-55644-9.

[3] The importance of this continuity is explained here: "Stromquist's 3 knives procedure". Math Overflow. Retrieved 14 September 2014.

[rw-4] Robertson, Jack; Webb, William (1998). Cake-Cutting Algorithms: Be Fair If You Can. Natick, Massachusetts: A. K. Peters. ISBN 978-1-56881-076-8. LCCN 97041258. OL 2730675W.

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