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Weighted component fairness for forest games
Mathematical Social Sciences 64 (2012) 144–151
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Weighted component fairness for forest games✩ Sylvain Béal a,∗ , Eric Rémila b , Philippe Solal c a
Université de Franche-Comté, CRESE, 30 Avenue de l’Observatoire, 25009 Besançon, France
b
Université de Lyon, LIP, UMR 5668 CNRS-ENS Lyon-Université Lyon 1, France
c
Université de Saint-Etienne, CNRS UMR 5824 GATE Lyon Saint-Etienne, France
article
info
Article history: Available online 30 March 2012
abstract We study the set of allocation rules generated by component efficiency and weighted component fairness, a generalization of component fairness introduced by Herings et al. (2008). Firstly, if the underlying TU-game is superadditive, this set coincides with the core of a graph-restricted game associated with the forest game. Secondly, among this set, only the random tree solutions (Béal et al., 2010) induce Harsanyi payoff vectors for the associated graph-restricted game. We then obtain a new characterization of the random tree solutions in terms of component efficiency and weighted component fairness. © 2012 Elsevier B.V. All rights reserved.
1. Introduction In this article we study TU-games where the cooperation possibilities are represented by the links of an undirected communication graph as in Myerson (1977). We assume that the communication graph is a forest, i.e. each component of the graph is a tree. A forest game is a pair consisting of a TU-game and a forest on the agent set. Herings et al. (2008) introduce the Average Tree solution for the class of forest games, which can be characterized by two axioms: component efficiency and component fairness. Component efficiency asserts that the members of a component ought to allocate to themselves the total worth available to them. When a link is severed in a component of a forest, this component breaks up into two new components that will be called the proper cones incident to the link. Component fairness asserts that deleting a link in a component yields for both resulting proper cones the same average change in payoff, where the average is taken over the players in each cone. Combining component efficiency with component fairness implies that the members of each proper cone receive the sum of two parts: the worth of their proper cone and a share of the surplus
✩ For helpful comments received, the authors want to thank an anonymous referee, Gerard van der Laan, Hans Peters and the participants at S.I.N.G. 5 in Amsterdam and at the conference ‘‘Bargaining, Evolution and Networks’’ in honor of Hans Haller. Financial support by the National Agency for Research (ANR) – research program ‘‘Models of Influence and Network Theory’’ (MINT) ANR.09.BLANC-0321.03 – and the ‘‘Mathématiques de la décision pour l’ingénierie physique et sociale’’ (MODMAD) project is gratefully acknowledged. ∗ Corresponding author. Tel.: +33 0 3 81 66; fax: +33 0 3 81 66. E-mail addresses: [email protected] (S. Béal), [email protected] (E. Rémila), [email protected] (P. Solal).
0165-4896/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.03.004
generated by the deletion of the link. This share is precisely the relative size of this proper cone. In a sense, one can consider these cones as two bargaining entities which agree upon the distribution of a surplus according to their relative sizes. Many other sharing systems can be considered. For instance, Brink et al. (2007) study line-graph games and their axiom of equal loss property implies that the two cones will get the same share of the surplus. In this article, we explore the consequences of this specific sharing system. As a further extension, we introduce a system of weights that determines the share of the surplus between the two proper cones incident to a link. Then, we keep component efficiency as an axiom, replace component fairness by the new axiom of weighted component fairness, and study three properties of the induced set of allocation rules. Firstly, we establish that for a fixed system of weights, component efficiency and weighted component fairness yield a unique allocation rule on the class of forest games. The expression of this rule reveals that it induces a payoff vector of the cone-restricted game associated with the forest game, which is defined as the restriction of the underlying TU-game on the set of cones of the forest. Secondly, we explore the relationships between the set of allocation rules generated by component efficiency and weighted component fairness, and the core of the cone-restricted games associated with the forest games. We provide a necessary and sufficient condition for the nonemptiness of the core of a conerestricted game and then show that it forms a polytope within each component. Then, we prove that a payoff vector belongs to the (nonempty) core of the cone-restricted game if and only if it is obtained by component efficiency and weighted component fairness for some system of weights. This result is encouraging because it points out that fairness and stability considerations are not always incompatible. As far as we know it is the first
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
characterization of the core in terms of fairness, although it is obtained on the cone-restricted game and not on the game itself. Furthermore, we find that the extreme points of the core are computed through the set of all orientations of a forest. This result is not an extension of the characterization of the core of convex TU-games in terms of the marginal vectors associated with all permutations of the player set (see Shapley, 1971), even if the orientations of a forest are used in a similar fashion as the permutations of the players. Thirdly, we establish a link between the random tree solutions of a forest game and the Harsanyi solutions of the associated cone-restricted game. Harsanyi solutions have been studied by Derks et al. (2000) for games where cooperation possibilities are unrestricted. The average tree solution can be computed as the average of marginal contribution vectors defined according to all rooted trees of a forest, where a rooted tree consists in directing all links away from a selected agent, called the root. A random tree solution consists in a weighted average of these marginal contribution vectors, where the weights are given by a probability distribution over the set of contribution vectors. We show that among the component efficient and weighted component fair allocation rules, only the random tree solutions generate Harsanyi payoff vectors in the associated cone-restricted games. Since core allocations of the cone-restricted games with a nonempty core are generated by component efficiency and weighted component fairness, it amounts to saying that the search for Harsanyi solutions among core allocations boils down to selecting the set of random tree solutions. As a corollary, we provide an alternative characterization of the random tree solutions in terms of component efficiency and weighted component fairness for suitable systems of weights. While Béal et al. (2010) characterize the set of random tree solutions by axioms that are similar to the axioms used by Shapley (1953) for the characterization of the Shapley value for TU-games, this result is more appealing in the framework of graph games since the considered axioms do not rely exclusively on the characteristic function of the games. This article is organized as follows. Section 2 contains definitions and a preliminary result. The axiom of weighted component fairness is introduced in Section 3, in which we establish that, for a given system of weights, component efficiency and weighted component fairness yield a unique allocation rule on the class of forest games. In Section 4, we investigate the structure of the core of the cone-restricted game. Section 5 is devoted to the relationship between the Harsanyi solutions of the cone-restricted game and the random tree solutions. Section 6 concludes. 2. Preliminaries 2.1. TU-games For a finite set of agents N = {1, . . . , n}, each S ∈ 2N is a coalition of cardinality denoted by s. In many practical situations some coalitions may not be meaningful, because of the lack of communication possibilities, or certain institutional constraints. Thus we shall consider a subcollection Ω ⊆ 2N of feasible coalitions with the convention that ∅ ∈ Ω (see Bilbao, 2000 and Grabisch, 2009). A TU-game on N is a characteristic function v : Ω −→ R such that v(∅) = 0. For each S ∈ Ω , v(S ) is interpreted as the worth of S. By C (Ω ), we denote the real linear space of all TU-games v on Ω , which can be identified with the corresponding linear sub-space of R|Ω | of dimension |Ω | − 1. A payoff vector x ∈ Rn of v ∈ C (Ω ) is a vector giving a payoff xi ∈ R to any agent i ∈ N. For each S ∈ 2N , the notation xS stands for i∈S xi . An allocation rule on C (Ω ) is a map f that assigns to each v ∈ C (Ω ) a payoff vector f (v) ∈ Rn .
