A strategic bankruptcy problem is a variant of a bankruptcy problem (also called claims problem) in which claimants may act strategically, that is, they may manipulate their claims or their behavior. There are various kinds of strategic bankruptcy problems, differing in the assumptions about the possible ways in which claimants may manipulate.[1]
There is a divisible resource, denoted by (=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 . Usually, , that is, the estate is insufficient to satisfy all the claims. The goal is to allocate to each claimant an amount such that .
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 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]
The proposed rule must satisfy the property of order-preservation (a claimant with a higher claim must have weakly-higher gain and weakly-higher loss).
All proposed rules are applied to the problem; each claimant's claim is replaced with the maximum amount awarded to him by a proposed rule.
The process repeats with the revised claims.
The process converges. Moreover, it has a unique Nash equilibrium, in which the payoffs are equal to the ones prescribed by the constrained equal awards rule.[3]
Herrero's game
Herrero[4] describes a dual game, in which, at each round, each claimant's claim is replaced with the minimum amount awarded to him by a proposed rule. This process, too, has a unique Nash equilibrium, in which the payoffs are equal to the ones prescribed by the constrained equal losses rule.
Corchon and Herrero[8] describe the following game. It is parametrized by a "compromise function" (for example: arithmetic mean).
Agents propose division vectors, which must be bounded by the claims vector.
The compromise function is used to aggregate the proposals.
A two-claimant rule is implementable in dominant strategies (using arithmetic mean) if-and-only-if it is strictly increasing in each claim, and the allocation of agnet i is a function of and . Rules for more than two claimants are usually not implementable in dominant strategies.[8]
Implementation game for downward-manipulation of claims
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.
Costly manipulations of claims
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.
Manipulation by pre-donations
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.
This page is based on this Wikipedia article Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.