Updating Claims in Bankruptcy Problems

Abstract

We reexamine the consistency axiom in bankruptcy problems and propose arguments in favor of an alternative definition of a reduced problem. The classical definition updates the size of the estate while keeping agents' claims unaffected. Instead, we suggest updating agents' claims along with the estate. The resulting consistency axiom characterizes the well-known Random Arrival rule as the unique bilaterally consistent extension of the Contested Garment rule to many agents. We also establish that our definition of a reduced bankruptcy problem corresponds to the definition of a reduced TU game proposed in Hart & Mas-Colell (Econometrica, 1989).

Full text

Page 1

UPDATING CLAIMS IN BANKRUPTCY PROBLEMS
M. J. ALBIZURI, J. LEROUX AND J. M. ZARZUELO
Abstract. We reexamine the consistency axiom in bankruptcy prob-
lems and propose arguments in favor of an alternative definition of a
reduced problem. The classical definition updates the size of the estate
while keeping agents’ claims unaffected. Instead, we suggest updating
agents’ claims along with the estate. The resulting consistency axiom
characterizes the well-known Random Arrival rule as the unique bilater-
ally consistent extension of the Contested Garment rule to many agents.
We also establish that our definition of a reduced bankruptcy problem
corresponds to the definition of a reduced TU game proposed in Hart &
Mas-Colell (Econometrica, 1989). JEL classification: C7.
1. Introduction
A bankruptcy problem refers to a situation in which a group of individ-
uals have rights over an estate, but the estate is not large enough to cover
their joint claims. Such problems appear for instance when a firm declares
bankruptcy and creditor claims cannot be satisfied. From a normative stand-
point, the main issue is to find rules which specify a division of the estate
according to the claims in a fair way via the formulation of desirable prop-
erties, or axioms, which transpose philosophical principles to the problem
at hand. Early work by O’Neill (1982) lays ground to the formalism we use
here to study bankruptcy problems. For an extensive review on this topic
the reader may also consult Thomson (2003).
A principle which has played a significant role in many economic allo-
cation problems is that of consistency. According to this principle, if a
Date: April 3, 2008.
Key words and phrases. Bankruptcy Problems, Consistency, Cooperative Games.
M. J. Albizuri: elpalirm@bs.ehu.es; J. M. Zarzuelo: elpzazaj@bs.ehu.es, Fac. of Eco-
nomics and Business Administration. Av. Lehendakari Aguirre, 83; 48015 Bilbao, Spain
J. Leroux: justin.leroux@hec.ca, HEC Montr´ eal and CIRP´EE, 3000 chemin de la Cˆ ote
Sainte-Catherine, Montr´ eal, QC H3T 2A7, Canada.
We wish to thank J´ er´ emy Laurent-Lucchetti for his comments to highlight the motiva-
tions of the paper. This research has been partially supported by the Basque Government
(project GICO07/155-IT-293-07), DGES Ministerio de Educaci´ on y Ciencia (project SEJ
2006-05455), and the Fonds Qu´ eb´ ecois de la Recherche sur la Soci´ et´ e et la Culture.
1

