Qin [J. Eco. Th., 1996] recently showed that in a game of endogenous formation of cooperation structure, if the underlying TU-game is superadditive, then the full cooperation structure is stable. In this note, we characterize the class of games that ensure the stability of the full cooperation structure, and show that this class is much larger than that of superadditive TU-games.Cooperation structure, graph, Myerson value, stability

Guillaume Haeringer

English

Research Papers in Economics

On the Stability of Cooperation Structures
Guillaume Haeringery
Departament d'Economia i d'Hist oria Econ omica and CODE
Universitat Aut onoma de Barcelona, Spain
November 22, 2000
Abstract
Qin (1996) recently showed that in a game of endogenous formation
of cooperation structure, if the underlying TU-game is superadditive,
then the full cooperation structure is stable. In this note, we character-
ize the class of games that ensure the stability of the full cooperation
structure, and show that this class is much larger than that of super-
additive TU-games.
Keywords Cooperation structure, graph, Myerson value, stability.
Journal of Economic Literature Classication: C71.
I wish to thank Bel en Jerez and Jordi Mass o for incisive and helpful comments. Fi-
nancial support from the European Commission, contract number HPMF-CT-1999-00284,
is gratefully acknowledged.
yTel.: (+34) 93 581 25 05, Fax: (+34) 93 581 24 61, Departament d'Economia
i d'Hist oria Econ omica and CODE, 08193 Bellaterra (Barcelona), Spain. E-mail:
ghaeringer@volcano.uab.es.
11 Introduction
In a seminal paper, Aumann and Myerson (1988) proposed a model of en-
dogenous cooperation structure formation. Cooperation structures represent
situations in which players have the possibility to cooperate, or communi-
cate, only with a subset of the population. An interesting feature of these
situations is that two players who are in cooperation may not be in coop-
eration with the same subset of players. Hence, these structures may be
represented by non-directed graphs. Aumann and Myerson proposed to use,
in an extensive-form game, the Myerson value (Myerson, 1977) to compute
players' payos in cooperation structures. A major drawback of their ap-
proach is that the order of play has to bet set exogenously. Recently, Qin
(1996) reformulated Aumann and Myerson's game as a normal-form game.
One of Qin's results is that if the underlying TU-game is superadditive,
then the full cooperation structure (each player cooperates with all the oth-
er players) is stable in the sense that no player can protably deviate by
severing some of her links.
In this note we characterize the class of TU-games that admit a stable
full cooperation structure, and show that it is much larger than the class of
superadditive games. Indeed, we show that the full cooperation structure
is stable if and only if the value of each player is individually rational and
the TU-game is superadditive for two-player coalitions only. In particular,
this implies that if, for some TU-game (N;v) and two players i and j, we
have v(fi;jg) < v(fig) + v(fjg), then the full cooperation structure is not
stable.
The paper is organized as follows. In section 2 we present the framework.
In section 3 we dene the game of endogenous formation of cooperation
structure, and state and prove our characterization result.
22 Denitions
A transferable utility game (or a TU-game) is dened by a couple (N;v)
where N = f1;:::;ng is the set of players, and v is the characteristic function
that maps the power set of N to the set of real numbers (with the convention
that v(?)  0). The number v(S) yields the worth of coalition S 2 2N. The
space of all TU-games with the set of players N is denoted by  N.
We assume that players are able to establish meaningful relationships
with a subset of players, and we call the set of all private relationships a
cooperation structure. A useful way to represent cooperation structures
is by a non-directed graph, whose vertices represent players and whose edges
represent the relationships that players have between each other.
We dene a graph, g, as a set of links between players. We write i:j 2 g
when i and j are linked in g. Two players i and j are indirectly linked in g if
there exists a sequence i1;::: ;ik in N, with k  3, such that i = i1, it 1:it
for all t = 2;::: ;k, and ik = j. The sequence i1;::: ;ik is called a path.
