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The principle of balanced contributions and hierarchies of cooperation
:'~
mathematical
soda/ sciences
ELSEVIER
Mathematical Social Sciences 31 (1996) 171-182
The principle of balanced contributions and hierarchies of cooperation Emilio Calvo a'*, J. Javier Lasaga a, Eyal W i n t e r b "Departamento de Economia Aplicada 1, Universidad del Pais Vasco, Avenida Lehendakari Aguirre 83, 48015 Bilbao, Spain bCenter for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Received December 1994; revised June 1995
Abstract The principle of balanced contributions has appeared repeatedly in the literature on the Shapley value. This principle is akin to the reciprocity properties shared by almost all cooperative solution concepts. We provide a new axiomatization for the level structure value. This axiomatization has the advantage that it can be applied to many important subdomains of TU games. We use the Hart-Mas-Colell potential function as a tool to prove our main result, and establish another interesting characterization for the value as a by-product.
Keywords: TU games; Shapley value; Balanced contributions; Potential function; Level structures
1. I n t r o d u c t i o n
Most cooperative solution concepts possess a reciprocity property between the players. This is at least the case for the kernel (Davis and Maschler, 1965), the bargaining set (Aumann and Maschler, 1964), the set of semi-stable demand vectors (Albers, 1979; Bennett, 1983; Selten, 1981), and the Shapley (1953) value. In some cases the reciprocity property simply means a symmetry axiom. In other cases, like the bargaining set and the kernel, it involves a more complicated * Corresponding author. 0165-4896/96/$15.00 (~) 1996 - Elsevier Science B.V. All rights reserved S S D I 0165-4896(95)00806-3
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symmetric binary relation on the set of players. Loosely speaking, the common idea behind these properties is that a reasonable solution concept must be balanced in the sense that each player i affects player j's payoff in the same manner that player j affects i's payoff. In the literature on the Shapley value a principle o f this sort was introduced by Myerson (1980)) Myerson's principle of balanced contributions asserts that, for any two players, the amount that each player would gain or lose by the other's withdrawal from the game should be equal. Myerson used this principle to extend the Shapley value for games with conference structures, but a special case of his results provides an axiomatization for the Shapley value. Our objective in this paper is to use the same principle to axiomatize the cooperation structure extension of the Shapley value, namely the level structure value, due to Owen (1977) and Winter (1989). A level structure describes a hierarchy of cooperation between the players. Formally, it is given by a sequence of partitions of the set of players, each coarser than the previous one. This reflects a situation in which players unite into groups to improve their bargaining power. These groups then form a coarser partition by creating larger groups, which can again extend even further. Consider, for example, a game involving the members of the European Parliament as players. The first level divides the set of players according to their party membership, but there is also a coarser partition, i.e. dividing them by nations, or an even coarser one based on cooperation at the international level. The value for games with a level structure, which we define below, will take into account this hierarchy of cooperation in determining the final payoff distribution. The level structure value was axiomatized by Winter (1989). The special case involving a single level was also axiomatized by Owen (1977) and Hart and Kurz (1983, 1984). We propose a different sort of axiomatization for this value. Unlike the characterizations mentioned above, ours involves basically one axiom (in addition to efficiency), namely the principle of balanced contributions. This makes the relationship between the level structure value, the Shapley value and Myerson's value more direct and apparent. Moreover, the fact that we avoid using an additivity axiom allows us to apply our characterization to many important subdomains without violating the uniqueness. Specifically, our axiomatization applies also to the following subdomains: (i) simple games; (ii) market games (totally balanced games); (iii)convex games; and (iv) super-additive games, as well as many others. This is not the case with the other axiomatizations mentioned above, since in most of these subdomains they fail to yield a unique value. This happens whenever the subdomain is not closed under the summation 1 Myerson (1977) uses a similar property for games with graphs, i.e. the fairness property. This property asserts that by breaking a link that connects two players, both players should lose the same amount.
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of games. For example, the sum of two simple games is no longer a simple game. We return to this issue in the concluding remarks. To obtain our axiomatization we make use of another important concept: the potential function, due to Hart and Mas-Colell (1989). As a by-product, we obtain another characterization for the level structure value, which we will then use as a tool to prove our main result. We start in Section 2 with some basic notations and present two explicit expressions for the level structure value. In Section 3 we use the potential function to characterize the value. Section 4 introduces the axiom of balanced contributions for games with a level structure; we then use this axiom, together with the results established in Section 3, to obtain our main axiomatic characterization for the value. Section 5 is devoted to some concluding remarks.
