www.elsevier.nl / locate / econbase
Potential maximizers and network formation
a ,* b c a
Marco Slikker , Bhaskar Dutta , Anne van den Nouweland , Stef Tijs
a
Department of Econometrics and CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
b
Indian Statistical Institute, 7SJS Sansanwal Marg, New Delhi 110016, India c
Department of Economics, 435 PLC, 1285 University of Oregon, Eugene, OR 97403-1285, USA
Received 16 December 1997; accepted 21 October 1998
Abstract
In this paper we study the formation of cooperation structures in superadditive cooperative TU-games. Cooperation structures are represented by hypergraphs. The formation process is modelled as a game in strategic form, where the payoffs are determined according to a weighted (extended) Myerson value. This class of solution concepts is the unique class resulting in weighted potential games. The argmax set of the weighted potential predicts the formation of the complete structure and structures payoff-equivalent to the complete structure. As by-products we obtain a representation theorem of weighted potential games in terms of weighted Shapley values and a characterization of the weighted (extended) Myerson values. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Network formation; Potential games; Potential maximizers; TU-games; Shapley values
1. Introduction
Several recent papers have modelled the distribution of payoffs in a cooperative game as a two-stage procedure. In the first stage, players decide on the extent and nature of cooperation with other players. During this period, players cannot enter into binding agreements of any kind, either on the nature of cooperation or on the subsequent division of payoffs. In the second period, the payoffs are given by an exogenously given allocation rule.
An innovative paper in this area was Hart and Kurz (1983). A situation is analyzed where the first stage consists of a game where strategies of the players are
announce-*Corresponding author. Tel.:131-13-4662340; fax:131-13-4663280. E-mail address: M.Slikker@kub.nl (M. Slikker)
ments of players with whom a particular player wants to form a coalition. In Monderer and Shapley (1996) a simpler model is studied, which is called participation model. In this model the players decide in the first stage whether or not to participate. In the second stage the players who chose not to participate receive some stand-alone value. The participating players form a coalition and receive payoffs determined by an efficient allocation rule applied to the cooperative game restricted to the participating players. In Monderer and Shapley (1996) it is shown that the Shapley value is the unique efficient allocation rule for which this participation game is a potential game.
1
Potential games were introduced in Monderer and Shapley (1996). A potential game is a game where the information that is sufficient to determine Nash equilibria can be incorporated in a single function on the strategy space, the potential function. In Monderer and Shapley (1996) the set of strategy profiles that maximize this potential is studied and it is pointed out that the potential maximizer can be used as an equilibrium refinement. In the current paper we will restrict ourselves to potential games.
In Qin (1996) the formation of cooperation structures rather than coalition structures is studied. A cooperation structure is a graph whose vertices are identified with the players. A link between two players means that two players can carry on meaningful and direct negotiations with each other. In Qin (1996) a game in strategic form is used as suggested in Myerson (1991). In the first stage of this game each player announces a set of players with whom he or she wants to form a link. Then a link is formed between players i and j if and only if both i and j want to form a link with each other. The resulting (undirected) graph divides the set of players into components. In Qin (1996) it is shown that there is a unique component efficient allocation rule that results in a potential game. This rule is the Myerson value, which coincides with the Shapley value of the graph-restricted game (cf. Myerson, 1977). Subsequently, it is shown in Qin (1996) that if the underlying cooperative game is superadditive then the full cooperation structure results from a potential maximizing strategy and, moreover, every potential maximizing strategy results in a cooperation structure that is payoff-equivalent to the full
2 cooperation structure.
The results in Monderer and Shapley (1996) and Qin (1996) suggest a relation between potential games and the Shapley value. This relation is studied in Ui (1996) and it is shown that every potential game can be seen as a two-stage model with some specific features. In the first stage every player chooses a strategy. In the second stage the players play a cooperative game. The specific game played in the second stage depends on the strategies chosen in the first stage. It is assumed that the players receive the Shapley value of the cooperative game they play in the second stage. In Ui (1996) it is shown that if the value of a coalition in the second stage cooperative games does not depend on the strategies that the players outside this coalition choose in the first stage then the two-stage game is a potential game. Moreover, it is shown that every potential game can be represented by a two-stage game in the way described above.
