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The Owen Value Applied to Games with Graph-Restricted Communication
GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.
12, 42–53 (1996)
0003
The Owen Value Applied to Games with Graph-Restricted Communication* MARGARITA VA´ZQUEZ-BRAGE Department of Statistics and OR, University of Vigo, 36271 Vigo (Pontevedra), Spain
IGNACIO GARCI´A-JURADO Department of Statistics and OR, University of Santiago, 15771 Santiago de Compostela, Spain AND
FRANCESC CARRERAS Department of Applied Mathematics, Polytechnic University of Catalunya, 08222 Terrassa (Barcelona), Spain Received May 4, 1993
We introduce an allocation rule for transferable utility games with graph-restricted communication and a priori unions, provide two characterizations of this rule, and apply it to the analysis of a real political example. Journal of Economic Literature Classification Numbers: 000,020,026. 1996 Academic Press, Inc.
1. INTRODUCTION The Owen value was introduced by Owen (1977) as a modification of the Shapley value (Shapley, 1953) for n-person cooperative games with transferable utility (TU games) with systems of unions, a system of unions being a partition of the set of players which describes its a priori cooperation structure. The Owen value allocates the total utility among the unions as * We thank the University of Santiago and the Xunta de Galicia for financial support through Projects 60902.25064(5060), XUGA20701B91, and XUGA20702B93. 42 0899-8256/96 $12.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
OWEN VALUE IN GRAPH-RESTRICTED GAMES
43
the Shapley value of the induced TU game played by the unions and within each union allocates its allotted utility among its members taking into account their possibilities of entering other unions. The formation of political coalitions has been analyzed by comparing the Owen values resulting from various possible union structures (see, for instance, Carreras and Owen, 1988 and 1991). The Myerson value (Myerson, 1977) can be interpreted as a modification of the Shapley value for TU games in which communication among the players is restricted by an undirected graph, each link of which indicates direct cooperative communication between the players represented by the nodes at each end. Since then, games with graph-restricted communication have been widely studied; for a review see Borm et al. (1991). In political situations, the communication links can be viewed as political affinities between players. In this paper we propose an allocation rule for TU games with both graph-restricted communication and systems of unions. The partition given by the system of unions and the partition defined by the connected components of the communication graph, which can both appear naturally in many political and economic situations (we show an example in Section 4), both affect the bargaining positions of the players and hence the final allocation of utility among them. The paper is organized as follows. In Section 2 we announce notation and preliminary definitions and state Owen’s (1977) theorem, in Section 3 we present two sets of fairness and efficiency conditions characterizing our allocation rule, in Section 4 we apply our rule to a real political situation by considering its outcome for two plausible union structures on a given graph of political affinities, and in Section 5 we prove the results stated in Section 3.
2. PRELIMINARIES An n-person cooperative game with transferable utility (a TU game) is a pair (N, v), where N 5 {1, . . . , n} is the set of players and v, the characteristic function, is a real function on 2 N 5 {S u S , N} with v(B) 5 0. We denote by G(N) the set of TU games with player set N, identify (N, v) [ G(N) with its characteristic function v when no confusion will be caused, and for every v [ G(N) write F(v) for its Shapley value. An undirected graph without loops on N is a (possibly empty) set B of unordered pairs of distinct elements of N. We call the pairs (i, j) [ B links. We denote by g(N) the set of all undirected graphs without loops on N and by B N the complete graph on N defined by B N 5 {(i, j) u i [ N, j [ N, i ? j}. For any S , N and any B [ g(N) we say that i, j [ S are connected
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´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
in S by B if B contains a path in S connecting i and j, and we denote by S/B the set of connected components of S determined by B, i.e., the set of maximal subsets of elements connected in S by B. S/B is a partition of S. For any set T, we denote by P(T) the set of partitions of T. A TU game with a system of unions and set of players N is a pair (v, P), where v [ G(N) and P [ P(N). We denote by U(N) the set of all such pairs. If (v, P) [ U(N), with P 5 {Pi u i [ M 5 {1, . . . , m}}, the quotient game v P is the TU game in G(M ) defined by v P(R) 5 v(
;R , M.
It is the TU game played by the unions. THEOREM 1 (Owen, 1977). There exists a unique map C: U(N) R R n with the following properties. 1. The Carrier Property (CP): for all (v, P) [ U(N), if T is a carrier of v (i.e., if v(S) 5 v(S > T), ;S , N), then oi[T Ci (v, P) 5 v(T). 2. Symmetry in the Unions (SU): for all (v, P) [ U(N), all Pk [ P, and all i, j [ Pk , if v(S < i) 5 v(S < j) for all S , N \{i, j}, then Ci (v, P) 5 Cj (v, P). 3. Symmetry in the Quotient (SQ): for all (v, P) [ U(N) and all Pk , Ps [ P 5 {P1 , . . . , Pm }, if v P(R < k) 5 v P(R < s) for all R , M \{k, s}, then oi[Pk Ci (v, P) 5 oi[Ps Ci (v, P). 4. Additivity (A): for all (v, P), (w, P) [ U(N), C(v 1 w, P) 5 C(v, P) 1 C(w, P). This unique map is given by the formula Ci (v, P) 5
O O
t!( pk 2 t 2 1)!r!(m 2 r 2 1)! (v(Q < T < i) 2 v(Q < T)), pk !m! R,M \ {k} T,Pk\ {i} (1)
where Pk [ P is the union containing i, Q 5
45
OWEN VALUE IN GRAPH-RESTRICTED GAMES
Pm } a union structure. We denote by S(N) the set of all such triplets (for a fixed N). Given (v, B, P) [ S(N), the graph-restricted game v B is defined by v B(S) 5
O
v(T),
T[S/B
;S , N.