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2M. J. ALBIZURI, J. LEROUX AND J. M. ZARZUELO
subgroup of agents decides to leave the others, the rewards the remain-
ing agents obtain by (re)applying the same sharing rule among themselves
should be unchanged. In bankruptcy problems the consistency axiom, or
its weaker version of bilateral consistency, has been used in the characteri-
zation of several rules rules; see for instance Aumann and Maschler (1985),
or Young (1987) among others. Customarily, in the reduced problem the
residual estate is defined as the original estate minus the sum of the rewards
paid to the departing agents, while all claims remain the same. So in the
reduced situation the estate is updated, while claims are not.
One can argue that this definition only partially handles the problem
reduction by omitting to update agents’ claims. Indeed, one can argue that
when agents face their reduced problem, they have already conceded part
of their claims to the departing agents, and vice versa. If so, their claims in
the reduced problem should be updated, and may differ from their original
ones. For instance, when agent i departs with her share, one may argue that
she forgoes a portion of her claim on the remainder of the estate she is not
receiving. To be consistent with this argument, one must further consider
that the remaining agents also give up a portion of their own claim on the
share of the estate agent i has secured. Following this intuition, we propose
an alternative definition of the reduced problem in which the estate and the
claims are simultaneously updated.
To calculate the share of the claim a departing agent i concedes to a
remaining agent k we proceed as follows. First, partition the estate into
two parts: agent i’s claim and the rest of the estate unclaimed by agent i,
if any. The other agents can negotiate this remainder between them in a
subproblem by applying the sharing rule in effect. The share of agent k in
this subproblem is not conceded as part of agent i’s claim. Therefore, the
part of his claim agent i concedes to agent k is the difference between agent
k’s share in the original problem and his agent k’s share in this subproblem.
Hence, such updated claims are typically less than the original ones. This
definition of updated claims leads to a definition of a reduced bankruptcy
problem different from the traditional one.
A key result in the literature on bankruptcy problems is that the Tal-
mudic solution is the unique bilaterally consistent—in the usual sense—of
the Contested Garment rule (Aumann & Maschler, 1985). By contrast, us-
ing our definition of a reduced problem, we characterize another important
sharing method, the Random Arrival rule (O’Neill, 1982), as the unique bi-
laterally consistent extension of the Contested Garment rule to many agents
(Theorem 2).

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BANKRUPTCY PROBLEMS 3
Finally, in Section 4 we examine the relationship between cooperative TU
games and bankruptcy problems. TU games are particularly suitable to ex-
amining bankruptcy problems, and there exists a natural way to associate
every bankruptcy problem with a TU game. The consistency property has
also received considerable attention in the context of TU games, and sev-
eral definitions of reduced games exist in this literature as well. We shall
focus on the definitions proposed by Davis & Maschler (1965), and Hart &
Mas-Colell (1989) and highlight the relationship between these definitions
of a reduced TU game and the definitions of reduced bankruptcy problem.
Aumann & Maschler (1985) establishes that the traditional way of reducing
a bankruptcy problem corresponds to the Davis & Maschler reduced game.
In turn, we find (Theorem 3) that the definition of a reduced bankruptcy
problem we introduced here corresponds to the Hart & Mas-Colell reduced
game for bilateral reduced problems. However, in general the Hart & Mas-
Colell reduced game of a bankruptcy problem is not a game associated to
any bankruptcy game for any symmetric rule.
In other words, our results suggest that the dichotomy of the consistency
axiom which is found in cooperative games, but also in cost-sharing prob-
lems1, arises in bankruptcy problems as well.
2. Preliminaries
Let U denote a set of potential agents. Given a non-empty finite subset N
of U, by RNdenote the |N|-dimensional Euclidean space with axes labeled
by the members of N, and RN
and x = (xi)i∈N∈ RN, then xS denotes the projection of x onto RS, i.
e., xS= (xi)i∈S∈ RS, and x(S) =?
+= {x ∈ RN: xi≥ 0}. If S ⊆ N, S ?= ∅,
i∈Nxi. Finally, if x ∈ R we denote
x+= max{x,0}.
A triple (N,E,c) is called a bankruptcy problem, if N is a non-empty
finite subset of U (the set of agents involved in the problem), E ∈ R+(the
estate), and c ∈ RN
will denote ¯ ci= min{E,ci}.
Let BUdenote the set of all bankruptcy problems with the foregoing
properties.
+(the vector of claims) is such that?
i∈Nci≥ E. We
1See, e.g., Sudh¨ olter (1998) for a comparison and Leroux (2007) for a discussion regard-
ing the Davis & Maschler and the Hart & Mas-Colell interpretations of the consistency
axiom in the cost-sharing literature and their respective characterizations of the nucleolus
and the Shapley rule.