The result in Ui (1996) is applied in Slikker (1999), where coalition formation games
1
See also Rosenthal (1982). 2
that are potential games are studied. In the model in Slikker (1999) every player announces a coalition he wants to form. Players prefering the same coalition will end up in a coalition together. Then an exogenous allocation rule is used to obtain the payoffs to the players. It is shown that the unique component efficient allocation rule that results in a potential game is the value of Aumann and Dreze (cf. Aufmann and Dreze, 1974). Furthermore, it is shown in Slikker (1999) that if the underlying game is superadditive then according to the potential maximizing strategies the grand coalition will be formed or a coalition structure that is payoff-equivalent to the grand coalition.
In the current paper we consider the problem of formation of conference structures, which are mathematically represented by hypergraphs. In a hypergraph the vertices are identified with the players and each edge represents a conference, which is a subset of players. Direct negotiations between players can only take place within conferences. In the words of Hwang and Lin (see Hwang and Lin, 1987): ‘A conference is defined as a meeting of two or more individuals to pool their ideas, experiences, and knowledge for solving problems.’ The use of hypergraphs to model cooperation possibilities between players was proposed in Myerson (1980). Note that since a graph can be viewed as a special hypergraph in which each conference is a pair of players, the model in the present paper is a generalization of the model in Myerson (1991). In van den Nouweland et al. (1992) hypergraph communication situations are illustrated by means of an example dealing with football teams. Football teams have committees and some members of those committees are also in the board of the national football associations of their country, which in turn have representatives in international football associations. Basic decisions can be made within each of those three types of organizations and interaction between organizations is possible if they have at least one member in common. In van den Nouweland et al. (1992) allocation rules for hypergraph communication situations that are based on the Shapley value are studied.
Our primary focus of interest is to see which hypergraphs turn out to result from
potential maximizing strategies. We allow for asymmetry between the players and hence
we study weighted Shapley values and weighted potential games. We show that the complete hypergraph results from a potential maximizing strategy. Conversely, every potential maximizing strategy results in a cooperation structure which is payoff-equivalent to the complete hypergraph.
In the process of proving the main result we generalize the representation theorem of Ui (1996) to the setting of weighted Shapley values and weighted potential games. We also show that under an efficiency requirement, the only allocation rule which results in a conference formation game being a weighted potential game is the weighted (extended) Myerson value. Finally, we find a characterization of the class of weighted (extended) Myerson values.
The plan of this paper is as follows. In Section 2 we will show that a non-cooperative game is a weighted potential game if and only if its payoff function coincides with weighted Shapley values of particular cooperative games indexed by the set of strategy profiles. Section 3 deals with hypergraph communication situations and provides an axiomatic characterization of the class of weighted (extended) Myerson values using
w-fairness and component efficiency. Conference formation games are discussed in
potential game are the weighted (extended) Myerson values. In Section 5 we show that the argmax set of the weighted potential corresponds to the full cooperation structure and payoff-equivalent structures. We conclude in Section 6.
2. Potential games
In Ui (1996) a representation theorem for potential games is given in terms of the Shapley value. In this section we will extend the result of Ui (1996) and provide a representation theorem for weighted potential games in terms of weighted Shapley values. We will first give some definitions.
A game in strategic form will be denoted by G 5(N; (S )i i[N; (pi i)[N), where
N5h1, . . . , nj denotes the player set, S the strategy space of player i[N, and i
p 5(p) the payoff function which assigns to every strategy-tuple s5(s ) [
i i[N i i[N
N
Pi[N Si5S a vector in R . For notational convenience we write s2i5(s )j j[N \hij and
sR5(s )i i[R.
The class of weighted potential games is formally defined in Monderer and Shapley N
(1996). Let w5(w ) [R be a vector of positive weights. A function i i[N 11
w
Q :P S →Ris called a w-potential forGif for every i[N, every s[S, and every i[N i
t [S it holds that i i
w w
pi(s , si 2i)2pi(t ,si 2i)5w (Q (s ,si i 2i)2Q (t ,si 2i)). (1) The game G is called a w-potential game if it admits a w-potential. G is called a
N weighted potential game if G is a w-potential game for some weights w[R .
11
In Monderer and Shapley (1996) it is pointed out that the argmax set of a weighted potential game does not depend on a particular choice of a weighted potential, and hence can be used as an equilibrium refinement. It is also remarked that this refinement is
3 supported by some experimental results.
The representation theorem in this section is in terms of cooperative games and weighted Shapley values. A cooperative game is an ordered pair (N, v), where
N
N5h1, . . . , njis the set of players, andvis a real-valued function on the family 2 of
all subsets of N withv(5)50. Denote the set of all cooperative games with player set N
N by TU .
