Fair division
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 (especially social choice theory), and dispute resolution.
can be of various types: Additionally, the set to be divided may be: Finally, it is common to make some assumptions about whether the items to be divided are: Based on these distinctions, several general types of fair division problems have been studied: Combinations and special cases are also common: Most of what is normally called a fair division is not considered so by the theory because of the use of arbitration.
The decisions in the Talmud on entitlement when an estate is bankrupt reflect the development of complex ideas regarding fairness.
Examples are: Based on these subjective value functions, there are a number of widely used criteria for a fair division.
If different participants have different entitlements (e.g., in a partnership where each partner invested a different amount), then the fairness criteria should be adapted accordingly.
In the real world people sometimes have a very accurate idea of how the other players value the goods and they may care very much about it.
A major part of the practical side of fair division is the devising and study of procedures that work well despite such partial knowledge or small mistakes.
An additional requirement is that the fair division procedure be strategyproof, i.e. it should be a dominant strategy for the participants to report their true valuations.
As a result, it is often weakened to incentive compatibility, which only requires players to report their true valuations if they behave according to a specified solution concept.
A fair division procedure lists actions to be performed by the players in terms of the visible data and their valuations.
A valid procedure is one that guarantees a fair division for every player who acts rationally according to their valuation.
A discrete procedure would for instance only involve one person at a time cutting or marking a cake.
Another type of continuous procedure involves a person assigning a value to every part of the cake.
Negotiations involving more than two people are also quite common, the Potsdam Conference is a notable recent example.
The theory of fair division dates back only to the end of the second world war.
It was devised by a group of Polish mathematicians, Hugo Steinhaus, Bronisław Knaster and Stefan Banach, who used to meet in the Scottish Café in Lvov (then in Poland).
This was attributed to Banach and Knaster by Steinhaus when he made the problem public for the first time at a meeting of the Econometric Society in Washington, D.C., on 17 September 1947.
