M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
139
4. Examples
4.1. Single-claim bankruptcy In this case T 5
R 2 h0j and maxt 5 t . A type is interpreted as an agent’s ‘claim’
1 i
i
on a property and a problem is interpreted as a ‘bankruptcy’ or ‘bequest’ problem. Below, such problems will be called, in short, ‘claims.’
The formal analysis of claims was pioneered by O’Neill 1982 who introduced the claims problem with a story from the Babylonian Talmud, discussed several rules, and
compared their properties. The mathematical analysis of claims was subsequently developed by Aumann and Maschler 1985, and Young 1987.
Aumann and Maschler’s analysis is focused on the Talmudic bequest problem. Marriage contracts for three wives of a deceased husband specify their claims as equal to
100, 200 and 300. How should the estate be divided if it is valued below the total of 600? In three discussed problems, the estate is worth 100, 200 and 300. The Talmudic
solutions are clearly different from proportional division Table 1. Aumann and Maschler noted that the Contested Garment CG solution from the other Mishna in the
]]]]]] Talmud makes recommendations for any pair of claimants and the amounts they
received, that are bilaterally consistent with the three-person problem. Aumann and Maschler reconstructed a hypothetic general rule that generated the
Talmudic numbers as a function of the estate. For two claimants, when the value of an estate rises continuously, CG divides the property equally up to the size of a smaller
claim, next gives everything to the larger claimant up to the size of a larger claim, and finally divides the remaining part equally. Formally, for two claims
t and t , t , t , the
i j
i j
1 1
] ]
CG rule is defined by the following formula: F t ,t ;g 5 g, g for 0 g t ;
i j
i 2
2 1
1 1
1 ]
] ]
]
F t ,t ;g 5 t ,g 2 t for t , g t ; Ft ,t ;g 5 [g 2 t 1 t ], [g 2 t 1 t ] for
i j
i i
i j
i j
j i
i j
2 2
2 2
t , g t 1 t . The CG rule can be compactly depicted as an appropriate ‘system of
j j
i
connected vessels’ Fig. 2. Some of Aumann–Maschler’s central results follow directly from Corollary 1. The
CG rule clearly satisfies B.AN and B.MON. The reader can easily check that, in
CG
addition, the underlying binary relation R is transitive. Thus, Corollary 1 implies the
existence of a unique extension of CG Aumann and Maschler’s Theorem A which is consistent Aumann and Maschler’s Corollary 3.1.
The CG rule successfully challenges the apparently universal charm of proportional division. The two rules appeal to different value systems but, as many other claims rules,
share the property of being hydraulic. Other examples of popular claims rules include
Table 1 Solutions to three claims problems recommended by the Talmud Aumann and Maschler, 1985
Estate Claim
100 200
300
1 1
1 ]
] ]
100 33
33 33
3 3
3
200 50
75 75
300 50
100 150
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.M. Kaminski Mathematical Social Sciences 40 2000 131 –155
Fig. 2. A hydraulic CG solution to the three claims problems from the Talmud. Note: the size of a claim is equal to the area of the corresponding vessel.
Talmudic constrained equal award and its vertical mirror image, constrained equal loss, ]]]]]]]
]]]]]] both by Maimonides Aumann and Maschler, 1985, pp. 202–203; Thomson, 1995
provides a review of claims rules. Fig. 3 shows hydraulic systems for a few popular rules.
4.2. Multi-claim bankruptcy
B k
k
Let us define the case of multi-claim bankruptcy as T 5 R 2 h0j and maxt 5 o
1 i
j 51
x , where x is a jth coordinate of t . A type is interpreted as a vector of claims of various
j j
i
priorities and a creditor cannot get more than the sum of his claims. Claims are a special case of multi-claim bankruptcy with k 5 1.
American bankruptcy law illustrates that multi-claim bankruptcy involves more complex information than a single claim. Every legal case in the United States
constitutes a bankruptcy problem: ‘‘Once a bankruptcy petition is filed . . . the
Fig. 3. Hydraulic systems for claims rules. Note: in Lexicographic System, vessels occupy different levels. In Constrained Equal Award and in Constrained Equal Loss, diameters are equal, but vertical positions of vessels
are reversed. In Proportional System, diameters are proportional to the claim.
M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
141
2
bankruptcy priority system becomes applicable. There is no discretion in the court . . . ’’ Types are defined implicitly in the bankruptcy code by enlisting various ‘priority
classes,’ since ‘‘[t]he general policy of the law is that there are no sub-classifications in priority classes.’’ The four major priority classes include the Federal Government,
]]]]]] trustees and two kinds of creditors. Thus, a type can be represented formally as a vector
]] ]]]
of non-negative reals with different coordinates corresponding to claims from different priority classes.
