We provide a general coalitional procedure that characterizes a family of rules for bankruptcy problems inspired by the Talmud.bankruptcy, coalitions, claims, Talmud.

Juan D. Moreno-Ternero

English

Research Papers in Economics

 
 
    Working papers series 
   Department of Economics 
 
 
   
 
 
WP ECON 10.15 
 
 
A coalitional procedure leading to a family of 
bankruptcy rules 
 
 
 
 
Juan D. Moreno-Ternero (U. Pablo de Olavide, U. de Málaga 
and CORE-Université catholique de Louvain) 
 
 
 
 
JEL Classification numbers: D63, C71. 
Keywords: bankruptcy, coalitions, claims, Talmud. 
 
 
 
 
 
 
 
 
  
 
 
 
 
http://www.upo.es/econ 
 
A coalitional procedure leading to a family of bankruptcy
rules
Juan D. Moreno-Terneroy
September 29, 2010
Abstract
We provide a general coalitional procedure that characterizes a family of rules for bankruptcy
problems inspired by the Talmud.
Keywords: bankruptcy, coalitions, claims, Talmud.
Universidad de M alaga, Universidad Pablo de Olavide, and CORE, Universit e catholique de Louvain.
yAddress for correspondence: Juan D. Moreno-Ternero. Universidad Pablo de Olavide. Department of
Economics. Carretera de Utrera, Km. 1. 41013, Seville, Spain. Phone: (34) 954977609. Fax: (34) 954349339.
e-mail: jdmoreno@upo.es
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  1 Introduction
A bankruptcy problem refers to a situation in which one has to distribute, among a group
of agents, a perfectly divisible commodity whose available amount is not enough to cover all
agents' demands on it. This is a classic allocation problem, which encompasses many dierent
situations like the bankruptcy of a rm, the division of an insucient estate, or the collection of a
given amount of taxes. Although instances of bankruptcy problems (and solutions for them) are
already documented in ancient sources, such as the Talmud, Aristotle's books or Maimonides'
essays, their formalization was not presented till the early eighties [6]. The reader is referred
to [7] for a review of the fast-expanding literature concerning this model. An early (and
inuential) contribution within this eld is due to Aumann and Maschler [1] who, among other
things, initiated the so-called game-theoretical approach to bankruptcy problems. Aumann and
Maschler also provided a specic formula to rationalize the (apparently unrelated) solutions to
several bankruptcy situations that appear in the Talmud. Their formula, which came to be
known as the Talmud rule, implements a basic principle by which individual rationing is of the
same type as collective rationing. More precisely, if the amount to divide is below half of the
aggregate claim then no agent gets more than half of her claim, whereas if the amount exceeds
half of the aggregate claim then no agent gets less than half of her claim. The same principle can
be implemented while considering all possible shares of the amount to divide in the aggregate
claim. That is, for any given value  2 [0;1], we may construct the rule that distributes the
amount accordingly so that nobody gets more than a fraction  of her claim if the amount
to divide is smaller than  times the aggregate claim and nobody gets less than a fraction 
of her claim if the amount to divide exceeds  times the aggregate claim [5]. The family, so
constructed, would have the Talmud rule as a focal element, but would also encompass, as
extreme cases, two classical rules that can be traced back to Maimonides; namely, the so-called
constrained equal awards and constrained equal losses rules.
We provide in this note a general coalitional procedure characterizing each of the rules
within the family described above. Our procedure is also inspired by a Talmudic principle,
regarding bankruptcy problems, according to which the creditors empower each other. For
instance, in a three-agent problem in which agents are increasingly ranked according to their
claims, the third empowers the second to deal with the rst [1].
The note is organized as follows. Section 2 introduces the model and basic concepts. Section
3 presents the coalitional procedure. Finally, Section 4 concludes.
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  2 Model and basic concepts
We study bankruptcy problems in a variable population model. The set of potential creditors,
or agents, is identied with the set of natural numbers N. Let N be the set of nite subsets
of N, with generic element N. Let n denote the cardinality of N. For each i 2 N, let ci 2 R+
be i's claim and c  (ci)i2N the claims prole. Without loss of generality, we assume that
c1  c2    cn. A bankruptcy problem is a triple consisting of a population N 2 N, a
claims prole c 2 Rn
+, and an amount to be divided E 2 R+ such that
P
i2N ci  E. Let
C 
P
i2N ci. To avoid unnecessary complication, we assume C > 0. Let DN be the set of
rationing problems with population N and D 
S
N2N DN.
Given a problem (N;c;E) 2 DN, an allocation is a vector x 2 Rn satisfying the following
three conditions:
 Boundedness: for each i 2 N, 0  xi  ci,
 Balance:
P
i2N xi = E, and
 Order Preservation: for each i;j 2 N such that ci  cj, then xi  xj, and ci xi  cj xj.
A bankruptcy rule on D, R: D !
S
N2N Rn, associates with each problem (N;c;E) 2 D an
allocation R(N;c;E) for the problem. Some classical rules are the constrained equal awards
rule, which distributes the amount equally among all agents, subject to no agent receiving more
than what she claims; the constrained equal losses rule, which imposes that losses are as equal
as possible, subject to no one receiving a negative amount; and the proportional rule, which
yields awards proportionally to claims. The following family of rules encompasses two of those
rules, while generalizing another one. The family is dened by means of a parameter  2 [0;1].
The rule R in the family resolves bankruptcy problems according to the following principle:
Nobody gets more than a fraction  of her claim if the amount to divide is less than  times
the aggregate claim and nobody gets less than a fraction  of her claim if the amount to divide
exceeds  times the aggregate claim. Formally:
The TAL-family consists of all rules with the following form: For some  2 [0;1], for all
(N;c;E) 2 D, and all i 2 N,
R