2. Notations and definitions
We start with some preliminary notations and the definitions of the level structure value. We then introduce some properties of the value which are essential for our current axiomatization. Let • be the set of natural numbers which stands for the universe of players. A game o (with side payments) is a function that assigns a real number to each subset of N. We say that a subset N C N is a support of v if v(S tq N ) = v(S). We denote by F N the set of all pairs (N, v), where v has a finite support on N. Given a finite set of players N, a level structure B is a sequence B = (B 0, B 1. . . . . Bk) of coalition structures (i.e. partitions of N) such that Bi÷ 1 is coarser than B/. That is to say, for each i > 1 , if S E B ~ , then S = Ure~,_t T, where/~/_1 C B i_t. The coalition structure B~ is called the ith level of B. We also assume that Bo is the coalition structure of all singletons, i.e. B 0 = {{i): i E N } and B k = {N}, i.e. B k consists of one bloc which is the grand coalition. The set of all possible B defined on N is denoted L N. We often refer to the coalitions of B~ as blocs or unions. The blocs of B 0 are thus the players in N. Let G N be the set of all triples (N, B, v), where v is a game on F N and B E L N. Let us take a triple (N, B, v), for S C N, and let vls be the game restricted to S, i.e. ols(T ) = o(T tq S) for all T C N. Note that vls is also an element of G s. This is because for all N D S, N is also a finite support of vl,. We use this fact repeatedly in what follows. To simplify the notation, we use N to denote the triple (N, B, v) and S to denote (N, B, vls ). A solution concept for games with a level structure is a function that assigns a payoff vector in R INI for each element in G N. The solution concept we are interested in here is the level structure value (Winter, 1989). Roughly speaking, this concept assigns to each player his expected marginal contribution with respect
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to a uniform distribution over all orders that are consistent with the level structure. We now formulate an explicit expression for this value. Let g2 be the set of all permutations on N. Each to ~ g2 is a bijection from N to N. Given a coalition structure B on N we set /2(B) = {to E / 2 s.t. Vh, m E S ~ B : to(h)lES}, i.e. ~ ( B ) consists of all the permutations in which players of the same coalition appear successively. k For a given level structure B we set I ] ( B ) = Cliff, g](Bj). The level structure value for h is now given by
tph(N, B, v) -
1
~
[o(Ph U h) - v(P~')],
where P:' = {j E N: t o ( ] ) < to(h)} ? A different explicit expression for the value, which we use later, is based on Harsanyi's (1963) idea of dividends. Let us consider a unanimity game U r with the carrier T, i.e. Ur(R ) = 1 if R D T and Ur(R)= 0 otherwise. The payoff for player h in Ur is determined in the following way: at level k - 1, each bloc that has representatives in T gets an equal share of the one-unit payoff. At level k - 2, each bloc that has representatives in T gets an equal share of the total payoff allocated to his super-bloc at level k - 1. This process continues until the lowest level is reached (i.e. the zero level). More precisely: For each h E N let Si(h) be the coalition containing h at level i. Let
Kr(h) =
I{s ~
B,: S c
S'+'(h)
and S f'l T # t~}l,
for i = 0 . . . . , k - 1,
and k-1
r (h) = 11
iffio
Player h's payoff is now given by
~ph(N,B, UT) = K ~ h ) ' if h e T , (0,
otherwise.
To extend the value for a general game v, recall that o can be expressed as a linear combination of unanimity games, i.e. v = ~ r ¢ ~ t a r" U r. Thus, using the additivity property of the value, we have
a÷ q'h(N'B'v)= rcuE Kr(h ) • bET
z We use T U h and T ~ for the m o r e c u m b e r s o m e T U {h} a n d T~{h} respectively.
(1)
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For a formal proof of this expression, see the second part of the proof of theorem 1 in Winter (1989). We now move to another important property of the level structure value, which we make use of later. This property determines the payoff for a bloc at some level by treating this bloc as a player. Let B = (Bt, B 2 , . . . , Bk) E L ~¢. For 1 ~
v(S u T). Let us denote by B~ = (Bi, Bi÷~ . . . . . Bk) the truncated level structure defined on the set of players [B~]. Intuitively, if blocs of B~ act as players, then the relevant level structure becomes B,. Winter (1989) shows that the total payoff for each bloc at level i is the same as the payoff for this bloc when considered as a single player in [Bi]. Formally, we have
~vlSl([Bi],B,,vi) = ~, ~ah(N,B,v)
(SEBi, i=O . . . . , k ) .
(2)
hES
3. The potential We now apply the notion of the potential to characterize the level structure value. Later we also make use of the potential to establish our main axiomatization for the value. The potential was first introduced by Hart and Mas-Colell (1989) to characterize the Shapley value. In their framework, a potential function assigns to each game (N, o) a number ~(N, o) such that the sum of players' marginal contributions to the potential is exactly the value of the grand coalition, i.e.
Y~ ~(N, v) - ~(N, vl,,~ ) = v(N) . hEN
Hart and Mas-Colell have shown that there exists a unique potential function, and that each player's marginal contribution to the potential is exactly his Shapley value in the underlying game. In Winter (1992) the potential was extended to games with a coalition structure, where it is shown that the Owen (1977) value can be characterized using the concept of the potential. We establish here a similar characterization for the level structure value. Let ~d be the set of all triples (N, B, v), i.e. ~ = l..J~c~ G N. Let ~ be a function that assigns to each triple ( N , B , v ) in ~3 a vector ~(N, B, v ) E R IB~I. If $ E B~, then we set ~S(N, B, v) for the component of associated with the bloc 3. For the sake of simplicity, in what follows we use [Bi]
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to denote the game ([Bi],B,,vi). Furthermore, for [S]E[Bi] we denote by [B,I\tS] the game ([B,I\[S], ~B, Given a triple (N, B, v) and h E S E B1, the marginal contribution of player h to the potential is given by D h ~ ( N ) = ~ S ( N ) - ~S(Nkh). In the same manner, if S E B,., i < k, S C T ~ Bi+ 1, then the marginal contribution of player [S] to the potential of the game [Bi] is
v;llB,],[Sl).
o lSl~([B,]) = ~r(IB,] ) - ~ r([B, lqSl). We now introduce the conditions that define the potential function. These conditions extend those imposed by Hart and Mas-Colell for the Shapley value, as well as those used by Winter to characterize the Owen value. is said to be a potential function for games with a level structure if the following three conditions are satisfied:
~. D h ~ ( N ) = Dts]~([B,])
(S ~ B , ) .
(3)
hES
Condition (3) requires that the sum of the marginal contributions of the players in some bloc at the first level be identical to the marginal contribution of this bloc when treated as a player in the game between the blocs. The second condition requires that the sum of marginal c°ntributi°ns to the potential be v(N):
Dh~(N) = v(N) .
(4)
hEN
Finally, we have
VS ~B~: ~)S(T) =0 for all T C N with T fqS =O.
(5)
The third condition is a normalization condition, which is analogous to the Hart and Mas-Colell requirement that the potential of a game on the empty set be zero. We are now ready to characterize the level structure value by means of the potential function. Theorem 1. There exists a unique potential function for games with a level structure. Furthermore, each player's marginal contribution to the potential function is identical to his level structure value, i.e. Dh~(N, B, v) = q~h(N, B, o) for all h E N.