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2.2. Graphs The restrictions on communication possibilities faced by the group of agents N can be represented by an undirected graph (N , L), where the set of nodes is the set N, and the set of links L is a subset of the set of unordered pairs of elements of N. For simplicity, we write ij for the link {i, j}. The set Li = {j ∈ N : ij ∈ L} is the set of links of i ∈ N. For each S ∈ 2N \ {∅}, L(S ) = {ij ∈ L : {i, j} ⊆ S } is the set of links between agents in coalition S. The graph (S , L(S )) is the subgraph of (N , L) induced by S. A sequence of distinct agents (i1 , . . . , ip ) is a path in (N , L) if ik ik+1 ∈ L for k = 1, . . . , p − 1. A graph is connected if there exists a path between any two agents, and it constitutes a tree if this path is unique for any two agents. A coalition S is connected if (S , L(S )) is connected. A coalition C ∈ 2N is a component of (N , L) if (C , L(C )) is connected and for each i ∈ N \ C , the subgraph (C ∪ {i}, L(C ∪ {i})) is not connected. The set of components of (N , L), denoted by N /L, forms a partition of N. A forest is a graph in which each component is a tree. The set of cones of a forest (N , L) consists of the set of components N /L, the set ∅, and for each C ∈ N /L and ij ∈ L(C ) the two components of (C , L(C ) \ {ij}). A cone strictly included in C is a proper cone of C . The unique agent of a nonempty proper cone K who has a link with the complement C \ K is called the head of K . Note that K ⊆ C is a cone if and only if the coalition C \ K is connected. In order to insist on the link ij, K(j,i) denotes the cone with head i and K(i,j) its complement with head j in the corresponding component. Let ∆L and ∆0L be the set of cones and the set of nonempty proper cones of (N , L) respectively. An orientation ◦ of a graph (N , L) is a directed graph (N , L◦ ) obtained from (N , L) by replacing each link ij by a directed link: either (i, j) or (j, i). Let ◦(ij) be the orientation of link ij ∈ L. 2.3. Solutions for graph games and cone-restricted games A graph game on N is a pair (v, L) such that v ∈ C (2N ) is a TU-game and (N , L) is a communication graph. In what follows, we only consider the set F of all forest games on N, i.e. the set of all pairs (v, L) where v ∈ C (2N ) and (N , L) is a forest. An allocation rule on F is a map f that assigns to each (v, L) ∈ F a payoff vector f (v, L) ∈ Rn . While for the class C (Ω ) it is explicitly assumed that only coalitions in Ω are feasible, a graph game (v, L) indicates that the entire set of coalitions 2N may be affected by (N , L). The payoff vector f (v, L) determines how the communication links affect both the formation of coalitions and the process of redistribution. Since the communication graph is meant to reduce the set of feasible coalitions, it is often convenient to express f (v, L) as a payoff vector g (v Ω ) on some class C (Ω ) where v Ω : Ω −→ R is the restriction of v to the set Ω , i.e. for all S ∈ Ω , v Ω (S ) = v(S ). The restricted game v Ω incorporates both the possible gains from cooperation as modeled by v and the restrictions on communication induced by (N , L) and reflected by Ω . The axioms that are considered can tell how the restrictions imposed by communication graph (N , L) shrinks the set of feasible coalitions from 2N to Ω . For instance, if one interprets the communication graph as the set of direct and indirect communication possibilities between the agents, then a minimal requirement is that Ω should be included in the set of connected coalitions. In the following, we will consider the set of cones of a forest as the set of feasible coalitions. For a forest (N , L), let C (∆L ) be the set of cone-restricted games. There are two justifications for considering the class C (∆L ). Firstly, the axioms that we are going to present on F will imply that the coalitions in ∆L are necessary and sufficient for computing a solution. Secondly, two complementary cones reflect the original idea of von Neumann and Morgenstern
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in which the set of players is split in two competing coalitions. An interpretation in terms of Nash bargaining will also be discussed later on in the article. On the class F , Herings et al. (2008) introduce a new allocation rule called the Average Tree solution, that can be characterized by component efficiency and component fairness. Component efficiency requires that the payoffs in a component add up to the worth of that component. Component efficiency. For each (v, L) ∈ F and each C ∈ N /L, it holds that: fC (v, L) = v(C ). Component fairness says that deleting a link between two agents yields for both resulting cones the same average change in payoffs. Component fairness therefore emphasizes that the gains associated with linking one cone to its complement in a component should be attributed to these two proper cones, rather than to the two agents whose link is deleted, and the gains should be proportional to the size of the cones. Component fairness. For each (v, L) ∈ F and each link ij ∈ L, it holds that: 1 fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) k(j,i)
=
1 fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}) .
k(i,j)
(1)
The Average Tree solution can be expressed as the average of marginal contribution vectors, which require some definitions to be described. By a rooted tree tr on the subgraph (C , L(C )), we mean an orientation that arises from a component C of a forest (N , L) by selecting agent r ∈ C , called the root, and directing all links of L(C ) away from r. For a given tree (C , L(C )), each agent r ∈ C is the root of exactly one rooted tree tr in (C , L(C )). Note also that for any rooted tree tr on C , any agent k ∈ C \ {r }, there is exactly one directed link (j, k); agent j is the unique predecessor of k and k is a successor of j in tr . Denote by sr (j) the possibly empty set of successors of agent j in tr . An agent k is a subordinate of j in tr if there is a directed path from j to k. The set Sr (j) denotes the union of the set of all subordinates of j in tr and {j}. Pick any (v, L) ∈ F , any component C ∈ N /L, any root r ∈ C , and consider the marginal contribution vector mr (v, L) on Rc defined as:
∀i ∈ C ,
mri (v, L) = v(Sr (i)) −
v(Sr (j)).