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4M. J. ALBIZURI, J. LEROUX AND J. M. ZARZUELO
A bankruptcy rule σ associates with each (N,E,c) ∈ BUa vector σ(N,E,c) ∈
RN
(a)?
(c) σ(N,E,c) = σ(N,E,¯ c).
The first condition is an efficiency property stating that a cost sharing
rule must exactly allocate the total value of the estate. The second property
requires that no agent be awarded more than her claim. Finally, the third
property is one of independence of claims higher than the estate according
to which agents’ claims are only valid to the extent that they do not exceed
the total value of the estate.
+satisfying:
i∈Nσi(N,E,c) = E,
(b) σi(N,E,c) ≤ ci,
In the most simple case where only 2 agents are involved, let us consider
the following principle that can be traced back to the Babylonian Talmud:
“Each claimant i concedes (E − ¯ ci)+to the other, and the remaining is
equally divided”. This is known as the Contested Garment principle (Au-
mann & Maschler, 1985). Accordingly define for each (N,E,c) ∈ BUsuch
that |N| = 2, the CG rule, for each i ∈ N by
CGi(N,E,c) = (E − cj)++E − (E − ci)+− (E − cj)+
Several rules have been suggested for more general problems extending
the CG rule to many agents (for a survey consult Thomson, 2003). For
instance the Talmud rule, T, introduced in Aumann & Maschler (1985), is
defined as follows:
For each (N,E,c) ∈ B and each i ∈ N,
1. If?
2. If?
2
=E + ¯ ci− ¯ cj
2
.
k∈N(ck/2) ≥ E, then Ti(N,E,c) ≡ min{ci/2,λ}, where λ is chosen
so that?
chosen so that?
We will mainly focus on a rule proposed by O’Neill (1982) commonly
called the Random Arrival rule, which we denote by RA. To define this rule
formally let ΠNdenote the class of permutations of N. For convenience,
denote for each (N,E,c) ∈ BU, i ∈ N, and π ∈ ΠN:
?(N,E,c),π?:= MCi(π) = min
k∈Nmin{ck/2,λ} = E,
k∈N(ck/2) ≤ E, then Ti(N,E,c) ≡ ci− min{ci/2,λ}, where λ is
k∈N
?ck− min{ci/2,λ}?= E.
MCi
?
¯ ci,max
?
E −
?
j∈N:
π(j)<π(i)
¯ cj
?
+
?
.
The permutation π represents an ordering on N. The real number MCi(π)
has the following interpretation. The first individual in this ordering receives

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BANKRUPTCY PROBLEMS 5
his claim, or the entire estate if his claim is greater. The next agent is given
his claim, or the estate that is left if his claim is greater, and so on. Then
MCi(π) is what agent i would receive if the agents were paid by using this
method. If we assume that all the orderings π are equally likely, then the
expected payoff of agent i is precisely what the RA rule prescribes. That is:
RAi(N,E,c) := E(MCi) =1
n!
?
π∈ΠN
MCi(π).
3. Consistency
The consistency axiom has played a significant role in the axiomatiza-
tion of bankruptcy rules (see Thomson, 2006, for a comprehensive survey).
Consistency can be described as follows. Let σ be a rule applied to a spe-
cific bankruptcy problem. This rule is said to be consistent if whenever a
subgroup of agents are paid according to σ—thus leaving the other agents
with a reduced problem—the payoffs of the remaining agents do not change
after (re)applying σ to the reduced problem. Several ways of defining a re-
duced problem exist, in turn giving rise to different interpretations of the
consistency axiom.
Given a rule σ, (N,E,c) ∈ BU, and a coalition S ⊂ N, define
ES,σ= E −
k∈N\S
That is, ES,σdenotes what is left to the members of S after paying to the
rest of the agents according to σ. The traditional definition of the reduced
game of a coalition S ⊆ N is written as?N,ES,σ,cS
Consequently, following Aumann & Maschler (1985) a rule σ is called
CG-bilateral consistent if for any (N,E,c) ∈ BU, and every S ⊆ N such
that |S| = 2, its restriction to S coincides with the Contested Garment rule
on the reduced problem of S:
?
σk(N,E,c).
?. Note that the size of
the estate is updated but claims are not.
σS(N,E,c) = CG(S,ES,σ,cS).
Theorem 1. (Aumann & Maschler, 1985) The Talmud rule is the unique
CG-bilateral consistent solution.
In the definition of CG-bilateral consistency presented above, it is as-
sumed that agents keep their claims invariant in the reduced problem. How-
ever, one can argue that this definition only partially handles the problem
reduction by omitting to update agents’ claims. Indeed, one can argue that
when agents face their reduced problem, they have already conceded part

