This paper strengthens the result of Sengupta and Sengupta (1996). We show that for the class of games with nonempty cores the core can be reached in a bounded number of proposals and counterproposals. Our result is more general than this: the boundedness holds for any two imputations with an indirect dominance relation between them.dynamic cooperative game, indirect dominance, core.

László Á. Kóczy

English

Research Papers in Economics

THE CORE CAN BE ACCESSED
IN A BOUNDED NUMBER OF STEPS
L´ ASZL´ O ´ A.K´ OCZY
Abstract. This paper strengthens the result of Sengupta and Sengupta (1996). We
show that for the class of TU games with non-empty cores the core can be reached via
a bounded number of proposals and counterproposals. Our result is more general than
this: the boundedness holds for any two imputations with an indirect dominance relation
between them.
1. Introduction
The core (Kannai 1992) is the collection of agreements that once proposed are never
abandoned. While this made the core popular enough it is not an irrelevant question to
ask whether the core can actually be reached via a sequence of blockings. This question is
answered aﬃrmatively by Sengupta and Sengupta (1996), where they introduce a recursive
algorithm that generates a sequence of imputations. This algorithm terminates with a core-
imputations. They do not, however make any claims about the number of steps required
to reach the core. The aim of this paper is to show that the number of steps required
is bounded. We do not make use of the algorithm proposed by Sengupta and Sengupta
(1996), but prove the existence of a sequence of bounded length.
Consider a 3-player gamewhere the grand coalition gets 3, pairs get 2, singletons get 0.
The core of this game consists of a single payoﬀ-vector (1;1;1). Now consider an initial
allocation
¡
1
2m;2 ¡ 1
2m;1
¢
and the following –rather ineﬃcient– process:
µ
1
2m;2 ¡
1
2m;1
¶
!
µ
1
2m¡1;1;2 ¡
1
2m¡1
¶
µ
1
2m¡1;1;2 ¡
1
2m¡1
¶
!
µ
1
2m¡2;2 ¡
1
2m¡2;1
¶
. . . µ
1
2
;2 ¡
1
2
;1
¶
or
µ
1
2
;1;2 ¡
1
2
¶
! (1;1;1)
Date: March 22, 2002.
Key words and phrases. Dynamic cooperative game, indirect dominance, core.
Discussions with Luc Lauwers have lead to considerable improvement of this paper. The support
(COE/01/02) of the Katholieke Universiteit Leuven is gratefully acknowledged.
12 L´ ASZL´ O ´ A.K´ OCZY
where the penultimate allocation depends on the parity of m. This process terminates in
the core in exactly m steps. As m can be chosen arbitrarily the number of steps required
has no upper bound. In this paper we show that given the result of Sengupta and Sengupta
(1996) the core can be reached in a bounded number of steps, moreover, our proof points
out where do such ineﬃcient processes make unnecessary detours.
The result we prove is not speciﬁc to the core. We show that if an imputation a can
be reached from another imputation b via a dominating sequence of imputations, then the
length of this sequence is bounded. Here we will only consider paths via imputations, that
is via eﬃcient and individually rational allocations. If this is not required, as in (Sengupta
and Sengupta 1994) our proof can be simpliﬁed signiﬁcantly.
Regarding the process of deviations and counter-deviations as an algorithm our result
shows that reaching the core is a primitive recursive algorithm; it can be programmed with
“for” loops only (Weisstein 2002), and the running time of such a program can be set in
advance. See P´ eter (1981) and P´ eter (1967) for more on primitive recursive algorithms.
The structure of the paper is as follows: First we introduce our notation and some
terminology. In Section 3 we state our results. The main part of this paper is the proof of
our key lemma which is presented in Section 4.
2. Preliminaries
Let (N;v) be a TU-game with player set N, and characteristic function v. Subsets of
N are coalitions and v(S) is the payoﬀ for coalition S. An imputation x is a payoﬀ-vector
in RN that is individually rational and eﬃcient, that is, xi ¸ v(fig) for all players i in N
and x(N) = v(N), where x(S) =
P
i2S xi. Let A(N;v) denote the set of imputations. The
projection of x onto S is denoted by xS; we write xS > ys if xi ¸ yi for each i 2 S, but
xS 6= yS.