(2)
j∈sr (i)
(v, L) of i ∈ C in tr is thus equal to the worth of the coalition consisting of agent i and all his subordinates in tr minus the sum of the worths of the coalitions consisting of any successor of i and all subordinates of this successor in tr . The Average Tree solution is the allocation rule AT on F which assigns to each (v, L) ∈ F the payoff vector in which agent i in a component C gets the average over r ∈ C of mri (v, L): The marginal contribution mri
∀C ∈ N /L, ∀i ∈ C ,
ATi (v, L) =
1 c r ∈C
mri (v, L).
(3)
For further developments on the Average Tree solution, see Herings et al. (2010) and Baron et al. (2011). Béal et al. (2010) introduce and characterize the set of Random Tree solutions on the class of forest games. An allocation rule on F is a Random Tree solution, denoted by RTq , if for each forest (N , L) and each component C ∈ N /L, there is a probability distribution qC = (qC (r ))r ∈C over the set of c rooted trees in (C , L(C )) such that:
∀C ∈ N /L, ∀i ∈ C ,
q
RTi (v, L) =
qC (r )mri (v, L).
r ∈C
Thus, AT is the Random Tree solution where qC (r ) = 1/c.
3. Weighted component fairness In this section, we introduce the axiom of weighted component fairness, which can be motivated by the following remark. Note that in (1), the weight of each proper cone K(j,i) ⊂ C , ij ∈ L(C ), can be measured by its relative size k(j,i) /c since (1) is equivalent to: k(i,j) fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) c k(j,i) fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}) . = c The ratio of the sizes k(i,j) /k(j,i) determines the ratio of the payoff variations. Theorem 1 will point out that k(j,i) /c can also be considered as the share of the surplus generated by the deletion of the link ij. Another interpretation in terms of Nash bargaining is provided below. We want to extend the axiom of component fairness by considering all possible weights for the proper cones. The induced solutions are also payoff vectors of the cone-restricted game. We will show that these payoff vectors have some interesting properties. Formally, a system of weights on the set of forests on N is a function α which assigns to each L the system of weights
α(L) = αK (L) : K ∈ ∆0L , αK (L) ∈ [0, 1], αK (L) + αC \K (L) = 1 , where C is the component containing cone K . Let A be the set of all systems of weights α . α -component fairness. For each α ∈ A, each (v, L) and each link ij ∈ L, it holds that:
αK(i,j) (L) fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) = αK(j,i) (L) fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}) . The axiom of component fairness as introduced in (1) corresponds to the special case where αK(i,j) (L) = k(i,j) /c, where c is the size of the component C and k(i,j) is the size of the proper cone K(i,j) ⊂ C . In case the payoff variations of the two cones and their weights are non null, α -component fairness can be rewritten as: fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) fK(i,j) (v, L) − fK(i,j) (v, L \ {ij})
=
αK(j,i) (L) αK(i,j) (L)
,
that is, the ratio of the payoff variations equals the ratio of weights. An alternative interpretation of the axiom can be formulated. Imagine that the coalitions K(j,i) and K(i,j) have to bargain on the consequences of creating the link ij. It seems natural to choose the stand-alone payoffs (fK(j,i) (v, L \ {ij}), fK(i,j) (v, L \ {ij})) obtained by the cones if they are not connected by the link ij as the disagreement point in this two-entity bargaining. Let us make the mild assumption that v ∆L is cone-modular, i.e. for each C ∈ N /L and each link ij ∈ L(C ), it holds that:
v(C ) ≥ v(K(j,i) ) + v(K(i,j) ). The set of feasible agreements is given by {(fK(j,i) (v, L), fK(i,j) (v, L)) ∈ R2 : fK(j,i) (v, L)+ fK(i,j) (v, L) ≤ u}, where u ∈ R is a constant reflecting the quantity of utility over which the coalitions K(j,i) and K(i,j) negotiate. For the purpose of the interpretation, the constant u is chosen sufficiently large to have the disagreement point in the interior of the set of feasible agreements. Furthermore, assume that the bargaining power of the coalitions during the negotiation is given by the weights αK(j,i) (L) and αK(i,j) (L). Then the asymmetric Nash bargaining solution of this bargaining problem is precisely given by our axiom of α -component fairness. In any (strict) conemodular game, the creation of the link ij connecting two cones K(j,i) and K(i,j) generates some surplus. By the axiom of component efficiency, the disagreement point (fK(j,i) (v, L \{ij}), fK(i,j) (v, L \{ij})) reduces to the pair of stand-alone worths (v(K(j,i) ), v(K(i,j) )), and the
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
constant u is equal to the worth v(C ) of the component C resulting from the merge of the two cones. Strict cone-modularity ensures that v(C ) > v(K(j,i) ) + v(K(i,j) ). As a consequence, the combination of strict cone-modularity and the axioms of component efficiency and α -component fairness imply that the two cones K(j,i) and K(i,j) bargain over the set of efficient distributions of the (positive) surplus produced by the creation of link ij with disagreement point (v(K(j,i) ), v(K(i,j) )). It will be shown in the next section that this surplus division belongs to the core of the cone-restricted game provided that the underlying game is cone-modular. In that sense, the bargaining which is inherent in the axiom α -component fairness leads to stability in the cone-restricted game. We show in Theorem 1 that the combination of α -component fairness and component efficiency generates a unique solution. It is obtained as the solution of a system of linear equalities which we describe before stating Theorem 1. Pick any component C ∈ N /L of a forest (N , L) and consider the directed subgraph (C , L◦ (C )) obtained from the orientation ◦. Define a collection of real numbers {bK : K ∈ ∆L \ {∅}} such that:
∀K ∈ ∆0L ,
b C = b K + b C \K ,
(4)
where C is the component containing cone K . From (C , L◦ (C )) and {bK : K ∈ ∆L \ {∅}}, construct the following system of c linear equations with c unknowns:
xk = bC
xk = bK(j,i)
for each directed link (j, i) ∈ L◦ (C ),
(5)
where K(j,i) denotes the proper cone whose head is i and obtained by deleting the directed link (j, i). Because the directed subgraph (C , L◦ (C )) has exactly c − 1 directed links, this procedure selects c − 1 proper cones plus the cone C . The lemma below will be important in proving Theorem 1. Lemma 1. For each collection {bK : K ∈ ∆L \ {∅}} and each orientation ◦ of (C , L(C )), the system (5) admits exactly one solution. Proof. Pick any collection {bK : K ∈ ∆L \ {∅}} satisfying (4), any orientation ◦ of (C , L(C )) and any (j, i) ∈ L◦ (C ). Substituting xk k∈K (j,i)
= bK(j,i) in (5) by the complementary equation
k∈K(i,j)
xk = bC −
bK(j,i) yields an equivalent system. Thus, it suffices to prove that a system of c linear equations with equation k∈C xk = bC and, for each (j, i) ∈ L◦ , either equation xk = bK(j,i)
k∈K(j,i)
or
xk = bC − bK(j,i)
k∈K(i,j)
has a unique solution. To construct such a system, we consider an ordering of the elements of C such that σ (i) is the agent at position i ∈ C in the ordering. We define the ordering σ such that σ (1) is a leaf of the tree (C , L(C )) and for each i ∈ C , σ (i) is a leaf of the (sub)tree obtained by deleting the set {σ (1), . . . , σ (i − 1)}. Let {σ (i), σ (j)} ∈ L(C ) and assume without loss of generality that i < j. We have the following facts: σ (i) is a leaf of the tree obtained by deleting the set {σ (1), . . . , σ (i − 1)}; σ (i) is linked to σ (j) in this tree; for each k ∈ C such that σ (k) ∈ K(σ (j),σ (i)) , it holds that k ≤ i. By construction, for each i ∈ C there is a unique cone Kσ (i) such that σ (i) is the head of Kσ (i) and k ≤ i for each σ (k) ∈ Kσ (i) . In particular, Kσ (c ) = C . Now, order the unknowns according to σ and choose the equation associated with the cone Kσ (i) for each σ (i). We get the following linear system:
∀i ∈ C ,
ful to consider the linear system of the form (5) with constant terms bC = v(C ) and for each directed link (j, i), bK(j,i) = v(K(j,i) ). Such a system will be called a ◦-system associated with (v, L) in component C . Denote by x◦ (v, L) its unique solution. Note that each marginal contribution vector mr (v, L) ∈ Rc as defined in (2) is the unique solution of the tr -system associated with (v, L) in component C , where tr is the orientation inducing the directed tree tr rooted at q r ∈ C . Hence, a Random Tree payoff vector (RTi (v, L))i∈C is a convex combination of these solutions over the c orientations that induce the c rooted trees of (C , L(C )). We are now equipped to state the first result of this section. Theorem 1. For each system of weights α ∈ A on the set of forests on N, there is a unique allocation rule f α on F that satisfies component efficiency and α -component fairness. Moreover, for each (v, L) ∈ F it holds that f α (v, L) = g α (v ∆L ) where for each C ∈ N /L and each i ∈ C, giα (v ∆L ) = v(C ) −
v(K(i,j) )
j∈Li
−
αK(i,j) (L) v(C ) − v(K(j,i) ) − v(K(i,j) ) .
(6)
and
k∈K(j,i)
For each forest game (v, L) ∈ F and each directed subgraph (C , L◦ (C )), C ∈ N /L, obtained from the orientation ◦, it is also use-
j∈Li
k∈C
147
xσ (k) =
bKσ (i)
σ (k)∈Kσ (i)
which is lower triangular with each diagonal term equal to 1. Thus, it admits a unique solution.
Proof. Consider an allocation rule f satisfying component efficiency and α -component fairness for some α ∈ A. From Lemma 1, we only have to show that component efficient and α -component fairness generate a system of type (5). Pick any (v, L) ∈ F and any C ∈ N /L. Component efficiency implies that fC (v, L) = v(C ).
(7)
By component efficiency, we also have for each ij ∈ L(C ), fK(i,j) (v, L\ {ij}) = v(K(i,j) ), fK(j,i) (v, L \{ij}) = v(K(j,i) ), and fK(j,i) (v, L) = v(C )− fK(i,j) (v, L). Therefore, using the fact that αK(i,j) (L) + αK(j,i) (L) = 1, α -component fairness reduces to fK(i,j) (v, L) = v(K(i,j) ) + αK(i,j) (L) v(C ) − v(K(j,i) ) − v(K(i,j) ) . (8) For each component C of size c, there is one linear equation (7) and c − 1 linear equations of type (8). It follows that Eqs. (7) and (8) constitute a linear system of c equations with c unknowns. This is a system of the type (5) where the constant terms are: bC = v(C ), and
∀ij ∈ L(C ),
bK(i,j) = v(K(i,j) ) + αK(i,j) (L) v(C ) − v(K(j,i) ) − v(K(i,j) ) .
Clearly, these constant terms satisfy (4). By Lemma 1, this linear system has a unique solution (fiα (v, L))i∈C . Continuing in this fashion for each component C of (N , L), we conclude that component efficiency and α -component fairness yield a unique solution f α (v, L). Next, pick any i ∈ C . By component efficiency, we have: fiα (v, L) = v(C ) −
j∈Li
fKα(i,j) (v, L).
Using the expression of fKα (v, L) given in (8), we obtain: (i,j)
fiα (v, L) = v(C ) −
v(K(i,j) )
j∈Li
−
αK(i,j) (L) v(C ) − v(K(i,j) ) − v(K(j,i) ) .
j∈Li
The right-hand side of this expression is precisely the right-hand side of (6). Since only cones of (N , L) are used to compute f α (v, L), the result follows.