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6M. J. ALBIZURI, J. LEROUX AND J. M. ZARZUELO
of their claims to the departing agents. If so, their claims in the reduced
problem should be updated, and may differ from the original claims. Al-
ternatively, when agent i departs with her share, one may argue that she
forgoes a portion of her claim on the remainder of the estate she is not re-
ceiving. In other words, her claim has been ”truncated” to equal exactly
her share on the residual problem where she is alone and her residual estate
equals her allotted share. To be consistent with this argument, one must
also consider that the remaining agents give up a portion of their own claim
on the share of the estate agent i has secured via σ.
Formally, we suggest the following way to update claims. To calculate the
updated claims of agents k ∈ N\{i} after agent i has left with her share we
proceed as follows. First, partition the estate into two parts: ¯ ci, i.e. agent
i’s claim, and E−¯ ci, which is the portion of the estate uncontested by agent
i. On the latter portion of the estate, every agent k ∈ N\{i} should obtain
σk(N\{i},E − ¯ ci,¯ cN\{i}). The remainder of agent k’s payoff, σk(N,E,¯ c) −
σk(N\{i},E − ¯ ci,¯ cN\{i}), is then obtained from ¯ ci. More generally, when
considering the reduced problem of a coalition S ⊂ N, we define the updated
claim of agent i ∈ S as follows:
(1)
cS,σ
i
= ¯ ci−
k∈N\S
Remark 1. Notice that if S = {i,j} then:
a) cS,σ
ij
≥ ES,σ, so (S,ES,σ,cS,σ) is a bankruptcy problem,
b) 0 ≤ cS,σ
c) c{i},σ
i
= σi(N,E,c).
?
(σk(N,E,¯ c) − σk(N\{i},E − ¯ ci,¯ cN\{i})).
+ cS,σ
i
≤ ES,σ, hence cS,σ
i
= cS,σ
i
, and
We say that a rule σ is CG-bilateral consistent∗if for every (N,E,c) ∈ BU
and every S ⊆ N such that |S| = 2 it holds
σS(N,E,c) = CG(S,ES,σ,cS,σ).
As it turns out, this version of bilateral consistency where claims are
updated in the subproblem characterizes the RA rule:
Theorem 2. The RA rule is the unique CG-bilateral consistent∗solution.
This theorem is consequence of the two following propositions.
Proposition 1. There is at most one CG-bilateral consistent∗solution.
Proof. Let σ and ψ be two CG-bilateral consistent∗rules. We check that
both coincide on every bankruptcy problem (N,E,c) ∈ BUby induction on
|N|, with |N| = 2 being the hypothesis.

