An Existence Result for Farsighted Stable Sets of Games in
Characteristic Function Form1
Anindya Bhattacharya
Victoria Brosi
Deparment of Economics
University of York, UK
June, 2009
Abstract
In this paper we show that every finite-player game in characteristic function form (not necessarily
with side payments) obeying an innocuous condition (that the set of individually rational pay-off
vectors is bounded) possesses a farsighted von-Neumann-Morgenstern stable set.
JEL Classification No.: C71, D71.
Keywords: coalitional games; farsighted stable sets.
1
We are indebted to Sylvain Beal, Jacques Durieu, Yuan Ju and Philippe Solal for their comments and
suggestions. Of course, however, the errors remaining are ours. Brosi thanks ESRC, UK for a
scholarship.
1
1 Introduction
In this paper we show that every finite-player game in characteristic function form
obeying an innocuous condition (that the set of individually rational pay-off vectors is
bounded) possesses a farsighted von-Neumann-Morgenstern stable set.
It is well-known that characteristic function form is the oldest and perhaps the most
common representation of game situations for analysing group behaviour in such
situations. One of the most famous results in game theory is that there exist games in
characteristic function form that admit no von-Neumann-Morgenstern solution
(Lucas, 1968) with respect to the classical direct dominance relation.
Studies on von-Neumann-Morgenstern solution (or von-Neumann-Morgenstern
stable sets) in the environment of general coalitional games in characteristic function
form (possibly without side payments) go as far back as Aumann and Peleg (1960).
One significant development in the literature on coalitional games was the idea of
indirect dominance introduced by Harsanyi (1974) to take into account the fact that
the players may be farsighted—while making a move, they may take into account the
further moves by other groups of players that may ensue. This idea was further
developed in Chwe (1994) which, in turn, generated a sizeable literature on analysing
game situations using indirect or farsighted dominance.
Recently Diamantoudi and Xue (2005) showed that the game in the famous Lucas
counter-example admits a stable set with indirect dominance relations. This was
pursued further in Beal et al. (2008) who showed that, in stark contrast to that for
direct dominance relation, at least one stable set with respect to the indirect (or
farsighted) dominance relation exists for every coalitional game in characteristic
function form with transferable pay-offs (or a TU game, in short).2 In this paper we
generalize this result of Beal et al. (2008) to show that even without side payments,
every game in characteristic function form obeying an innocuous condition (that the
set of individually rational pay-off vectors is bounded) possesses a von-NeumannMorgenstern stable set with respect to the farsighted dominance relation.
There exists a series of recent works studying farsighted stable sets or some
analogues of them in different environments without transferable pay-offs. These
include Diamantoudi and Xue (2003), Kamijo and Muto (2007), Mauleon et al.
(2008), Vartiainen (2008) etc. However, the natures of the environments they studied
are quite different to that for games in characteristic function form.
The present work is closest to Greenberg et al. (2002) who studied farsighted stable
sets for exchange economies. However, the dominance notion they used was the first
notion (of the two) introduced in Harsanyi (1974) which is not intuitively very
appealing.
The next section gives the preliminary definitions and notation. The results are given
in Section 3. Section 4 contains some concluding comments.
2 Preliminary definitions, assumptions and notation
2
Beal et al. (2008), quite naturally, worked with the set of imputations as the set of outcomes under
consideration for their analysis. However, we have checked that one can get similar existence results
(using their own proof-technique) as theirs even when the set of outcomes under consideration is
expanded to the set of, say, efficient pay-off vectors or the set of all feasible pay-off vectors.
2
NTU games
A finite-player coalitional game in characteristic function form (without sidepayments) or in short, a non-transferable utility or NTU-game ( N ,V ) consists of a
finite player set N (of cardinality, say, n ) and a function V (⋅) that assigns to every
nonempty subset S of N , called a coalition, a set V (S ) ⊆ R . It is assumed that
V (∅ ) = ∅ .
For any coalition S, any x ∈ V (S ) specifies the payoff for every i ∈ S in the pay-off
vector x. We assume that for each coalition S, V (S ) is a nonempty, closed and proper
S
subset of R S . For each coalition S , V (S ) is also comprehensive, i.e., if x ∈V (S )
and y ≤ x , then y ∈V (S ) . The relation y ≤ x means that either yi < xi or yi = xi
T
holds for every i ∈ S . Given T ⊆ N , S ⊆ T and x ∈ R , denote the S coordinates
of x by x S .
Let bi := max{yi y ∈ V ({i})} < ∞ . For a set A , by bdA we denote the boundary of A
and by int A we denote the interior of A .