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Lemma 1 and Theorem 1 are also useful to construct new allocations rules. By Eq. (8) in Theorem 1, each allocation rule satisfying component efficiency and α -component fairness distributes to the members of the cone K(i,j) the sum of two parts. One part is the worth of K(i,j) and the second part is a share αK(i,j) (L) of the surplus v(C ) − (v(K(j,i) ) + v(K(i,j) )) generated by the creation/deletion of the link ij. For instance, the allocation rule AT distributes this surplus according to the relative size of the cone. Instead, consider the case where this surplus is equally distributed to the members of the two cones incident to the link ij. The associated system of weights, denoted by e, is such that eK (L) = 1/2 for each forest (N , L) and each proper cone K ∈ ∆0L . From this, we derive the following axiom of e-component fairness.1 e-component fairness. For each (v, L) ∈ F and each link ij ∈ L, it holds that: fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) = fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}).
(9)
As for AT, the allocation rule f e is equal to the average of the solutions of a set of linear systems of the form (5). More precisely, we have the following result. Theorem 2. For each forest game (v, L) ∈ F , each component C ∈ N /L and each agent i ∈ C , the payoff fie (v, L) is equal to the average over the set of all orientations of the solutions of the ◦-systems associated with (v, L) in component C . Proof. Pick any (v, L) ∈ F , any C ∈ N /L. From (8), we have:
∀ij ∈ L(C ),
fKe(j,i) (v, L) = v(K(j,i) ) +
v(C ) − v(K(j,i) ) − v(K(i,j) ) 2
.
On the other hand, consider the solutions x◦ (v, L) = (x◦k (v, L))k∈C of the ◦-systems associated with (v, L) in component C . There are exactly 2c −1 such linear systems, one for each orientation ◦ of the subgraph (C , L(C )). Define the payoff vector (AOk (v, L))k∈C as the average of the solutions x◦ (v, L) over the set of all orientations:
∀k ∈ C ,
AOk (v, L) =
1
2c −1
◦
x◦k (v, L).
Proceeding in this way for each C ∈ N /L, we obtain a unique payoff vector AO(v, L) ∈ Rn . Pick any link ij ∈ L(C ) and compute the aggregate payoff vector AOK(j,i) (v, L). We get: AOK(j,i) (v, L) =
1 2c −1
+
v(K(j,i) )
◦:◦(ij)=(j,i)
v(C ) − v(K(i,j) ) .
Obviously, there are as much orientations such that ◦(ij) = (j, i) as orientations such that ◦(ij) = (i, j). Thus, we have:
1 c −2 c −2 2 v( K ) + 2 v( C ) − v( K ) ( j , i ) ( i , j ) 2c −1 = fKe(j,i) (v, L).
AOK(j,i) (v, L) =
(i,j)
AOK(j,i) (v, L)
= v(K(j,i) ) +
v(C ) − v(K(j,i) ) − v(K(j,i) )
= AOK(j,i) (v, L \ {ij}) +
2
v(C ) − AOK(j,i) (v, L \ {ij}) − AOK(i,j) (v, L \ {ij}) 2
,
and so
1 AOK(j,i) (v, L) − AOK(j,i) (v, L \ {ij}) 2 1 = AOK(i,j) (v, L) − AOK(i,j) (v, L \ {ij}) , 2 which is precisely the axiom of e-component fairness in (9). Therefore, AO satisfies component efficiency and e-component fairness on F . By Theorem 1, we conclude that f e = AO. In view of Theorem 2, the only difference between AT(v, L) and f e (v, L) is the subset of orientations from which the average of the solutions x◦ (v, L) is taken. Besides the fact that f e coincides with the Average Orientation solution AO on F , it possesses several interesting features. Firstly, it is worth noting that x◦ (x, L) can equivalently be obtained on F by component efficiency and α -component fairness for a system of weights α ∈ A such that αK(j,i) (L) = 0 if (j, i) ∈ L◦ (C ) and αK(j,i) (L) = 1 otherwise. Secondly, even if the Average Orientation solution is the average of 2c −1 payoff vectors for each component C , it can be obtained by much simpler computations for some special classes of graph games. As an example, for line-graph games, e-component fairness coincides with the equal loss property introduced by Brink et al. (2007) and the Average Orientation solution resumes to the average of only two payoff vectors, corresponding to the two rooted spanning trees where the two pending players of the line are the roots (see Theorem 3.3 in Brink et al., 2007). Thirdly, in case Ω = 2N , Shapley (1971) shows that the Shapley value is the center of gravity of the core – the average of its extreme points – of each supermodular game belonging to C (2N ). Tijs et al. (2011) show that the Alexia value coincides with the Shapley value in supermodular games. It is possible to extend this result to the class of games C (∆L ): e-component fairness and component efficiency on F selects the Alexia value in each cone-modular v ∆L associated with the forest game (v, L), and the Alexia value is the center of gravity of this cone-modular game.2 4. Fairness and the cone-restricted core
◦:◦(ij)=(i,j)
In the same way, we obtain AOK(i,j) (v, L) = fKe
each link ij ∈ L(C ), AO satisfies component efficiency, from which we also obtain:
(v, L). By compo-
nent efficiency of f e , we get: AOK(i,j) (v, L) + AOK(j,i) (v, L) = v(C ). Since this equality holds for each (v, L) ∈ F , each C ∈ N /L and
For a fixed L, Theorem 1 indicates that for each system of weights α , the map v −→ f α (v, L) determines an allocation rule g α : C (∆L ) −→ Rn . In this section, we study the properties of these solutions. A payoff vector x ∈ Rn is acceptable with respect to the set ∆L if xK ≥ v(K ) for each K ∈ ∆L . The core is the most well-known solution for TU-games. Given a cone-restricted game v ∆L , the core of v ∆L is the set of payoff vectors that are both acceptable and component efficient: Core(v ∆L ) = x ∈ Rn : ∀C ∈ N /L, xC = v(C ),
and ∀K ∈ ∆0L , xK ≥ v(K ) .
1 Another weighted extension of component fairness, called α -TIBS Fairness, is proposed in Brink et al. (2012) for the distribution of welfare in international river basins. Contrary to our system of weights, in the α -TIBS Fairness the weight of a cone is obtained by summing the weights of its members.
