Steven J. Brams is an American game theorist and political scientist at the New York University Department of Politics. Brams is best known for using the techniques of game theory, public choice theory, and social choice theory to analyze voting systems and fair division. He is one of the independent discoverers of approval voting, as well as extensions of approval voting to multiple-winner elections to give proportional representation of different interests.
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.
Equity, or economic equality, is the concept or idea of fairness in economics, particularly in regard to taxation or welfare economics. More specifically, it may refer to a movement that strives to provide equal life chances regardless of identity, to provide all citizens with a basic and equal minimum of income, goods, and services or to increase funds and commitment for redistribution.
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 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.
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.
Adjusted Winner (AW) is a procedure for envy-free item allocation. Given two agents and some goods, it returns a partition of the goods between the two agents with the following properties:
- Envy-freeness: Each agent believes that his share of the goods is at least as good as the other share;
- Equitability: The "relative happiness levels" of both agents from their shares are equal;
- Pareto-optimality: no other allocation is better for one agent and at least as good for the other agent;
- At most one good has to be shared between the agents.
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.
Alan Dana Taylor is an American mathematician who, with Steven Brams, solved the problem of envy-free cake-cutting for an arbitrary number of people with the Brams–Taylor procedure.
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 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.
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:
Rental harmony is a kind of a fair division problem in which indivisible items and a fixed monetary cost have to be divided simultaneously. The housemates problem and room-assignment-rent-division are alternative names to the same problem.
The AL procedure is a procedure for fair item assignment between two people. It finds an envy-free item assignment of a subset of the items. Moreover, the resulting allocation is Pareto efficient in the following sense: there is no other envy-free allocation which is better for one person and not worse for the other person.
Various experiments have been made to evaluate various procedures for fair division, the problem of dividing resources among several people. These include case studies, computerized simulations, and lab experiments.