Two players are said to be connected if they are (indirectly) linked. A
coalition S is said to be connected if any pair of players i;j 2 S is connected
by a path involving only players in S. We denote by GRN the set of all
graphs and by gN the complete graph, i.e. the graph in which each player
is linked with all other players,
gN = fi:j j i 2 N ; j 2 N ; i 6= jg; (1)
GRN = fg j g  gNg: (2)
For any graph g, gnfi:jg denotes the graph in which the link i:j has been
deleted, and g [fi:jg is the graph to which the link i:j has been added. We
denote the graph g restricted to the set of vertices S by g(S),
g(S) = fi:j j i;j 2 S; i:j 2 gg:
For any subset of players S, there is a unique partition of players that
groups together players connected by g(S). Such a partition is denoted
3by S=g. In other words, for any i 2 S, the element of the partition S=g
containing i also contains all other players that are connected to i by g(S).
A formal denition of this partition is dened as follows,
S=g = ffi 2 S j i and j are connected in S by gg j j 2 Sg:
3 The game
Qin's (1996) model works as follows. Given a TU-game (N;v), we allow
players to choose who they want to cooperate, or communicate with. Hence,
each player can have private relationships only with a subset of the grand
coalition. More formally, we dene a non-cooperative game
G (N;v)
 =< N;(i)i2N;(hi)i2N) >;
where the TU-game (N;v) is called the underlying game of G
(N;v)
 . As in
(N;v), the set of players is N and i is the strategy space of player i. A
strategy i for player i is dened as a n-dimensional vector whose coordinates
are either 0 or 1. When i(j) = 0 player i does not wish to form a link with
player j, while he would be glad to create such a link when i(j) = 1. The
ith coordinate of i is assumed to be equal to zero, i(i)  0. In other
words, no player can make a link with herself. A link between players i and
j is formed only if both i and j wish to create it: i(j) = j(i) = 1. We
denote the n-tuple of strategies by  = (1;::: ;n). Each  2 (i)i2N gives
a unique cooperation structure g(),
g() = fi:j j i(j) = j(i) = 1g:
For each cooperation structure g, players are rewarded using the Myerson
value of the game (N;v) on g, denoted by the n-dimensional vector (v;g).
The Myerson value is an allocation rule for TU-games with graphs. To
compute the Myerson value for some game (N;v) and some cooperation
4structure g, we rst need to dene a characteristic function embedded on
graphs,
8 S  N; (v=g)(S) =
X
T2S=g
v(T): (3)
The Myerson value is the unique allocation  rule that satises the Com-
ponent Eciency axiom,
X
i2N
i(v;g) =
X
S2N=g
v(N);
and the Fairness axiom,
i(v;g)   i(v;gni:j) = j(v;g)   j(v;gni:j); 8 g 2 GRN; 8 i:j 2 g:
Myerson proved that his value is closely related to the Shapley value.
Indeed, we have
(v;g) = '(v=g); 8 (N;v) 2  N; 8 g 2 GRN;
where ' is the Shapley value. It is straightforward to see that for the full
cooperation structure, v=g = v, and hence (v;gN) = '(v). Clearly, if some
player i is not connected to anyone in the cooperation structure g, then
i(v;g) = v(fig). Note that it may happen that for some game (N;v) and
some g, we have i(v;g) < v(fig).
Thus, the utility of a player in the game G
(N;v)
 under the strategy prole
 is her Myerson value of the corresponding cooperation structure,
hi()  i(v;g()): (4)
Consider now a cooperation structure g such that players i and j are not
linked, i.e., i:j = 2 g. Then, player i desires to link up with player j if and
only if
i(v;g [ fi:jg)  i(v;g): (5)
5Given that the Myerson value satises the Fairness axiom, it is readily ver-
ied that if (5) holds true for player i, then it also holds true for player j.1
A cooperation structure g is stable in Qin's sense if there is a strategy
proles  such that g() = g and  is a Nash equilibria of the game G
(N;v)
 .