2 The proof of this claim is available from the corresponding author.
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
149
Note that Core(v ∆L ) is a convex polyhedron. A payoff vector x ∈ Rn is an extreme point of the core if there do not exist distinct payoff vectors z and y in Core(v ∆L ) and a ∈]0, 1[ such that x = az + (1 − a)y. The result of this section establishes that a game v ∆L ∈ C (∆L ) has a nonempty core if and only if it is cone-modular. Moreover, the core of a cone-modular game v ∆L coincides with the set of all allocations g α (v ∆L ) obtained by component efficiency and α -component fairness for all possible α ∈ A on F . In this sense, the core of a cone modular game is fair. Finally, we prove that the core of a cone-modular game is a polytope, and that its profile is constituted by the allocations x◦ (v, L) associated with all the possible orientations ◦ of (N , L). As a consequence, combining Theorems 2 and 3 yields that the Average Orientation solution is the barycenter of the core of any cone-modular game.
and x◦K
Theorem 3. A game v ∆L ∈ C (∆L ) has a nonempty core if and only if it is cone-modular. Moreover, for each cone-modular game v ∆L ∈ C (∆L ), the core is a polytope given by:
To complete the proof of the claim, let x be an extreme point of Core(v ∆L ). We show that at least one of the following equalities holds for each link ij ∈ L : xK(j,i) = v(K(j,i) ) or xK(i,j) = v(K(i,j) ). By contradiction, suppose there exists at least one link ij ∈ L such that both equalities are violated. Consider any such link ij ∈ L, choose any ϵ ∈ R and construct the payoff vector xϵ as follows: xϵi = xi + ϵ, xϵj = xj − ϵ and xϵk = xk for each k ∈ N \ {i, j}. For |ϵ|
Core v ∆L = x ∈ Rn : x = g α (v ∆L ), α ∈ A . Proof. Pick any game v ∆L ∈ C (∆L ). It is obvious that Core v ∆L = ∅ if v ∆L is not cone-modular. So, assume that v ∆L is cone-modular. Since the core is component decomposable, i.e. the payoff of each player i ∈ N on any core allocation is completely determined within the component to which he belongs, it is sufficient to focus the analysis on the case where (N , L) has a single component. So, assume without loss of generality that N is the unique component of (N , L). We proceed in four steps. (a) We show that Core v ∆L is a polytope. The core is a convex polyhedron, so it remains to verify that it constitutes a bounded set. Consider any i ∈ N and any payoff vector x ∈ Core v ∆L . To show: there exist ai ∈ R and bi ∈ R such that ai ≤ xi ≤ bi . For each j ∈ Li , we have: xKji ≥ v(Kji ). Let li be the cardinality of Li . Summing over the neighbors of agent i, we get:
xKji ≥
j∈Li
v(Kji ),
j∈Li
v(Kji ) −
sufficiently small, it easy to see that xϵ and x−ϵ belong to Core(v ∆L ) and x = (xϵ + x−ϵ )/2, which contradicts the premise that x is an extreme point of Core(v ∆L ). Thus the claim is true. It immediately follows that we can orientate all links in such a way that for each directed link (j, i) of L◦ , xK(j,i) = v(K(j,i) ). By definition and Lemma 1, x = x◦ (v, L). Therefore, the profile of Core(v ∆L ) coincides with the set in (11). (c) We claim that:
Core v ∆L ⊆ x ∈ Rn : x = g α (v ∆L ), α ∈ A . For each pair α 1 and α 2 in A, and each a ∈ [0, 1], we have aα 1 + (1 − a)α 2 ∈ A and 1 2 1 2 ag α (v ∆L ) + (1 − a)g α (v ∆L ) = g aα +(1−a)α (v ∆L ),
x ∈ Rn : x = g α (v ∆L ), α ∈ A
j∈Li
v(Kji ) − (li − 1)xN \{i} .
(12)
is a convex set of payoff vectors. On the other hand, we see that each element x◦ (v, L) belonging to the profile of Core v ∆L is obtained on F by component efficiency and α -component fairness, where α ∈ A is such that αK(j,i) (L) = 0 if (j, i) ∈ L◦ (C ) and
αK(ji) (L) = 1 otherwise. So, x◦ (v, L) = g α (v ∆L ) for such a system of weights α ∈ A. Since Core v ∆L is a convex hull of its profile by
Finally, by efficiency of x we obtain the following lower bound for xi :
(v, L). If (j, i) is an ele(j,i) (v, L) = v(K(j,i) ), which contradicts the fact (j,i) that x ∈ Core(v ∆L ). Thus, (i, j) is necessarily an element of L◦ and x◦K (v, L) = v(K(i,j) ). By component efficiency, we have xK(i,j) > (i,j) x◦K (v, L) = v(K(i,j) ). It follows that yK(i,j) < x◦K (v, L) = v(K(i,j) ), (i,j) (i,j) which contradicts the fact that y ∈ Core(v ∆L ). Therefore, the solution x◦ (v, L) is an extreme point of Core(v ∆L ). ment of L◦ , then x◦K
xKji \{i} .
j∈Li
xi ≥
(j,i)
without loss of generality that xK(j,i) < x◦K
where the equality follows from Eq. (6) in Theorem 1. This means that
Because any agent in N \ {i} belongs to all but one of the cones with head i, we can rewrite the previous inequality as follows: li xi ≥
(v, L) = v(N ) − v(K(j,i) ) ≥ v(K(i,j) ) by component effi-
or equivalently,
j∈Li
li xi ≥
(i,j)
ciency of x◦ (v, L) and cone-modularity of v ∆L . Assume, by way of contradiction, that x◦ (v, L) is not an extreme point of Core(v ∆L ). Then, we can choose x and y in Core(v ∆L ) such that x ̸= x◦ (v, L) and x◦ (v, L) = (x + y)/2. There necessarily exists a cone K(j,i) for which xK(j,i) ̸= x◦K (v, L). One can assume
v(Kji ) − (li − 1)v(N ).
(10)
j∈Li
The upper bound for xi is obtained by replacing in the equation of efficiencythe payoff xk of any agent k ∈ N \ {i} by (10). Conclude that Core v ∆L is a polytope and thus is defined as the convex hull of its extreme points. (b) We determine the profile of Core v ∆L , i.e. the collection of all extreme points of Core v ∆L . We claim that the profile of
∆ Core v L is given by: x ∈ Rn : x = x◦ (v, L) for some orientation ◦ .