Proof. For each ( N , B , v ) ~ G N, for all S E B 1 and for all T C N such that T N S ~ 0 , we define Kr(S ) = Kr(h), where h E T fqS and Kr(h ) are the coefficients defined in Section 2.3 Let s Kr(S ) is well defined because Kr(h ) = Kr(1) for all h, I E T N S.
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at= ~ RCT
(-1)'-'v(R)
be the unique coefficients of the decomposition of v using unanimity games. We now explicitly define the potential. Let us consider the function ~ ( N ) E R tsxl such that aT
VSEB,: ~S(N)= TCN E Kr(S). First we prove that the marginal contribution to this function yields the value. Let h E S E B~, then aT aT aT Dh~(N)= ~S(N)- ~S(N~h)= ~u K-~S) r~,~ K'-~S) - ~u K~S) TAS#~
TNS#~
= q~h(N),
hET
(6)
where the last equality follows from expression (1). We now show that the function ~, defined above, satisfies the conditions for potential functions. Condition (5) follows directly from the definition of ~S(T). Eq. (4) follows from the efficiency of the level structure value, using (6). To verify (3), we apply the potential function to N and [Bt] together with (2) at the zero level to obtain
DtSl~([Bl])= ~ptSl([B1]) =
~] q~h(N)= ~] h•S
Ohm(N).
hES
The proof is now completed by showing the uniqueness. Note that (3) can be rewritten as ~S(N) : . ~ .1
[DiSl~([B,])+n~s29aS(N~)]
•
(7)
We now continue by induction on the number of levels to prove the uniqueness. Note, first, that for k = 1, ~(N) reduces to the Hart and Mas-Colell potential, which we know is unique. This means that 9~([Bk_~]) is uniquely determined. Suppose now that uniqueness is established for all level structures with k - 1 levels. By the induction assumption, ~([B~]) is uniquely determined. We now have to show that for all S E B~, ~S(N) also is uniquely determined. We proceed by induction on the number of players in S that belong to the support of the game. Let T be a finite support of v. If [S n T[ = 1, then S = {h} and thus ~ S ( N ~ ) = 0, by (5). Using (7) we obtain that ~S(N) is uniquely determined. Let us assume now that the uniqueness of ~S(N) is proved for IS n T[ = m - 1. Under this induction assumption 9~S(N~h) are uniquely determined for all h E S. Moreover, using our previous induction assumption (on the levels), DIsI~([B~]) is uniquely determined as well, which means by (7) that ~S(N) is uniquely
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determined for IS N TI = m. This means that uniqueness is proved for all games with finite support on N. Q.E.D.
4. Balanced contributions This section provides our main axiomatization for the level structure value based on the principle of balanced contributions. We start with the following axiom: Axiom 1. A solution ~: G N - ' > R N is said to satisfy the principle of balanced contributions if, for all S , T E B~ s.t. S , T C R E Bi+ 1 (i = O, 1 . . . . . k - 1), we have d/S(N) - g/S(NNT) = qjT(N) -- Cr(NLS), where oS(N) = E g/(N). iES
The condition of balanced contributions is imposed on any two blocs, S and T, that belong to the same union at higher levels. For such blocs, the condition asserts that the contribution of T to the total payoff of the members in S must be equal to the contribution of S to the total payoff of the members in T. Let us return to the example in Section 1 of the European Parliament as a case of a level structure. The underlying game is a voting game, and the level structure value is a power index that takes into account various levels of cooperation. In this context, our balanced contributions axiom means the following: consider two parties of the same nationality that are represented in the European Parliament (such as the German Social Democrats (SPD) and the German Christian Democrats (CDU)). 4 Since both parties belong to the same bloc at a higher level (they belong to the same nation), our balanced contributions axiom imposes that the total payoff for the SPD should be affected by the presence of the CDU in the same manner that the CDU affects the total payoff of the SPD. Note that such a requirement will make sense especially in cases where the SPD has more in common with the CDU than with any other social democrat party in Europe. This is indeed the case when the underlying issue involves representation in European Community institutions or in various economic decisions. On environmental issues, or the immigration policy, this may not be the case, so partisanship may play a much greater role. 4 SPD and CDU stand for Sozialistische Partei Deutschland and Christliche Demokratische Union, respectively.
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A special case of the property of balanced contributions was used by Myerson (1980) to characterize fair allocation rules for games with conference structures. Myerson (1992) showed that this property is related to the A-equity condition of the Nash bargaining solution. Theorem 2 provides an axiomatization of the level structure value by means of the principle of balanced contributions and efficiency: Axiom 2. A solution qJ is said to be efficient if, for any game v with a finite support N C N, we have EhE N d/h(N) = v(N). Theorem 2. There exists a unique solution for games with a level structure that satisfy balanced contributions and efficiency. This solution is the level structure value.
Proof. Let ~0 denote the level structure value. We start by showing existence. Since ~ satisfies efficiency, it suffices to show that q~ satisfies balanced contributions. Let us take a triple, (N, B, v) E G N, a level, i < k, and suppose that S, T E B i such that S, T C R E B,+~. Recall that for all S E Bi we have ~S(N) = ~lSl([B~]) by Eq. (2) and DIsI~([B~])= ~olsl([Bi]) by Theorem 1. Therefore, we have
~oS(N) - ~oS(NkT) = q~ISl([B,]) - q~tSl([B,]\[T]) = DtSl~([Bi]) - ulSl~([B, N T ] ) = ~ R([B,]) - ~ R ( [ B , ] X [ S l )
= Dtrl~([B,])
- ~ R ( [ B , I \ [ T I ) + ~ R ( [ B , lX{[Sl, [ T I } )
- D[r]~([B,]k[S])
= ~o[r]([Bi]) - ~otr]([B,]X[S])
= ,p ~(N) - ~ ~ ( N ~ ) . Thus, balanced contributions is satisfied. To prove uniqueness, suppose that there exist two solutions, ~b and 4), that satisfy Axioms 1 and 2. We show that ~bS(N) = ¢kS(N) for all blocs S E B~ and for all levels 0 <~i < k. We note first that
~bS(N) = ~ SEB i
~bh(N)=v(N).
hES SEB i
Furthermore, if v has a finite support R C N and R N S = 0 for S E B~, then efficiency implies that OS(N)= O. We now use induction on the levels starting with level k. Because B k = {N} we have, by efficiency, ~ N ( N ) = ~N(N). Let us assume now that the uniqueness is established for levels t, t ~> i, i.e. ~bR(N) = ~bS(N) for all R E B , with k>~t>~i.
