Fair cake-cutting

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, e.g., 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 "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 prototypical procedure for fair cake-cutting is divide and choose, which is mentioned in the book of Genesis to resolve Abraham and Lot's conflict. This procedure solves the fair division problem for two people. The modern study of fair cake-cutting was initiated during World War II, when Hugo Steinhaus asked his students Stefan Banach and Bronisław Knaster to find a generalization of divide-and-choose to three or more people. They developed the last diminisher procedure. Today, fair cake-cutting is the subject of intense research in mathematics, computer science, economics and political science.

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 subjective value functions over C. Each person i has a 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 Lebesgue measure. 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. In many cases, the value functions are assumed to be sigma additive.
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, and.
The n people have equal rights to C. I.e., there is no dispute over the rights of the people – everyone agrees that everyone else is entitled to a fair share. The only problem is how to divide the cake such that each person receives a fair share.
In the following examples the following cake will be used 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 requirements

Proportionality

The original and most common criterion for justice is proportionality. In a proportional cake-cutting, 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, and the other 5/9 of the chocolate to Alice. In symbols:
For n people with additive valuations, a proportional division always exists. The most common protocols are:

Last diminisher, a protocol that can guarantee that the n pieces are connected. 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 actions.
The Dubins–Spanier Moving-knife procedure is a continuous-time version of Last diminisher.
Fink protocol 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 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 actions, but guarantees only a partially proportional division, 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.

See proportional cake-cutting for more details and complete references.
The proportionality criterion can be generalized to situations in which the rights of the people are not equal. For example, in proportional cake-cutting with different entitlements, 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 :
Where the w_i are weights that sum up to 1.

Envy-freeness

Another common criterion is envy-freeness. In an envy-free cake-cutting, each person receives a piece that he values at least as much as every other piece. In symbols:
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
2	general
3+	additive
3+	general

The divide and choose protocol finds an allocation that is always EF. If the value functions are additive then this division is also PR; otherwise, proportionality is not guaranteed.
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.

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.
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, it returns a division in which the value of each partner is at least the largest value minus, in time.
Three different procedures, one by Brams and Taylor and one by Robertson and Webb and one by Pikhurko, produce an envy-free division for n people. Both algorithms require a finite but unbounded number of cuts.
A procedure by Aziz and Mackenzie 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 queries. There is a large gap between this result and the runtime complexity of the best known procedure.
See envy-free cake-cutting for more details and complete references.

Other criteria

A third, less common criterion is equitability. 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:
A fourth criterion is exactness. If the entitlement of each partner i is w_i, then an exact division is a division in which:
If the weights are all equal then the division is called perfect and:

Probabilistic fair division

Traditional fair division methods typically allocate fixed shares of a resource to each participant in a deterministic manner. In contrast, probabilistic fair division assigns shares based on probabilities determined by participants’ attributes, such as contributions, needs, or individual scores.
One example is the Boltzmann fair division approach, which applies the Boltzmann distribution from statistical mechanics. In this framework, each participant's share is determined according to an exponential function of their score. The allocation rule is as follows:
Here, is the share assigned to participant, is the score or merit of participant, and is a parameter that controls the balance between equality and merit-based allocation. When, the method reduces to equal division. As increases, the allocation becomes more heavily weighted toward participants with higher scores.
This method seeks to maximize entropy under the constraints given by the participants’ scores, producing allocations that can interpolate between strict equality and strong meritocracy. The probabilistic approach can be applied in various contexts, including the division of divisible or indivisible resources, and can be adjusted to reflect different societal preferences for fairness or efficiency.
Other probabilistic division mechanisms, such as random lotteries or assignment by chance, are also used, particularly in cases involving indivisible goods or when deterministic solutions are difficult to implement.