The imputation x directly dominates y via coalition S, or x ÂS
D y if xS > yS, x(S) = v(S)
and x 2 A(N;v). We say that x indirectly dominates y and we write x ÂI y if there
exists a ﬁnite sequence of imputations
©
y0;:::;y¸(´)
ª
and a ﬁnite collection of coalitions
©
Y1;:::;Y¸(´)
ª
, such that x = y¸(´), y = y0, and yj Â
Yj
D yj¡1.
Deﬁnition 1 (Path). The sequence f(yj;Yj)g
¸(´)
j=1 is a (dominance) path from y to x if
² y¸(´) = x,
² yj Â
Yj
D yj¡1 for j > 1, and
² y1 Â
Y1
D y,
and we denote it by ´.
Paths will be denoted by Greek letters: »;´;³ with the corresponding small (x;y;z), and
capital letters (X;Y;Z) denoting the respective imputations and blocking coalitions. The
length of a path ´ is denoted by ¸(´). Finally, an index 0 refers to the imputation that the
path originates from, so in this case y0 = y.MINIMAL DOMINANCE PATHS ARE BOUNDED 3
Deﬁnition 2 (Concatenation of paths). Pats » and ´ can be concatenated if » is a con-
tinuation of ´, formally if x1 Â
X1
D y¸(´), and we write this as ³ = » ^ ´ to mean that
¸(³) = ¸(») + ¸(´);
Zj =
(
Xj if j · ¸(»)
Yj otherwise,
zj =
(
xj if j · ¸(»)
yj otherwise.
Note that in general » + ´ 6= ´ + », in fact, they may or may not be deﬁned.
Deﬁnition 3 (Class). Imputations x and y belong to the same class of imputations if
(1) x(S) ¸ v(S) iﬀ y(S) ¸ v(S) for all coalitions S and
(2) x(R) +
P
i2SnR v(fig) ¸ v(S) iﬀ y(R) +
P
i2SnR v(fig) ¸ v(S) for all S µ N and
R µ S and
(3) for all pairs S and C of coalitions such that C \S 6= ; while C [S = N and for all
partitions R of all coalitions R ½ C \ S we have that
v(S) > v(N) ¡ v(C) +
X
T2R
v(T) + x(C \ S n R) (2.1)
iﬀ v(S) > v(N) ¡ v(C) +
X
T2R
v(T) + y(C \ S n R):
Members of a class are safe against the same blockings (Condition 1), even after certain
modiﬁcations (Conditions 2 & 3), where the initial diﬀerences are limited to a subset of
the players. The ﬁniteness of the number of classes is a trivial, but crucial property.
Deﬁnition 4 (Set of Winners). Given a dominance path ´ the set of winners, Wj(´) is
the set of players who proﬁt from each subsequent deviation, that is
Wj(´) =
¸(´) \
i=j+1
Yi:
Deﬁnition 5 (Composite Player). A composite player is a set of players, a blocking coali-
tion that stays together and forms part of blocking coalitions until examined, but may
break up in subsequent blockings. Formally S is a composite player in path ´ at time j if
S = Yh for some h · j and S µ Yi for all h < i · j. Should the composite player contain
smaller composite players, the term will refer to the largest one.
3. Results
Lemma 6. Let (N;v) be a game. Then there exists an integer M, such that for all a;b
pairs in A(N;v) such that a ÂI b there exist a dominance path ¼ from b to a such that
¸(¼) < M.
We give the proof of this result in the next section.4 L´ ASZL´ O ´ A.K´ OCZY
Lemma 7 (Sengupta and Sengupta (1996)). Let (N;V ) be a game with a non-empty
core C(N;V ). Let a be an imputation outside C(N;V ). Then there exists an imputation
c 2 C(N;V ) such that c ÂI a.
This lemma, combined with Lemma 6 gives the following theorem:
Theorem 8. Let (N;V ) be a game with a non-empty core that we denote C(N;V ). Then
there exists an integer M such that for all imputations a 2 A(N;V ) there exists an impu-
tation c 2 C(N;V ) and a path ¼ from a to c such that ¸(¼) < M.