A vector x ∈ R N is efficient if x ∈ bd (V (N ) ) .
An NTU game is essential if ∃ y ∈V (N ) such that y i ≥ bi ∀i ∈ N and y i > bi for at
least one i ∈ N . It is inessential otherwise. We work with the class of essential games.
For any any S ⊆ N and any x, y ∈V ( N ) y p S x means that yi < xi for every i ∈ S .
Let IR ( N , V ) be the set of individually rational pay-off vectors, that is,
IR ( N , V ) = { x ∈ V ( N ) xi ≥ bi ∀i ∈ N } .
A pay-off vector is an imputation if it is individually rational as well as efficient. The
set of imputations is denoted by I(N,V).
The set of imputations is the set of outcomes or allocations that we shall study.
Farsighted dominance
A coalition S can enforce an imputation x from an imputation z, denoted
as, z →S x , if and only if x S ∈ V (S ) .
Remark 1: This notion of enforcement is common in literature for analysing such
environments. (This notion is implicit in the definition of stable sets with respect to
the classical myopic dominance relation as well as the farsighted dominance relation
(see Harsanyi (1974), Chwe (1994), Beal et al. (2008) for example). However, we do
not consider this to be very satisfactory because this implies that a coalition can
enforce pay-offs even for the members outside itself. However, we are not aware of
any simple and immediate way of resolving this while remaining within this
environment of characteristic functions. So, we stick to this. However, of course, the
comprehensive body of literature on non-cooperative foundation of coalition
formation and coalitional behaviour addresses this issue.
The notion of indirect or farsighted dominance we use is given below.
3
Definition 1 An imputation z is indirectly dominated by another imputation x ,
denoted as x >> z , if there exist a finite collection of imputations
z = z 0 , z 1 , z 2 ,..., z m = x and a finite collection of coalitions S 0 , S1 , S 2 ,..., S m −1 such
that z i → Si z i +1 and z i p Si x for i = 0,1,2,..., m − 1 .
Recall that this dominance notion captures farsightedness on the part of the players,
because, when initiating a chain of enforcements, each coalition active at any outcome
of the chain compares the pay-offs at the final outcome of the chain to the immediate
pay-offs and not the very next outcome in the chain.
Stable sets with respect to farsighted dominance
Definition 2 Let ( N , V ) be an NTU-game. A subset K of I ( N , V ) is a farsighted
stable set if (a) for all x, z ∈ K neither x >> z nor z >> x (internal stability), and (b)
for all z ∈ I ( N , V ) \ K there exists x ∈ K such that x >> z (external stability).
An assumption
Throughout the following analysis we assume the following.
Assumption 1: (Boundedness of the set of individually rational pay-off vectors) The
set IR(N,V) is bounded.
This assumption, of course, is quite common in the literature; some notable examples
being Scarf (1967) or Peleg (1992). Representation of any reasonable economy by
characteristic function form should also satisfy this.
3 Farsighted stable sets for NTU games
Recall that we restrict our analysis to the set of imputations.
Below we show that for every such NTU game, a farsighted stable set exists. To show
this, first we prove a proposition which is a generalization of Theorem 1 in Beal et al.
(2008). Next we prove the main theorem for which this proposition is useful.
Proposition 1: Suppose x, z ∈ I ( N ,V ) . Then x >> z if and only If
(i) ∃i( x, z) ∈ N such that xi ( x , z ) > zi ( x, z )
(ii) and ∃S ⊆ N such that 1 < S < N with x S ∈ V (S ) and xi > bi for all i ∈ S .
The proof of this proposition uses the following lemmata. This proof uses the idea
behind the proof of Theorem 1 in Beal et al. (2008).
Lemma 1: Let x, z ∈ I ( N ,V ) be such that x >> z . Then ∃S ⊆ N such that
1 < S < N with x s ∈ V (S ) . And for every i ∈ S the relation xi > bi holds.
4
Proof: The relation x s ∈ V (S ) must hold for a coalition S to be able to enforce an
allocation x .
Suppose N is the only coalition that fulfills x s ∈ V (S ) . Then N acts in the last step
of the chain z → S0 x1 ...x k −1 → Sk −1 x . Then x k −1 p N x . As x ∈ I ( N , V ) , x k −1 would
have to be in the interior of V ( N ) . But this contradicts our assumption that x k −1 is an
imputation. So, S k −1 < N .