An allocation rule for graph-restricted games with systems of unions is a map c : S(N) R R n.
3. AN ALLOCATION RULE FOR GRAPH-RESTRICTED GAMES SYSTEMS OF UNIONS
WITH
Given (v, B, P) [ S(N), we define c (v, B, P) as the Owen value of the graph-restricted game v B:
c (v, B, P) 5 C(vB, P),
;(v, B, P) [ S(N).
Note that, as one might expect, c (v, B N, P) 5 C(v, P), for all v [ G(N) and all P [ P(N); and c (v, B, P) 5 F(v B ) (the Myerson value of (v, B)) if P 5 {N} or P 5 {{1}, . . . , {n}}, for all v [ G(N) and all B [ g(N). Hence c can be considered as generalizing the Owen and Myerson values. Given P 5 {P1 , . . . , Pm } [ P(N), Pk , Ps [ P, i [ Pk and B [ g(N), we define P2i [ P(N) by P2i 5 {P1 , . . . , Pk21 , Pk\{i}, Pk11 , . . . , Pm , {i}}; B2i [ g(N) by B2i 5 {( j, k) [ B u j ? i, k ? i}; and B\(Pk , Ps ) [ g(N) by B\(Pk , Ps ) 5 B\{(i, j) [ B u i [ Pk , j [ Ps }. We assert that c has the following properties for all (v, B, P) [ S(N). 1. Component Efficiency (CE): for all T [ N/B,
O c (v, B, P) 5 v(T); i
i[ T
i.e., the total worth of every connected component in N is distributed among its members. 2. Balanced Contributions for the Graph (BCG): for all Pk [ P and all i, j [ Pk ,
ci (v, B, P) 2 ci (v, B2j , P) 5 cj (v, B, P) 2 cj (v, B2i , P); i.e., if i and j are in same union, each loses (or gains) the same amount if the other is isolated. A similar property holds for the Myerson value (Myerson, 1980; van den Nouweland, 1993).
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´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
3. Balanced Contributions for the Unions (BCU): for all Pk [ P and all i, j [ Pk ,
ci (v, B, P) 2 ci (v, B, P2j ) 5 cj (v, B, P) 2 cj (v, B, P2i ); i.e., if i and j are in same union, each loses (or gains) the same amount if the other leaves the union. 4. Fairness in the Quotient (FQ): for all Pk , Ps [ P,
O c (v, B, P) 2 O c (v, B\(P , P ), P) 5 O c (v, B, P) 2 O c (v, B\(P , P ), P); i
i
i[Pk
k
s
i[Pk
i
i[Ps
i
k
s
i[Ps
i.e., each of two unions loses (or gains) the same amount as the other when all the links joining them are severed. THEOREM 2. There is a unique allocation rule satisfying CE, BCG, and FQ, and it is c. THEOREM 3. There is a unique allocation rule satisfying CE, BCU, and FQ, and it also is c.
4. A POLITICAL EXAMPLE The Parliament of Arago´n, one of the Spain’s seventeen autonomous communities, is constituted by 67 members. Since most decisions are taken by majority rule, the characteristic function of the game played by the parties with parliamentary representation is unity for any party or coalition summing 34 members, and zero for the rest. Following elections in 1991, the parliament was composed of 30 members of the socialist party PSOE, 17 members of the conservative PP, 17 members of the middle-of-the-road regionalist party PAR, and 3 members of IU, a coalition of communists and other left-wing parties. The affinities among these parties can quite plausibly be represented by a graph B consisting of just three links: (PP, PAR), (PAR, PSOE), and (PSOE, IU). Given the election results, there were just three minimal coalitions of nonzero worth, i.e., with enough elected deputies to be able to govern: {PP, PAR}, {PAR, PSOE}, and {PP, PSOE}. Only two of these coalitions are connected under the affinities graph.
47
OWEN VALUE IN GRAPH-RESTRICTED GAMES
TABLE I THE PARLIAMENT OF ARAGO´ N Party
Seats
F(v)
C(v, P1)
C(v, P2)
F(v B )
c (v, B, P1)
c (v, B, P2)
PSOE PP PAR IU
30 17 17 3
1/3 1/3 1/3 0
1/2 0 1/2 0
0 1/2 1/2 0
1/6 1/6 2/3 0
1/4 0 3/4 0
0 1/4 3/4 0
Table I lists, for each party i, its election results, its Shapley value Fi (v), its Owen value Ci (v, P1) given the system of unions P1 5 {{PP}, {PAR, PSOE}, {IU}}, its Owen value Ci (v, P2) given the system of unions P2 5 {{PP, PAR}, {PSOE}, {IU}}, its Myerson value Fi (v B ), ci (v, B, P1), and ci (v, B, P2). The Owen values predict that the two parties in the ruling coalition will share power equally, while the Myerson value predicts that the most powerful party will be PAR. Each of these predictions is based on only part of the available information. The allocation rule c uses all the available information to predict that PAR will predominate in the ruling coalition, regardless of which of the possible government coalitions it chooses to form. This prediction appears to have been borne out by events: PP and PAR formed a ruling coalition in which the presidency was occupied by a member of PAR.
