Fair cake-cutting

If a cake with a selection of toppings is simply divided into equal slices, different people will receive different toppings, and some may not consider this "fair"

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 subjectively fair, in that each person should receive a piece that he or she believes to be a fair share.

The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time.

The cake-cutting problem was introduced by Hugo Steinhaus after World War II^[1] and is still the subject of intense research in mathematics, computer science, economics and political science.^[2]

Assumptions

There is a cake C, which is usually assumed to be either a finite 1-dimensional segment, a 2-dimensional polygon or a finite subset of the multidimensional Euclidean plane R^d.

There are n people with equal rights to C.^[3]

C has to be divided to n disjoint subsets, such that each person receives a disjoint subset. The piece allocated to person i is called $X_{i}$, and ${\displaystyle C=X_{1}\sqcup \cdots \sqcup X_{n}}$.

Each person should get a piece with a "fair" value. Fairness is defined based on subjective value functions. Each person i has a subjective value function V_i which maps subsets of C to numbers.

All value functions are assumed to be absolutely continuous with respect to the length, area or (in general) Lebesgue measure.^[4] This means that there are no "atoms" – there are no singular points to which one or more agents assign a positive value, so all parts of the cake are divisible.

Additionally, in some cases the value functions are assumed to be sigma additive (the value of a whole is equal to the sum of the values of its parts).

Example cake

In the following lines we will use the following cake as an illustration.

• The cake has two parts: chocolate and vanilla.
• There are two people: Alice and George.
• Alice values the chocolate as 9 and the vanilla as 1.
• George values the chocolate as 6 and the vanilla as 4.

Justice criteria

The original and most common criterion for justice is proportionality (PR). In a proportional division, each person receives a piece that he values as at least 1/n of the value of the entire cake. In the example cake, a proportional division can be achieved by giving all the vanilla and 4/9 of the chocolate to George (for a value of 6.66), and the other 5/9 of the chocolate to Alice (for a value of 5). In symbols:

${\displaystyle \forall {i}:\ V_{i}(X_{i})\geq 1/n}$

The proportionality criterion can be generalized to situations in which the rights of the people are not equal. For example, the cake belongs to shareholders such that one of them holds 20% and the other holds 80% of the cake. This leads to the criterion of Weighted proportionality (WPR):

${\displaystyle \forall i:\ V_{i}(X_{i})\geq w_{i}}$

Where the w_i are weights that sum up to 1.

Another common criterion is envy-freeness (EF). In an envy-free division, each person receives a piece that he values at least as much as every other piece. In symbols:

${\displaystyle \forall i,j:\ V_{i}(X_{i})\geq V_{i}(X_{j})}$

In some cases, there are implication relations between proportionality and envy-freeness, as summarized in the following table:

Agents	Valuations	EF implies PR?	PR implies EF?
2	additive	yes	yes
2	general	no	yes
3+	additive	yes	no
3+	general	no	no

A third, less common criterion is equitability (EQ). In an equitable division, each person enjoys exactly the same value. In the example cake, an equitable division can be achieved by giving each person half the chocolate and half the vanilla, such that each person enjoys a value of 5. In symbols:

${\displaystyle \forall i,j:\ V_{i}(X_{i})=V_{j}(X_{j})}$

A fourth criterion is exactness. If the entitlement of each partner i is w_i, then an exact division is a division in which:

${\displaystyle \forall {i,j}:\ V_{i}(X_{j})=w_{j}}$

If the weights are all equal (to 1/n) then the division is called perfect and:

${\displaystyle \forall i,j:\ V_{i}(X_{j})=1/n}$

Additional criteria

In addition to justice, it is also common to consider the economic efficiency of the division; see Efficient cake-cutting.

In some cases, the pieces allocated to the partners must satisfy some geometric constraints, in addition to being fair.

• The most common constraint is connectivity: each piece must be a connected space. In case the "cake" is a 1-dimensional interval, this translates to the requirement that each piece is also an interval.
• Another constraint is adjacency. This constraint applies to the case when the "cake" is a disputed territory that has to be divided among neighboring countries. In this case, it may required that the piece allocated to each country is adjacent to its current territory; this constraint is handled by Hill's land division problem.

In addition to the desired properties of the final partitions, there are also desired properties of the division process. One of these properties is truthfulness (aka incentive compatibility), which comes in two levels.

• Weak truthfulness means that if the partner reveals his true value measure to the algorithm, he is guaranteed to receive his fair share (e.g. 1/n of the value of the entire cake, in case of proportional division), regardless of what other partners do. Even if all other partners make a coalition with the only intent to harm him, he will still receive his guaranteed proportion. Most cake-cutting algorithms are truthful in this sense.^[1]
• Strong truthfulness means that no partner can gain from lying. I.e., telling the truth is a dominant strategy. Most cake-cutting protocols are not strongly truthful; building a strongly truthful division protocol is a difficult task.^[5]

Results

2 people – proportional and envy-free division

For $n=2$ people, an EF division always exists and can be found by the Divide and choose protocol. If the value functions are additive then this division is also PR; otherwise, proportionality is not guaranteed.

