In this paper interval bankruptcy games arising from bankruptcy situations with interval claims are introduced. For this class of cooperative games two (marginal-based) Shapley-like values are considered and the relation between them is studied.

Dinko Dimitrov

Rodica Branzei

Stef Tijs

English

Research Papers in Economics

Shapley−like values for interval bankruptcy games 
Rodica Branzei
Faculty of Computer Science, Alexandru Ioan Cuza University, Iasi, Romania
Dinko Dimitrov Stef Tijs
CentER and Department of Econometrics and
Operations Research, Tilburg University, The
Netherlands
CentER and Department of Econometrics and
Operations Research, Tilburg University, The
Netherlands
Abstract
In this paper interval bankruptcy games arising from bankruptcy situations with interval
claims are introduced. For this class of cooperative games two (marginal−based)
Shapley−like values are considered and the relation between them is studied.
The work on this paper started while the authors were research fellows at ZiF (Bielefeld) for the project "Procedural Approaches
to Conflict Resolution", 2002. We thank our hosts for their hospitality. The work of Dinko Dimitrov was partially supported by
a Marie Curie Research Fellowship of the European Community programme "Improving the Human Research Potential and the
Socio−Economic Knowledge Base" under contract number HPMF−CT−2002−02121.
Citation: Branzei, Rodica, Dinko Dimitrov, and Stef Tijs, (2003) "Shapley−like values for interval bankruptcy games."
Economics Bulletin, Vol. 3, No. 9 pp. 1−8
Submitted: May 26, 2003.  Accepted: June 2, 2003.
URL: http://www.economicsbulletin.com/2003/volume3/EB−03C70012A.pdf1I n t r o d u c t i o n
In a classical bankruptcy situation one has to divide a certain amount of
money (estate) among some people (claimants) who have individual claims on
the estate, and the total claim is weakly larger than the estate (cf. Aumann
and Maschler (1985), O’Neill (1982)).
This paper deals with a generalized version of classical bankruptcy situ-
ations where claimants are facing uncertainty regarding their eﬀective rights
and, as a result, individual claims can be rather expressed in the form of
closed intervals, the estate being still weakly smaller than the sum of the
lower claims. Some appealing procedures leading to eﬃcient and reason-
able solutions for division problems of this type are described in Branzei et
al. (2002). Another extension of classical bankruptcy situations, namely
bankruptcy situations with references, is treated in Pulido et al. (2002a, b)
by introducing two classical coalitional games for which compromise values
are considered.
In this paper we tackle bankruptcy situations with interval claims by
introducing a new type of cooperative games, that we call interval bankruptcy
games, for which two values based on the Shapley value (Shapley (1953)) are
considered. To be more precise, our Shapley-like values use the formula of the
Shapley value which is based on marginal vectors. However, other well-known
formulae (and axiomatizations) for the Shapley value exist in the literature.
We mention here Harsanyi’s formula based on dividends (Harsanyi (1959))
and Owen’s formula based on the multilinear extension of a TU-game (Owen
(1972)). To have an idea about how much attention has been captured by
this interesting solution concept, the reader is referred to Roth (1988).
The rest of the paper is organized as follows. In Section 2 bankruptcy
problems with interval claims are introduced and the corresponding interval
games are constructed. Section 3 describes two Shapley-like values for in-
terval bankruptcy games and shows that they are related to each other via
the set inclusion. We conclude in Section 4 with some remarks on further
research.
2 Interval bankruptcy games
Let N = {1,...,n} be the set of claimants among which an estate E has to
be divided. We denote by Ii =[ ai,b i] the claim interval of claimant i ∈ N,
1where ai is the lower bound of the claim interval, while bi is the upper bound.
Let = be the family of closed intervals in <+ and =N be the set of all
vectors of the form I =( I1,...,I n).A bankruptcy problem with interval
claims is a pair (E,I),w h e r e0 <E≤
P
i∈N ai.W ed e n o t eb y=DN the set
of all bankruptcy problems of the form (E,I).
Note that if all claim intervals Ii, i ∈ N are degenerated intervals, i.e. Ii =
[ai,a i], the problem (E,I) coincides with the classical bankruptcy problem
(E,a) with a =( a1,...,a n) and 0 <E≤
P
i∈N ai. Moreover, all division
problems on N with sharp claims w.r.t. the available amount E,o ft h ef o r m
(E,d) with d =( d1,...,d n) and 0 <E≤
P
i∈N ai ≤
P
i∈N di appear as
particular cases of a problem (E,I) ∈ =D
N. In the following we use the
notation DN to refer to the family of classical division problems related to a
division problem with interval claims.
To each problem in =D
N one can associate a set of problems in DN
b a s e do nt h ei d e at oc o m p r o m i s et h ei n t e r v a lc l a i m so fe a c hp l a y e ri ∈ N by
weighting the upper bound of his claim interval with ti ∈ [0,1] and the lower
bound with (1 − ti).
Let I =( I1,...,I n) be the vector of interval claims in the problem
(E,I) ∈ =D
N and t =( t1,t 2,...,t n),w h e r eti ∈ [0,1] for each i ∈ N.W ei n -
troduce the t-compromise claim ct =
¡
c
t1
1 ,c
t2
2 ,...,c tn
n
¢
by c
ti
i = tibi+(1 − ti)ai
for each i ∈ N.G i v e nt h ea m o u n tE,f o re a c ht-compromise claim ct,w ec a n
consider the division problem (E,ct) ∈ DN,w h i c hw ec a l lt h et-compromise
division problem.
Now, to each standard (point claims) division problem (E,ct) ∈ DN,o n e
can associate a corresponding cooperative game as follows.
For x ∈ <,d e n o t ex+ =m a x ( x,0). The associated division game
hN,vE,cti has the characteristic function vE,ct :2 N → < where
vE,ct (S)=