Weighted Shapley values can easily be defined using unanimity games. For every 4
R7N the unanimity game (N, u ) is defined byR
1, if R7T
u (T )R 5
H
0, otherwise. (2)Unanimity games were introduced in Shapley (1953). It is shown that every cooperative
3
In Monderer and Shapley (1996) it is pointed out that this may be a mere coincidence. See also van Huyck et al. (1990) and Crawford (1991).
4
game can be written as a linear combination of unanimity games in a unique way,
v5o a u , where (a ) are called the unanimity coordinates of (N,v).
R7N R R R R7N N
Let w5(w ) [R be a vector of positive weights. For all R7N define i i[N 11
w N
w :5o w . The weighted Shapley value F of a cooperative game (N, v)[TU
R i[R i
with unanimity coordinates (aR R) 7N is then defined by
wi w
]
F (N,v)5
O
a . (3)i w R
R R7N,i[R
To represent weighted potential games in terms of weighted Shapley values we need the 5
following interaction between cooperative and non-cooperative games.
Consider a player set N5h1, . . . ,njand strategy space S5 Pi[NS . Assume that oncei the players have chosen a strategy profile s[S they face the cooperative game (N,v).
s Furthermore, assume that the players have made a pre-play agreement on the allocation rule that determines their payoffs for any chosen cooperative game. If the players have agreed on allocation ruleg this implies that player i obtainsg(N,v) if strategy profile
i s
s[S is played.
In Ui (1996) it is shown that in analyzing potential games it suffices to consider collections of cooperative games where the value of a coalition does not depend on the strategies of the players outside this coalition:v(R) only depends on s . We will show a
s R
similar result for weighted potential games. First we define the following set of collections of cooperative games (cf. Ui, 1996):
N S
& :5hh(N,v)j [(TU ) uv(R)5v(R) if s 5t for all s, t[S, R7Nj. (4)
N,S s s[S s t R R
s
Denote the unanimity coordinates of the game v by (a ) . It can be shown that the
s R R7N
condition in definition (4) can be rewritten in terms of these unanimity coordinates,
N S s t
& 5hh(N,v)j [(TU ) ua 5 a if s 5t for all s, t[S, R7Nj. (5)
N,S s s[S R R R R
We can now state the main result of this section, which states that the class of weighted potential games can be represented in terms of weighted Shapley values. In Ui 6 (1996) this theorem is shown for (unweighted) potential games and Shapley-values.
N Theorem 2.1. Let G 5(N; (S ) ; (p) ) be a game in strategic form and w[R .
i i[N i i[N 11
G is a w-potential game if and only if there exists h(N, v)j [& such that
s s[S N,S w
p(s)5 F (v), for all i[N and all s[S. (6)
i i s
Proof. First we will prove the if-part of the theorem. Assume there exists w
h(N, v)j [& withp(s)5 F (v), for all i[N and s[S. Define
s s[S N,S i i s
5
Note that we consider weighted potentials for non-cooperative games, as opposed to Hart and Mas–Colell (1989) where weighted Shapley values are characterized using weighted potentials for cooperative games.
6 w w
s
aR
w
]
Q (s):5
O
. (7)wR R7N,R±5
w
We will show that Q is a w-potential ofG. Let i[N, s[S, and t [S , then i i
w w
p(s)2p(t , s )5 F (v)2 F (v )
i i i 2i i s i (t ,si 2i)
s (t ,si 2i)
aR aR
] ]]
5wi
O
w 2wiO
wR R
R7N,i[R R7N,i[R
s (t ,si 2i)
aR aR
] ]]
5wi
O
w 2wiO
wR R
R7N,R±5 R7N,R±5
w w
5w (Q (s)i 2Q (t , si 2i)), where the third equality follows from (5).
w To prove the only-if-part assumeG is a w-potential game, with potential Q . Define for all s[S and all R7N
pi(s) w
]]
wR
H
O
S D
w 2(n21)Q (s) ,J
if R5Ni i[N s
p(s)
a 5R i w (8)
]]
w
H
2 1Q (s) ,J
if R5N\hij, i[NR w
5
i0, otherwise
s
which determine v 5o a u for all s[S. s R7N R R
We will show thath(N,v)j [& . Let R7N, s, t[S with s 5t . For R5N or R
s s[S N,S R R
s t
with uRu#n22 we immediately find that a 5 aR R. It remains to consider R with
w w
uRu5n21. Let i[N and R5N\hij then pi(s)2pi(t)5w (Q (s)i 2Q (t)) so
pi(s) pi(t)
s w w t
]] ]]
a 5R wR
H
2 w 1Q (s)J H
5wR 2 w 1Q (t)J
5aR.i i
So,h(N, v)j [& .