5. PROOFS Proof of Theorem 2. (1) Uniqueness. If c 1 and c 2 are two different allocation rules satisfying CE, BCG, and FQ, then there exist v [ G(N), P 5 {P1 , . . . , Pm } [ P(N), and B [ g(N) such that c 1(v, B, P) ? c 2(v, B, P). We may suppose that for the pair (v, P), B is the graph with fewest links for which this inequality holds. Let B P be the graph induced by B on M B P 5 {(k, s) u '(i, j) [ B with i [ Pk , j [ Ps }. Clearly, for every R [ M/B P, PR 5
O (c (v, B, P) 2 c (v, B, P)) 5 0. 1 i
i[PR
2 i
(2)
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´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
Also, as all the elements in R are connected in R by B P, the minimality of B, and the fact that c 1 and c 2 both satisfy FQ together imply that, for all r [ R,
O (c (v, B, P) 2 c (v, B, P)) 5 c , 1 i
2 i
R
(3)
i[Pr
where cR is a constant depending only on R. Equations (2) and (3) imply that, for all Pk [ P,
O c (v, B, P) 5 O c (v, B, P). 1 i
2 i
i[Pk
(4)
i[Pk
Now take Pk [ P, and j [ P Bk , the set of B-unisolated elements of Pk ( j [ N is B-unisolated if j : l [ B for some l [ N). Since c 1 and c 2 both satisfy BCG, the minimality of B implies that
c 1j (v, B, P) 2 c 2j (v, B, P) 5 ck ,
(5)
where ck is a constant depending only on Pk . Then, by (4) and (5),
O (c (v, B, P) 2 c (v, B, P)) 5 O c 1 O (c (v, B, P) 2 c (v, B, P)).
05
1 i
2 i
i[Pk
1 i
k
i[P kB
(6)
2 i
i[Pk\ P kB
Note that, if i [ Pk\P Bk , {i} [ N/B and then, as c 1 and c 2 both satisfy CE, c 1i (v, B, P) 5 c 2i (v, B, P). Hence, from (6), 05
Oc. k
i[P kB
Thus, ck 5 0, c 1(v, B, P) 5 c 2(v, B, P), and the uniqueness is proved. (2) Existence. We show that c satisfies CE, BCG, and FQ. We shall use the abbreviations CP, SU, SQ, and A defined in Section 2 for the properties of the Owen value. Given (v, B, P) [ S(N), consider, for every S [ N/B, the game u S given by
49
OWEN VALUE IN GRAPH-RESTRICTED GAMES
u S(T) 5
O
v(R)
R[(T>S)/B
;T , N.
Clearly, S is a carrier for u S, so CP implies that, for every S, T [ N/B,
O C (u , P) 5 H i
T
u S(N)
if S 5 T
0
if S ? T.
i[S
Also, v B 5 oS[N/B u S, so A implies that
O C (v , P) 5 O O i
B
i[S
i[S T[N/B
5 u S(N) 5
Ci (u T, P) 5
O
R[S/B
O O C (u , P) i
T
T[N/B i[S
v(R) 5 v(S).
Hence, c satisfies CE. To show that c satisfies BCG, take Pk [ P and i, j [ Pk , and consider the game w given by w 5 v B 2 v B2i 2 v B2j 1 v(i)ui 1 v( j)uj , where ui and uj denote the unanimity games with carriers {i} and { j}, respectively. For any S , N \{i, j}, w(S < i) 5 w(S < j) 5 2v B(S), so SU implies that Ci (w, P) 5 Cj (w, P). By the linearity of the Owen value, then Ci (v B, P) 2 Ci (v B2j, P) 2 Ci (v B2i 2 v(i)ui 2 v( j)uj , P) 5 Cj (v B, P) 2 Cj (v B2i, P) 2 Cj (v B2j 2 v(i)ui 2 v( j)uj , P), and since i is a dummy for v B2i 2 v(i)ui 2 v( j)uj and j is a dummy for v B2j 2 v(i)ui 2 v( j)uj , CP implies that Ci (v B, P) 2 Ci (v B2j, P) 5 Cj (v B, P) 2 Cj (v B2i, P), i.e., c satisfies BCG. Finally, to show that c satisfies FQ, take Pk , Ps [ P and consider the game z 5 v B 2 v B \ (Pk ,Ps). For all
50
´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
R , M \{k, s} z P(R < k) 5 z P(R < s) 5 0. Thus oi[Pk Ci (z, P) 5 oi[Ps Ci (z, P), by SQ, and A therefore implies that
O C (v , P) 2 O C (v B
i
i
i[Pk
, P) 5
B \ (Pk ,Ps)
i[Pk
O C (v , P) 2 O C (v i
B
i[Ps
i
B \ (Pk ,Ps)
, P).
i[Ps
This concludes the proof. n To demonstrate Theorem 3 we need to prove the following interesting property of the Owen value. LEMMA 1. For all (v, P) [ U(N), all Pk [ P, and all i, j [ Pk , Ci (v, P) 2 Ci (v, P2j ) 5 Cj (v, P) 2 Cj (v, P2i ). Proof. Take (v, P) [ U(N) (with P 5 {P1 , . . . , Pm }), Pk [ P, and i, j [ Pk , By (1), Ci (v, P) 5
O O
t!( pk 2 t 2 1)!r!(m 2 r 2 1)! (v(Q < T < i) 2 v(Q < T)) pk!m! R,M \ {k} T,Pk\ {i, j} 1
O O F(t 1 1)!( p 2 tp2!m!2)!r!(m 2 r 2 1)! k
R,M \ {k} T,Pk\ {i, j}
k
G
3 (v(Q < T < j < i) 2 v(Q < T < j)) , where Q is the union
O O
R,M 9\ {k}
t!( pk 2 t 2 2)!r!(m 2 r)! (v(Q < T < i) 2 v(Q < T)) ( pk 2 1)!(m 1 1)! k \ {i}
T,P 9
51
OWEN VALUE IN GRAPH-RESTRICTED GAMES
5
O O
R,M \ {k} T,Pk\ {i, j}
1
t!( pk 2 t 2 2)!r!(m 2 r)! (v(Q < T < i) 2 v(Q < T)) ( pk 2 1)!(m 1 1)!