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Let R be a bloc in B~, and let K C N be a finite support of v. We denote R~_ ~ = {S E B~_1: S C R } and R~_~(K) = {S E R~_1: S n K ~ I~}. Suppose now that IR~_~(K)I= 1. Then, by efficiency, d/r(N) = ¢hr(N) = 0 for all T E Ri_ 1 such that T O K = t~. So ~bS(N) = dpS(N), where {S} = Ri_l(K ). This is because the induction hypothesis yields
~bS(N) = ~ SERi- 1
¢kS(N)
for all R E B,.
SERi_ 1
We again use an induction argument on the number of blocs that have a non-empty intersection with the support of the game. Let us assume that uniqueness holds for I R , _ , ( K ) I - - m - 1 and consider a game s.t. IR,_ (K)I = m. Again, by efficiency, we have for all T ER~_ 1 with T A K = ~ that qjr(N)=
6 r(N) = O. Let us take S, T ~ R~_I(K ). By the induction assumption, qjS(N~T) = ~bS(N~T) and qjr(N~S)= qbr(N~). This is so because the games OINXT and VJN~S have a support on / ~ T and ~ S , respectively, and thus IR~_~(K~T)I= IRi_t(K~)l = m-1. Using this fact and the balanced contributions property we find that d/S(N)q~S(N) = d/r(N)- ~br(N). Since these differences depend only on R, we denote d(R) = ~OS(N) - 4~S(N) for S E R,_I(K ). Using the first induction hypothesis on levels, which implies that ~bR(N)= 4,R(N), we obtain
0=
E
~bS(N) =
SERi_I(K)
E
¢kS(N) = IR,_,(K)I'd(R).
S~-Ri_I(K)
Thus, d(R)=0 and hence qjS(N)= dpS(N) for all S ~R~_~(K). This yields the uniqueness for blocs at level i - 1. Going down to level 0 with the induction argument, we obtain ~b = ~b. Q.E.D.
5. Concluding remarks (1) We have shown how the principle of balanced contributions, when applied successively to each level of cooperation, can determine the level structure value. We use the domain of all TU games for this axiomatization, but we could have used many other subdomains, provided only that such a subdomain should be closed under restrictions. That is to say, if o is in the domain, then for every coalition S, Ols must be in the domain as well. The domains we mentioned in the introduction, i.e. simple games, totally balanced games, and convex games, all satisfy this condition and thus qualify as domains for which the same axiomatization applies, using precisely the same proof as that of Theorem 2. Our characterization of the value by means of the potential has, in fact, the
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s a m e d e s i r a b l e p r o p e r t y . H o w e v e r , t h e s t a n d a r d a x i o m a t i z a t i o n s for t h e v a l u e , i.e. H a r t a n d K u r z (1983) a n d W i n t e r (1989), d o n o t satisfy this p r o p e r t y b e c a u s e t h e y use an a d d i t i v i t y a x i o m . ( 2 ) N o t e t h a t o u r a x i o m a t i z a t i o n b a s i c a l l y uses a single p r o p e r t y ( b e s i d e s e f f i c i e n c y ) , i.e. t h e p r i n c i p l e o f b a l a n c e d c o n t r i b u t i o n s a p p l i e d to a c o l l e c t i o n o f c o a l i t i o n s . P o s s i b l y , b y i m p o s i n g this p r o p e r t y o n a d i f f e r e n t c o l l e c t i o n o f c o a l i t i o n s , we m i g h t c o m e u p with a n e w i n t e r e s t i n g c o o p e r a t i o n s t r u c t u r e e x t e n s i o n for t h e v a l u e .
Acknowledgements We are grateful for the constructive suggestions made by two anonymous r e f e r e e s , w h i c h h e l p e d us i m p r o v e an e a r l i e r v e r s i o n o f this p a p e r . T h e a u t h o r s a c k n o w l e d g e financial s u p p o r t f r o m t h e U n i v e r s i d a d d e l P a l s V a s c o ( p r o y e c t o UPV 036.321-HA127/93).