s s[S N,S
w
Finally, we will show that for all i[N and s[S it holds that F (v)5p(s).
i s i
Therefore, let i[N and s[S. Then s
aR
w
] F (v)5w
O
i s i w
R R7N,i[R
pj(s) w pj(s) w
]] ]]
5wi
H S D
O
w 2(n21)Q (s)1O
S
2 w 1Q (s)DJ
j j
j[N j[N, j±i
pi(s)
]]
5wi
H J
w 5pi(s). iThis completes the proof. h
s
w aR s
]
Q (s)5oR7N,R±5 wR for all s[S, where (aR R) 7N,s[S are the unanimity coordinates of
s w 7
h(N, v)j [& for which p(s)5 F (v) for all i[N and all s[S.
s[S N,S i i s
3. Networks
In this section we will first introduce hypergraphs. After that we will discuss hypergraph communication situations and characterize a class of allocation rules for these situations.
A hypergraph is a pair (N,*) with N the player set and* a family of subsets of N.
An element H[* is called a conference. The interpretation of a hypergraph is as
follows: communication between players in a hypergraph can only take place within a conference. Furthermore, communication via this conference cannot take place between a proper subset of this conference, i.e. all players of the conference have to participate in the communication. Note that a hypergraph is a generalization of a graph, which consists only of conferences with exactly two players.
Next, we consider hypergraph communication situations, first introduced in Myerson (1980). Formally, a hypergraph communication situation is a triple (N, v, *), where
(N, v) is a cooperative game and (N, *) a hypergraph. By assuming that every player
can communicate with himself we can restrict our attention to hypergraphs (N,*) with
N
*7hH[2 uuHu$2j. We will denote the class of all these hypergraph communication N
situations with player set N by HCS .
In a hypergraph communication situation a coalition S7N can effect communication
in conferences in *(S ):5hH[*uH7Sj. Further we define interaction sets of
(S, *(S )):
1. every hij7S is an interaction set.
2. every H[*(S ) is an interaction set.
3. if T and T are interaction sets with T >T ±5, then T <T is an interaction set.
1 2 1 2 1 2
A set T7S is a maximal interaction set of S if T is an interaction set of (S, *(S )) and
there exists no interaction set T9of (S,*(S )) with T,T9. Following the tradition set in earlier papers we will also refer to maximal interaction sets as components. We will denote the resulting partition of S in maximal interaction sets by S /*.
Conform this partition we define the value of coalition S7N in (N, v, *) by
*
v (S ):5
O
v(C ).C[S /* *
We call (N,v ) the hypergraph-restricted game. An allocation ruleg is a function that
N N
assigns to every (N,v,*)[HCS an element ofR . If there is no ambiguity about the
game (N, v) we will write g(*) instead of g(N, v, *). For a positive weight-vector
N w
w5(w ) [R the weighted (extended) Myerson value, m , is the allocation rule i i[N 11
7 w w w
It holds that Q (s)5P (N,v), where P denotes the weighted potential for cooperative games as used in the
s
which assigns to every (N, v, *) the w-weighted Shapley value of the
hypergraph-*
restricted game (N, v ),
w w *
m (N,v,*):5 F (N,v ).
We will characterize the w-weighted (extended) Myerson value by two properties,
component efficiency and w-fairness. Consider for an allocation rule g these two properties:
Component efficiency: For all hypergraph communication situations (N, v, *)[ N
HCS it holds for all C[N /*:
O
g(*)5v(C ).i i[C
N
w-Fairness: For all (N, v, *)[HCS , all H7N and all i, j[H
1 1
](g(*)2g(*\hHj))5](g(*)2g(*\hHj)).
i i j j
wi wj
Component efficiency states that the players in a maximal interaction set divide the value v(C ) amongst themselves. The property w-fairness is an extension of the fairness
property of Myerson (1980). In Myerson (1980) the (extended) Myerson value is characterized by the properties component efficiency and fairness. The weights represent the strength of the players: the changes in payoffs for two players as a consequence of forming an additional conference in which they are both involved are proportional to their weights.