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BANKRUPTCY PROBLEMS7
We assume by contradiction that x = σ(N,E,c) ?= ψ(N,E,c) = y for
some (N,E,c) ∈ BUwith |N| > 2. By efficiency, there exist S = {i,j} ⊂ N
such that xi> yiand xj< yj. By definition, for any ? ∈ S it holds:
cS,σ
?
=¯ c?−
k∈N\S
=¯ c?− E + xi+ xj+
k∈N\S
Similarly,
?
By the induction hypothesis, we have:
?
?σk(N,E,¯ c) − σk(N\{?},E − ¯ c?,¯ cN\{?})?
?
σk(N\{?},E − ¯ c?,¯ cN\{?}).
cS,ψ
?
= ¯ c?− E + yi+ yj+
k∈N\S
ψk(N\{?},E − ¯ c?,¯ cN\{?}).
σk(N\{?},E − ¯ c?,¯ cN\{?}) = ψk(N\{?},E − ¯ c?,¯ cN\{?}).
Hence
cS,σ
i
− cS,ψ
i
= (xi+ xj) − (yi+ yj) = cS,σ
j
− cS,ψ
j
,
and consequently
cS,σ
i
− cS,σ
j
= cS,ψ
i
− cS,ψ
j
.
which by Remark 1 implies
cS,σ
i
− cS,σ
j
= cS,ψ
i
− cS,ψ
j
.
But then by definition of the CG rule, we have for each ? ∈ S:
CG?(S,ES,σ,cS,σ) = CG?(S,ES,ψ,cS,ψ) +ES,σ− ES,ψ
2
,
that is
x?= y?+ES,σ− ES,ψ
2
,
which is in contradiction with the fact that xi> yiand xj< yj.
To prove the next proposition the following lemma will be useful. We
introduce the following notation: Given a permutation π of N, and ? ∈ N,
we will denote by π?the permutation such that:
i) π(k) < π(k?) implies π?(k) < π?(k?) for every k,k??= ?, and
ii) π?(k) > π?(?) for every k ?= ?.
(Thus, in the ordering π?, agents others than ? keep their relative order
invariant while ? becomes first). Also we will denote by Π?the set of permu-
tations in which ? comes first; i.e. Π?= {π ∈ ΠN: π(k) > π(?) for every k ?=
?}.
Moreover if π ∈ ΠN, and i,j ∈ N we denote by πij the permutation in
ΠNsuch that:

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8M. J. ALBIZURI, J. LEROUX AND J. M. ZARZUELO
i) πij(i) = π(j) and πij(j) = π(i), and
ii) πij(k) = π(k) for every k ?= i,j.
(i.e., πijinterchanges the location of i and j in the ordering π while leaving
the rest of the ordering unaltered).
Finally we denote πi
ij= (πij)i.
Lemma 1. If π ∈ ΠNand i,j ∈ N are such that π(i) < π(j), then
MCi(π) − MCi(πj) = MCj(πij) − MCj(πi
ij).
Proof. Notice that for every ? ∈ N, and every π ∈ ΠNit holds
?
MC?(π) = min
E,
?
π(k)≤π(?)
ck
?
− min
?
E,
?
π(k)<π(?)
ck
?
.
Then we have
MCi(π) − MCi(πj) =
?
?
min
?
?
E,
?
?
?
?
π(k)≤π(i)
ck
?
− min
?
E,
?
π(k)<π(i)
ck
??
−
min
E,
πj(k)≤πj(i)
ck
?
?
?
− min
?
?
?
E,
?
?
?
πj(k)<πj(i)
ck
??
=
?
?
min
?
?
E,
πi
ij(k)<πi
ij(j)
ck
− min
E,
πij(k)<πij(j)
ck
??
??
−
min
E,
πi
ij(k)≤πi(j)
ck
− min
E,
πij(k)≤πij(j)
= MCj(πij) − MCj(πi
ck
ij).
And the proof is complete.
Proposition 2. RA is CG-bilateral consistent∗.
Proof. Let S = {i,j} ⊂ N, and π be any permutation of N, then we have:
• If π(i) > π(j): then MCi(π) = MCi(πj).
• If π(i) < π(j): then MCi(π) = MCi(πj) +?MCi(π) − MCi(πj)?.