References W. Albers, Core- and kernel-variants based on imputations and demand profiles, in: O. Moeschin and D. Pallaschke, eds., Game Theory and Related Topics (North-Holland, Amsterdam, 1979). R.J. Aumann and M. Maschler, The bargaining set for cooperative games, in: M. Dresher, L.S. Shapley and A.W. Tucker, eds., Advances in Game theory (Princeton University Press, Princeton, NJ, 1964), 443-447. E. Bennett, The aspiration approach to predicting coalition formation and payoff distribution in sidepayment games, Int. J. Game Theory 12 (1983) 1-18. M. Davis and M. Maschler, The kernel of a cooperative game, Nay. Res. Logist. Quart. 12 (1965) 223-259. J.C. Harsanyi, A simplified bargaining model for n-person cooperative game, Int. Econ. Rev. 4 (1963) 194-220. S. Hart and M. Kurz, Endogenous formation of coalitions, Econometrica 51 (1983) 1047-1064. S. Hart and M. Kurz, Stable coalition structures, in: M.T. Holler, ed., Coalition and Collection Action (Physica, W/irzburg, 1984), 235-258. S. Hart and A. Mas-Colell, Potential, value and consistency, Econometrica 3 (1989) 589-614. R.B. Myerson, Graphs and cooperation in games, Math. Oper. Res. 2 (1977) 225-229. R.B. Myerson, Conference structures and fair allocation rules, Int. J. Game Theory 9 (1980) 169-182. R.B. Myerson, Fictitious-transfer solutions in cooperative game theory, in: R. Selten, ed.. Rational Interaction (Springer, Berlin, 1992). G. Owen, Values of games with a priori unions, in: R. Henn and O. Moeschlin, eds., Mathematical Economics and Game Theory (Springer, Berlin, 1977). R. Selten, A non cooperative model of characteristic function bargaining, in: V. Bohm and H. Nachtbamp, eds., Essays in Game Theory and Mathematical Economics in Honor of Oscar Morgenstern (Bibliographisches Institut, Mannheim, 1981), 131-151. L.S. Shapley, A value for n-person games, in: H.W. Kuhn and A.W. Tucker, eds., Contributions to the Theory of Games, II (Princeton University Press, Princeton, NJ, 1953), 307-317.
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mathematical
soda/ sciences
ELSEVIER
Mathematical Social Sciences 31 (1996) 171-182
The principle of balanced contributions and hierarchies of cooperation Emilio Calvo a'*, J. Javier Lasaga a, Eyal W i n t e r b "Departamento de Economia Aplicada 1, Universidad del Pais Vasco, Avenida Lehendakari Aguirre 83, 48015 Bilbao, Spain bCenter for Rationality and Interactive Decision Theory, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Received December 1994; revised June 1995
Abstract The principle of balanced contributions has appeared repeatedly in the literature on the Shapley value. This principle is akin to the reciprocity properties shared by almost all cooperative solution concepts. We provide a new axiomatization for the level structure value. This axiomatization has the advantage that it can be applied to many important subdomains of TU games. We use the Hart-Mas-Colell potential function as a tool to prove our main result, and establish another interesting characterization for the value as a by-product.
Keywords: TU games; Shapley value; Balanced contributions; Potential function; Level structures
1. I n t r o d u c t i o n
Most cooperative solution concepts possess a reciprocity property between the players. This is at least the case for the kernel (Davis and Maschler, 1965), the bargaining set (Aumann and Maschler, 1964), the set of semi-stable demand vectors (Albers, 1979; Bennett, 1983; Selten, 1981), and the Shapley (1953) value. In some cases the reciprocity property simply means a symmetry axiom. In other cases, like the bargaining set and the kernel, it involves a more complicated * Corresponding author. 0165-4896/96/$15.00 (~) 1996 - Elsevier Science B.V. All rights reserved S S D I 0165-4896(95)00806-3
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symmetric binary relation on the set of players. Loosely speaking, the common idea behind these properties is that a reasonable solution concept must be balanced in the sense that each player i affects player j's payoff in the same manner that player j affects i's payoff. In the literature on the Shapley value a principle o f this sort was introduced by Myerson (1980)) Myerson's principle of balanced contributions asserts that, for any two players, the amount that each player would gain or lose by the other's withdrawal from the game should be equal. Myerson used this principle to extend the Shapley value for games with conference structures, but a special case of his results provides an axiomatization for the Shapley value. Our objective in this paper is to use the same principle to axiomatize the cooperation structure extension of the Shapley value, namely the level structure value, due to Owen (1977) and Winter (1989). A level structure describes a hierarchy of cooperation between the players. Formally, it is given by a sequence of partitions of the set of players, each coarser than the previous one. This reflects a situation in which players unite into groups to improve their bargaining power. These groups then form a coarser partition by creating larger groups, which can again extend even further. Consider, for example, a game involving the members of the European Parliament as players. The first level divides the set of players according to their party membership, but there is also a coarser partition, i.e. dividing them by nations, or an even coarser one based on cooperation at the international level. The value for games with a level structure, which we define below, will take into account this hierarchy of cooperation in determining the final payoff distribution. The level structure value was axiomatized by Winter (1989). The special case involving a single level was also axiomatized by Owen (1977) and Hart and Kurz (1983, 1984). We propose a different sort of axiomatization for this value. Unlike the characterizations mentioned above, ours involves basically one axiom (in addition to efficiency), namely the principle of balanced contributions. This makes the relationship between the level structure value, the Shapley value and Myerson's value more direct and apparent. Moreover, the fact that we avoid using an additivity axiom allows us to apply our characterization to many important subdomains without violating the uniqueness. Specifically, our axiomatization applies also to the following subdomains: (i) simple games; (ii) market games (totally balanced games); (iii)convex games; and (iv) super-additive games, as well as many others. This is not the case with the other axiomatizations mentioned above, since in most of these subdomains they fail to yield a unique value. This happens whenever the subdomain is not closed under the summation 1 Myerson (1977) uses a similar property for games with graphs, i.e. the fairness property. This property asserts that by breaking a link that connects two players, both players should lose the same amount.
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of games. For example, the sum of two simple games is no longer a simple game. We return to this issue in the concluding remarks. To obtain our axiomatization we make use of another important concept: the potential function, due to Hart and Mas-Colell (1989). As a by-product, we obtain another characterization for the level structure value, which we will then use as a tool to prove our main result. We start in Section 2 with some basic notations and present two explicit expressions for the level structure value. In Section 3 we use the potential function to characterize the value. Section 4 introduces the axiom of balanced contributions for games with a level structure; we then use this axiom, together with the results established in Section 3, to obtain our main axiomatic characterization for the value. Section 5 is devoted to some concluding remarks.