The following lemma shows that the w-weighted (extended) Myerson value satisfies the two properties component efficiency and w-fairness. In the proof we use some results of Kalai and Samet (1988). It is shown that the w-weighted Shapley value satisfies the
dummy property, additivity, and partnership consistency. The dummy property states
w
that F (N,v)5v(hij) for all (N,v) with v(S<hij)5v(S )1v(hij) for all S7N\hij.
i
w w w
Additivity states thatF (N,v1z)5 F (N, v)1 F (N, z) for all cooperative games (N, v) and (N, z). To describe partnership consistency we need the notion of partnership. A
coalition S7N is a partnership in (N,v) if for all T,S and all R7N\S, v(R<T )5 w
v(R). Partnership consistency ofF states that for every partnership S in (N,v) it holds
that
w w w
F (v)5 F F
s
(v) u , for every id
[S,i i S S
w w
whereF (v)5o F (v).
S j[S j
w
Lemma 3.1. The w-weighted (extended) Myerson value, m , satisfies component
efficiency and w-fairness.
w N
Proof. First we will show thatm satisfies component efficiency. Let (N,v,*)[HCS
C N \C
and C a maximal interaction set of (N, *). We define two games (N,v ) and (N,v ).
C *
v (T ):5v (T>C ),
N \C *
v (T ):5v (T \C ).
* C N \C
Since C is a maximal interaction set of (N, *) it holds that v 5v 1v . Since all
N \C
i[C are dummy players in the game (N, v ), we conclude from the dummy player
w N \C
property of the w-weighted Shapley value, thatF (v )50 for all i[C. In the same i
w C
way we find for all i[N\C that F (v )50. Using this and the additivity of the
i
w-weighted Shapley values we find
w * w C w N \C
O
F (v )5O
F (v )1O
F (v )i i i
i[C i[C i[C
w C w C C *
5
O
F (v )5O
F (v )5v (N )5v (C )5v(C ),i i
i[C i[N
where the fourth equality follows from the efficiency of the w-weighted Shapley value. Secondly, we will show that the w-weighted (extended) Myerson value satisfies
N * *9
w-fairness. Let (N,v,*)[HCS and H[*. Define*9:5*\hHjandv9:5v 2v .
For all T7N with H≠T we then have
v9(T )5
O
v(R)2O
v(R)50R[T /* R[T /*9
since T /*5T /*9. This means that H is a partnership in v9. From partnership
w
consistency of F , it follows for all i[H that
wi
w w w w
]]
F (v9)5 F
O
F (v9) u 5O
F (v9)i i
SS
jD D
HS
jD
j[H
O
w j[Hj j[H So, for all i, j[H
w w F (v9)
F (v9) j
i
]]5]].
wi wj
From this we find
w w w
w w w
F (v9) m (*)2m (*9) m (*)2m (*9) F (v9) j j j
i i i
]]]]]5]]5]]5]]]]],
wi wi wj wj
where the first and third equalities follow from the definition of the game (N,v9) and the
w
additivity of the w-weighted Shapley values. Hence, m satisfies w-fairness. h
The following theorem shows that the w-weighted (extended) Myerson value is the unique rule that is component efficient and w-fair.
w
Theorem 3.1. The w-weighted (extended) Myerson value m is the unique rule that satisfies component efficiency and w-fairness.
w
w
We only need to show here thatm is the unique solution concept which satisfies these properties.
1 2
Suppose there are two rules g and g which satisfy component efficiency and
w-fairness. Let (N, v, *) be a communication situation with a minimum number of
1 2
conferences such thatg (*)±g (*). By component efficiency it follows that*±5. 1
Let H[* andhi, jj7H. From w-fairness ofg we then find
1 1 1 1 1 1
]
s
g (*)2g (*\hHj)d
5]s
g (*)2g (*\hHj) .d
i i j j
wi wj
2
Using this, the minimality of *, and the w-fairness ofg respectively, we find
1 1 1 1
wg (*)2wg (*)5wg (*\hHj)2wg (*\hHj)
j i i j j i i j
2 2
5wg (*\hHj)2wg (*\hHj)
j i i j
2 2
5wg (*)2wg (*).
j i i j
So
1 2
1 2
g (*)2g (*)
g (*)2g (*) j j
i i
]]]]]5]]]]].
wi wj
This expression is valid for all pairs hi, jj for which there exists an H[* with
hi, jj7H. Hence, it is also valid for all pairs hs, tj that are in the same maximal interaction set.