1 1)!(m 2 r 2 1)! O O Ft!( p 2 t(2p 2)!(r 2 1)!(m 1 1)! k
R,M \ {k} T,Pk\ {i, j}
k
G
3 (v(Q < T < j < i ) 2 v(Q < T < j)) . Hence, Ci (v, P) 2 Ci (v, P2j ) 5
O O A (v(Q < T < i) 2 v(Q < T)) 1 O O A (v(Q < T < j < i) 2 v(Q < T < j)), 1
R,M \ {k} T,Pk\ {i, j}
2
R,M \ {k} T,Pk\ {i, j}
where A1 5
t!( pk 2 t 2 1)!r!(m 2 r 2 1)! t!( pk 2 t 2 2)!r!(m 2 r)! 2 pk!m! ( pk 2 1)!(m 1 1)!
and A2 5
(t 1 1)!( pk 2 t 2 2)!r!(m 2 r 2 1)! pk!m! 2
t!( pk 2 t 2 2)!(r 1 1)!(m 2 r 2 1)! . ( pk 2 1)!(m 1 1)!
Using some elementary algebra, A1 5 2A2 5
S
D
t!( pk 2 t 2 2)!r!(m 2 r 2 1)! rpk 1 pk 2 mt 2 m 2 t 2 1 . pk!m! m11
Thus, Ci (v, P) 2 Ci (v, P2j ) 5
O O
R,M \ {k} T,Pk\ {i, j}
[A1(v(Q < T < i) 2 v(Q < T)
1 v(Q < T < j) 2 v(Q < T < j < i))].
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´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
Since the right-hand side of the last equality depends on i in the same way as it depends on j, Ci (v, P) 2 Ci (v, P2j ) 5 Cj (v, P) 2 Cj (v, P2i ), and the proof is concluded. n Proof of Theorem 3. (1) Uniqueness. The proof is analogous to that of Theorem 2. We suppose that there exist two different allocation rules, c 1 and c 2, satisfying CE, BCU, and FQ and consider v [ G(N), B [ g(N), and P 5 {P1 , . . . , Pm} [ P(N) such that (a) c 1(v, B, P) ? c 2(v, B, P), (b) for this v, B is the graph with fewest links for which (a) holds, and (c) for the given v and B, P is the partition with most unions for which (a) holds. As in the proof of Theorem 2, it can be shown that
O c (v, B, P) 5 O c (v, B, P) 1 i
i[Pk
2 i
(7)
i[Pk
for any Pk [ P. Now, select Pk [ P. If Pk 5 {i}, then from (7), c 1i (v, B, P) 5 c 2i (v, B, P). If Pk has more than one member, take i, j [ Pk . Since c 1 and c 2 satisfy BCU.
c ih (v, B, P) 2 c ih (v, B, P2j ) 5 c jh (v, B, P) 2 c jh (v, B, P2i ), for all h [ {1, 2}. Hence, the maximality of P implies that
c 1i (v, B, P) 2 c 2i (v, B, P) 5 ck
;i [ Pk ,
(8)
where ck is a constant depending only on Pk . Together, (7) and (8) imply that c 1i (v, B, P) 5 c 2i (v, B, P) for all i [ Pk , and the uniqueness follows. (2) Existence. In the proof of Theorem 2 we showed that c satisfies CE and FQ. From Lemma 1 it is straightforward that c also satisfies BCG. n
REFERENCES BORM, P., VAN DEN NOUWELAND, A., AND TIJS, S. (1991). ‘‘Cooperation and Communication Restrictions: A Survey,’’ in Economic Behaviour in an Imperfect Environment (R. Gilles and P. Ruys, Eds.). to appear. CARRERAS, F., AND OWEN, G. (1988). ‘‘Evaluation of the Catalonian Parliament, 1980–1984,’’ Math. Soc. Sci. 15, 87–92.
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CARRERAS, F., AND OWEN, G. (1991). ‘‘An Analysis of the Euskarian Parliament,’’ in Coalition Theory and Coalition Governments (N. Schofield, Ed.). Dordrecht: Kluwer. to appear. MYERSON, R. B. (1977). ‘‘Graphs and Cooperation in Games,’’ Math. Oper. Res. 2, 225–229. MYERSON, R. B. (1980). ‘‘Conference Structures and Fair Allocation Rules,’’ Int. J. Game Theory 9, 169–182. OWEN, G. (1977). ‘‘Values of Games with a Priori Unions,’’ in Mathematical Economics and Game Theory (R. Henn and O. Moeschlin, Eds.). Berlin: Springer-Verlag. SHAPLEY, L. S. (1953). ‘‘A Value for n-person Games,’’ in Contributions to the Theory of Games II (A. W. Tucker and H. Kuhn, Eds.), pp. 307–317, Annals of Mathematics Studies, Vol. 28. Princeton, NJ: Princeton Univ. Press. NOUWELAND, A. (1993). ‘‘Games and Graphs in Economic Situations,’’ Ph.D. dissertation, Tilburg University, The Netherlands. WINTER, E. (1992). ‘‘The Consistency and Potential for Values of Games with Coalition Structure,’’ Games Econ. Behav. 4, 132–144.