(11)
n Pick any orientation ◦ of (N , L). The solution x◦ (v, ∆L) ∈ R of the L ◦-system associated with (v, L) belongs to Core v . To see this, note that by construction, the solution x◦ (v, L) is component efficient, and for each (j, i) ∈ L◦ , it holds that x◦K (v, L) = v(K(j,i) ), (j,i)
point (a), the claim follows by convexity of the set (12). (d) It remains to show the reverse inclusion:
x ∈ Rn : x = g α (v ∆L ), α ∈ A ⊆ Core v ∆L .
Pick any payoff vector g α (v ∆L ). By Eq. (8) in Theorem 1, we know that:
∀K ∈ ∆0L ,
gKα (v ∆L ) = v(K ) + αK (L) v(N ) − v(N \ K ) − v(K ) .
By cone-modularity of v ∆L , v(N ) − v(N \ K ) − v(K ) ≥ 0, and so gKα (v ∆L ) ≥ v(K ), which means that g α (v ∆L ) is acceptable for all proper cones of (N , L). By component efficiency of g α (v ∆L ), we conclude that g α (v ∆L ) isacceptable for all cones of (N , L). Thus we get g α (v ∆L ) ∈ Core v ∆L . Even if Theorem 3 states that the core of the restricted game coincides with the set of component efficient and α -component fair payoff vectors, it remains to examine whether some payoff vectors in the core are more plausible than others. The requirement than an allocation is obtained through a Harsanyi solution is one way to do that. The next section studies such a core selection.
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S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
By (6), g α is a linear allocation rule on C (∆L ) so that for each
5. Harsanyi solutions In this section, we consider the class of Harsanyi solutions proposed by Vasil’ev (1982) and applied recently by Brink et al. (2012, 2011) to graph-restricted games.3 Each v ∈ C (Ω ) can be expressed in a unique way as:
v=
av (T )uT ,
T ∈Ω
where av (T ) ∈ R is called the Harsanyi dividend of T in v and where {uT : Ω −→ R : T ∈ Ω \ {∅}} is the collection of unanimity games given by: uT (S ) = 1 if T ⊆ S and uT (S ) = 0 otherwise. A sharing system on Ω is a system p = (pS )S ∈Ω \{∅} , where pS is an s-dimensional vector to each player i ∈ N a share assigning S pSi ∈ [0, 1] such that i∈S pi = 1. For a sharing system p, the corresponding Harsanyi solution f p induces the Harsanyi payoffs defined as:
∀v ∈ C (Ω ), ∀i ∈ N ,
p
fi (v) =
pSi av (S ).
α
We prove that an allocation rule g , α ∈ A, on C (∆L ) induced by the allocation rule f α (·, L), is a Harsanyi solution if and only if it is obtained from component efficiency and α -component fairness for some ‘‘credible’’ system of weights α . Imagine that each agent i ∈ N sums the shares of the surpluses collected through his links. His payoff is increasing is these shares αK(j,i) since (6) is decreasing in the complementary shares αK(i,j) . The system of weights α is ‘‘credible’’ if the computed total share is less than or equal to one unit. For such a system, this means that if an agent has a large number of links, then he must divide this unit to specify his share for numerous surpluses. At the same time, an agent with a unique link can enjoy the whole surplus created by his unique link. Thus, his sole neighbor will not benefit from this surplus, but can get some surpluses through his other links. In other words, the search for Harsanyi solutions boils down to selecting systems of weights that possess interesting properties. For instance, this result enables to show that an allocation rule g α is a Harsanyi solution if and only if it is induced by a Random Tree solution. As a corollary, we provide a new characterization of the Random Tree solutions on F in terms of component efficiency and α -component fairness. Let H be the set of Harsanyi solutions on C (∆L ). Note that H is a convex set. Theorem 4. For a given L, let g α , α ∈ A, be an allocation rule on C (∆L ) induced by the allocation rule f α (·, L). The following assertions are equivalent. 1. The allocation rule g α belongs to H . 2. The system of weights α is such that for each i ∈ N, it holds that
αK(i,j) (L) ≤ 1.
(13)
j∈Li
3. There is q such that g α (v ∆L ) = RTq (v, L) for each v ∈ C (2N ). Proof. (1) ⇐⇒ (2). Pick any α ∈ A and define the system p = (pK )K ∈∆L \{∅} as follows: for each nonempty cone K ∈ ∆L , pK is a k-dimensional real vector assigning to each i ∈ K the value pKi = 1 −
αK(i,j) (L).
K ∈∆L \{∅}
Consider a nonempty cone K ⊆ C , C ∈ N /L. Observe that for each i ∈ C \ K and each link ij ∈ L(C ), either K(i,j) ⊇ K or K(j,i) ⊇ K , and there is exactly one link ij ∈ L(C ) such that K(i,j) ⊇ K . From this observation and from (6) in Theorem 1, we immediately get: ∆
giα (uK L ) =
pKi 0
if i ∈ K if i ∈ N \ K .
In case C = K , we see from the expression of pC that every agent i ∈ C gets precisely the payoff expressed in (13): ∆
giα (uC L ) = 1 −
αK(i,j) (L).
j∈Li
Therefore, we can rewrite g α (v ∆L ) as follows:
S ∈Ω : i∈S
v ∆L ∈ C (∆L ), we have: ∆ av ∆L (K )g α (uK L ). g α (v ∆L ) =
(14)
j∈Li ∩K
Note that the system p depends only on the forest (N , L).
3 The set of all Harsanyi solutions is also called the selectope (see Derks et al., 2000).
giα (v ∆L ) =
∀i ∈ N ,
pKi av ∆L (K ).