Suppose a singleton coalition {i} is the only coalition that fulfils x s ∈ V (S ) . Then
{i} acts in the last step of the chain z → S0 x1 ...x k −1 → Sk −1 x . Then x k −1 p i x . As x k −1
is an imputation, i.e. x kj −1 ≥ b j ∀j ∈ N , and as p →{i} q only if pi ≤ bi , we have a
contradiction.
To prove the last part of the lemma, consider again that xik −1 < xi ∀i ∈ S k −1 holds as
we have x k −1 p Sk −1 x . As bi is the lower bound of a player’s payoff in any imputation,
bi < xi ∀i ∈ S k −1 holds.
Lemma 2: For every i ∈ N , ∃xi ≥ bi such that the vector
(b1 , b2 ,...,bi −1 , xi , bi +1 ,...,bN ) , i.e., ( xi , (b j ) j∈N \i ) (also denoted by ( xi , b−i )) is in
bdV ( N ) .
Proof: By essentialness and comprehensiveness, b = (b1 ,...,bN ) ∈V ( N ) . By
Assumption 1, ∃y large enough, such that ( y, b−i ) ∉ V ( N ) . Therefore, since V (N ) is
closed, ∃xi such that ( xi , b−i ) ∈ bdV ( N ) .
Proof of Proposition 1:
Necessity: Part (i) Suppose not. Then z ≥ x and obviously, x > / > z .
Part (ii) The necessity of the part (ii) in the statement of Proposition 1 has been
established in Lemma 1.
Sufficiency: Of course, below we show the following. Suppose x, z ∈ I ( N ,V ) are
such that ∃i( x, z) ∈ N for which xi ( x , z ) > zi ( x, z ) . Moreover, ∃S ⊆ N such that
1 < S < N with x S ∈ V (S ) and xi > bi for all i ∈ S . Then x >> z .
We consider two cases: Case 1: N \ S = {i( x, z )} ; Case 2: N \ S ≠ {i( x, z )} .
Case 1: Consider the following chain of moves from z to x . First z →{i ( x , z )} y1 with
yi1 1 = xi 1 for some iy1 ∈ S (recall the definition of xi ( x, z ) Lemma 2) and yi1 = bi for
y
y
2
2
every i ≠ i y1 . Then y 1 →{ j} y 2 with j ∈ S \ {i y1 } , yi ( x , z ) = xi ( x , z ) and y k = bk for all
k ∈ S . Finally, y 2 → S x . This enforcement chain is an indirect dominance as
x > i ( x , z ) z , x > j y 1 and x > S y 2 .
5
Case 2: Consider the following chain of moves from z to x . First z →{i ( x , z )} y1 with
yi1 = bi for every i ≠ iy1 for some iy1 ∉ S and yi1 1 = xi 1 . Then y 1 → S x . The
y
y
enforcement chain is an indirect dominance as x > i ( x , z ) z and x > S y 1 .
The following corollary is immediate from the proof of Proposition 1.
Corollary 1: Suppose x ∈ I ( N ,V ) be such that for some y ∈ I ( N ,V ) x >> y and
z ∈ I ( N ,V ) is such that for some i( x, z) ∈ N , xi ( x, z ) > zi ( x, z ) . Then x >> z .
Now we prove the main theorem.
Theorem 1: For any game (N,V) satisfying Assumption 1, a farsighted stable set
exists.
Proof: Suppose I(N,V) be such that it has no two elements x and y for which x >> y .
Then, I(N,V) itself is a farsighted stable set.
Suppose otherwise, i.e., suppose that there exists an imputation x such that x >> y for
some y ∈ I ( N ,V ) . Let G( x) = { y ∈ I ( N , V ) y ≥ x} \ {x}
and D( x) = { y ∈ G ( x) y >> x} . We consider three cases.
Case 1: Suppose G( x) = φ . Then, by Corollary 1 the singleton set {x} is a farsighted
stable set. (Note that the class of TU games falls into this case.)
Case 2: Suppose G( x) ≠ φ , but D(x) = φ .Then, we show below that {x} ∪ G( x) is a
farsighted stable set.
External stability: For no z ∈ I ( N ,V ) \ G( x) , is it the case that z ≥ x . Therefore,
∀z ∈ I ( N ,V ) \ G( x) there is some i( x, z) ∈ N such that xi ( x, z ) > zi ( x, z ) . Then, by
Proposition 1, external stability of {x} ∪ G( x) is given.
1
2
Internal stability: To check that G(x) is internally stable take y , y ∈ G( x) and
2
1
suppose y >> y . By the definition of indirect dominance there must exist a chain
y 1 → S0 z 1 ...z k −1 → Sk −1 y 2 and y 2j > y1j for every j ∈ S 0 . Since y1 ∈ G( x) , for every
j ∈ S 0 , y 2j > x j . But then y 2 >> x which is a contradiction as D(x) = φ .
