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Resolving Bankruptcy ClaimsThere are surprisingly many approaches to finding ways of distributing money to bankruptcy claimants...
New companies are born regularly. They represent the optimism of their creators that they can market a product or service better than others. Such companies raise capital and open their doors, or at least as likely these days, their cell phones and computers. They might hire workers, rent space, and order things from other businesses. If all goes well these companies thrive and are successful, but all too often things turn sour and the companies suffer a cash flow problem. They have too many creditors and not enough money in the bank. So the company goes bust - bankrupt. This means that while the company may have some assets, the value of these assets is exceeded by the claims against these assets. This situation is handled by laws in every individual country. However, this situation also gives rise to very interesting mathematics. It turns out there are surprisingly many approaches to finding ways of "distributing money" to the claimants which obey a variety of equity and fairness principles. Here is a two-person example where, unless the problem is carefully chosen, it is common for the different methods to yield the same solutions. However, with more claimants, different methods are more likely to yield different solutions.
In the simplest form of a bankruptcy problem we have a collection of claimants: C1, C2, C3, ...., Cn, with verified claims c1, c2, c3, ..., cn. The remaining assets E have also been verified and are to be distributed by a "wise person" or judge. (The letter E is used to suggest the word "estate.") If someone leaves a will, it may turn out that the estate is not large enough to make the suggested dispersals. In this case, we have a "bankruptcy" problem where we treat the desired amounts to be dispersed as the claimant amounts and E is to be used to pay off these claims. (Later, I will mention a variety of ways that other problems can be recast in ways that either resemble bankruptcy problems or where the ideas used in studying bankruptcy problems provide insight into these other questions. In the recent economics literature, bankruptcy problems are often defined in a narrower way than I am using here.) Assume that the amount to be distributed is strictly less than the amount which is being claimed. Thus, c1 + c2 + c3 + .... + cn > E. We would like to be able to advise the judge about how to distribute the money using the best insights about fairness and equity principles. We are assuming that the claimants are isolated from each other and do not "bargain" or "negotiate" with each other regarding the amounts they might get from the judge. One can imagine that one could have a "game" where the claimants would get nothing if they could not agree how to split the estate, but if they could agree to share all of E in some way, they would be collectively allocated this amount. In this type of situation negotiations among the claimants would be required.
Perhaps the first approach to solving equity problems is to treat individuals with "total equality." It is this thought that governs the famous dictum with regard to voting: "one person, one vote." If we do this in our example, we take the estate E, which amounts to $210 and divide by 2, giving $105. This amount would be given to each of the claimants. This may seem strange because we are not taking into account the size of the claims to get this value, only the number of claimants and the value of E being used. In particular, for the numbers here this means giving Claimant 2 more than he/she claimed! Can you think of a real world situation where this might actually be reasonable?
Should we decide that the above method is unreasonable, we might adopt the following second method: Equalize the claims of the claimants as much as possible but never give a claimant more than is requested. This notation of solving a bankruptcy has very old roots, having been in essence suggested by the great medieval philosopher Moses Maimonides (1135-1204) (often referred to only as Maimonides or as Rambam).
The method of Maimonides
In modern mathematical terminology what we have here is a constrained optimization problem. Our desire is to make the amount given to each of the claimants as equal as possible but not to have any claimant receive more than his/her claim. This means in the mathematical formulation that certain inequalities would have to hold for a solution. Here is a geometrical way to solve this kind of problem easily without converting it to highly symbolic mathematical form. Imagine that the money in the estate to be distributed is a blue fluid. We begin to fill up two "bins" or "tanks" of size 300 and 60 (the claim sizes) with a small bit of fluid in each, keeping the amounts as equal as possible, as illustrated in Figure 1 below. Remember that the size of the tanks corresponds to the size of the claims.
We keep filling the two containers equally until we fill up the smaller of the two bins, which amounts to completely filling this claim. The situation is now as shown in Figure 2 below:
How much of the estate has been used up at this stage? The answer is 2(60) or $120. This leaves $210 - $120 = $90 to distribute and this all goes to Claimant 1 since the complete claim of Claimant 2 has been met. Since $60 + $90 is $150 the final settlement gives Claimant 1 $150 and Claimant 2 $60. Figure 3 shows this solution:
This geometric approach works very well for a large number of claimants. Typically the claim of the smallest claimant can be fulfilled and then if there is more "estate fluid" to distribute this is done until the next smallest claimant's claim is fulfilled. The process continues until all the estate "fluid" is gone.
