Bankruptcy Models (Spring, 2016)

Prepared by:

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


web page:

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, there might be two claimants A and B with verified claims of 20 and 140 respectively, with only assets of 50 available. Or for a larger number of claimants, 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. Thus, one might want to pay off claims by workers that they have not been paid before the claims of other 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" cannot 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? The situation that has occurred after Hurricane Sandy is a good example. The claims of damages far exceed the amounts that State governments and the Federal government have appropriated to relieve the victims of the hurricane.

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 contexts 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.

Here are some brief remarks about some of the methods that have been developed to solve bankruptcy problems. Let us the first example above to illustrate these these methods.


                         A    B       Estate = 50

Verified claims 20 140

Method 1 (Entity equity; gains)

A gets 25, and B gets 25.

Comment: This method seems strange since one of the claimants got more than he/she claimed. So usually, we assume in bankruptcy problems that no claimant gets more than he/she claimed.

Method 2 (Maimonides gain) (Idea: treat claimants as equally as possible but don't give any claimant more than he/she claimed.)

A gets 20 and B gets 30.

Method 3 (Entity equity, loss) (Idea: Claimants incur the same loss even it means augmenting E to make losses equal)

A gets -35 and B gets 85 (each loses 55)

Many reject this approach because it requires a claimant to help "subsidize" the settlement.

Method 4 (Maimonides loss) (Idea: Claimants are teated as equally as possible from the point of view of equalizing loss)

A gets nothing and B gets 50.

Method 5 (Contested garment rule, sometimes also called concede and divide) (Each claimant given their uncontested claim and whatever is left if anything is split equally.)

A's uncontested claim against B is 0. B's uncontested claim against A is 30.

So: A gets 10 and B gets 40.

Method 6 (Shapley value:) Idea: average over all permutations of the claimants given their "rewards" reading the permutation from LEFT to RIGHT .

Order BA. B gets 50 and A gets nothing
Order AB, A gets 20 and B gets 30.

So A's mean is (0 + 20)/2 = 10
B's mean is (50 + 30)/2 = 40.

A gets 10 and B gets 40.

Method 7: (Proportionality of Gain)

A gets (20/160)(50) and B gets (140/160)(50)

Method 8 (Proportionality of Loss)

Gives same result as proportionality of gains. (This is a general theorem.)

Method 9 (Return c cents per dollar.)

Use algebra to find how much each claimant is given if they are returned c cents for every dollar claimed.

Method 10 (Find present value of the estate E allowed to grow in value to where it will pay off all the claims)

Method 11 (Talmudic method)

Pay off half of the claims using Maimonides gain, and if there are still funds left over pay off the remaining half of the claims using Maimonides loss.

One can compute how long (how much time) it will take at the current interest rate for E to grow to the size of the sum of the claims. Now pay off claimants based on the present value of the amounts claimed at that future time.

Unlike apportionment where only Hamilton and Webster's methods are likely to be discovered by students on their "own," students typically have a great variety of ideas about how to solve bankruptcy problems.

Additional interesting insights can be obtained (case of 2 claimants) by plotting the amount given to A and B on the x and y axis and plotting appropriate lines and inequalities.


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.

Thomson, W., Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update, Mathematical Social Sciences 74 (2015): 41-59.