Let C[N /* and i[C. For all j[C we now have
1 1 2 1 1 2
]
s
g (*)2g (*)d
5]s
g (*)2g (*) .d
j j i i
wj wi
1 2 1 2
1
]
Let d:5
s
g (*)2g (*) . Then for all jd
[C :g (*)2g (*)5w d. Componenti i j j j
wi
1 2
efficiency of g andg gives us
1 2
O
g (*)5O
g (*)5v(C ).j j
j[C j[C Thus,
1 2
05
O
s
g (*)2g (*)d
5O
w d.j j j
j[C j[C
N
Since w[R it follows that d50. Since C was chosen arbitrarily, we conclude that
11
1 2
g (*)5g (*). h
4. Network formation
We will show that the only component efficient allocation rules that result in a weighted potential game are the weighted (extended) Myerson values.
Let g be an allocation rule and (N, v) a cooperative game. Define the conference
g
formation game G(N, v,g) determined by the tuple (N; (S ) ; ( f ) ) where for all
i i[N i i[N
i[N
N \hij
S :i 5hTuT72 j
represents the strategy set of player i . A strategy of player i denotes the set of coalitions player i wants to join to form conferences. A strategy profile s5(s , . . . , s )[P S ,
1 n i[N i
induces a set of conferences *(s) given by
*(s):5hHu uHu$2; H\hij[s , i[Hj. i
The interpretation is that a conference is formed if and only if all players in this
g g
conference are willing to form it. The payoff function f 5( f )i i[Nis then defined as the allocation rule applied to the conference structure formed,
g
f (s)5g(*(s)).
In case there is no ambiguity about the underlying cooperative game we will simply writeG(g) instead of G(N,v,g). In the remainder we will consider an arbitrary game
(N, v).
In order to prove that weighted (extended) Myerson values are the only allocation rules that are component efficient and that lead to conference formation games which are weighted potential games, we need two lemmas.
N
Lemma 4.1. Let g be an allocation rule and w[R . If the conference formation
11
game G(g) is a w-potential game, then for all hypergraphs (N, *), all H7N and all
i, j[H
1 1
](g(*)2g(*\hHj))5](g(*)2g(*\hHj)). (9)
i i j j
wi wj
w
Proof. SinceG(g) is a w-potential game,G(g) has a w-potential P . We will show that
g satisfies equation (9).
N
Let (N, *) be a hypergraph, so (N, v, *)[HCS . Define for all k[N,
s :5hH\hkjuH[*, k[Hj. k
Then it holds that *(s)5*. Let H[* and i[H, then for all j[H\hij we get
w w w
P (s \ihH\hijj, s , sj 2ij)5P (s \ihH\hijj, s \hj H\hjjj, s2ij)5P (s , s \i jhH\hjjj, s2ij), since the three strategy tuples all result in the formation of the same conferences, the conferences in *\hHj, and hence, they all result in the same payoffs.
1 1 g g
](g(*)2g(*\hHj))5]( f (s)2f (s \hH\hijj, s ))
i i i i i 2i
wi wi
w w
5P (s)2P (s \ihH\hijj, s2i)
w w
5P (s)2P (s \jhH\hjjj, s2j)
1 g g
]
5w ( f (s)j 2f (s \j jhH\hjjj, s2j)) j
1
]
5 (g(*)2g(*\hHj)).
j j
wj
This completes the proof. h
The following lemma shows that the conference formation game corresponding to an arbitrary cooperative game with a weighted (extended) Myerson value used as an allocation rule is a weighted potential game.
w
Lemma 4.2. The conference formation game G(m ) is a w-potential game.
Proof. Consider the following set of cooperative games, indexed by the set of strategy
w *(s) *(s)
profiles of G(m ), h(N, v )j . Let R7N and s5(s , s )[S. Since v (R)5
s[S R N \R
*(s)
o v(C ) and R /*(s) does not depend on s it follows that v (R) does not
C[R /*(s) N \R w
*(s) m w
depend on s . This implies that h(N, v )j [& . Since f (s)5m (*(s))5
N \R s[S N,S i i
w *(s) w
F (v ) by definition, it follows by Theorem 2.1 thatG(m ) is a w-potential game. h
i
In Qin (1996) it is shown that there is a unique allocation rule that is component efficient and results in a link formation game with a potential in the restricted case where only bilateral communication between the players is possible, i.e.uHu52 for all H[*.