Proportional division

For n people with additive valuations, a PR division always exists. Moreover, a weighted proportional division is also guaranteed to exist for every set of weights (Corollary 1.1 of ^[6]).

The most common protocols for unweighted PR division are:

• Last diminisher, a protocol that can guarantee that the n pieces are connected (i.e. no person gets a set of two or more disconnected pieces). In particular, if the cake is a 1-dimensional interval then each person receives an interval. This protocol is discrete and can be played in turns. It requires O(n²) actions.
• The Dubins–Spanier Moving-knife procedure is a continuous-time version of Last diminisher.^[6]
• Fink protocol (also known as successive pairs or lone chooser) is a discrete protocol that can be used for online division: given a proportional division for n − 1 partners, when a new partner enters the party, the protocol modifies the existing division so that both the new partner and the existing partners remain with 1/n. The disadvantage is that each partner receives a large number of disconnected pieces.
• The Even–Paz protocol, based on recursively halving the cake and the group of agents, requires only O(n log n) actions. This is fastest possible deterministic protocol for proportional division, and the fastest possible protocol for proportional division which can guarantee that the pieces are connected.
• Edmonds–Pruhs protocol is a randomized protocol that requires only O(n) actions, but guarantees only a partially proportional division (each partner receives at least 1/an, where a is some constant), and it might give each partner a collection of "crumbs" instead of a single connected piece.
• Beck land division protocol can produce a proportional division of a disputed territory among several neighbouring countries, such that each country receives a share that is both connected and adjacent to its currently held territory.
• Woodall's super-proportional division protocol produces a division which gives each partner strictly more than 1/n, given that at least two partners have different opinions about the value of at least a single piece.

None of the protocols described above guarantees that the division is envy-free.

Envy-free division

An EF division for n people exists even when the valuations are not additive, as long as they can be represented as consistent preference sets. EF division has been studied separately for the case in which the pieces must be connected, and for the easier case in which the pieces may be disconnected.

For connected pieces the major results are:

• Stromquist moving-knives procedure produces an envy-free division for 3 people, by giving each one of them a knife and instructing them to move their knives continuously over the cake in a pre-specified manner.
• Simmons' protocol can produce an approximation of an envy-free division for n people with an arbitrary precision. If the value functions are additive, the division will also be proportional. Otherwise, the division will still be envy-free but not necessarily proportional. The algorithm gives a fast and practical way of solving some fair division problems.^[7][8]

Both these algorithms are infinite: the first is continuous and the second might take an infinite time to converge. In fact, envy-free divisions of connected intervals to 3 or more people cannot be found by any finite protocol.^[9]

For possibly-disconnected pieces the major results are:

• Selfridge–Conway discrete procedure produces an envy-free division for 3 people using at most 5 cuts.
• Brams–Taylor–Zwicker moving knives procedure produces an envy-free division for 4 people using at most 11 cuts.
• A reentrant variant of the Last Diminisher protocol finds an additive approximation to an envy-free division in bounded time. Specifically, for every constant $\epsilon >0$, it returns a division in which the value of each partner is at least the largest value minus $\epsilon$, in time $O(n^{2}/\epsilon )$.
• Three different procedures, one by Brams and Taylor (1995) and one by Robertson and Webb (1998) and one by Pikhurko (2000), produce an envy-free division for n people. Both algorithms require a finite but unbounded number of cuts.
• A procedure by Aziz and Mackenzie (2016) finds an envy-free division for n people in a bounded number of queries.

The negative result in the general case is much weaker than in the connected case. All we know is that every algorithm for envy-free division must use at least Ω(n²) queries.^[10] There is a large gap between this result and the runtime complexity of the best known procedure.

Efficient division

When the value functions are additive, UM divisions exist. Intuitively, to create a UM division, we should give each piece of cake to the person that values it the most. In the example cake, a UM division would give the entire chocolate to Alice and the entire vanilla to George, achieveing a utilitarian value of 9 + 4 = 13.

This process is easy to carry out when the value functions are piecewise-constant, i.e. the cake can be divided to pieces such that the value density of each piece is constant for all people. When the value functions are not piecewise-constant, the existence of UM divisions is based on a generalization of this procedure using the Radon–Nikodym derivatives of the value functions; see Theorem 2 of.^[6]

When the cake is 1-dimensional and the pieces must be connected, the simple algorithm of assigning each piece to the agent that values it the most no longer works. In this case, the problem of finding a UM division is NP-hard, and furthermore no FPTAS is possible unless P = NP. There is a 8-factor approximation algorithm, and a fixed-parameter tractable algorithm which is exponential in the number of players.^[11]

For every set of positive weights, a WUM division can be found in a similar way. Hence, PE divisions always exist.