2. Notations and definitions
We start with some preliminary notations and the definitions of the level structure value. We then introduce some properties of the value which are essential for our current axiomatization. Let • be the set of natural numbers which stands for the universe of players. A game o (with side payments) is a function that assigns a real number to each subset of N. We say that a subset N C N is a support of v if v(S tq N ) = v(S). We denote by F N the set of all pairs (N, v), where v has a finite support on N. Given a finite set of players N, a level structure B is a sequence B = (B 0, B 1. . . . . Bk) of coalition structures (i.e. partitions of N) such that Bi÷ 1 is coarser than B/. That is to say, for each i > 1 , if S E B ~ , then S = Ure~,_t T, where/~/_1 C B i_t. The coalition structure B~ is called the ith level of B. We also assume that Bo is the coalition structure of all singletons, i.e. B 0 = {{i): i E N } and B k = {N}, i.e. B k consists of one bloc which is the grand coalition. The set of all possible B defined on N is denoted L N. We often refer to the coalitions of B~ as blocs or unions. The blocs of B 0 are thus the players in N. Let G N be the set of all triples (N, B, v), where v is a game on F N and B E L N. Let us take a triple (N, B, v), for S C N, and let vls be the game restricted to S, i.e. ols(T ) = o(T tq S) for all T C N. Note that vls is also an element of G s. This is because for all N D S, N is also a finite support of vl,. We use this fact repeatedly in what follows. To simplify the notation, we use N to denote the triple (N, B, v) and S to denote (N, B, vls ). A solution concept for games with a level structure is a function that assigns a payoff vector in R INI for each element in G N. The solution concept we are interested in here is the level structure value (Winter, 1989). Roughly speaking, this concept assigns to each player his expected marginal contribution with respect
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to a uniform distribution over all orders that are consistent with the level structure. We now formulate an explicit expression for this value. Let g2 be the set of all permutations on N. Each to ~ g2 is a bijection from N to N. Given a coalition structure B on N we set /2(B) = {to E / 2 s.t. Vh, m E S ~ B : to(h)
tph(N, B, v) -
1
~
[o(Ph U h) - v(P~')],
where P:' = {j E N: t o ( ] ) < to(h)} ? A different explicit expression for the value, which we use later, is based on Harsanyi's (1963) idea of dividends. Let us consider a unanimity game U r with the carrier T, i.e. Ur(R ) = 1 if R D T and Ur(R)= 0 otherwise. The payoff for player h in Ur is determined in the following way: at level k - 1, each bloc that has representatives in T gets an equal share of the one-unit payoff. At level k - 2, each bloc that has representatives in T gets an equal share of the total payoff allocated to his super-bloc at level k - 1. This process continues until the lowest level is reached (i.e. the zero level). More precisely: For each h E N let Si(h) be the coalition containing h at level i. Let
Kr(h) =
I{s ~
B,: S c
S'+'(h)
and S f'l T # t~}l,
for i = 0 . . . . , k - 1,
and k-1
r (h) = 11
iffio
Player h's payoff is now given by
~ph(N,B, UT) = K ~ h ) ' if h e T , (0,
otherwise.
To extend the value for a general game v, recall that o can be expressed as a linear combination of unanimity games, i.e. v = ~ r ¢ ~ t a r" U r. Thus, using the additivity property of the value, we have
a÷ q'h(N'B'v)= rcuE Kr(h ) • bET
z We use T U h and T ~ for the m o r e c u m b e r s o m e T U {h} a n d T~{h} respectively.
(1)
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For a formal proof of this expression, see the second part of the proof of theorem 1 in Winter (1989). We now move to another important property of the level structure value, which we make use of later. This property determines the payoff for a bloc at some level by treating this bloc as a player. Let B = (Bt, B 2 , . . . , Bk) E L ~¢. For 1 ~
v(S u T). Let us denote by B~ = (Bi, Bi÷~ . . . . . Bk) the truncated level structure defined on the set of players [B~]. Intuitively, if blocs of B~ act as players, then the relevant level structure becomes B,. Winter (1989) shows that the total payoff for each bloc at level i is the same as the payoff for this bloc when considered as a single player in [Bi]. Formally, we have
~vlSl([Bi],B,,vi) = ~, ~ah(N,B,v)
(SEBi, i=O . . . . , k ) .
(2)
hES
3. The potential We now apply the notion of the potential to characterize the level structure value. Later we also make use of the potential to establish our main axiomatization for the value. The potential was first introduced by Hart and Mas-Colell (1989) to characterize the Shapley value. In their framework, a potential function assigns to each game (N, o) a number ~(N, o) such that the sum of players' marginal contributions to the potential is exactly the value of the grand coalition, i.e.
Y~ ~(N, v) - ~(N, vl,,~ ) = v(N) . hEN
Hart and Mas-Colell have shown that there exists a unique potential function, and that each player's marginal contribution to the potential is exactly his Shapley value in the underlying game. In Winter (1992) the potential was extended to games with a coalition structure, where it is shown that the Owen (1977) value can be characterized using the concept of the potential. We establish here a similar characterization for the level structure value. Let ~d be the set of all triples (N, B, v), i.e. ~ = l..J~c~ G N. Let ~ be a function that assigns to each triple ( N , B , v ) in ~3 a vector ~(N, B, v ) E R IB~I. If $ E B~, then we set ~S(N, B, v) for the component of associated with the bloc 3. For the sake of simplicity, in what follows we use [Bi]
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to denote the game ([Bi],B,,vi). Furthermore, for [S]E[Bi] we denote by [B,I\tS] the game ([B,I\[S], ~B, Given a triple (N, B, v) and h E S E B1, the marginal contribution of player h to the potential is given by D h ~ ( N ) = ~ S ( N ) - ~S(Nkh). In the same manner, if S E B,., i < k, S C T ~ Bi+ 1, then the marginal contribution of player [S] to the potential of the game [Bi] is
v;llB,],[Sl).
o lSl~([B,]) = ~r(IB,] ) - ~ r([B, lqSl). We now introduce the conditions that define the potential function. These conditions extend those imposed by Hart and Mas-Colell for the Shapley value, as well as those used by Winter to characterize the Owen value. is said to be a potential function for games with a level structure if the following three conditions are satisfied:
~. D h ~ ( N ) = Dts]~([B,])
(S ~ B , ) .