If we combine the results of the lemmas above we can extend the result of Qin (1996) in two directions. First, we consider hypergraphs and not only graphs, and second, we allow for asymmetric players.
Theorem 4.1. Letg be a solution concept that is component efficient. The conference
formation game G(g) is a weighted potential game if and only if g coincides with a weighted (extended) Myerson value for all hypergraphs (N, *).
Proof. Suppose that the conference formation gameG(g) is a weighted potential game. From Lemma 4.1 it follows that there exist weights w for whichg satisfies equation (9). Since g is component efficient, it then follows, by the proof of Theorem 3.1, that g
w
coincides with m for all hypergraphs (N, *).
5. Potential maximizing strategies
In this section we will consider potential maximizing strategies in the conference w
formation game G(m ). Throughout this section we will assume that the underlying cooperative game (N,v) is superadditive, i.e. v(R<T )$v(R)1v(T ), for all disjoint R, T7N. We will show that the strategy resulting in the complete conference structure, the
structure with all subsets of players in the set of conferences, is a potential maximizing strategy. Furthermore, we will show that all potential maximizing strategies result in the same payoffs as the strategy corresponding to the full cooperation structure.
First we need some notation to denote the structures that will result according to the conference formation game with a weighted (extended) Myerson value used as
N \hij
] ] ] ]
allocation rule. Let s5(s , . . . ,s ) be the strategy tuple with s 52 for all i[N.
1 n i
This strategy implies that player i is willing to cooperate with all subsets of the other
] ]
players. The corresponding set of conferences will be denoted by *:5*(s)5hT[ N
2 uuTu$2j. A set of conferences* is called essentially complete with regard to solution
] ]
conceptg iff* and* are payoff-equivalent, i.e.g(*)5g(*). To facilitate the proof
of the main theorem in this section we will first prove two lemmas. The first lemma states that a player is never worse off forming an additional conference, extending a result of Myerson (1977).
N N
Lemma 5.1. Let (N, v, *)[HCS , H7N and w[R . For all i[H it holds that
11
w w
m (*<hHj)$m (*).
i i
*<hHj *
Proof. Let v9:5v 2v . The superadditivity of v implies that v9(R)$0 for all R7N since every maximal interaction set in (N,(*<hHj)(R)) is the union of one or more maximal interaction sets in (N, *(R)). For all R with H≠R it follows that
*<hHj *
v (R)5v (R) and hencev9(R)50.
Let i[H. Since v9(R)50 for all R7N\hij we have
v9(R<hij)$v9(R), for all R7N\hij. (10)
w N \hij From Weber (1988) it follows that there exists a probability distribution pi on 2 such that
w w
F (v9)5
O
p (R)(v9(R<hij)2v9(R)). (11)i i
R7N \hij
*<H * w Combining equations (10) and (11) completes the proof sincev9 5v 2v andF
satisfies additivity. h
N 9
Lemma 5.2. Let w[R . Then for all i[N, all s [S , and all s , s [S with
11 2i 2i i i i
w w w w
m m m m
9 9 9
si7s , it holds that if fi i (s , si 2i)5fi (s , si 2i) then f (s , si 2i)5f (s , si 2i). 9
Proof. If *(s , s )5*(s , s ), then the statement in the theorem is obviously true.
i 2i i 2i
N
9
Otherwise, since s 7s there exist k[N and H , . . . ,H [2 with i[H for all
i i 1 k j
9 9
j[h1, . . . kj such that *(s , s )5*(s , s )<hH , . . . ,H j. Define* :5*(s , s )
i 2i i 2i 1 k 0 i 2i
and for all j[h1, . . . ,kj
9
* :5*(s , s )<hH , . . . ,Hj.
j i 2i 1 j
w w w w
Since m (* )5m (* ), it follows from lemma 5.1 that m (* )5m (* ) for all
i 0 i k i j21 i j
j[h1, . . . ,kj.
