Constrained equal awards

Constrained equal awards (CEA), also called constrained equal gains, is a division rule for solving bankruptcy problems. According to this rule, each claimant should receive an equal amount, except that no claimant should receive more than his/her claim. In the context of taxation, it is known as leveling tax.

Formal definition

There is a certain amount of money to divide, denoted by $E$ (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by $c_{i}$. Usually, $\sum _{i=1}^{n}c_{i}>E$, that is, the estate is insufficient to satisfy all the claims.

The CEA rule says that each claimant i should receive $\min(c_{i},r)$, where r is a constant chosen such that $\sum _{i=1}^{n}\min(c_{i},r)=E$. The rule can also be described algorithmically as follows:

• Initially, all agents are active, and all agents get 0.
• While there are remaining units of the estate:
  • The next estate unit is divided equally among all active agents.
  • Each agent whose total allocation equals its claim becomes inactive.

Examples

Examples with two claimants:

• $CEA(60,90;100)=(50,50)$; here $r=50$. In general, when all claims are at least $E/n$, each claimant receives exactly $E/n$.
• $CEA(40,80;100)=(40,60)$; here $r=60$.

Examples with three claimants:

• $CEA(50,100,150;100)=(33.333,33.333,33.333)$; here $r=33.333$.
• $CEA(50,100,150;200)=(50,75,75)$; here $r=75$.
• $CEA(50,100,150;300)=(50,100,150)$; here $r=150$.

Characterizations

The CEA rule has several characterizations. It is the only rule satisfying the following sets of axioms:

• Equal treatment of equals, invariance under truncation of claims, and composition up;
• Conditional full compensation, and composition down;
• Conditional full compensation, and claims-monotonicity.

Dual rule

The constrained equal losses (CEL) rule is the dual of the CEA rule, that is: for each problem $(c,E)$, we have $CEL(c,E)=c-CEA(c,\sum c-E)$.