(3)
hES
Condition (3) requires that the sum of the marginal contributions of the players in some bloc at the first level be identical to the marginal contribution of this bloc when treated as a player in the game between the blocs. The second condition requires that the sum of marginal c°ntributi°ns to the potential be v(N):
Dh~(N) = v(N) .
(4)
hEN
Finally, we have
VS ~B~: ~)S(T) =0 for all T C N with T fqS =O.
(5)
The third condition is a normalization condition, which is analogous to the Hart and Mas-Colell requirement that the potential of a game on the empty set be zero. We are now ready to characterize the level structure value by means of the potential function. Theorem 1. There exists a unique potential function for games with a level structure. Furthermore, each player's marginal contribution to the potential function is identical to his level structure value, i.e. Dh~(N, B, v) = q~h(N, B, o) for all h E N.
Proof. For each ( N , B , v ) ~ G N, for all S E B 1 and for all T C N such that T N S ~ 0 , we define Kr(S ) = Kr(h), where h E T fqS and Kr(h ) are the coefficients defined in Section 2.3 Let s Kr(S ) is well defined because Kr(h ) = Kr(1) for all h, I E T N S.
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at= ~ RCT
(-1)'-'v(R)
be the unique coefficients of the decomposition of v using unanimity games. We now explicitly define the potential. Let us consider the function ~ ( N ) E R tsxl such that aT
VSEB,: ~S(N)= TCN E Kr(S). First we prove that the marginal contribution to this function yields the value. Let h E S E B~, then aT aT aT Dh~(N)= ~S(N)- ~S(N~h)= ~u K-~S) r~,~ K'-~S) - ~u K~S) TAS#~
TNS#~
= q~h(N),
hET
(6)
where the last equality follows from expression (1). We now show that the function ~, defined above, satisfies the conditions for potential functions. Condition (5) follows directly from the definition of ~S(T). Eq. (4) follows from the efficiency of the level structure value, using (6). To verify (3), we apply the potential function to N and [Bt] together with (2) at the zero level to obtain
DtSl~([Bl])= ~ptSl([B1]) =
~] q~h(N)= ~] h•S
Ohm(N).
hES
The proof is now completed by showing the uniqueness. Note that (3) can be rewritten as ~S(N) : . ~ .1
[DiSl~([B,])+n~s29aS(N~)]
•
(7)
We now continue by induction on the number of levels to prove the uniqueness. Note, first, that for k = 1, ~(N) reduces to the Hart and Mas-Colell potential, which we know is unique. This means that 9~([Bk_~]) is uniquely determined. Suppose now that uniqueness is established for all level structures with k - 1 levels. By the induction assumption, ~([B~]) is uniquely determined. We now have to show that for all S E B~, ~S(N) also is uniquely determined. We proceed by induction on the number of players in S that belong to the support of the game. Let T be a finite support of v. If [S n T[ = 1, then S = {h} and thus ~ S ( N ~ ) = 0, by (5). Using (7) we obtain that ~S(N) is uniquely determined. Let us assume now that the uniqueness of ~S(N) is proved for IS n T[ = m - 1. Under this induction assumption 9~S(N~h) are uniquely determined for all h E S. Moreover, using our previous induction assumption (on the levels), DIsI~([B~]) is uniquely determined as well, which means by (7) that ~S(N) is uniquely
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determined for IS N TI = m. This means that uniqueness is proved for all games with finite support on N. Q.E.D.
4. Balanced contributions This section provides our main axiomatization for the level structure value based on the principle of balanced contributions. We start with the following axiom: Axiom 1. A solution ~: G N - ' > R N is said to satisfy the principle of balanced contributions if, for all S , T E B~ s.t. S , T C R E Bi+ 1 (i = O, 1 . . . . . k - 1), we have d/S(N) - g/S(NNT) = qjT(N) -- Cr(NLS), where oS(N) = E g/(N). iES
The condition of balanced contributions is imposed on any two blocs, S and T, that belong to the same union at higher levels. For such blocs, the condition asserts that the contribution of T to the total payoff of the members in S must be equal to the contribution of S to the total payoff of the members in T. Let us return to the example in Section 1 of the European Parliament as a case of a level structure. The underlying game is a voting game, and the level structure value is a power index that takes into account various levels of cooperation. In this context, our balanced contributions axiom means the following: consider two parties of the same nationality that are represented in the European Parliament (such as the German Social Democrats (SPD) and the German Christian Democrats (CDU)). 4 Since both parties belong to the same bloc at a higher level (they belong to the same nation), our balanced contributions axiom imposes that the total payoff for the SPD should be affected by the presence of the CDU in the same manner that the CDU affects the total payoff of the SPD. Note that such a requirement will make sense especially in cases where the SPD has more in common with the CDU than with any other social democrat party in Europe. This is indeed the case when the underlying issue involves representation in European Community institutions or in various economic decisions. On environmental issues, or the immigration policy, this may not be the case, so partisanship may play a much greater role. 4 SPD and CDU stand for Sozialistische Partei Deutschland and Christliche Demokratische Union, respectively.
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A special case of the property of balanced contributions was used by Myerson (1980) to characterize fair allocation rules for games with conference structures. Myerson (1992) showed that this property is related to the A-equity condition of the Nash bargaining solution. Theorem 2 provides an axiomatization of the level structure value by means of the principle of balanced contributions and efficiency: Axiom 2. A solution qJ is said to be efficient if, for any game v with a finite support N C N, we have EhE N d/h(N) = v(N). Theorem 2. There exists a unique solution for games with a level structure that satisfy balanced contributions and efficiency. This solution is the level structure value.