i (N;c;E) =
8
<
:
minfci;g if E  C
maxfci;ci   g if E  C
,
where  > 0; > 0 are chosen so that
P
i2N R
i (N;c;E) = E.
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The constrained equal losses rule corresponds to the case  = 0, whereas the constrained
equal awards rule corresponds to the case  = 1. The so-called Talmud rule [1] is obtained for
 = 1
2 .
One can visualize the rule R as follows. First, it applies equal division until the creditor
with the smallest claim has obtained a fraction  of her claim. Then, that agent stops receiving
additional units and the remaining amount is divided equally among the other agents until the
creditor with the second smallest claim gets the fraction  of her claim. The process continues
until every agent has received a fraction  of her claim, or the available amount is distributed.
If there is still something left after this process, agents are invited back to receive additional
shares. Now agents receive additional amounts sequentially starting with those with larger
claims and applying equal division of their losses.
It is interesting to provide the following alternative denition of the rules within the family
for the two-agent case. Formally, given  2 [0;1], each rule in the TAL-family has the following
expression, in the two-agent case:
R
 (N;c;E) =
8
> > > <
> > > :
 
E
2 ; E
2

if E  2c1
(c1;E   c1) if 2c1  E  c2   c1 + 2c1
 
c1   C E
2 ;c2   C E
2

if c2   c1 + 2c1  E
(1)
It is straightforward to show from (1) that any possible allocation x, for a given two-agent
problem (N;c;E), can be obtained as a realization of a given rule within the family. Formally,
there exists  2 [0;1] such that R (N;c;E) = x.
It is also worth mentioning that (1) can also be seen as a two-stage allocation process. In
the rst stage, agents' claims are weighted to reect exogenous factors that do not appear in
our benchmark model of bankruptcy problems. For instance, the liquidation of a rm might
have a cost per se, which should be borne by the creditors. Alternatively, if two heirs agree
on a procedure to divide a bequest, without resorting to an outside authority, they might be
saving part of their awards. The former case could be reected by reducing (equally) the claims
of both agents. The second one could be reected by increasing (equally) the claims of both
agents. In the second stage, the \standard solution" (conceding each agent what the other does
not claim, and dividing the remainder equally) is applied to the resulting problem.
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  3 A characterization
In this section, we design coalition formation mechanisms leading to the outcomes of the rules
in the TAL-family. More precisely, x some  2 [0;1], and consider the following procedure.
First, in the case of a two-agent problem, we apply the solution (1). Suppose now that we have
a problem with three creditors. Then, we proceed in the following way. First, creditors 2 and 3
pool their claims an act as a single agent vis-a-vis 1. The solution (1) of the resulting problem
yields awards to agent 1, and to the coalition of agents 2 and 3; to divide its award among its
members, the coalition again applies solution (1). The result is order preserving if and only
if 3c1  E  C   3(1   )c1. To see this, note that if 3c1 > E, then the award of creditor
1, c1, would be strictly greater than the one of creditor 2, which is
E c1