E −
X
i∈N\S
c
ti
i


+
for each S ∈ 2
N,( 1 )
i.e. vE,ct (S) is the amount of the estate that is left if all players outside
coalition S receive their claim (cf. Curiel et al. (1987)). The set of all
division games hN,vE,cti is denoted by FN.
Let c1 =( c1
1,c 1
2,...,c 1
n) and c0 =( c0
1,c 0
2,...,c 0
n). Now we construct the
cooperative interval game hN,vE,Ii that corresponds to a division problem
with interval claims (E,I) ∈ =D
N. This game has the characteristic function
2vE,I :2 N → = where, for each S ∈ 2N, vE,I (S)=
n
vE,ct(S) | t ∈ [0,1]
N
o
or,
equivalently,
vE,I (S)=



E −
X
i∈N\S
bi


+
,

E −
X
i∈N\S
ai


+

 =[ vE,c1 (S),v E,c0 (S)].
(2)
So, in an interval bankruptcy game each coalition S can receive an amount
which is between the amount of the estate that is left if all players outside
coalition S receive their upper claim and the amount of the estate that is
left if all players outside coalition S receive their lower claim. The set of all
cooperative interval games hN,vE,Ii is denoted by GN.
3 Two Shapley-like values
In this section we introduce two Shapley-like values for n-person interval
bankruptcy games. The ﬁrst Shapley-like value is an indirect one in the
sense that it is built on the standard (marginal-based) Shapley value for each
cooperative game corresponding to a t-compromise division problem. The
second Shapley-like value is constructed directly on an interval bankruptcy
game and it associates to each such a game one vector of intervals in =N.
As proved in Proposition 1 the intervals generated via the indirect value are
always included in the corresponding intervals generated via the direct value.
Moreover, Example 2 shows that these two values are diﬀerent.
We start by introducing for games hN,vE,cti ∈ FN and permutations
σ : N → N marginal vectors mσ (vE,ct).L e t
Pσ (i)=
©
r ∈ N | σ
−1 (r) <σ
−1 (i)
ª
be the set of predecessors of i in σ.T h e n
m
σ
i (vE,ct)=vE,ct (Pσ (i) ∪ {i}) − vE,ct (Pσ (i))
for each i ∈ N. The Shapley value ϕ(vE,ct) is equal to the average of the
marginal vectors (cf. Shapley (1953))
ϕi (vE,ct)=
1
n!
X
σ∈π(N)
m
σ
i (vE,ct) for all i ∈ N,
3where π(N) is the set of all permutations π on N.
We introduce the indirect Shapley-like value ϕ(vE,I) for an interval bankruptcy
game hN,vE,Ii ∈ GN as having as i-th component the set
ϕi (vE,I)=
n
ϕi (vE,ct) | t ∈ [0,1]
N
o
=[ m i n ( ϕi (vE,c0),ϕ i (vE,c1)),max(ϕi (vE,c0),ϕ i (vE,c1))]
for all i ∈ N.
The second Shapley-like value is based on the marginal vectors mσ
i (vE,I)
constructed directly on games hN,vE,Ii ∈ GN, i.e.
m
σ
i (vE,I)=vE,I (Pσ (i) ∪ {i}) − vE,I (Pσ (i)) (3)
for each i ∈ N.
Note that by using (2) formula (3) can be rewritten as
m
σ
i (vE,I)=[ vE,c1 (Pσ (i) ∪ {i}),v E,c0 (Pσ (i) ∪ {i})] − (4)
[vE,c1 (Pσ (i)),v E,c0 (Pσ (i))]
for each i ∈ N.
The direct Shapley-like value Φ(vE,I) of a game hN,vE,Ii ∈ GN is the
average of the marginal vectors of the game, i.e.
Φ(vE,I)=
1
n!
X
σ∈π(N)
m
σ (vE,I).
By using (4) this formula can be rewritten (cf. Moore (1979)) as
Φ(vE,I)=
h
Φ(vE,I),Φ(vE,I)
i
,
where
Φ(vE,I) =
1
n!
X
σ∈π(N)