VAN DEN
12, 42–53 (1996)
0003
The Owen Value Applied to Games with Graph-Restricted Communication* MARGARITA VA´ZQUEZ-BRAGE Department of Statistics and OR, University of Vigo, 36271 Vigo (Pontevedra), Spain
IGNACIO GARCI´A-JURADO Department of Statistics and OR, University of Santiago, 15771 Santiago de Compostela, Spain AND
FRANCESC CARRERAS Department of Applied Mathematics, Polytechnic University of Catalunya, 08222 Terrassa (Barcelona), Spain Received May 4, 1993
We introduce an allocation rule for transferable utility games with graph-restricted communication and a priori unions, provide two characterizations of this rule, and apply it to the analysis of a real political example. Journal of Economic Literature Classification Numbers: 000,020,026. 1996 Academic Press, Inc.
1. INTRODUCTION The Owen value was introduced by Owen (1977) as a modification of the Shapley value (Shapley, 1953) for n-person cooperative games with transferable utility (TU games) with systems of unions, a system of unions being a partition of the set of players which describes its a priori cooperation structure. The Owen value allocates the total utility among the unions as * We thank the University of Santiago and the Xunta de Galicia for financial support through Projects 60902.25064(5060), XUGA20701B91, and XUGA20702B93. 42 0899-8256/96 $12.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
OWEN VALUE IN GRAPH-RESTRICTED GAMES
43
the Shapley value of the induced TU game played by the unions and within each union allocates its allotted utility among its members taking into account their possibilities of entering other unions. The formation of political coalitions has been analyzed by comparing the Owen values resulting from various possible union structures (see, for instance, Carreras and Owen, 1988 and 1991). The Myerson value (Myerson, 1977) can be interpreted as a modification of the Shapley value for TU games in which communication among the players is restricted by an undirected graph, each link of which indicates direct cooperative communication between the players represented by the nodes at each end. Since then, games with graph-restricted communication have been widely studied; for a review see Borm et al. (1991). In political situations, the communication links can be viewed as political affinities between players. In this paper we propose an allocation rule for TU games with both graph-restricted communication and systems of unions. The partition given by the system of unions and the partition defined by the connected components of the communication graph, which can both appear naturally in many political and economic situations (we show an example in Section 4), both affect the bargaining positions of the players and hence the final allocation of utility among them. The paper is organized as follows. In Section 2 we announce notation and preliminary definitions and state Owen’s (1977) theorem, in Section 3 we present two sets of fairness and efficiency conditions characterizing our allocation rule, in Section 4 we apply our rule to a real political situation by considering its outcome for two plausible union structures on a given graph of political affinities, and in Section 5 we prove the results stated in Section 3.
2. PRELIMINARIES An n-person cooperative game with transferable utility (a TU game) is a pair (N, v), where N 5 {1, . . . , n} is the set of players and v, the characteristic function, is a real function on 2 N 5 {S u S , N} with v(B) 5 0. We denote by G(N) the set of TU games with player set N, identify (N, v) [ G(N) with its characteristic function v when no confusion will be caused, and for every v [ G(N) write F(v) for its Shapley value. An undirected graph without loops on N is a (possibly empty) set B of unordered pairs of distinct elements of N. We call the pairs (i, j) [ B links. We denote by g(N) the set of all undirected graphs without loops on N and by B N the complete graph on N defined by B N 5 {(i, j) u i [ N, j [ N, i ? j}. For any S , N and any B [ g(N) we say that i, j [ S are connected
44
´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
in S by B if B contains a path in S connecting i and j, and we denote by S/B the set of connected components of S determined by B, i.e., the set of maximal subsets of elements connected in S by B. S/B is a partition of S. For any set T, we denote by P(T) the set of partitions of T. A TU game with a system of unions and set of players N is a pair (v, P), where v [ G(N) and P [ P(N). We denote by U(N) the set of all such pairs. If (v, P) [ U(N), with P 5 {Pi u i [ M 5 {1, . . . , m}}, the quotient game v P is the TU game in G(M ) defined by v P(R) 5 v(
;R , M.
It is the TU game played by the unions. THEOREM 1 (Owen, 1977). There exists a unique map C: U(N) R R n with the following properties. 1. The Carrier Property (CP): for all (v, P) [ U(N), if T is a carrier of v (i.e., if v(S) 5 v(S > T), ;S , N), then oi[T Ci (v, P) 5 v(T). 2. Symmetry in the Unions (SU): for all (v, P) [ U(N), all Pk [ P, and all i, j [ Pk , if v(S < i) 5 v(S < j) for all S , N \{i, j}, then Ci (v, P) 5 Cj (v, P). 3. Symmetry in the Quotient (SQ): for all (v, P) [ U(N) and all Pk , Ps [ P 5 {P1 , . . . , Pm }, if v P(R < k) 5 v P(R < s) for all R , M \{k, s}, then oi[Pk Ci (v, P) 5 oi[Ps Ci (v, P). 4. Additivity (A): for all (v, P), (w, P) [ U(N), C(v 1 w, P) 5 C(v, P) 1 C(w, P). This unique map is given by the formula Ci (v, P) 5
O O
t!( pk 2 t 2 1)!r!(m 2 r 2 1)! (v(Q < T < i) 2 v(Q < T)), pk !m! R,M \ {k} T,Pk\ {i} (1)
where Pk [ P is the union containing i, Q 5
45
OWEN VALUE IN GRAPH-RESTRICTED GAMES
Pm } a union structure. We denote by S(N) the set of all such triplets (for a fixed N). Given (v, B, P) [ S(N), the graph-restricted game v B is defined by v B(S) 5
O
v(T),
T[S/B
;S , N.
