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The bankruptcy problem,^[1] also called the claims problem,^[2] is the problem of distributing a homogeneous divisible good (such as money) among people with different entitlements. The focus is on the case where the amount is insufficient to satisfy all the claims.

The canonical example is a bankrupt firm that is to be liquidated. The firm owes different amounts of money to different creditors, but the total worth of the company's assets is smaller than its total debt. The problem is how to divide the scarce existing money among the creditors. Another example would be the division of an estate amongst several heirs, particularly when the estate cannot meet all the deceased's commitments.

There are at least three simple methods for solving bankruptcy problems in practice, but each is deficient in one or more ways. The methods are:^[1]

The proportional rule: divide the estate proportionally to each agent's claim. For example, if the estate is worth 100 and the claims are 60 and 90, then the first creditor gets 40 and the second creditor gets 60.
The constrained equal-awards rule: divide the estate equally among the agents, ensuring that nobody gets more than their claim.
The constrained equal-losses rule: divide equally the difference between the aggregate claim and the estate, ensuring that no agent ends up with a negative transfer.

Another rule, appearing already in the Babylonian Talmud, is the contested garment rule.^[3]

References

^ ^a ^b Alcalde, José; Peris, Josep E. (2017-02-17). "Equal Awards vs. Equal Losses in Bankruptcy Problems". SSRN.
^ "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey". Mathematical Social Sciences. 45 (3): 249–297. 2003-07-01. doi:10.1016/S0165-4896(02)00070-7. ISSN 0165-4896.
^ Dagan, Nir (1996). "New characterizations of old bankruptcy rules". Social Choice and Welfare.

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 {{more footnotes|date=May 2019}}
 }}
-In [[mathematical sociology]], and especially [[game theory]], the '''bankruptcy problem''' is a distribution or [[entitlement (fair division)|entitlement]] problem involving the allocation of a given amount of a [[Infinite divisibility#In economics|perfectly divisible]] [[good (economics)|good]] among a group of agents. The focus is on the case where the amount is insufficient to satisfy all their demands.
+The '''bankruptcy problem''',<ref name=":0" /> also called the '''claims problem''',<ref>{{Cite journal|date=2003-07-01|title=Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey|url=https://www.sciencedirect.com/science/article/abs/pii/S0165489602000707|journal=Mathematical Social Sciences|language=en|volume=45|issue=3|pages=249–297|doi=10.1016/S0165-4896(02)00070-7|issn=0165-4896}}</ref> is the problem of distributing a [[Fair division of a single homogeneous resource|homogeneous divisible good]] (such as money) among people with different [[entitlement (fair division)|entitlement]]<nowiki/>s. The focus is on the case where the amount is insufficient to satisfy all the claims.
-Problems of the [[bankruptcy]] type arise in many real life situations. The [[canon (basic principle)|canonical]] example would be that of a bankrupt [[business|firm]] that is to be [[liquidation|liquidated]]. Another example would be the division of an estate amongst several [[heir]]s, particularly when the estate cannot meet all the deceased's commitments.
+The [[canon (basic principle)|canonical]] example is a [[Bankruptcy in the United States|bankrupt]] [[business|firm]] that is to be [[liquidation|liquidated]]. The firm owes different amounts of money to different [[creditors]], but the total worth of the company's assets is smaller than its total debt. The problem is how to divide the scarce existing money among the creditors. Another example would be the division of an estate amongst several [[heir]]s, particularly when the estate cannot meet all the deceased's commitments.
-There are at least three simple methods for solving bankruptcy problems in practice, but each is deficient in one or more ways. The methods are:<ref>{{Cite journal|title=Equal Awards vs. Equal Losses in Bankruptcy Problems|journal=SSRN|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2919582|last=Alcalde|first=José|date=2017-02-17|last2=Peris|first2=Josep E.}}</ref>
+There are at least three simple methods for solving bankruptcy problems in practice, but each is deficient in one or more ways. The methods are:<ref name=":0">{{Cite journal|title=Equal Awards vs. Equal Losses in Bankruptcy Problems|journal=SSRN|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2919582|last=Alcalde|first=José|date=2017-02-17|last2=Peris|first2=Josep E.}}</ref>
-# The proportional rule: divide the estate proportionally to each agent's claim.
+# The ''proportional rule'': divide the estate proportionally to each agent's claim. For example, if the estate is worth 100 and the claims are 60 and 90, then the first creditor gets 40 and the second creditor gets 60.
 # The constrained equal-awards rule: divide the estate equally among the agents, ensuring that nobody gets more than their claim.
 # The constrained equal-losses rule: divide equally the difference between the aggregate claim and the estate, ensuring that no agent ends up with a negative transfer.