Proof. Let ~0 denote the level structure value. We start by showing existence. Since ~ satisfies efficiency, it suffices to show that q~ satisfies balanced contributions. Let us take a triple, (N, B, v) E G N, a level, i < k, and suppose that S, T E B i such that S, T C R E B,+~. Recall that for all S E Bi we have ~S(N) = ~lSl([B~]) by Eq. (2) and DIsI~([B~])= ~olsl([Bi]) by Theorem 1. Therefore, we have
~oS(N) - ~oS(NkT) = q~ISl([B,]) - q~tSl([B,]\[T]) = DtSl~([Bi]) - ulSl~([B, N T ] ) = ~ R([B,]) - ~ R ( [ B , ] X [ S l )
= Dtrl~([B,])
- ~ R ( [ B , I \ [ T I ) + ~ R ( [ B , lX{[Sl, [ T I } )
- D[r]~([B,]k[S])
= ~o[r]([Bi]) - ~otr]([B,]X[S])
= ,p ~(N) - ~ ~ ( N ~ ) . Thus, balanced contributions is satisfied. To prove uniqueness, suppose that there exist two solutions, ~b and 4), that satisfy Axioms 1 and 2. We show that ~bS(N) = ¢kS(N) for all blocs S E B~ and for all levels 0 <~i < k. We note first that
~bS(N) = ~ SEB i
~bh(N)=v(N).
hES SEB i
Furthermore, if v has a finite support R C N and R N S = 0 for S E B~, then efficiency implies that OS(N)= O. We now use induction on the levels starting with level k. Because B k = {N} we have, by efficiency, ~ N ( N ) = ~N(N). Let us assume now that the uniqueness is established for levels t, t ~> i, i.e. ~bR(N) = ~bS(N) for all R E B , with k>~t>~i.
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Let R be a bloc in B~, and let K C N be a finite support of v. We denote R~_ ~ = {S E B~_1: S C R } and R~_~(K) = {S E R~_1: S n K ~ I~}. Suppose now that IR~_~(K)I= 1. Then, by efficiency, d/r(N) = ¢hr(N) = 0 for all T E Ri_ 1 such that T O K = t~. So ~bS(N) = dpS(N), where {S} = Ri_l(K ). This is because the induction hypothesis yields
~bS(N) = ~ SERi- 1
¢kS(N)
for all R E B,.
SERi_ 1
We again use an induction argument on the number of blocs that have a non-empty intersection with the support of the game. Let us assume that uniqueness holds for I R , _ , ( K ) I - - m - 1 and consider a game s.t. IR,_ (K)I = m. Again, by efficiency, we have for all T ER~_ 1 with T A K = ~ that qjr(N)=
6 r(N) = O. Let us take S, T ~ R~_I(K ). By the induction assumption, qjS(N~T) = ~bS(N~T) and qjr(N~S)= qbr(N~). This is so because the games OINXT and VJN~S have a support on / ~ T and ~ S , respectively, and thus IR~_~(K~T)I= IRi_t(K~)l = m-1. Using this fact and the balanced contributions property we find that d/S(N)q~S(N) = d/r(N)- ~br(N). Since these differences depend only on R, we denote d(R) = ~OS(N) - 4~S(N) for S E R,_I(K ). Using the first induction hypothesis on levels, which implies that ~bR(N)= 4,R(N), we obtain
0=
E
~bS(N) =
SERi_I(K)
E
¢kS(N) = IR,_,(K)I'd(R).
S~-Ri_I(K)
Thus, d(R)=0 and hence qjS(N)= dpS(N) for all S ~R~_~(K). This yields the uniqueness for blocs at level i - 1. Going down to level 0 with the induction argument, we obtain ~b = ~b. Q.E.D.
5. Concluding remarks (1) We have shown how the principle of balanced contributions, when applied successively to each level of cooperation, can determine the level structure value. We use the domain of all TU games for this axiomatization, but we could have used many other subdomains, provided only that such a subdomain should be closed under restrictions. That is to say, if o is in the domain, then for every coalition S, Ols must be in the domain as well. The domains we mentioned in the introduction, i.e. simple games, totally balanced games, and convex games, all satisfy this condition and thus qualify as domains for which the same axiomatization applies, using precisely the same proof as that of Theorem 2. Our characterization of the value by means of the potential has, in fact, the
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s a m e d e s i r a b l e p r o p e r t y . H o w e v e r , t h e s t a n d a r d a x i o m a t i z a t i o n s for t h e v a l u e , i.e. H a r t a n d K u r z (1983) a n d W i n t e r (1989), d o n o t satisfy this p r o p e r t y b e c a u s e t h e y use an a d d i t i v i t y a x i o m . ( 2 ) N o t e t h a t o u r a x i o m a t i z a t i o n b a s i c a l l y uses a single p r o p e r t y ( b e s i d e s e f f i c i e n c y ) , i.e. t h e p r i n c i p l e o f b a l a n c e d c o n t r i b u t i o n s a p p l i e d to a c o l l e c t i o n o f c o a l i t i o n s . P o s s i b l y , b y i m p o s i n g this p r o p e r t y o n a d i f f e r e n t c o l l e c t i o n o f c o a l i t i o n s , we m i g h t c o m e u p with a n e w i n t e r e s t i n g c o o p e r a t i o n s t r u c t u r e e x t e n s i o n for t h e v a l u e .
Acknowledgements We are grateful for the constructive suggestions made by two anonymous r e f e r e e s , w h i c h h e l p e d us i m p r o v e an e a r l i e r v e r s i o n o f this p a p e r . T h e a u t h o r s a c k n o w l e d g e financial s u p p o r t f r o m t h e U n i v e r s i d a d d e l P a l s V a s c o ( p r o y e c t o UPV 036.321-HA127/93).
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