Number of cuts
Straightforward use of the algorithm would generate pieces, but in fact only about are needed as each partner only really needs to do cuts when the th partner comes along.
The Fink protocol [1] (also known as Successive Pairs [2] or Lone Chooser [3] ) is a protocol for proportional division of a cake.
Its main advantage is that it can work in an online fashion, without knowing the number of partners in advance. When a new partner joins the party, the existing division is adjusted to give a fair share to the newcomer, with minimal effect on existing partners.
Its main disadvantage is that, instead of giving each partner a single connected piece, it gives each partner a large number of "crumbs".
We describe the protocol inductively for an increasing number of partners.
When partner #1 enters the party, he just takes the entire cake. His value is thus 1.
When partner #2 comes, partner #1 cuts the cake to two pieces that are equal in his eyes. The new partner chooses the piece that is better in his eyes. The value of each partner is thus at least 1/2 (just like in the divide and choose protocol).
When partner #3 joins, both partners #1 and #2 cut their share to 3 pieces that are equal in their eyes. The new partner chooses one piece from each partner. The value of each of partners #1 and #2 is at least 2/3 of their previous value, which was 1/2. Hence their new value is at least 1/3. The value of partner #3 is at least 1/3 from the piece of #1 and 1/3 from the piece of 2, which gives him at least 1/3 of the total cake.
In general, when partner #i enters the party, the previous i-1 partners divide their share to i equal pieces and the new partner picks a piece from each pile. Again it is possible to prove that the value of each partner is at least 1/n of the total, so the division is proportional.
Straightforward use of the algorithm would generate pieces, but in fact only about are needed as each partner only really needs to do cuts when the th partner comes along.
Fink's protocol is used in a subroutine in other cake-cutting protocols:
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, dispute resolution, etc. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods.
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. They allocate each of n partners, 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.
Exact division, also called consensus 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:
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 that he or she believes 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.
Edmonds–Pruhs protocol is a protocol for fair cake-cutting. Its goal is to create a partially proportional division of a heterogeneous resource among n people, such that each person receives a subset of the cake which that person values as at least 1/an of the total, where is some sufficiently large constant. It is a randomized algorithm whose running time is O(n) with probability close to 1. The protocol was developed by Jeff Edmonds and Kirk Pruhs, who later improved it in joint work with Jaisingh Solanki.
Efficient cake-cutting is a problem in economics and computer science. It involves a heterogeneous resource, such as a cake with different toppings or a land with different coverings, 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, etc. The allocation should be economically efficient. Several notions of efficiency have been studied:
A strongly-proportional division is a kind of a fair division. It is a division of resources among n partners, in which the value received by each partner is strictly more than his/her due share of 1/n of the total value. Formally, in a strongly-proportional division of a resource C among n partners, each partner i, with value measure Vi, receives a share Xi such that
.
The fair pie-cutting problem is a variation of the fair cake-cutting problem, in which the resource to be divided is circular.
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:
A proportional cake-cutting is a kind of fair cake-cutting. It is a division of a heterogeneous resource ("cake") that satisfies the proportionality criterion, namely, that every partner feels that his allocated share is worth at least 1/n of the total.
In the fair cake-cutting problem, the partners often have different entitlements. For example, the resource may belong to two shareholders such that Alice holds 8/13 and George holds 5/13. This leads to the criterion of weighted proportionality (WPR): there are several weights that sum up to 1, and every partner should receive at least a fraction of the resource by their own valuation.
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.
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.
Truthful cake-cutting is the study of algorithms for fair cake-cutting that are also truthful mechanisms, i.e., they incentivize the participants to reveal their true valuations to the various parts of the cake.
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.
