Notes on Modeling: Session XIII: Bankruptcy

Prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451


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The bankruptcy model involves situations where claims are being made against a "resource" where the size of the resource is not large enough to pay off all the claims completely. What is a fair way to repay the claimants?

For example, suppose A, B, and C have verified claims of 80, 120, 200 respectively, but assets E of only 100 are available. What is a fair way to pay off the claims?

The letter E is often chosen for the assets to be assigned "collectively" to the claimants because one type of bankruptcy problem occurs when an estate E has less value than the claims that are being made against it. The assets that are being distributed are usually assumed to be homogenous (like water or money) and infinitely divisible. (Money is not strictly speaking infinitely divisible but the smallest unit of money is sufficiently small that as a modeling assumption we can assume that E can represent money.)

Bankruptcy problems also occur when a company does not have enough cash on hand to pay off all of those people (firms) that have claims against it. In practice the laws that govern bankruptcy often create priority classes for how the remaining assets are to be returned to creditors. In a general way, bankruptcy questions arise when something is in short supply, and lots of this item are desired.


1. Water

The amount of water that is available in rivers such as the Jordan or Colorado is much smaller than the neighboring countries (states) could use. The amounts of use are often specified by treaty. What is a fair way of distributing the water when the amount specified by the "treaty" can not be met due to reduced flow?

2. Emergency funds

After a hurricane or flood an amount of money may be put aside as emergency assistance to those adversely affected. However, the claims often far exceed the amounts that have been set aside. How can the claims be treated fairly?

3. Limited medical supplies

What is a fair way to give out limited medical supplies? For some medical situations there is a complexity that when one gives a small amount of medicine to lots of claimants that a threshold of having enough medicine to be effective may not be reached. In many cases the amount E is not infinitely divisible. The resource being distributed may be vaccine shots, kidneys, hearts, antibiotics. Often situations of this kind are handled by creating a market where people pay for the scarce resource. However, is this fair to the poor, who would like access to medical treatment just as rich people would?

4. Tax collection

One of the most interesting context to which the bankruptcy model can be applied is tax collection. Suppose that the government of a country must raise E dollars by taxation. Different income classes in the country have available various amounts of money. (There may not be very many rich people in a particular country. If one takes the average money available for these individuals multiplied by their number, one gets the amount this "rich group" can contribute. Poor people may have much less to contribute on average but there are many more such people.) What should be the "obligation:" of each of the income classes towards collecting the required E amount of taxes.

There are many axioms that have been suggested as fairness conditions for the methods (functions, algorithms) that are used to distribute E to the claimants. Here are some examples.

a. Monotonicity for E

As the amount E increases the amount that each claimant gets should not go down.

b. If two claimants have identical claims they should be assigned equal amounts.

There are also axioms that compare the consistency with which a bankruptcy settlement method will treat two similar but not identical bankruptcy problems. For example, if two problems are identical except for the fact that claimant i is entitled to more in a second bankruptcy situation than in the first situation, fairness would mean that i would not get less in the second situation than in the first. There are also issues such as if two claimants are in fact a "split" version of one claimant (one claimant pretends to be two) will the split claimants get more or less than they would if they were treated separately?

Recent interest in bankruptcy problem has stemmed in part from the fact that such questions were being looked at nearly 1000 years ago in the Talmud. Recent work (1980's) by the game theorists Barry O'Neill (political science), Robert Aumann, and Michael Maschler (recently deceased) showed a new solution to the bankruptcy problem which is very elegant. It obeys the fairness condition that if one looks at how the algorithm developed by Aumann and Maschler resolves bankruptcies, it has the property that the amount given collectively to two claimants X and Y (there are many) divides that amount given to X and Y in the way it would be distributed using the "contested garment rule." The idea behind the algorithm is that it divides the claims be asked by each claimant in half. At first an attempt is made to pay off this amount (half the claims) with respect to Maimonides method (gains). If there is still more left to distribute, it is used to try to equalize loses with respect to the Maimonides loss method.


Aumann, Robert J. and Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213

O'Neill, Barry, 1982. "A problem of rights arbitration from the Talmud," Mathematical Social Sciences, Elsevier, vol. 2(4), pages 345-371,

William Thomson, 2001. "On the axiomatic method and its recent applications to game theory and resource allocation," Social Choice and Welfare, Springer, vol. 18(2), pages 327-386.

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