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BANKRUPTCY PROBLEMS9
Hence
RAi(N,E,c) =1
n!
?
π∈Π:
π(i)>π(j)
MCi(π) +1
n!
?
π∈Π:
π(i)<π(j)
?MCi(πj) +?MCi(π) − MCi(πj)?
MCi(π) +1
n!
π∈Π:
π(i)<π(j)
MCi(π)
=1
n!
?
π∈Π:
π(i)>π(j)
MCi(πj) +1
n!
?
π∈Π:
π(i)<π(j)
=
1
(n − 1)!·
?
π∈Πj
?
?MCi(π) − MCi(πj)?
Analogously
RAj(N,E,c) =
1
(n − 1)!·
?
π∈Πi
MCj(π) +1
n!
?
π∈Π:
π(j)<π(i)
?MCj(π) − MCj(πi)?.
By Lemma 1 and the definition of the RA rule we have, and the definition
of the CG rule
RAi(N,E,c) − RAj(N,E,c) =
1
(n − 1)!·
?N\{j},E − ¯ cj,cN\{j}
RAk(N\{j},E−¯ cj,¯ cN\{j})−(E−¯ ci)+
??
π∈Πj
MCi(π) −
?− RAj
?
?N\{i},E − ¯ ci,cN\{i}
π∈Πi
MCj(π)
?
=
RAi
?
= cS,RA
i
?=
E−¯ cj−
k∈N\S
?
k∈N\S
RAk(N\{i},E−¯ ci,¯ cN\{i})
− cS,RA
j
= CGi(N,ES,RA,cS,RA) − CGj(N,ES,RA,cS,RA).
Hence
RAi(N,E,c)−RAj(N,E,c) = RAi(N,ES,RA,cS,σ)−RAj(N,ES,RA,cS,σ).
On other hand by efficiency
RAi(N,E,c) + RAj(N,E,c) = RAi(N,ES,RA,cS,σ) + RAj(N,ES,RA,cS,σ).
From the last two equalities
RAi(N,E,c) = RAi(N,ES,RA,cS,σ) and RAj(N,E,c) = RAj(N,ES,RA,cS,σ),
and the proof is done.

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10M. J. ALBIZURI, J. LEROUX AND J. M. ZARZUELO
4. Cooperative Games and Bankruptcy Problems
A (TU) game on a finite set of players N is a mapping v associating a
real number v(S) with every subset S ⊆ N, such that v(∅) = 0. The subsets
S ⊆ N are called coalitions. The real number v(S) represents the worth of
coalition S, interpreted as the total amount that this coalition can obtain
alone, without the rest of the players.
A solution concept is a mapping σ associating with each game in a specific
class a payoff vector, i.e. a vector x = σ(N,v) ∈ RN, whose components
represent the payoffs to the players and add up to v(N). Two of the most
prominent solution concepts are the Shapley value and the nucleolus.
There is a natural way to associate a game with a bankruptcy problem (see
O’Neill, 1982). The worth of a coalition S is what is left, after the other
players receive their claims. Thus given a bankruptcy problem (N,E,c)
define for every S ⊆ N
v(N,E,c)(S) =
E −
?
?
k∈N\S
ck
?
+.
Having associated a game to every bankruptcy problem, every solution σ
for TU games automatically induces a rule for bankruptcy problems, which
will also denote by σ by abusing notations slightly. It is well known that the
Shapley value gives place to the Random Arrival rule, and the nucleolus to
the Talmud rule (0’Neill, 1982; and Aumann & Maschler, 1985, respectively).
The consistency property has been extensively analyzed in the context
of cooperative games as well. Several interpretations exist there as well
and, just like in bankruptcy problems, also depend on the definition of a
reduced situation. It is instructive to compare the consistency property for
bankruptcy problems with the one for cooperative games.
Following Davis & Maschler (1965), given a game v on N, a coalition S,
and a payoff vector x, the reduced game vS,xon S is defined as follows
?
(2) vS,x(T) =
x(T)
max?v(Q ∪ T) − x(Q) : Q ⊆ N\S?
A different notion was suggested by Hart & Mas-Colell (1989). Given a
game v on N, a coalition S, and a TU solution σ, they define the reduced
game vS,σfor all T ⊂ S as
vS,σ(T) = v(T ∪ Sc) −
if T = S or T = ∅,
if T ⊂ S, T ?= ∅,S.
?
i∈Sc
σi(T ∪ Sc,v).