An allocation rule for graph-restricted games with systems of unions is a map c : S(N) R R n.
3. AN ALLOCATION RULE FOR GRAPH-RESTRICTED GAMES SYSTEMS OF UNIONS
WITH
Given (v, B, P) [ S(N), we define c (v, B, P) as the Owen value of the graph-restricted game v B:
c (v, B, P) 5 C(vB, P),
;(v, B, P) [ S(N).
Note that, as one might expect, c (v, B N, P) 5 C(v, P), for all v [ G(N) and all P [ P(N); and c (v, B, P) 5 F(v B ) (the Myerson value of (v, B)) if P 5 {N} or P 5 {{1}, . . . , {n}}, for all v [ G(N) and all B [ g(N). Hence c can be considered as generalizing the Owen and Myerson values. Given P 5 {P1 , . . . , Pm } [ P(N), Pk , Ps [ P, i [ Pk and B [ g(N), we define P2i [ P(N) by P2i 5 {P1 , . . . , Pk21 , Pk\{i}, Pk11 , . . . , Pm , {i}}; B2i [ g(N) by B2i 5 {( j, k) [ B u j ? i, k ? i}; and B\(Pk , Ps ) [ g(N) by B\(Pk , Ps ) 5 B\{(i, j) [ B u i [ Pk , j [ Ps }. We assert that c has the following properties for all (v, B, P) [ S(N). 1. Component Efficiency (CE): for all T [ N/B,
O c (v, B, P) 5 v(T); i
i[ T
i.e., the total worth of every connected component in N is distributed among its members. 2. Balanced Contributions for the Graph (BCG): for all Pk [ P and all i, j [ Pk ,
ci (v, B, P) 2 ci (v, B2j , P) 5 cj (v, B, P) 2 cj (v, B2i , P); i.e., if i and j are in same union, each loses (or gains) the same amount if the other is isolated. A similar property holds for the Myerson value (Myerson, 1980; van den Nouweland, 1993).
46
´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
3. Balanced Contributions for the Unions (BCU): for all Pk [ P and all i, j [ Pk ,
ci (v, B, P) 2 ci (v, B, P2j ) 5 cj (v, B, P) 2 cj (v, B, P2i ); i.e., if i and j are in same union, each loses (or gains) the same amount if the other leaves the union. 4. Fairness in the Quotient (FQ): for all Pk , Ps [ P,
O c (v, B, P) 2 O c (v, B\(P , P ), P) 5 O c (v, B, P) 2 O c (v, B\(P , P ), P); i
i
i[Pk
k
s
i[Pk
i
i[Ps
i
k
s
i[Ps
i.e., each of two unions loses (or gains) the same amount as the other when all the links joining them are severed. THEOREM 2. There is a unique allocation rule satisfying CE, BCG, and FQ, and it is c. THEOREM 3. There is a unique allocation rule satisfying CE, BCU, and FQ, and it also is c.
4. A POLITICAL EXAMPLE The Parliament of Arago´n, one of the Spain’s seventeen autonomous communities, is constituted by 67 members. Since most decisions are taken by majority rule, the characteristic function of the game played by the parties with parliamentary representation is unity for any party or coalition summing 34 members, and zero for the rest. Following elections in 1991, the parliament was composed of 30 members of the socialist party PSOE, 17 members of the conservative PP, 17 members of the middle-of-the-road regionalist party PAR, and 3 members of IU, a coalition of communists and other left-wing parties. The affinities among these parties can quite plausibly be represented by a graph B consisting of just three links: (PP, PAR), (PAR, PSOE), and (PSOE, IU). Given the election results, there were just three minimal coalitions of nonzero worth, i.e., with enough elected deputies to be able to govern: {PP, PAR}, {PAR, PSOE}, and {PP, PSOE}. Only two of these coalitions are connected under the affinities graph.
47
OWEN VALUE IN GRAPH-RESTRICTED GAMES
TABLE I THE PARLIAMENT OF ARAGO´ N Party
Seats
F(v)
C(v, P1)
C(v, P2)
F(v B )
c (v, B, P1)
c (v, B, P2)
PSOE PP PAR IU
30 17 17 3
1/3 1/3 1/3 0
1/2 0 1/2 0
0 1/2 1/2 0
1/6 1/6 2/3 0
1/4 0 3/4 0
0 1/4 3/4 0
Table I lists, for each party i, its election results, its Shapley value Fi (v), its Owen value Ci (v, P1) given the system of unions P1 5 {{PP}, {PAR, PSOE}, {IU}}, its Owen value Ci (v, P2) given the system of unions P2 5 {{PP, PAR}, {PSOE}, {IU}}, its Myerson value Fi (v B ), ci (v, B, P1), and ci (v, B, P2). The Owen values predict that the two parties in the ruling coalition will share power equally, while the Myerson value predicts that the most powerful party will be PAR. Each of these predictions is based on only part of the available information. The allocation rule c uses all the available information to predict that PAR will predominate in the ruling coalition, regardless of which of the possible government coalitions it chooses to form. This prediction appears to have been borne out by events: PP and PAR formed a ruling coalition in which the presidency was occupied by a member of PAR.