This result was also obtained in a similar framework by Dutta, van den
Nouweland, and Tijs (1998).2 We show now that this result can be extended
to a larger class of games.
Proposition 1 Let (N;v) be a TU-game and  the Myerson value. Then
the full cooperation structure is stable if and only if i(v;gN)  v(fig) for
all i 2 N, and
v(fi;jg)  v(fig) + v(fjg); for all i;j 2 N: (6)
Proposition 1 says that if the Shapley value of each player in the game
(N;v) is individually rational then gN is stable if and only if the game (N;v)
is \superadditive" for two-player coalitions. Hence, the two conditions |
individual rationality and (6) | fully characterize the class of TU-games
that ensure the stability of the full cooperation structure.
As the following example shows, superadditivity of the game (N;v) is
not necessary for the stability of the full cooperation structure.
Example 1 Consider the following TU game (N;v) with N = fa;b;c;dg,
1Reciprocity may not hold if player i wants to link up with several players at a time.
2Dutta, van den Nouweland, and Tijs (1998) employed several equilibrium concepts,
(undominated Nash equilibrium, coalition-proof Nash equilibrium and Strong Nash equi-
librium) and did not use a specic solution concept to analyse to equilibria of the game.
6and v dened by
v(S) =
8
> > > > > > > > > <
> > > > > > > > > :
0 if jSj = 1;
20 if jSj = 2;
12 if jSj = 3;
16 if jSj = 4:
The Myerson value of each player for the full cooperation structure is
i(v;gN) = 4, for i = a;b;c;d, which is individually rational. Obviously,
condition (6) is satised. Hence, Proposition 1 implies that the full coopera-
tion structure is stable. Yet, it is obvious that the full cooperation structure
is Pareto dominated, i.e. there is another cooperation structure (for instance
g0 = fa:b; c:dg) that yields a strictly higher value to all players.3
In order to prove the proposition, we introduce more denitions.
For any cooperation structure g and players i and j such that i:j 2 g,
dene the set
S(g;i:j) = fS  N j S=g = S and S=(gnfi:jg) 6= Sg:
In other words, S(g;i:j) is the set of coalitions S such that S is connected in
g(S) but is non-connected in g(S)nfi:jg. When no confusion is possible, we
write S(i:j) instead of S(g;i:j). In other words, S(i:j) represents the set of
coalitions S that are connected thanks to i:j when the graph is restricted to
S.4 Thus, for any S 2 S(i:j), if i:j is deleted, S=(gnfi:jg = fSi;Sjg), with
Si and Sj being the sets in S=(gnfi:jg) that contain i and j respectively.
Proof of Proposition 1 Observe that there is a unique strategy prole
 such that g() = gN. Hence,  is a Nash equilibrium if and only if
3A similar result was obtained by Jackson and Wolinsky (1996), who showed that
some games G
(N;v)
 may have a stable cooperation structures that are not ecient, i.e.
structures that do not maximize the worth of coalition(s).
4In graph terminology, we say that S(g;i:j) is the set of all subgraphs ^ g of g such that
the link i:j is critical in ^ g | see Bollob as (1998).
7no player breaks one or more links. Hence, it suces to compare, for any
player i, her payos under the strategy prole  and under the strategy
prole (0
i;
 i), where 0
i(j) = 0 for at least one j 2 Nnfig.
Clearly, if the value of some player is not individually rational, then that
player has an incentive to break up all her links (or any part thereof), and
then the full cooperation structure is not stable. In other words, individual
rationality is a necessary condition for the full cooperation structure to be
stable. We now show that if the value of each player is individual rational,
then stability is equivalent to condition (6).