Are you happy with the Maimonides solution to the bankruptcy problem? Unlike the total equality solution, it takes into account the size of the claims . However, let us look, for example, at how much of what each claimant hoped for, failed to get recovered. For C2 this amount is 0, while for C1 this amount is $150. This does not seem to spread the pain of "loss" very fairly. How much do the claimants collectively lose? Since E = $210 and the claimants are claiming $300 and $60 respectively, the loss L = $360 - $ 210 = $150. This notion suggests a new method. Why not spread the loss equally? This would mean assigning a loss of $75 to each claimant. For C1 this amounts to giving him/her $300 - $75 or $225, while for C2 this amounts to giving him/her $60 - $75 = - $15! Although $225 + (-$15) adds to $210, the amount E the judge must distribute, something seems wrong here! The problem is that Claimant 2 is being asked to "subsidize" the settlement. The -$15 that Claimant 2 coughs up is given to Claimant 1 along with all of the $210 available to the judge. This total of $225 makes it possible to cut Claimant 2's loss to $75, which is equal to that of Claimant 1. However, many people will consider this unfair because the pain of Claimant 2 is made worse by having to subsidize the settlement. (Can you think of a real world situation where it might not seem totally unreasonable to ask players to subsidize the settlement of the bankruptcy to achieve some goal?)
Like the contrast between "total equality" and Maimonides, one can consider the analogue for loss of Maimonides. The idea is to equalize loss as much as possible without any claimant's loss becoming negative as a result. To do this we must reduce the loss of the player with the largest claim to that of the person with the second largest claim, if this is possible. In this case, if we give C1 $210 this will bring his loss to only $90; to reduce the loss further requires more money than is available in E. Thus, we accept the solution of c1 = $210 and c2 = $0.
Suppose we have three claimants with claims of $100, $80, and $60, and there is an estate E of $210. We can give $20 to the first claimant reducing his/her he "current loss" to $80. Now we can give $20 to each of claimants 1 and 2 which reduces all the claimants to a current loss of $60. At this point $60 of the estate has been used. This leaves $210 - $60 = $150. By giving each claimant $50 of this we can equalize the losses. Thus Claimant 1 gets $90, Claimant 2 gets $70 and Claimant 3 gets $50. These numbers add to $210 as required and give each claimant a loss of $10. Of course, in this problem one can also conceptualize as follows. Since the claims are $240, and E = $210, each claimant of the three will sustain a loss of $10. This means that $10 less than each claim is given to the claimant.
Another natural approach to settling a bankruptcy is to award the claimants an amount proportional to their claims. This would entail in our prime example giving C1 the amount (300/360)(210) = $175 and C2 the amount (60/360)(210) = $35. This seems a very natural approach because it uses the size of the claims to decide how to divide what is given to each claimant. We might also look at settling the bankruptcy by proportionality of loss. The loss in this example is $150. Computing C1's loss we get (5/6)(150) = $125 and C2's loss would be (1/6)(150) = $25. Thus, we would have c1 = $300 - $ 125 = $175 and c2 = $60 - $25 = $35. This is the same solution as when we assign the gains proportionally. Is this an accident? No! We can use a bit of algebra to see that this result holds in general.
Since (7/12)(300) = $175, and (7/12)(60) = $35 we see that the proportional solution gives the same solution as the "fixed percentage on the dollar" solution for this example. This situation is again not an accident; it holds in general.
More solution concepts
Having seen so many different approaches to settling bankruptcies, it might seem that we had exhausted the intuitively appealing ways to solve such problems. In fact, this turns out not to be true. Here are a few more interesting approaches to these questions.
Another important solution idea for bankruptcy problems is based on a truly central solution concept for n-person games that is due to Lloyd Shapley, and is known as the Shapley Value. There are various ways to think of the Shapley Value but here I will work out an example which involves three rather than two claimants to make the ideas for one approach clearer. Assume that the claims are $120, $60, and $30 respectively, and the size E of the estate is $150. Imagine that the judge announces that the claimants should line up to get a share of the funds. Suppose the claimants form a line in front of the judge in the order C1C2C3.