5. PROOFS Proof of Theorem 2. (1) Uniqueness. If c 1 and c 2 are two different allocation rules satisfying CE, BCG, and FQ, then there exist v [ G(N), P 5 {P1 , . . . , Pm } [ P(N), and B [ g(N) such that c 1(v, B, P) ? c 2(v, B, P). We may suppose that for the pair (v, P), B is the graph with fewest links for which this inequality holds. Let B P be the graph induced by B on M B P 5 {(k, s) u '(i, j) [ B with i [ Pk , j [ Ps }. Clearly, for every R [ M/B P, PR 5
O (c (v, B, P) 2 c (v, B, P)) 5 0. 1 i
i[PR
2 i
(2)
48
´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
Also, as all the elements in R are connected in R by B P, the minimality of B, and the fact that c 1 and c 2 both satisfy FQ together imply that, for all r [ R,
O (c (v, B, P) 2 c (v, B, P)) 5 c , 1 i
2 i
R
(3)
i[Pr
where cR is a constant depending only on R. Equations (2) and (3) imply that, for all Pk [ P,
O c (v, B, P) 5 O c (v, B, P). 1 i
2 i
i[Pk
(4)
i[Pk
Now take Pk [ P, and j [ P Bk , the set of B-unisolated elements of Pk ( j [ N is B-unisolated if j : l [ B for some l [ N). Since c 1 and c 2 both satisfy BCG, the minimality of B implies that
c 1j (v, B, P) 2 c 2j (v, B, P) 5 ck ,
(5)
where ck is a constant depending only on Pk . Then, by (4) and (5),
O (c (v, B, P) 2 c (v, B, P)) 5 O c 1 O (c (v, B, P) 2 c (v, B, P)).
05
1 i
2 i
i[Pk
1 i
k
i[P kB
(6)
2 i
i[Pk\ P kB
Note that, if i [ Pk\P Bk , {i} [ N/B and then, as c 1 and c 2 both satisfy CE, c 1i (v, B, P) 5 c 2i (v, B, P). Hence, from (6), 05
Oc. k
i[P kB
Thus, ck 5 0, c 1(v, B, P) 5 c 2(v, B, P), and the uniqueness is proved. (2) Existence. We show that c satisfies CE, BCG, and FQ. We shall use the abbreviations CP, SU, SQ, and A defined in Section 2 for the properties of the Owen value. Given (v, B, P) [ S(N), consider, for every S [ N/B, the game u S given by
49
OWEN VALUE IN GRAPH-RESTRICTED GAMES
u S(T) 5
O
v(R)
R[(T>S)/B
;T , N.
Clearly, S is a carrier for u S, so CP implies that, for every S, T [ N/B,
O C (u , P) 5 H i
T
u S(N)
if S 5 T
0
if S ? T.
i[S
Also, v B 5 oS[N/B u S, so A implies that
O C (v , P) 5 O O i
B
i[S
i[S T[N/B
5 u S(N) 5
Ci (u T, P) 5
O
R[S/B
O O C (u , P) i
T
T[N/B i[S
v(R) 5 v(S).
Hence, c satisfies CE. To show that c satisfies BCG, take Pk [ P and i, j [ Pk , and consider the game w given by w 5 v B 2 v B2i 2 v B2j 1 v(i)ui 1 v( j)uj , where ui and uj denote the unanimity games with carriers {i} and { j}, respectively. For any S , N \{i, j}, w(S < i) 5 w(S < j) 5 2v B(S), so SU implies that Ci (w, P) 5 Cj (w, P). By the linearity of the Owen value, then Ci (v B, P) 2 Ci (v B2j, P) 2 Ci (v B2i 2 v(i)ui 2 v( j)uj , P) 5 Cj (v B, P) 2 Cj (v B2i, P) 2 Cj (v B2j 2 v(i)ui 2 v( j)uj , P), and since i is a dummy for v B2i 2 v(i)ui 2 v( j)uj and j is a dummy for v B2j 2 v(i)ui 2 v( j)uj , CP implies that Ci (v B, P) 2 Ci (v B2j, P) 5 Cj (v B, P) 2 Cj (v B2i, P), i.e., c satisfies BCG. Finally, to show that c satisfies FQ, take Pk , Ps [ P and consider the game z 5 v B 2 v B \ (Pk ,Ps). For all
50
´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
R , M \{k, s} z P(R < k) 5 z P(R < s) 5 0. Thus oi[Pk Ci (z, P) 5 oi[Ps Ci (z, P), by SQ, and A therefore implies that
O C (v , P) 2 O C (v B
i
i
i[Pk
, P) 5
B \ (Pk ,Ps)
i[Pk
O C (v , P) 2 O C (v i
B
i[Ps
i
B \ (Pk ,Ps)
, P).
i[Ps
This concludes the proof. n To demonstrate Theorem 3 we need to prove the following interesting property of the Owen value. LEMMA 1. For all (v, P) [ U(N), all Pk [ P, and all i, j [ Pk , Ci (v, P) 2 Ci (v, P2j ) 5 Cj (v, P) 2 Cj (v, P2i ). Proof. Take (v, P) [ U(N) (with P 5 {P1 , . . . , Pm }), Pk [ P, and i, j [ Pk , By (1), Ci (v, P) 5
O O
t!( pk 2 t 2 1)!r!(m 2 r 2 1)! (v(Q < T < i) 2 v(Q < T)) pk!m! R,M \ {k} T,Pk\ {i, j} 1
O O F(t 1 1)!( p 2 tp2!m!2)!r!(m 2 r 2 1)! k
R,M \ {k} T,Pk\ {i, j}
k
G
3 (v(Q < T < j < i) 2 v(Q < T < j)) , where Q is the union
O O
R,M 9\ {k}
t!( pk 2 t 2 2)!r!(m 2 r)! (v(Q < T < i) 2 v(Q < T)) ( pk 2 1)!(m 1 1)! k \ {i}
T,P 9
51
OWEN VALUE IN GRAPH-RESTRICTED GAMES
5
O O
R,M \ {k} T,Pk\ {i, j}
1
t!( pk 2 t 2 2)!r!(m 2 r)! (v(Q < T < i) 2 v(Q < T)) ( pk 2 1)!(m 1 1)!
