divideunequal

DivideUnequal is a concept in the study of fair division and resource allocation describing methods for partitioning a divisible good when participants have unequal entitlements or valuations. It focuses on allocating shares in proportion to specified weights rather than dividing equally, with the aim of reflecting different entitlements, needs, or preferences among recipients.

Definition and goals

In a DivideUnequal framework, each participant i is assigned an entitlement e_i, a nonnegative value that sums to one across all participants. The objective is to allocate portions x_i of the divisible resource so that each participant receives a share aligned with their entitlement, typically ensuring that the value of x_i to i meets or exceeds e_i times the total value of the resource to i. Common fairness criteria include proportionality (each x_i reflects e_i) and efficiency (no waste or preventable envy in the allocation).

Procedures and methods

DivideUnequal procedures adapt traditional cake-cutting techniques to unequal entitlements. Steps often include: (1) determining entitlements e_i, (2) selecting a division protocol (moving-knife, cut-and-choose, or discrete division) tailored to weighted fairness, and (3) resolving disputes or envy through adjustments or compensations. In practice, methods may vary when valuations differ among participants, requiring iterative refinement to balance fairness with efficiency and strategy resistance.

Example

With two participants A and B and entitlements e_A = 2/3 and e_B = 1/3, a simple equal-valued cake can be cut into pieces of sizes 2/3 and 1/3. A receives the larger piece, B the smaller, ensuring proportional allocation by entitlement when both value the cake similarly. When valuations differ, more sophisticated procedures may be used to preserve the intended weights.

See also

Fair division, cake-cutting, proportional allocation, weighted envy-freeness, resource allocation.