Once the size of the estate and the types of creditors are reconstructed, the division of the estate is guided by the following ‘priority system.’ Among the four major classes of
claims, the priority system is lexicographic. The highest priority is accorded to governmental
taxes. These
are followed,
in order,
by secured
claims, ]]
]] administrative expenses of the trustee, and unsecured claims. Within the second and
]]] ]]]]]]]
fourth classes of claims, the assets are being divided proportionally because the principle of ‘equal treatment of creditors’ is interpreted to be ‘a pro rata distribution.’ Thus, the
actual American bankruptcy rule is mixed lexicographic–proportional. In addition to four main classes of claims, a number of code sections describe a new
category of claims. These ‘exceptions’ represent the introduction to the standard of some rarely met categories like lessors of property, whose priority is located between
]]]]] administrative expenses and unsecured claims. As another exception, the ‘super-priority
is confined to lien-holders,’ but ‘super super-priority’ to administrative expenses, ‘if a trustee is unable to obtain unsecured credit otherwise.’ Clearly, the code is intended to
handle, in a somewhat awkward manner, priorities of all imaginable types of creditors and to set a corresponding lexicographic–proportional standard. The main idea of this
complex system can be depicted in a remarkably simple way Fig. 4.
The properties of bankruptcy laws around the world remain a question for empirical study. If a given bankruptcy code defines a rule that is symmetric, continuous, and
consistent with respect to some set of types, Theorem 2 receives a remarkable empirical
Fig. 4. A hydraulic solution to a bankruptcy problem according to American bankruptcy law. Note: different levels represent four basic classes of claims. At levels 1 and 3, there is a single creditor. At levels 2 and 4, the
rule is proportional division.
2
The quotations and terms in these three consecutive paragraphs come from Cowans, 1989, Section 12.
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.M. Kaminski Mathematical Social Sciences 40 2000 131 –155
interpretation. It says that the code can be rewritten, and possibly simplified, so that it would provide an explicit list of types and their numerical priorities.
4.3. Social choice rationing
I
In this case T 5 hu: R
→ R s.t. u is continuous and strictly increasingj and no
1
restriction is assumed. The following bilateral correspondences along with their multi- person versions are frequently considered in social choice theory:
2
UTILITARIAN LEXIMIN: L u ,u ; g 5 Argmax min
hu x,u g 2 xj,
i j
i j
for 0 x g.
2
NASH PRODUCT: N u ,u ; g 5 Argmax [u x 2 u 0][u g 2 x 2 u 0],
i j
i i
j j
for 0 x g.
2
UTILITARIAN SUMMATION: U u ,u ; g 5 Argmax
hu x 1 u g 2 xj,
i j
i j
for 0 x g. One can ask under which conditions the bilateral correspondences are single-valued
and can be consistently extended to multi-person rules. This question can be answered with Corollary 1.
2 2
2
L is single-valued and satisfies L u ,u ; g 1 L u ,u ; g 5 g for the original type
i j
i i
j j
I 2
I
space T , which means that L is a bilateral rule on T . The other two rules are
single-valued on narrower domains:
WC I
WEAK CONCAVITY: T
5 hu [ T : ;x,y [ R ; p [ 0,1
1
u px 1 1 2 py pux 1 1 2 pu y j;
SC I
STRICT CONCAVITY: T
5 hu [ T : ;x,y [ R ; p [ 0,1
1
x ± y →
u px 1 1 2 py . pux 1 1 2 pu y j.
2 2
WC SC
2
N and U are bilateral rules on T
and T , respectively. It is also clear that L ,
2 2
N , and U are anonymous, non-decreasing in g, and that the corresponding binary
relations R are transitive. Thus, Corollary 1 can be applied to obtain the next result.
Theorem 3. The following bilateral rules have unique consistent extensions for respective spaces of types:
2 I
a L for T ;
2 WC
b N for T ;
2 SC
c U for T
. Theorem 3b resembles Harsanyi’s 1959, 1963 characterization of the extension of
the Nash 1950 bargaining solution to multi-person bargaining problems. The relation-
M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
143
ship between social choice problems and bargaining seems to be close indeed. The precise extent to which the results from one framework can be applied to the other
framework depends on the assumed space of types. With every social choice problem u; g, where u 5 u , . . . ,u and g . 0, the
1 n
following bargaining problem can be associated: Bu; g 5 hu g , . . . ,u g : all
1 1
n n
]]]]]]
n
g 0 and o
g g j. Young 1994, pp. 210–212 noted that Lensberg’s 1987
i i 51
i
results obtained for bargaining can be applied in the social choice setting, and he offered a preliminary discussion of the properties of anonymous, continuous and bilaterally
WC
consistent rules for T with respect to maximization of separately additive objective
functions. However, the results for bargaining cannot be mechanically restated for the social choice setting, and vice versa, since the function B is not 1–1 and onto Billera
and Bixby, 1973. Moreover, almost all results in bargaining theory are obtained under the assumption of weak convexity of the bargaining set. This assumption rules out from
consideration the utilitarian summation since for a non-strictly convex bargaining set, U may be multi-valued and makes the domain of L unnecessarily narrow since some
I
bargaining problems corresponding to T are non-convex. More work remains to be done on the formal relationship between the two frameworks.
5. Related results
Kategori