Strategic bankruptcy problem

There is a divisible resource, denoted by ${\displaystyle E}$ (=Estate or Endowment). There are n people who claim this resource or parts of it; they are called claimants. The amount claimed by each claimant i is denoted by ${\displaystyle c_{i}}$. Usually, ${\displaystyle \sum _{i=1}^{n}c_{i}>E}$, that is, the estate is insufficient to satisfy all the claims. The goal is to allocate to each claimant an amount ${\displaystyle x_{i}}$ such that ${\displaystyle \sum _{i=1}^{n}x_{i}=E}$.

O'Neill[2] describes the following game.

• The estate is divided to small units (for example, if all claims are integers, then the estate can be divided into E units of size 1).
• Each claimant i chooses some ${\displaystyle c_{i}}$ units.
• Each unit is divided equally among all agents who claim it.

Naturally, the agents would try to choose units such that the overlap between different agents is minimal. This game has a Nash equilibrium. In any Nash equilibrium, there is some integer k such that each unit is claimed by either k or k+1 claimants. When there are two claimants, there is a unique equilibrium payoff vector, and it is identical to the one returned by the contested garment rule.[2]

Sonn[5][1] describes the following sequential game.

• Claimant 1 proposes an amount to claimant 2.
• If claimant 2 accepts, he leaves with it and claimant 1 then proposes an amount to claimant 3, etc.
• If a claimant k rejects, then claimant 1 moves to the end of line, the claimant k starts making offerts to the next claimant.
• The offer made to each claimant i must be at most ${\displaystyle c_{i}}$, and at most the remaining amount.
• The process continues until one claimant remains; that claimant gets the remaining estate.

Sonn proves that, when the discount factor approaches 1, the limit of payoff vectors of this game converges to the constrained equal awards payoffs.

Dagan, Serrano and Volij[9] consider a setting in which the claims are private information. Claimants may report false claims, as long as they are lower than the true ones. This assumption is relevant in taxation, where claimants may report incomes lower than the true ones. For each rule that is consistent and strictly-claims-monotonic (a person with higher claim gets strictly more), they construct a sequential game that implements this rule in subgame-perfect equilibrium.

Landsburg[10][1]^{: 42, ftn.67} considers a setting in which claims are private information, and claimants may report false claims, but this manipulation is costly. The cost of manipulation increases with the magnitude of manipulation. In the special case in which the sum of claims equals the estate, there is a single generalized rule that is a truthful mechanism, and it is a generalization of constrained equal losses.

Sertel[11] considers a two-claimant setting in which a claimant may manipulate by pre-donating some of his claims to the other claimant. The payoff is then calculated using the Nash Bargaining Solution. In equilibrium, both claimants receive the payoffs prescribed by the contested garment rule.