2 , as a result of the
awards sharing in the coalition of creditors 2 and 3. Analogously, if E > C   3(1   )c1, then
the loss of creditor 1, (1 )c1, would be greater than
c2+c3 E+c1
2 , the resulting loss associated
to creditor 2, after dividing the awards in the coalition. If one divides the awards equally when
E  3c1, and the losses equally when E  C  3(1 )c1, it is obtained, precisely, the solution
provided by the rule R, over the entire range 0  E  C.
By using induction, one may generalize this in a natural way to an arbitrary n. Suppose we
already know the solution for (n   1)-agent problems. Depending on the values of the amount
to divide and the vector of claims, we treat a given n person problem in one of the following
three ways:
(i) Divide E between f1g and M = f2;:::;ng, in accordance with the solution (1) to the
two-agent problem (f1;Mg;E;(c1;c2 +:::+cn)), and then use the (n 1)-agent rule, which we
know by induction, to divide the amount assigned to the coalition M between its members.
(ii) Assign equal awards to all creditors.
(iii) Assign equal losses to all creditors.
Specically, (i) is applied whenever it yields an order-preserving result, which is precisely,
when nc1  E  C   n(1   )c1. We apply (ii) when E  nc1. Finally, we apply (iii) when
E  C   n(1   )c1. We call this generalization, the  coalitional procedure. In the particular
case of  = 1
2, the -coalitional procedure corresponds to the coalitional procedure stated by
Aumann and Maschler [1]. To summarize the previous discussion, we can state the following
result:
Theorem 1 For each  2 [0;1], and for each bankruptcy problem, the  coalitional procedure
and the rule R in the TAL-family yield the same solution to the problem.
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Theorem 1 describes an orderly step-by-step process, which by its very denition must lead
to a unique result, therefore characterizing the TAL-family of rules. We now illustrate the
process by means of some examples.
Let n = 3, ci = 100  i, E = 200, and  = 1
3. At the rst stage, the coalition f2;3g forms,
and its joint claim is 500. Applying rule (1), when  = 1
3, it yields 100
3 to creditor 1, and 500
3 to
the coalition. Now, 500
3 is shared among creditors 2 and 3 as rule (1), providing 200
3 to creditor
2 and 100 to creditor 3. Therefore, the nal result is order preserving, and it coincides with
the outcome that the corresponding member of the TAL-family R
1
3, yields for the problem at
stake.
Let n = 5, ci = 100  i and E = 510. If  = 1
2, they show that the  coalitional procedure
yields the solution (50;100;120;120;120). Suppose now, that  = 2
3. At the rst stage, the
coalition f2;3;4;5g forms, and its joint claim is 1400. If we apply rule (1), then creditor 1
obtains 100   = 200
3 , and the coalition, 510   100   = 1330
3 . If we would split again the
coalition, among creditor 2 and the remaining ones, an apply the same rule, then the allocation
would not be order preserving, due to the fact that 4  200 = 1600
3 > 510   100   = 1330
3 .
Thus, the  coalitional procedure would yield for equal awards. As a result, the proposed
allocation would be
 
200
3 ; 665
6 ; 665
6 ; 665
6 ; 665
6

, which is order preserving. Now, if  = 1
3, it yields
100
3 to creditor 1, and 510   100
3 to the coalition. It is straightforward to see that we can apply
(1) until the last step, obtaining the order preserving allocation,
 