(vE,c1 (Pσ (i) ∪ {i}) − vE,c0 (Pσ (i))),
Φ(vE,I)=
1
n!
X
σ∈π(N)
(vE,c0 (Pσ (i) ∪ {i}) − vE,c1 (Pσ (i))).
4Proposition 1 Let hN,vE,Ii ∈ GN.T h e n
ϕi (vE,I) ⊂ Φi (vE,I) for all i ∈ N.
Proof. Let i ∈ N and take a t ∈ [0,1]
N. By using (1), we have
vE,ct (Pσ (i) ∪ {i}) − vE,ct (Pσ (i))
≥ vE,c1 (Pσ (i) ∪ {i}) − vE,ct (Pσ (i))
≥ vE,c1 (Pσ (i) ∪ {i}) − vE,c0 (Pσ (i)).
Then
1
n!
X
σ∈π(N)
(vE,ct (Pσ (i) ∪ {i}) − vE,ct (Pσ (i)))
≥
1
n!
X
σ∈π(N)
(vE,c1 (Pσ (i) ∪ {i}) − vE,c0 (Pσ (i))),
which is equivalent to
ϕi (vE,ct) ≥ Φi (vE,I).
Similarly, one obtains
ϕi (vE,ct) ≤ Φi (vE,I).
The last two inequalities imply ϕi (vE,I) ⊂ Φi (vE,I).
Example 2 Let N = {1,2}, E =1 0 , I =( I1,I 2)=( [ 4 ,6],[8,9]).S o ,t h e
division problem with interval claims is (10,[4,6],[8,9]) and the character-
istic function of the corresponding cooperative interval game (see (2)) is as
follows:
vE,I (∅)=[ 0 ,0],
vE,I ({1})=[ 1 ,2],
vE,I ({2})=[ 4 ,6],
vE,I ({1,2})=[ 1 0 ,10].
For the marginal vectors of the corresponding point-claims games we have
m
(12) (vE,c0)=m
(12) ¡
v10,(4,8)
¢
=( 2 ,8),
m
(21) (vE,c0)=m
(21) ¡
v10,(4,8)
¢
=( 4 ,6),
m
(12) (vE,c1)=m
(12) ¡
v10,(6,9)
¢
=( 1 ,9),
m
(21) (vE,c1)=m
(21) ¡
v10,(6,9)
¢
=( 6 ,4),
5and the marginal vectors of the interval game are
m
(12) ¡
v10,([4,6],[8,9])
¢
=( [ 1 ,2],[10,10] − [1,2]) = ([1,2],[8,9]),
m
(21) ¡
v10,([4,6],[8,9])
¢
=( [ 1 0 ,10] − [4,6],[4,6]) = ([4,6],[4,6]).
The two Shapley-like values are then
ϕ
¡
v10,([4,6],[8,9])
¢
=
¡£
ϕ1
¡
v10,(4,8)
¢
,ϕ 1
¡
v10,(6,9)
¢¤
,
£
ϕ2
¡
v10,(6,9)
¢
,ϕ 2
¡
v10,(4,8)
¢¤¢
=
µ·
1
2
(2 + 4),
1
2
(1 + 6)
¸
,
·
1
2
(9 + 4),
1
2
(8 + 6)
¸¶
=( [ 3 ,3.5],[6.5,7])
and
Φ
¡
v10,([4,6],[8,9])
¢
=
µ
1
2
([1,2] + [4,6]),
1
2
([8,9] + [4,6])
¶
=( [ 2 .5,4],[6,7.5]).
It is easy to see that for each i ∈ {1,2}, ϕi
¡
v10,([4,6],[8,9])
¢
is strictly included
in Φi
¡
v10,([4,6],[8,9])
¢
and, hence, diﬀerent from it.
4 Final remarks
The Shapley value has been inspiring for constructing Shapley-like values on
diﬀerent classes of cooperative games (cf., for example, Timmer et al. (2002))
where diﬀerent formulae for the Shapley value are considered. It could be a
topic of further research to apply also diﬀerent formulae of the Shapley value
on interval bankruptcy games and to study their interrelations. Another
possibility is connected with the fact that uncertainty is just as likely to aﬀect
the estate as to aﬀect claims. Hence, one could assume that the estate is also
expressed in the form of a closed interval, and to analyze the connections
among the corresponding point claims and interval games.
References
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7

A bargaining model for the cooperative n-person game, In:

A problem of rights arbitration from the Talmud,

A value for n-person games, In:

Bankruptcy games,

Dividing university money: compromise solutions,

Game theoretic analysis of a bankruptcy problem from the Talmud,

Game theoretic techniques for university management: an extended bankruptcy model,

Shapley-like values for interval bankruptcy games

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