Case 3: Suppose D(x) ≠ φ .
Pick some z ∈ D(x) as follows (with possible relabelling of the players) such that
z1 ≥ y1∀y ∈ D( x)
z2 ≥ y2∀y ∈{w ∈ D( x) w1 = z1}
z3 ≥ y3∀y ∈{w ∈ D( x) w1 = z1; w2 = z2 }
and in general, for any k less than or equal to n,
6
zk ≥ yk ∀y ∈{w ∈ D( x) wl = zl ∀l < k} .
Note that if such a z exists, then by construction, z ∈ I ( N , V ) and z >> x .
Claim 1: Such a z exists.
Proof of Claim 1: Since x >> y , for some y ∈ I ( N ,V ) , by Corollary 1 ∃S ⊆ N with
1 < S < N such that x S ∈ V (S ) and xi > bi ∀i ∈ S . Without loss of generality,
label players such that S = {1,..., s} and so, N \ S = {s + 1,..., n} .
Step 1: Pick n . Consider mn := sup{y n y ∈ D( x)} . Below we show that ∃z ∈ D( x)
such that zn = mn . Suppose not. Then
mn ≠ xn and also D(x) is not finite (if
mn = xn or if mn > xn but D(x) is finite, then the contradiction is immediate). Then
∃y such that y is the limit point of a sequence of elements from D(x) such
that yn = mn . (The reader should not confuse this with the notation yi used in Lemma
2 as the contexts are quite different.) Since V (N ) is closed and y is the limit point of
a sequence of imputations, y ∈ I ( N ,V ) . Now yn > xn and yi ≥ xi ∀i ∈ N . Then
take z ∈ R N such that z S = x and z
S
N \S
=y
N \S
. This z must be an imputation (as
z ≥ x , z ≤ y ). Now, by construction, yn = z n > xn and z S = x ∈ V (S ) .
S
Therefore, by Proposition 1, z >> x . Therefore, ∃z ∈ D( x) such that z n ≥ yn
∀y ∈ D(x) , contrary to the supposition.
n −1
Step 2: Pick n − 1 . Consider D ( x) = {w ∈ D( x) wn = mn } . Let
mn −1 := sup{ yn −1 y ∈ D n−1 ( x)} . Again, below we show that ∃z ∈ Dn−1 ( x) such that
zn−1 = mn−1 . Again, suppose this is not true. Then mn−1 ≠ xn−1 and also D n −1 ( x) is
not finite (if mn−1 = xn−1 or if mn−1 > xn−1 but D ( x) is finite, then the
contradiction is immediate). Then ∃y such that y is the limit point of a sequence of
n −1
elements from
D n−1 ( x) such that yn−1 = mn−1 . Since V (N ) is closed and yn−1 is the
limit point of a sequence of imputations, yn−1 ∈ I ( N ,V ) . Now yn ≥ xn , yn−1 > xn−1
and yi ≥ xi ∀i ∈ N . Then take z ∈ I ( N ,V ) such that z S = x and z
S
N \S
=y
N \S
.
This z must be an imputation (as z ≥ x , z ≤ y ). Now, y n−1 = z n−1 > xn−1 and
z S = x ∈ V (S ) . Therefore, again, by Proposition 1, z >> x . Therefore,
S
∃z ∈ D n−1 ( x) such that z n−1 ≥ yn−1 ∀y ∈ D n−1 ( x) .
s
Proceeding in this way recursively, construct D ( x) .
s
Step 3: Let ms := sup{ y s y ∈ D s ( x)} . Once again, below we show that ∃z ∈ D ( x)
such that zs = ms . Again, suppose this is not true. Then, again, ms ≠ xs and D ( x)
is not finite (again, note that if either of these cases are true, the contradiction is
t
immediate). Then ∃y such that y is the limit point of a sequence of elements {r }
s
s
from D (x) such that y s = ms . Moreover, since N is finite (and as, consequently,
7
there exist only finitely many coalitions), ∃T ⊆ N such that 1 < T < N , and a
q
t
q
subsequence of imputations { y } of {r } from such that y → y and for every q ,
y q ∈ V (T ) and yiq > bi for every i ∈ T (by Proposition 1). Since V (T ) is
T
closed, y T ∈ V (T ) . Now we claim that for each i ∈ T , yi > bi . If i ∈ S ∩ T , then it is
yi ≥ xi > bi . Take i ∈ ( N \ S ) ∩ T . Since, by construction, for every
y of the subsequence, yiq = mi > bi for each i ∈ N \ S , yi = mi as well. Then
true, since
q
ys > xs and y T ∈ V (T ) with yi > bi for every i ∈ T . So, by Proposition 1, y >> x .
s
Therefore, ∃z ∈ D ( x) such that zs = ms .