4. Proof of Lemma 6
4.1. Outline. We prove the result by contradiction.
We assume that there exists a game (N;v), such that for all M0 > M there exists a
pair a and b in A(N;v), such that a ÂI b but the shortest path from b to a has length
¸ > M0. We show that for M0 suﬃciently large there exists a shorter path, giving the
required contradiction. In our argument we ﬁnd two suﬃciently similar imputations in the
path. These two imputations divide the path into three parts. We show that the middle
part can be removed, and with some modiﬁcations the end part can be reattached.
r
p0
- P1 r
p1
- P2 r
p2
r
pk
| {z }
»
r
pK
| {z }
´
r
p¸(¼)¡1
- P¸(¼) r
p¸(¼)
| {z }
³
? ?
r
x0
- X1 r
x1
- X2 r
x2
r
| {z }
»
x¸(») = z0
0
r
z0
¸(³)¡1
- Z¸(³) r
z0
¸(³)
| {z }
³0
| {z }
¼0
Figure 1. Sematic picture of the proof.
Formally the outline of the proof is as follows: Given a dominance path ¼ there exists a
break-up of the path into 3 sections, ¼ = »^´^³, such that there exists a modiﬁcation of ³
denoted by b ³ such that the path b ¼ = »^b ³ leads from b to a as well. Moreover b ³ has the same
same blocking coalitions, and hence the same length as ³. Clearly ¸(» ^ ´ ^ ³) > ¸(» ^ b ³),
which gives the required contradiction. Observe that ³ is a path from y¸(´) to a, while b ³ is
from x¸(») to a. Therefore in order to be able to deﬁne b ³, the imputations y¸(´) and x¸(»)
must be suﬃciently similar. What remains is to deﬁne “similar” for the two imputations,MINIMAL DOMINANCE PATHS ARE BOUNDED 5
deﬁne b ³ and show that the coalitions
n
b Zj
o¸(b ³)
j=1
= fZjg
¸(b ³)
j=1
are indeed blocking.
4.2. Existence of two similar imputations. Let k and K be such that the correspond-
ing imputations pk and pK
(1) belong to the same class, and
(2) satisfy that Wk(¼) = WK(¼).
The number of classes is ﬁnite, and there are also a ﬁnite number of possible winner sets.
Thus for suﬃciently high M0, that is, in a suﬃciently long path ¼ the existence of such two
imputations is guaranteed. Setting x¸(») = pk, and y¸(´) = pK we get a suitable division.
Outcomes pk and pK break the path into 3 sections as follows:
pj =)
8
> <
> :
xj if j · k
yj¡k if k < j · K
zj¡K if K < j:
In the rest of the proof we attach the ³ path to the » path, cut ´, thereby shortening
the path. However, since x¸(»), and y¸(´) are not identical the path ³ has to be adjusted.
We keep the deviating coalitions, but deﬁne a new path.
4.3. The new path.
4.3.1. Creating the new imputations. Let b z0 = x¸(») and b z¸(³) = a. Given the imputations
up to b zj¡1, where 0 < j < ¸(³) we deﬁne b zj. In the deﬁnition we have to take care about
the following:
² Blockings are proﬁtable, so blocking players cannot be worse oﬀ than prior to the
blocking.
² Players who are blocking in the next round must have a suﬃciently low payoﬀ so
that the blocking, which is ﬁxed in advance gives them an improvement.
² Winners increase their payoﬀs monotonically, so it cannot exceed the end-payoﬀ.
² Eﬃciency for N and for the blocking coalition and individual rationality for all
players must hold.
The imputations we propose satisfy these conditions. Note that non-blocking players’
payoﬀs have their eﬀects only later, so they can be chosen with little restriction.
(1) If Zj [ Zj+1 = N all players have blocked or will block next.
(a) If Zj \ Zj+1 = ; then none of the blocking players block in the next round,
thus their payoﬀs can be set freely. We select the payoﬀs for the others (who
are all blocking in the next round) so that their blocking is proﬁtable.
b z
i
j+1 =
(
b zi
j + ±i if i 2 Zj
zi
j+1 otherwise.6 L´ ASZL´ O ´ A.K´ OCZY
where ± 2 RZj is the proﬁt-division vector such that ±i > 0 for all i 2 Zj and
±(Zj) = v(Zj) ¡ b zj(Zj).
(b) otherwise (Zj \Zj+1 6= ;) we make sure that the blocking players who are also
blocking in the next round get the minimum possible: the surplus is divided
among the rest of the players1.
b z
i
j+1 =
8
> <
> :
b zi
j if i 2 Zj \ Zj+1
b zi
j + ±i if i 2 Zj n Zj+1
zi
j+1 otherwise.
where ± 2 RZjnZj¡1 is the proﬁt-division vector such that ±i > 0 for all i 2
Zj n Zj¡1 and ±(Zj n Zj+1) = v(Zj) ¡ b zj(Zj).