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BANKRUPTCY PROBLEMS11
A question arises: do these definitions of a reduced bankruptcy game
corresponds to the definitions of reduced bankruptcy problem seen in the
previous sections. The answer concerning the definition of Davis & Maschler
for a reduced game (Expression (2)) is given in the following lemma, where
it is shown to correspond to the ”traditional” reduced bankruptcy problem.
Lemma 2 (Aumann & Maschler, 1985). Let (N,E,c) be a bankruptcy prob-
lem and σ a rule such that x = σ(N,E,c) with 0 ≤ xi≤ ci. Then for any
coalition S,
v(S,ES,σ,cS)= vS,x
(N,E,c)
Alternatively, it turns out that the definition of a reduced bankruptcy
problem introduced in the previous section (Expression (1)) corresponds
to the Hart & Mas-Colell reduced game, but only for the case of bilateral
reduced problems, as we show in the next lemma, the proof of which is
straightforward and will be omitted.
Lemma 3. Let (N,E,c) be a bankruptcy problem, σ a rule, and S ⊆ N
such that |S| = 2, then
v(S,ES,σ,cS,σ)= vS,σ
(N,E,c).
That is, the following diagrams are commutative
(N,E,c)
↓
(S,ES,x,cS)
→
v(N,E,c)
↓
→
v(S,ES,x,cS)= vS,x
(N,E,c)
(N,E,c)
↓
?S,ES,σ,cS,σ?
→
v(N,E,c)
↓
→
v(S,ES,σ,cS,σ)= vS,σ
(N,E,c)
(if |S| = 2).
However, if |S| > 2 the game vS,σ
a bankruptcy problem. That is, in general the Hart & Mas-Colell reduced
game of a bankruptcy problem is not a game associated to any bankruptcy
game, as it is shown in the following example. Let N = {1,2,3,4,5}, E =
100, and ci = 40 for i = 1,2,3,4,5. Also let S = {1,2,3,4}, and σ a
(N,E,c)does not correspond in general to

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12M. J. ALBIZURI, J. LEROUX AND J. M. ZARZUELO
symmetric rule. Then
vS,σ
(N,E,c)(T) =













80
35
40/3
0
if T = S,
if |T| = 3,
if |T| = 2,
if |T| = 1,
which clearly is not a game associated to any bankruptcy problem. So the
second diagram is not necessarily commutative in the many-agent case.
Nevertheless, Theorem 2 suggests a characterization of the Shapley value
by means of bilateral consistency ` a la Hart & Mas-Colell as follows.
We say that a solution σ on TU games satisfies 2- consistency ` a la Hart
& Mas-Colell (2-CO) if for every game v on N, and every S ⊂ N, such that
|S| = 2, it holds
σ?vS,σ?= σS(v).
We call a solution σ on TU games standard for 2-person games if for every
game v on {i,j} it holds
σi(v) = v?{i}?+ (1/2)
Theorem 3. The Shapley value, for the class of TU games, is the unique
single-point solution concept which is standard for 2-person games and sat-
isfies 2-CO.
?
v?{i,j}?− v?{i}?− v?{j}??
.
Proposition 1 can be easily adapted to prove the uniqueness part of this
theorem. The existence part is an immediate corollary of Theorem B in
Hart & Mas-Colell (1989) or Theorem 2 in Maschler & Owen (1989). Notice
finally that the uniqueness in Theorem 3 does not follow from any of these
two results since in both works the respective authors use an induction
argument on the whole class of games.
References
AUMANN, R. J. and MASCHLER, M. (1985). Game theoretic analysis of
a bankruptcy problem from the Talmud. Journal of Economic Theory, 36,
195–213.
DAVIS, M. and MASCHLER, M. (1965). The kernel of a cooperative game.
Naval Research Logistic Quarterly, 12, 223–259.
HART, S. and MAS-COLELL, A. (1989). Potential, value and consistency.
Econometrica, 57, 589-614.

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BANKRUPTCY PROBLEMS 13
LEROUX, J. (2007). A note on consistency in cost-allocation problems.
mimeo HEC Montr´ eal.
MASCHLER, M. and OWEN, G. (1989). The consistent Shapley value for
hyperplane games. International Journal of Game Theory, 18, 389-407.
O’NEILL, B. (1982). A problem of rights arbitration from the Talmud.
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SUDH¨OLTER, P. (1998). Axiomatizations of Game Theoretical Solutions
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THOMSON, W. (2003). Axiomatic and Game Theoretic Analysis of Bank-
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THOMSON, W. (2006).Consistent allocation rules. Mimeo, University of
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