Step 4: Proceeding recursively as in Step 3 for the remaining players in S , we find
that the z with the desired property exists. Thus, Claim 1 is proved.
Next we show that K = {z} ∪ { y ∈ G( x) \ D( x) y ≥ z} is a farsighted stable set.
Internal stability: Of course, by the definition of the set K, z > / > y since y ≥ z
1
2
holds for every y ∈ K . Now suppose, if possible, y , y ∈ { y ∈ G( x) \ D( x) y ≥ z}
2
1
such that y >> y . By the definition of indirect dominance there must exist a chain
y 1 → S0 z 1 ...z k −1 → Sk −1 y 2 and y 1j < y 2j for every j ∈ S 0 . Since
y 1 ∈ { y ∈ G( x) \ D( x) y ≥ z} , y1 ≥ x . Then y 1j ≥ x j for every j ∈ S 0 and y 2j > x j
2
for every j ∈ S 0 . But then y >> x which is a contradiction as
y 2 ∈ { y ∈ G( x) \ D( x) y ≥ z} and therefore y 2 > / > y1 .
External stability: Subcase 1: Suppose y ∈ I ( N ,V ) \ G( x) . Then there exists some
i( x, y) ∈ N such that zi ( x , y ) ≥ xi ( x , y ) > yi ( x , y ) . Then, z >> y by Corollary 1.
Subcase 2: Suppose y ∈ D(x) . Then by the construction of z there exists some
i( z, y) ∈ N such that zi ( z , y ) > yi ( z , y ) . Then, by Corollary 1, again we are done.
Subcase 3: Suppose y ∈ {G( x) \ D( x)} but it is not the case that y ≥ z . Then there
exists some i( z, y) ∈ N such that z i ( z , y ) > yi ( z , y ) . Then, by Corollary 1 we are again
done.
Remark 2: The assumption that the set of individually rational pay-offs is bounded is
not a necessary condition for the existence of a farsighted stable set. We illustrate this
in the following (slightly non-trivial) example.
Let the player set be N = {1,2,3} and define the set
K := {x ∈ R 3 x1 + x2 + x3 = 1} ∪ {x ∈ R 3 x1 = x2 = 0, x2 ≥ 1} .
V ({1,2,3}) = {x ∈ R 3 ∃y ∈ K such that x ≤ y} and b1 = b2 = b3 = 0.
V ({i, j}) = {x ∈ R{i , j} xi + x j ≤ 0.2} ∀i, j ∈ N . Note that I ( N ,V ) = K .
Obviously, this game violates Assumption 1.
It is easy to check that x = (0.1,0.1,0.8) forms a singleton farsighted stable set of this
game.
8
An even more trivial example illustrating this point is as follows.
Let the player set be N = {1,2} and the set of efficient vectors
K := {x ∈ R 2 xi > 0 ∀i and x1 x2 = 1} . V ({1,2}) = {x ∈ R 2 ∃y ∈ K such that x ≤ y}
and b1 = b2 = 0 . Note that I ( N ,V ) = K .
Obviously, this game violates Assumption 1. However, it is easy to see that by
Proposition 1, for this game no imputation can indirectly dominate another
imputation. Thus, the whole K is a farsighted stable set for this game.
Thus, the existence result in this paper can be more generalized. However, we
consider Assumption 1 to be quite innocuous and as we remarked earlier, almost all
reasonable economic situations should satisfy this assumption.
4 Concluding Remarks
Since a farsighted stable set is contained in the largest consistent set (Chwe,1994), the
result above also shows that the largest consistent set is non-empty for games in
characteristic function form (obeying the condition specified in this paper) which is an
interesting fact by itself, given the importance of this solution concept in analysing
farsighted behaviour.
In this work we have only provided an existence theorem; we have not attempted to
study in any detail the structure of the farsighted stable sets. Some facts, though, are
immediate. For example, for TU games, a farsighted stable set is either the whole set
of imputations or it is singleton (Beal et al., 2008). However, with general NTU
games, as Theorem 1 shows, this is no more necessarily true. But a proper exploration
of the structural properties of the farsighted stable sets in this environment still
remains to be done.
9
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