(2) Otherwise (if Zj [ Zj+1 6= N) there exist players outside Zj who do not block
in the next round either, so we can give the soon-blocking outsiders a minimum
(individually rational) payoﬀ and the rest can share the remaining payoﬀ. Note that
in zj+1, which is an imputation, they share no less, so there exists a distribution
where each get at least its individually rational payoﬀ.
(a) If all blocking players are winners we have to take special care of their payoﬀs:
b z
i
j+1 =
8
> <
> :
b zi
j + ±i if i 2 Zj
v(fig) if i 2 (N n Zj) \ Zj+1
v(fig) + °i otherwise.
where ± 2 RZjnZ¤ is the proﬁt-division vector such that 0 < ±i < zi
j+1 ¡ b zi
j
for all i 2 Zj and ±(Zj) = v(Zj) ¡ b zj(Zj). Moreover ° 2 RNn(Zj[Zj+1) is
a vector to divide the left-over such that °i > 0 for all i = 2 Zj [ Zj+1 and
°(N n (Zj [ Zj+1)) = v(N) ¡ vZj ¡
P
i= 2Zj v(fig).
(b) Otherwise at some point some of the blocking players will not block. It is
particularly interesting if the coalition breaks up. If this happens then we
want the blocking players to have the minimal (non-increasing) payoﬀ and the
rest having the whole proﬁt. Let Z¤ = Zh where h is the smallest integer such
that h > j, and Zj * Zh. Z¤ is well-deﬁned.
b z
i
j+1 =
8
> > > <
> > > :
b zi
j if i 2 Zj \ Z¤
b zi
j + ±i if i 2 Zj n Z¤
v(fig) if i 2 (N n Zj) \ Zj+1
v(fig) + °i otherwise.
where ± 2 RZjnZ¤ is the proﬁt-division vector such that ±i > 0 for all i 2 Zj
and ±(Zj n Z¤) = v(Zj) ¡ b zj(Zj) and ° 2 RNn(Zj[Zj+1) is a vector to divide
the left-over such that °i > 0 for all i = 2 Zj [ Zj+1 and °(N n (Zj [ Zj+1)) =
v(N) ¡ vZj ¡
P
i= 2Zj v(fig).
1Such “rest” will always exist as the grand coalition cannot be blocking.MINIMAL DOMINANCE PATHS ARE BOUNDED 7
4.3.2. Proﬁtability of deviations. It remains to prove that the blocking coalitions Zj (which
we have ﬁxed in advance) are indeed blocking, that is, none of their members lose in the
blocking and at least one of them gets better oﬀ. Before starting our proof we give a simple
deﬁnition and state some simple properties.
Proposition 9. In the path constructed gains from a blocking are temporary, that is,
b zi
j+1  b zi
j¡1 except for two cases:
(1) If player i belongs to the just-created composite player Yj.
(2) If b zi
j¡1 = v(fig) and b zi
j+1 = zi
j+1.
Proof. By construction the proﬁt is shared among those players who leave the coalition
next. In the next step, however, these players get either an individually rational payoﬀ
or their payoﬀ in the ³ path, so they loose the surplus. The only exception is when the
coalition stays together and keeps on blocking: if it is a composite player. ¤
Corollary 10. The pre-blocking payoﬀ of a blocking player i is either
² its starting payoﬀ in the b ³-path, b zi
0, or
² its individually rational payoﬀ v(fig), or
² its payoﬀ in the ³-path, zi
j or
² player i belongs to a composite player.
Corollary 11. The pre-blocking payoﬀ of a player (composite or individual) is less in the
new path than in the original, b zi
h > zi
h unless zi
h = zi
0.
In the following part the four types of blockings are reexamined from the point of view
of proﬁtability. This part of the proof is by induction.
Step 1. First blocking. The blocking by PK+1 is proﬁtable in the path ¼, so v(PK+1) ¸
pK(PK+1). We have assumed that Z1 = PK+1. The imputations pk = z0 and pK are in the
same class, so by Cond. (1) in Defn. 3 v(Z1) ¸ z0(Z1) and the ﬁrst blocking is proﬁtable.
