Lone divider

Last updated January 02, 2024

The lone divider procedure is a procedure for proportional cake-cutting. It involves a 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 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.

Description

For convenience we normalize the valuations such that the value of the entire cake is n for all agents. The goal is to give each agent a piece with a value of at least 1.

Step 1. One player chosen arbitrarily, called the divider, cuts the cake into n pieces whose value in his/her eyes is exactly 1.

Step 2. Each of the other n − 1 partners evaluates the resulting n pieces and says which of these pieces he considers "acceptable", i.e., worth at least 1.

Now the game proceeds according to the replies of the players in step 3. We present first the case n = 3 and then the general case.

Steinhaus' procedure for the case n = 3

There are two cases.

Case A: At least one of the non-dividers marks two or more pieces as acceptable. Then, the third partner picks an acceptable piece (by the pigeonhole principle he must have at least one); the second partner picks an acceptable piece (he had at least two before, so at least one remains); and finally the divider picks the last piece (for the divider, all pieces are acceptable).
Case B: Both other partners mark only one piece as acceptable. Then, there is at least one piece that is acceptable only for the divider. The divider takes this piece and goes home. This piece is worth less than 1 for the remaining two partners, so the remaining two pieces are worth at least 2 for them. They divide it among them using divide and choose.

The procedure for any n

There are several ways to describe the general case; the shorter description appears in ^[5] and is based on the concept of envy-free matching – a matching in which no unmatched agent is adjacent to a matched piece.

Step 3. Construct a bipartite graph G = (X + Y, E) in which each vertex in X is an agent, each vertex in Y is a piece, and there is an edge between an agent x and a piece y iff x values y at least 1.

Step 4. Find a maximum-cardinality envy-free matching in G. Note that the divider is adjacent to all n pieces, so |N_G(X)| = n ≥ |X| (where N_G(X) is the set of neighbors of X in Y). Hence, a non-empty envy-free matching exists.

Step 5. Give each matched piece to its matched agent. Note that each matched agent has a value of at least 1, and thus goes home happily.

Step 6. Recursively divide the remaining cake among the remaining agents. Note that each remaining agent values each piece given away at less than 1, so he values the remaining cake at more than the number of agents, so the precondition for recursion is satisfied.

Query complexity

At each iteration, the algorithm asks the lone divider at most nmark queries, and each of the other agents at most neval queries. There are at most n iterations. Therefore, the total number of queries in the Robertson-Webb query model is O(n²) per agent, and O(n³) overall. This is much more than required for last diminisher (O(n) per agent) and for Even-Paz (O(log n) per agent).

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, 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.

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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.

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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:

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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 V_i, receives a share X_i 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 $j$ :

Envy-freeness, also known as no-envy, is a criterion for fair division. It says that, when resources are allocated among people with equal rights, each person should receive a share that is, in their eyes, at least as good as the share received by any other agent. In other words, no person should feel envy.

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.

Utilitarian cake-cutting is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different cardinal utility functions, such that the sum of the utilities of the partners is as large as possible. It is a special case of the utilitarian social choice rule. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with fair cake-cutting.

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 $that sum up to 1, and every partner should receive at least a fraction of the resource by their own valuation.$

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.

References

↑ Steinhaus, Hugo (1948). "The problem of fair division". Econometrica. 16 (1): 101–4. JSTOR 1914289.
↑ Kuhn, Harold (1967), "On games of fair division", Essays in Mathematical Economics in Honour of Oskar Morgenstern, Princeton University Press, pp. 29–37, archived from the original on 2019-01-16, retrieved 2019-01-15
↑ Brams, Steven J.; Taylor, Alan D. (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. ISBN 0-521-55644-9.
↑ 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.
↑ Segal-Halevi, Erel; Aigner-Horev, Elad (2022). "Envy-free matchings in bipartite graphs and their applications to fair division". Information Sciences. 587: 164–187. arXiv: 1901.09527 . doi:10.1016/j.ins.2021.11.059. S2CID 170079201.

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[steinhaus48-1] Steinhaus, Hugo (1948). "The problem of fair division". Econometrica. 16 (1): 101–4. JSTOR 1914289.

[2] Kuhn, Harold (1967), "On games of fair division", Essays in Mathematical Economics in Honour of Oskar Morgenstern, Princeton University Press, pp. 29–37, archived from the original on 2019-01-16, retrieved 2019-01-15

[bt96-3] Brams, Steven J.; Taylor, Alan D. (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. ISBN 0-521-55644-9.

[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.

[5] Segal-Halevi, Erel; Aigner-Horev, Elad (2022). "Envy-free matchings in bipartite graphs and their applications to fair division". Information Sciences. 587: 164–187. arXiv: 1901.09527 . doi:10.1016/j.ins.2021.11.059. S2CID 170079201.

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