While the Shapely Value can be found relatively easily for small games, the computational work for large games is considerable.
Contested garment rule
Finally, let us jump from the relatively recent Shapley Value to a solution concept that goes back hundreds of years. This solution idea is discussed in a "document" known as the Babylonian Talmud, which initially consisted of oral materials handed down from one generation to another. (There is also a Jerusalem Talmud.). It was Barry O'Neil who called attention in modern times to the fact that the Babylonian Talmud treats various examples of bankruptcy problems. In modern accounts the technique described in the Talmud has come to be known as the contested garment rule, since the method was applied to a situation where two individuals claimed portions of a single garment.
Notice that the contested garment rule in this two-claimant example gives the same solution as the Shapley Value. This is not an accident. The two solutions will always coincide for two claimants.
A mathematical detective story
One of the most interesting episodes in the modern understanding of bankruptcy problems arises from a detective story related to the ideas in the preceeding discussion concerning bankruptcy ideas originating in the Talmud. Robert Aumann and Michael Maschler, Israeli game theorists, were the mathematical detectives in this case. Biblical scholars presented them with the issue of explaining the data in the following table. The source of the data is the Talmud. Note there are three questions where the size of the claims are held fixed (columns) but the size of the estate varies from 100, to 200, and then to 300 (rows). The Talmud offers the numbers in the "body" of the table as the proposed answers.
The biblical scholars could make sense of the first and last lines of the table but were puzzled by the middle line. What was the method being used here? Was there a "copying error?" Over the years had some data been changed which resulted in the second line's being erroneous? The story is that initially Aumann and Maschler did not see how to explain the second line. However, to avoid a hasty conclusion they had a game theory package compute different solutions to various games. To their surprise one of the solution concepts coincided with the second line! This solution concept is known as the nucleolus.
The original definition of nucleolus was given in 1969 by David Schmeidler, who did his doctoral work at Hebrew University in Jerusalem and now is at Ohio State University. It grows in a natural way out of solution concepts of n-person games which for general games are rather complex. It seemed unlikely that early scholars were using these ideas as far back as when the Babylonian Talmud was put together. Aumann and Maschler realized that perhaps there was an approach to the nucleolus for the special class of games that arose in bankruptcy problems that would have made it possible to solve them with a clever use of arithmetic skills. Eventually they found a line of reasoning that did just that. The place they began was in the Middle Ages, when thoughts about bankruptcy centered not around the use of the proportional solution but along the lines of the contested garment rule. They knew that Maimonides and other scholars were concerned with both the gain and loss that a claimant received. Aumann and Maschler realized that viewed through a modern perspective, for every solution concept that involved gain there was a "dual" method which involved loss.
When people think of the power of axiom systems they invariably think of Euclid's Elements. The Elements set in motion a mathematical paradigm where, starting from a few basic undefined terms and rules (axioms), a large system of theorems was deduced using basic ideas of logic. It was one of the great milestones in intellectual history when Janos Bolyai and N. Lobachevsky independently showed that substituting a different axiom for Euclid's famous parallel postulate resulted in an alternative geometric world to Euclidean geometry.
Kenneth Arrow turned the situation on its head by writing down appealing fairness axioms and examining which election and voting methods obeyed his axioms. He showed that based on some reasonable assumptions there was no method that obeyed all the desirable conditions. Equally important, he inspired other workers to find fairness principles (axioms) which were uniquely obeyed by particular methods. This pattern of moving from intuitively appealing algorithms or techniques for solving fairness problems to finding axioms that characterize individual methods has become standard methodology. This approach has been followed for apportionment problems, fair division problems, bargaining problems (game theory solution concepts in general), and bankruptcy. New fairness principles to study are being developed all the time and new characterizations of important methods are being found.
Thomson's goal, like that of other game theorists, mathematicians, and economists who study bankruptcy problems, has been to get a list of axioms which characterize individual solution methods and families of methods for bankruptcy problems. Since no method can obey all the fairness axioms for bankruptcy problems that one might like to have satisfied, Thomson's efforts have gone far in clarifying the tradeoffs between different choices. At this point quite a few important methods have one or more characterizations. New models of bankruptcy and new axioms continue to be explored.
Arin, J. and E. Iñarra, A characterization of the nucleolus for convex games, Games and Economic Behavior, 23 (1998) 12-24.
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