Step 2. Inductive assumption. We assume that the imputation b zj¡1 has been reached via
successive proﬁtable blockings.
Step 3. The blocking by Zj. We discuss four cases depending on the blocking by Zj¡1.
1(a): By construction b z
Zj
j¡1 = z
Zj
j¡1, so v(Zj) > zj¡1(Zj) implies v(Zj) > b zj¡1(Zj).
1(b): We look at the players in Zj \ Zj¡1. By Corollary 10
(4.1) b zj¡1(Zj) = v(N) ¡ v(Zj¡1) +
X
Z2R
v(Z) + b z0(Zj \ Zj¡1 n R);
where R is the set of players in Zj¡1 who do not have their original payoﬀs, and
R is a partition of this set into composite players including singleton “composite”
players, that is, players who have their individually rational payoﬀs. Corollary 11
combined this with the proﬁtability of the blocking Zj gives
(4.2) v(Zj) > zj(Zj) > v(N) ¡ v(Zj¡1) +
X
Z2R
v(Z) + z0(Zj \ Zj¡1 n R):
Since z0 = pK and b z0 = pk are in the same class, by 2.1 we have v(Zj) > b zj(Zj).8 L´ ASZL´ O ´ A.K´ OCZY
2(a): By Corollary 11 we focus on the set R of players with their initial payoﬀs. Such
a player i 2 R had a “winning streak”, that is i 2
Tj¡1
h=1 Zh. Since in this case
we assume that Wj¡1(³) = Zj, the player i belongs to W0(³) as well. Going back
to ¼ this means that i 2 WK(¼). By assumption then i 2 Wk(¼). Due to the
monotonicity of the payoﬀs of winning players pi
k · pi
K, and this argument holds
for all i 2 R. Then by the proﬁtability of the blocking Zj in path ³ we have
v(Zj) > v(N) ¡ v(Zj¡1) +
X
Z2R
v(Z) + z0(Zj \ Zj¡1 n R);
where R is a partition of this set into composite players including singletons, that
is, players with their individually rational payoﬀs. Our argument combined with
Corollary 11 and the fact that z0 and b z0 belong to the same class gives
v(N) ¡ v(Zj¡1) +
X
Z2R
v(Z) + z0(Zj \ Zj¡1 n R)
> v(N) ¡ v(Zj¡1) +
X
Z2R
v(Z) + b z0(Zj \ Zj¡1 n R):
But we have
(4.3) v(N) ¡ v(Zj¡1) +
X
Z2R
v(Z) + b z0(Zj \ Zj¡1 n R) = b zj¡1(Zj);
hence we can conclude v(Zj) > b zj¡1(Zj).
2(b): This case combines the previous ones, so we will not repeat our arguments. By
Corollary 11 troubles are only caused by players having a too high initial payoﬀ.
By the fact that z0 and b z0 belong to the same class Condition (3) of Deﬁnition 3
proves the proﬁtability of Zj.
4.3.3. The ﬁnal touch. The only part that remains perhaps unclear is that b z¸(b ³) = a.
Observe however, that the last blocking is of type 2(a) from the point of view of the
blocking, since all the blocking players are winners. Indeed, our condition on ± ensures
that all blocking players get the right payoﬀ provided their payoﬀs are not too high. Our
argument as in 2(a) shows that this condition holds. Since a is itself an imputation the
payoﬀs for the non-blocking players can be chosen so that b z
NnZ¸
¸ = aNnZ¸.
This completes our proof.
References
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North Holland, Amsterdam pp. 355–395
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P´ eter R (1981) Recursive Functions in Computer Theory. Computers and their Applications Akad´ emiai -
Ellis Horwood, Budapest - Chichester
Sengupta A, Sengupta K (1994) Viable proposals. International Economic Review 35(2):347–359
Sengupta A, Sengupta K (1996) A property of the core. Games and Economic Behavior 12:266–273
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http://mathworld.wolfram.com/PrimitiveRecursiveFunction.html

The core and balancedness. In: Aumann RJ, Hart S (eds.) Handbook of game theory, North Holland, Amsterdam pp. 355–395 P´ eter R

The Core Can Be Accessed in a Bounded Number of Steps

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The Core Can Be Accessed in a Bounded Number of Steps

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