*j *j21 w w
For j[h1, . . . , kj, define v9:5v 2v . Since m (* )5m (* ) it follows
i j i j21 from (10) and (11) that
v9(R<hij)5v9(R), for all R7N\hij, (12)
w 8
since p (R).0 for all R7N\hij. Consider an arbitrary l[N\hij and S7N\hlj. Using i
equation (12) and the fact thatv9(T )50 for every T with HÆT we have
j
v9(R<hlj)5v9((R<hlj)\hij)50 (13)
and
v9(R)5v9(R\hij)50. (14)
w
It follows thatF (v9)50 and hence, by the additivity of the weighted Shapley values
l that
w w *j w *j21 w
m (* )5 F (v )5 F (v )5m (* ). (15)
l j l l l j21
We conclude that
w w
m 9 w w w m
f (s , s )5m (* )5m (* )5 . . . 5m (* )5f (s , s ). (16)
i 2i 0 1 k i 2i
This completes the proof. h
We can now state our main theorem. It states that if a weighted Myerson value is used as an allocation then potential maximizing strategies result in essentially complete hypergraphs.
w
Theorem 5.1. Let w be a vector of positive weights and let P be a weighted potential
w ] w
for the conference formation game *(m ). Then s[argmax P . Further, if t[
w w
argmax P then*(t) is essentially complete form .
1 Proof. Let i[N, s [S and s [S . Define the following conference sets: * :5
i i 2i 2i
2 2 1
] ]
*(s , s ) and* :5*(s , s ). From s 7s we conclude that* 7* . Furthermore,
i 2i i 2i i i
8
1 2
note that if H[* \* , then i[H. If we apply Lemma 5.1 repeatedly for all 1 2
H[* \* then
w w
m ] w 1 w 2 m
f (s , s )5m (* )$m (* )5f (s , s ). (17)
i i 2i i i i i 2i
]
We conclude that s is a weakly dominant strategy.
]
Consider the n-tuple of weakly dominant strategies s and an arbitrary n-tuple of
i n i ] ]
strategies t. Construct a sequence (s )i50 with s 5(s , . . . , s , t1 i i11, . . . , t ). Thisn
0 n ] ]
construction implies that s 5t and s 5s. Since s is a weakly dominant strategy it holds
w i11 w i w i11 w i
for all i[h0, . . . , n21jthatm (*(s ))$m (*(s )), so P (s )$P (s ). Thus
i11 i11
w] w n w n21 w 1 w 0 w
P (s)5P (s )$P (s )$ . . . $P (s )$P (s )5P (t). (18)
This completes the proof of the first part of the theorem. w] w
Furthermore, since P (s)$P (t) for all strategy-tuples t[S it follows that if t is a w] w
potential maximizing strategy then P (s)5P (t). But then every inequality in (18) has
to hold with equality for this strategy-tuple t. Since 1
w k w k21 w k w k21
]
P (s )2P (s )5
s
m (*(s ))2m (*(s ))d
$0k k
wk
w k w k21
for all k[h1, . . . , nj it follows that m (*(s ))5m (*(s )) for all k[h1, . . . ,nj.
k k
w k w k21
From Lemma 5.2 we then conclude thatm (*(s ))5m (*(s )) for all k[h1, . . . ,nj and hence,
w w 0 w n w ]
m (*(t))5m (*(s ))5 . . . 5m (*(s ))5m (*(s)).
w
We conclude that if t[argmax P then it holds that *(t) is essentially complete for
w
m . h
If we drop the superadditivity assumption in Theorem 5.1 the full cooperation structure may fail to form. This follows easily by considering a non-superadditive two-person cooperative game. Then the potential maximizing strategy profiles result in the formation two one-player coalitions.
6. Concluding remarks
In this paper we have extended the model introduced in Myerson (1991), a strategic form game that describes a link formation process resulting in a graph. Here we describe a strategic form game, called the conference formation game, resulting in a hypergraph. In this paper we restrict ourselves to superadditive games and to conference formation games that are weighted potential games. It turns out that, under an efficiency requirement, the class of allocation rules generating conference formation games that are weighted potential games is the class of weighted (extended) Myerson values. Further-more, the argmax set of the conference formation games that are weighted potential games predicts the formation of the full cooperation structure or a structure that is payoff equivalent to the full cooperation structure.
conferences. This is shown in a much more technical way, not using the representation theorem of Section 2, in Dutta et al. (1995), a preliminary version of Dutta et al. (1998). The results in the current paper and related papers point in the direction of the formation of a full cooperation structure. However, this result is sensitive to the superadditivity assumption, the allocation rule and the equilibrium refinement used, as is shown in Dutta et al (1998). In Aumann and Myerson (1988) it is shown that the complete structure need not form if the cooperation structure is formed sequentially rather than simultaneously.
Acknowledgements
The authors are grateful to Henk Norde and two anonymous referees for their helpful suggestions and comments.
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