A strategic bankruptcy problem is a variant of a
bankruptcy problem
A bankruptcy problem, also called a claims problem, is a problem of distributing a homogeneous divisible good (such as money) among people with different claims. The focus is on the case where the amount is insufficient to satisfy all the claims.
...
(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.
Definitions
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
.
Unit-selection game
O'Neill
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 ''
'' 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 game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...
. 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
The contested garment (CG) rule, also called concede-and-divide, is a division rule for solving problems of conflicting claims (also called "bankruptcy problems"). The idea is that, if one claimant's claim is less than 100% of the estate to divide ...
.
Rule-proposal games
Chun's game
Chun
describes the following game.
* Each claimant proposes a division rule.
* 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 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 ...
rule.
Herrero's game
Herrero 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.
Amount-proposal game
Sonn
describes the following
sequential game
In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. The other players must have information on the first player's choice so that the difference in time has no strategic effect. Sequen ...
.
* 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 ''
'', 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
Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee.See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Efficient ...
approaches 1, the limit of payoff vectors of this game converges to the
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 ...
payoffs.
Division-proposal games
Serrano's game
Serrano describes another sequential game of offers. It is parametrized by a two-claimant rule ''R''.
* The highest claimant (say, claimant 1) suggests a division.
* Each other claimant can either accept or reject the offer.
** Any claimant that accepts the offer, leaves with it.
** Any claimant ''k'' that rejects the offer, receives the outcome of rule ''R'' on the two-claimant problem for ''k'' and 1, on the sum of the offers for ''k'' and 1.
* The highest claimant receives the remainder.
* The process is repeated with all the rejecters.
If ''R'' satisfies
resource monotonicity
Resource monotonicity (RM; aka aggregate monotonicity) is a principle of fair division. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM pri ...
and
super-modularity, then the above game has a unique
subgame perfect equilibrium
In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every ...
, at which each agent receives the amount recommended by the consistent extension of ''R''.
Corchon and Herrero's game
Corchon and Herrero
describe the following game. It is parametrized by a "compromise function" (for example:
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
).
* 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.
Implementation game for downward-manipulation of claims
Dagan, Serrano and Volij 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
A tax is a compulsory financial charge or some other type of levy imposed on a taxpayer (an individual or legal person, legal entity) by a governmental organization in order to fund government spending and various public expenditures (regiona ...
, 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
In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. The other players must have information on the first player's choice so that the difference in time has no strategic effect. Sequen ...
that implements this rule in
subgame-perfect equilibrium
In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every ...
.
Costly manipulations of claims
Landsburg
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 In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about ...
, and it is a generalization of
constrained equal losses Constrained equal losses (CEL) is a division rule for solving bankruptcy problems. According to this rule, each claimant should lose an equal amount from his or her claim, except that no claimant should receive a negative amount. In the context of t ...
.
Manipulation by pre-donations
Sertel
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 Cooperative bargaining is a process in which two people decide how to share a surplus that they can jointly generate. In many cases, the surplus created by the two players can be shared in many ways, forcing the players to negotiate which division o ...
. In equilibrium, both claimants receive the payoffs prescribed by the
contested garment rule
The contested garment (CG) rule, also called concede-and-divide, is a division rule for solving problems of conflicting claims (also called "bankruptcy problems"). The idea is that, if one claimant's claim is less than 100% of the estate to divide ...
.
References
{{reflist
*
Mechanism design