fairdivison

Fair division is a field of study in economics and mathematics concerned with allocating resources among individuals or groups in a way that is considered fair according to formal criteria. The resources may be divisible, such as a cake or land, or indivisible, such as houses orUnique items. Each participant is assumed to have a valuation function that assigns a value to any portion of the resource.

Common fairness criteria include envy-freeness, proportionality, and equitability. Envy-freeness requires that no participant prefers someone else’s allocated portion to their own. Proportionality demands that each of n participants receives at least 1/n of the total value according to their own assessment. Equitability requires that all participants obtain equal value from their allocations. Efficiency, or Pareto efficiency, requires that no other allocation can make someone better off without making someone else worse off.

Problems with divisible goods, often framed as cake-cutting, typically seek proportional or envy-free divisions. For any number of participants, proportional allocations can be achieved by various division procedures; envy-free divisions exist in theory for any number of agents, though practical constructive methods become more complex as the group grows. For three agents, procedures like the Selfridge-Conway method implement envy-freeness. General existence results are known, but finite, simple procedures are limited.

When goods are indivisible, perfect envy-freeness may be impossible, and fairness goals shift to approximate envy-freeness, proportionality, or max-min fairness. Methods may combine discrete allocations with monetary transfers or use procedures such as adjusted-winner for two agents.

Fair division has applications in inheritance, divorce settlements, public resource allocation, and multi-agent systems, and remains an active area of research in economics, computer science, and operations research.