1 1)!(m 2 r 2 1)! O O Ft!( p 2 t(2p 2)!(r 2 1)!(m 1 1)! k
R,M \ {k} T,Pk\ {i, j}
k
G
3 (v(Q < T < j < i ) 2 v(Q < T < j)) . Hence, Ci (v, P) 2 Ci (v, P2j ) 5
O O A (v(Q < T < i) 2 v(Q < T)) 1 O O A (v(Q < T < j < i) 2 v(Q < T < j)), 1
R,M \ {k} T,Pk\ {i, j}
2
R,M \ {k} T,Pk\ {i, j}
where A1 5
t!( pk 2 t 2 1)!r!(m 2 r 2 1)! t!( pk 2 t 2 2)!r!(m 2 r)! 2 pk!m! ( pk 2 1)!(m 1 1)!
and A2 5
(t 1 1)!( pk 2 t 2 2)!r!(m 2 r 2 1)! pk!m! 2
t!( pk 2 t 2 2)!(r 1 1)!(m 2 r 2 1)! . ( pk 2 1)!(m 1 1)!
Using some elementary algebra, A1 5 2A2 5
S
D
t!( pk 2 t 2 2)!r!(m 2 r 2 1)! rpk 1 pk 2 mt 2 m 2 t 2 1 . pk!m! m11
Thus, Ci (v, P) 2 Ci (v, P2j ) 5
O O
R,M \ {k} T,Pk\ {i, j}
[A1(v(Q < T < i) 2 v(Q < T)
1 v(Q < T < j) 2 v(Q < T < j < i))].
52
´ ZQUEZ-BRAGE, GARCI´A-JURADO, AND CARRERAS VA
Since the right-hand side of the last equality depends on i in the same way as it depends on j, Ci (v, P) 2 Ci (v, P2j ) 5 Cj (v, P) 2 Cj (v, P2i ), and the proof is concluded. n Proof of Theorem 3. (1) Uniqueness. The proof is analogous to that of Theorem 2. We suppose that there exist two different allocation rules, c 1 and c 2, satisfying CE, BCU, and FQ and consider v [ G(N), B [ g(N), and P 5 {P1 , . . . , Pm} [ P(N) such that (a) c 1(v, B, P) ? c 2(v, B, P), (b) for this v, B is the graph with fewest links for which (a) holds, and (c) for the given v and B, P is the partition with most unions for which (a) holds. As in the proof of Theorem 2, it can be shown that
O c (v, B, P) 5 O c (v, B, P) 1 i
i[Pk
2 i
(7)
i[Pk
for any Pk [ P. Now, select Pk [ P. If Pk 5 {i}, then from (7), c 1i (v, B, P) 5 c 2i (v, B, P). If Pk has more than one member, take i, j [ Pk . Since c 1 and c 2 satisfy BCU.
c ih (v, B, P) 2 c ih (v, B, P2j ) 5 c jh (v, B, P) 2 c jh (v, B, P2i ), for all h [ {1, 2}. Hence, the maximality of P implies that
c 1i (v, B, P) 2 c 2i (v, B, P) 5 ck
;i [ Pk ,
(8)
where ck is a constant depending only on Pk . Together, (7) and (8) imply that c 1i (v, B, P) 5 c 2i (v, B, P) for all i [ Pk , and the uniqueness follows. (2) Existence. In the proof of Theorem 2 we showed that c satisfies CE and FQ. From Lemma 1 it is straightforward that c also satisfies BCG. n
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CARRERAS, F., AND OWEN, G. (1991). ‘‘An Analysis of the Euskarian Parliament,’’ in Coalition Theory and Coalition Governments (N. Schofield, Ed.). Dordrecht: Kluwer. to appear. MYERSON, R. B. (1977). ‘‘Graphs and Cooperation in Games,’’ Math. Oper. Res. 2, 225–229. MYERSON, R. B. (1980). ‘‘Conference Structures and Fair Allocation Rules,’’ Int. J. Game Theory 9, 169–182. OWEN, G. (1977). ‘‘Values of Games with a Priori Unions,’’ in Mathematical Economics and Game Theory (R. Henn and O. Moeschlin, Eds.). Berlin: Springer-Verlag. SHAPLEY, L. S. (1953). ‘‘A Value for n-person Games,’’ in Contributions to the Theory of Games II (A. W. Tucker and H. Kuhn, Eds.), pp. 307–317, Annals of Mathematics Studies, Vol. 28. Princeton, NJ: Princeton Univ. Press. NOUWELAND, A. (1993). ‘‘Games and Graphs in Economic Situations,’’ Ph.D. dissertation, Tilburg University, The Netherlands. WINTER, E. (1992). ‘‘The Consistency and Potential for Values of Games with Coalition Structure,’’ Games Econ. Behav. 4, 132–144.
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