Efficient fair division

For n people with additive value functions, a PEEF division always exists.^[12]

If the cake is a 1-dimensional interval and each person must receive a connected interval, the following general result holds: if the value functions are strictly monotonic (i.e. each person strictly prefers a piece over all its proper subsets) then every EF division is also PE.^[13] Hence, Simmons' protocol produces a PEEF division in this case.

If the cake is a 1-dimensional circle (i.e. an interval whose two endpoints are topologically identified) and each person must receive a connected arc, then the previous result does not hold: an EF division is not necessarily PE. Additionally, there are pairs of (non-additive) value functions for which no PEEF division exists. However, if there are 2 agents and at least one of them has an additive value function, then a PEEF division exists.^[14]

If the cake is 1-dimensional but each person may receive a disconnected subset of it, then an EF division is not necessarily PE. In this case, more complicated algorithms are required for finding a PEEF division.

If the value functions are additive and piecewise-constant, then there is an algorithm that finds a PEEF division.^[15] If the value density functions are additive and Lipschitz continuous, then they can be approximated as piecewise-constant functions "as close as we like", therefore that algorithm approximates a PEEF division "as close as we like".^[15]

An EF division is not necessarily UM.^[16][17] One approach to handle this difficulty is to find, among all possible EF divisions, the EF division with the highest utilitarian value. This problem has been studied for a cake which is a 1-dimensional interval, each person may receive disconnected pieces, and the value functions are additive.^[18]

See also

• Fair pie-cutting
• A list of books about fair division
• A list of research papers about fair division

References

1. Steinhaus, Hugo (1948). "The problem of fair division". Econometrica.
2. Ariel Procaccia, "Cake Cutting Algorithms". Chapter 13 in: Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016). Handbook of Computational Social Choice. Cambridge University Press. ISBN 9781107060432. (free online version)
3. I.e. there is no dispute over the rights of the people – the only problem is how to divide the cake such that each person receives a fair share.
4. Hill, T. P.; Morrison, K. E. (2010). "Cutting Cakes Carefully". The College Mathematics Journal. 41 (4): 281. doi:10.4169/074683410x510272.
5. Chen, Yiling; Lai, John K.; Parkes, David C.; Procaccia, Ariel D. (2013). "Truth, justice, and cake cutting". Games and Economic Behavior. 77 (1): 284–297. doi:10.1016/j.geb.2012.10.009.
6. Dubins, Lester Eli; Spanier, Edwin Henry (1961). "How to Cut a Cake Fairly". The American Mathematical Monthly. 68: 1. doi:10.2307/2311357. JSTOR 2311357.
7. "The Fair Division Calculator".
8. Ivars Peterson (March 13, 2000). "A Fair Deal for Housemates". MathTrek.
9. Stromquist, Walter (2008). "Envy-Free Cake Divisions Cannot be Found by Finite Protocols". The electronic journal of combinatorics.
10. Procaccia, Ariel (2009). "Thou Shalt Covet Thy Neighbor's Cake". IJCAI.
11. Aumann, Yonatan; Dombb, Yair; Hassidim, Avinatan (2013). Computing Socially-Efficient Cake Divisions. AAMAS.
12. Weller, D. (1985). "Fair division of a measurable space". Journal of Mathematical Economics. 14: 5. doi:10.1016/0304-4068(85)90023-0.
13. Berliant, M.; Thomson, W.; Dunz, K. (1992). "On the fair division of a heterogeneous commodity". Journal of Mathematical Economics. 21 (3): 201. doi:10.1016/0304-4068(92)90001-n.
14. Thomson, W. (2006). "Children Crying at Birthday Parties. Why?". Economic Theory. 31 (3): 501. doi:10.1007/s00199-006-0109-3.
15. Reijnierse, J. H.; Potters, J. A. M. (1998). "On finding an envy-free Pareto-optimal division". Mathematical Programming. 83: 291. doi:10.1007/bf02680564.
16. Caragiannis, I.; Kaklamanis, C.; Kanellopoulos, P.; Kyropoulou, M. (2011). "The Efficiency of Fair Division". Theory of Computing Systems. 50 (4): 589. doi:10.1007/s00224-011-9359-y.
17. Aumann, Y.; Dombb, Y. (2010). "The Efficiency of Fair Division with Connected Pieces". Internet and Network Economics. Lecture Notes in Computer Science. 6484. p. 26. doi:10.1007/978-3-642-17572-5_3. ISBN 978-3-642-17571-8.
18. Cohler, Yuga Julian; Lai, John Kwang; Parkes, David C; Procaccia, Ariel (2011). Optimal Envy-Free Cake Cutting. AAAI.