100
3 ; 200
3 ; 300
3 ; 400
3 ; 530
3

. Finally,
if  = 1
4, then the procedure gives subsequently ci to each of the rst three creditors. Now,
it remains 360, to be divided among the coalition f4;5g. In order to make this division order
preserving, then we have to assign equal losses to both creditors. Therefore, the allocation
would be (25;50;75;130;230). In each case, the  coalitional procedure yields a division of
the amount to divide which is order preserving, and it coincides with the allocation proposed
for this problem by the corresponding member of the TAL-family R.
4 Final remarks
We have presented in this note a coalitional procedure that characterizes a one-parameter fam-
ily of bankruptcy rules encompassing three of the most well known rules. The normative appeal
of such family seems unquestionable as it has been shown that it satises a wide variety of prop-
erties reecting ethical and operational principles [5]. There is, however, no characterization
result for the family as a whole, although independent characterizations results for the three
6 
 
 
 
 
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focal members of the family abound in the literature [4], [7], [8]. This note helps to ll that
gap.
To conclude, it is worth mentioning the connections between this work and another interest-
ing topic within the literature on bankruptcy problems that refers to coalitional manipulations
of bankruptcy rules [2], [3].
If we let agents the possibility of consolidating their claims and be treated as a single
creditor or, conversely, we let a particular creditor to divide her claim and be considered as
several dierent creditors then the resulting awards may be altered. It is then worth identifying
the precise cases for which creditors will not be able to manipulate the outcomes in their interest
via merging or splitting their claims. To do so, let (N;c;E) = E
C stand for the share of the
amount to divide in the aggregate claim of a given problem and dene D = f(N;c;E) 2 D :
 (N;c;E) = g, for each  2 (0;1). In other words, D is the set of problems whose ratio
between the amount to divide and the aggregate claim is . Obviously, fDg2(0;1) is a partition
of D. It is natural to consider this collection of sets in the context of coalitional manipulation,
as if a problem belongs to a particular set D then any resulting problem after merging or
splitting agents' claims belongs to the same set.
The following result is obtained:
Proposition 1 Let  2 [0;1], and  2 (0;1) be given. The following statements hold:
(a) If  < , the  coalitional procedure is a non manipulable by splitting mechanism, when
restricted to bankruptcy problems on D.
(b) If  > , the  coalitional procedure is a non manipulable by merging mechanism, when
restricted to bankruptcy problems on D.
(c) If  = , the  coalitional procedure is a non manipulable (by merging or splitting)
mechanism, when restricted to bankruptcy problems on D.
It follows from Proposition 1 that if  = 0 then the corresponding coalitional mechanism is
non manipulable by merging for the unrestricted domain of bankruptcy problems. Similarly,
if  = 1 then the corresponding coalitional mechanism is non manipulable by splitting for the
unrestricted domain of bankruptcy problems.
Acknowledgements
Thanks are due to Dunia L opez-Pintado, William Thomson and Antonio Villar for helpful
comments and suggestions. Financial support from the Spanish Ministry of Science and Inno-
vation (ECO2008-03883) and from Junta de Andaluc a (SEJ-04154) is gratefully acknowledged.
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  References
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the Talmud, Journal of Economic Theory, 36: 195{213.
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[3] Ju, B.-G., Miyagawa E., Sakai T. (2007), Non-manipulable division rules in claim problems
and generalizations, Journal of Economic Theory 132, 1-26.
[4] Moreno-Ternero J, Villar A (2004) The Talmud rule and the securement of agents' awards.
Mathematical Social Sciences 47, 245-257.
[5] Moreno-Ternero J, Villar A (2006) The TAL-family of rules for bankruptcy problems.
Social Choice and Welfare 27, 231-249
[6] O'Neill B (1982) A problem of rights arbitration from the Talmud, Mathematical Social
Sciences, 2: 345{371.
[7] Thomson W (2003) Axiomatic and game-theoretic analysis of bankruptcy and taxation
problems: a survey, Mathematical Social Sciences, 45: 249-297.
[8] Yeh CH (2008) Secured lower bound, composition up, and minimal rights rst for
bankruptcy problems. Journal of Mathematical Economics 44, 925-932.
8

A characterization of the proportional rule in multi-issue allocation situations,

A problem of rights arbitration from the Talmud,

Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey,

Game theoretic analysis of a bankruptcy problem from the Talmud,

Non-manipulable division rules in claim problems and generalizations,

Secured lower bound, composition up, and minimal rights  for bankruptcy problems.

Villar A

A coalitional procedure leading to a family of bankruptcy rules

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A coalitional procedure leading to a family of bankruptcy rules

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