We claim that for any graph g, if a player contemplates breaking a link,
yielding g0, the change of her payo (i.e., her Myerson value) will only depend
on the worth of the coalitions that are connected in g and not connected in
g0. To see this, consider the value of player i for a cooperation structure g
and gnfi:jg.
i(v;g) =
X
S3i
S ((v=g)(S)   (v=g)(Sni))
i(v;gnfi:jg) =
X
S3i
S ((v=gnfi:jg)(S)   (v=gnfi:jg)(Sni));
where S =
(n 1)!(n s)!
n! . Notice that
(i) g=(Snfig) = (gnfi:jg)(Snfig),
(ii) (gnfi:jg)=S 6= g=S when i:j connects S in g, i.e., when S 2 S(i:j),
(iii) (gnfi:jg)=S = g=S when i:j does not connects S in g.
Using (3) we deduce
(a) (i) ) (v=g)(Snfig) = (v=gnfi:jg)(Snfig); 8S 3 i,
(b) (ii) ) (v=gnfi:jg)(S) 6= (v=g)(S) when S 2 S(i:j),
(c) (iii) ) (v=gnfi:jg)(S) = (v=g)(S) when S = 2 S(i:j),
8Thus,
i(g)   i(gnfi:jg)  0
,
X
S2S(i:j)
S(v(S)   v(Si)   v(Sj))  0;
(7)
which proves the claim. Observe that if g = gN, then S(i:j) = ffi;jgg. In
other words, in the full cooperation structure, if a player i breaks a link with
player j, the only coalition that becomes non-connected is fi;jg.
More generally, suppose that some player i severs her links in gN with
players j 2 S, where S is some subset of Nnfig. Then we have fS(i:j)gj2N =
ffi;jgj2Sg. This implies that i severs her links with players j 2 S  Nnfig
if, and only if
X
j2S
fijg(v(fijg)   v(fig)   v(fjg)) < 0: (8)
As (8) holds for any S  Nnfig, for any i 2 N, the result follows. 
Qin (1996) showed that if the underlying TU-game is superadditive,
then for any partition P of N, there is a Nash equilibrium  game such
that N=g() = P | see Qin (1996, remark 2). We now show that this
result can be extended.5
Proposition 2 Let (N;v) be a superadditive TU-game. For any g  gN,
there exist a strategy prole  such that g = () = g and  is a Nash
equilibrium of G
(N;v)
 .
Proof Consider any superadditive game (N;v) and any cooperation struc-
ture g. Hence, (7) always holds and we can deduce that no player wishes to
break a link. Consider the following strategy prole. For each player i 2 N,
let i(j) = 0 if i:j = 2 g and 
i (j) = 1 if i:j 2 g. Because players do not sever
links, if a player i deviates with the strategy ^ , it should be the case that
5See Dutta, van den Nouweland, and Tijs (1998, proposition 1) for a similar result.
9for some j we have ^ i(j) = 1 and 
i (j) = 0. But deviations are individuals,
which implies that 
j(i) = 0. Hence, g(^ i;
 i) = g, and player's i payo
remains unchanged. Thus, i has no incentives to deviates. 
10References
Aumann, R. J., and R. B. Myerson (1988): \Endogenous Formation of
Links between Players and of Coalitions: an Application of the Shapley
Value," in The Shapley value, ed. by A. Roth, pp. 175{191. Cambridge
Univ. Press, Cambridge.
Bollob as, B. (1998): Modern graph theory. Springer-Verlag, New York.
Dutta, B., A. van den Nouweland, and S. Tijs (1998): \Link Forma-
tion in Cooperative Situations," International Journal of Game Theory,
27, 245{256.
Jackson, M. O., and A. Wolinsky (1996): \A Strategic Model of Social
and Economic Networks," Journal of Economic Theory, 71, 44{74.
Myerson, R. B. (1977): \Graphs and Cooperation in Games," Mathemat-
ics of Operation Research, 2, 225{229.
Qin, C.-Z. (1996): \Endogeneous Formation of Cooperation Structures,"
Journal of Economic Theory, 69, 218{226.
11

On the Stability of Cooperation Structures

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