K ∈∆L :
i∈K
By component efficiency of g α , we obtain:
i∈C
∆
giα (uK L ) =
i∈K
∆
giα (uK L ) =
pKi = 1.
i∈K
Thus, p is a sharing system if and only if for each nonempty cone K and each i ∈ K , pKi ∈ [0, 1], i.e. if and only if (13) holds. We conclude that g α is a Harsanyi solution on C (∆L ) if and only if (13) holds. (1) ⇐⇒ (3). To show: for any v ∆L ∈ C (∆L ), g α (v ∆L ) is a Harsanyi payoff vector if and only if g α (v ∆L ) is a Random Tree payoff vector. The set of payoff vectors g α (v ∆L ), α ∈ A, is a convex set by Theorem 3. The profile of this convex set, given by (11), is generated by the following systems of weights. For L and each component C ∈ N /L, there is an orientation ◦ of (C , L(C )) such that for each orientated link (j, i), αK(i,j) (L) = 1, and so αK(j,i) (L) = 0. Using component efficiency and α -component fairness on F , the induced payoff vectors g α (v ∆L ) are such that giα (v ∆L ) = x◦i (v, L), i ∈ C , C ∈ N /L. If an orientation ◦ is not a rooted tree, some agent has at least two predecessors in the induced orientated subgraph and so (13) is violated. In case ◦ is a rooted tree, each agent belonging to this rooted tree has at most one predecessor and (13) is satisfied. Thus, the systems of weights constructed in this manner satisfy (13) on L if and only if the orientations ◦ on the components of L are rooted trees. In such a case, by (6) in Theorem 1, giα (v ∆L ) = mri (v, L), i ∈ C , C ∈ N /L. By the preceding point, we know that assertion 2 is equivalent to assertion 1. Therefore, we conclude that among the elements of the profile (11), only the marginal contribution vectors induce Harsanyi payoffs vectors, where the sharing system p is constructed as in (14) from the vector of weights α(L) defined above. Because a convex combination of Harsanyi payoff vectors is still a Harsanyi payoff vector, the set of Random tree payoff vectors – the convex hull of the marginal vectors – belongs to the set of Harsanyi payoff vectors. It remains to show that the set of Harsanyi payoff vectors of the form g α (v ∆L ), α ∈ A, is included in the set of Random tree payoff vectors. It suffices to show that the marginal contribution vectors constitute the profile of the convex set of these Harsanyi payoff vectors. Suppose, by way of contradiction, that this assertion is false. Thus, there exists a Harsanyi payoff vector g α (v ∆L ) in this profile such that for at least one link ij ∈ L we have αK(j,i) and αK(i,j) in ]0, 1[. Let (i1 , i2 , . . . , iq ) be a maximal path in L such that for any link ik ik+1 , k ∈ {1, 2, . . . , q − 1}, on this path, αK(i ,i ) (L) and k+1 k
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
αK(ik ,ik+1 ) (L) in ]0, 1[. Since the path is maximal, αK(i1 ,j) ∈ {0, 1} for each j ∈ Li1 \ {i2 } and αK(iq ,j) ∈ {0, 1} for each j ∈ Liq \ {iq−1 }. Let ε ∈ R. For |ε| sufficiently small, define the system of weights α ε which differs from α only on L, and where – for each link ik ik+1 , k ∈ {1, 2, . . . , q − 1}, on the path,
αKε(i
k+1 ,ik )
(L) = αK(ik+1 ,ik ) (L) + ε and
αKε(i
k ,ik+1 )
(L) = αK(ik ,ik+1 ) (L) − ε;
– for any other link ij ∈ L, αKε
αK(i,j) (L).
(j,i)
(L) = αK(j,i) (L), and so αKε(i,j) (L) =
As g α (v ∆L ) is a Harsanyi vector payoff, α(L) satisfies (13) since assertion 1 is equivalent to assertion 2, and the sharing system p ε is constructed as in (14). Thus, for |ε| sufficiently small, g α (v ∆L ) is also a Harsanyi payoff vector since
j∈Li
αKε(j,i ) (L) = 1
αKε(j,i ) (L) =
q
j∈Liq
αK(j,i1 ) (L) + ε ≤ 1,
j∈Li
1
1
αK(j,iq ) (L) − ε ≤ 1
j∈Liq
j∈Li
ε
αK(j,i) (L) =
Cones also form an interesting family of feasible coalitions. The set of cones is in fact an union stable system as introduced by Algaba et al. (2001), i.e. a set of coalitions such that the union of any two elements in the set with a nonempty intersection belongs to the set. Furthermore, the set of cones satisfies two properties which are in the same spirit as the properties used by Brink (forthcoming) in order to characterize the sets of feasible coalitions arising from communication graphs and hierarchies. A collection of feasible coalitions P are called accessible from a feasible coalition S if P is a partition of S \ {i} for some i ∈ N. Such an agent i is called a starting point of S. Player j is a called successor of i, denoted by i ◃ j, if i is a starting point of some S and j is a starting point of some coalition accessible from S. The set of cones satisfies path property and accessibility. The path property is an adaptation of the 2-path property used in Brink (forthcoming) and requires the uniqueness of a ‘‘◃-path’’ between any two players. Accessibility requires that for any player of a feasible coalition, there exists at most one successor of this player in the complementary coalition. The set of cones is the unique set of feasible coalitions that satisfies the path property and accessibility.4 References
and for each other link ij ∈ L,
151
αK(j,i) (L) ≤ 1.
j∈Li ε
It follows that the convex combination g α (v ∆L ) + g α (v ∆L ) /2 is also a Harsanyi payoff vector for |ε| sufficiently small, which contradicts the initial assumption. Thus, the marginal contribution vectors constitute the profile of the Harsanyi payoff vectors of the form g α (v ∆L ), α ∈ A.
−ε
From Theorems 1 and 4, we derive a new characterization of the Random Tree solutions. Theorem 5. An allocation rule on F is a Random Tree solution if and only if it satisfies component efficiency and α -component fairness for some α satisfying (13) for each L. 6. Conclusion Let us motivate further the restriction to the cone-restricted core and the use of cones of a forest. It is true that the cone-restricted core of a forest game can be much larger than the usual core of the Myerson-restricted game. Nevertheless, it satisfies desirable properties. Firstly, efficiency of each element of the cone-restricted core is trivially met. Secondly, part (a) of proof of Theorem 3 shows that the payoff of each player in the cone-restricted core is bounded in spite of the fact that several singleton coalitions are not feasible. The latter property is not guaranteed for other restricted cores (see Grabisch, 2009 for a review of this literature). Thirdly, it can be proved that any payoff vector in the cone-restricted core satisfies the null player property introduced in Béal et al. (2010).
Algaba, E., Bilbao, J.M., Borm, P., López, J.J., 2001. The Myerson value for union stable structures. Mathematical Methods of Operations Research 54, 359–371. Baron, R., Béal, S., Rémila, E., Solal, P., 2011. Average tree solutions and the distribution of Harsanyi dividends. International Journal of Game Theory 40, 331–349. Béal, , Rémila, E., Solal, P., 2010. Rooted-tree solutions for tree games. European Journal of Operational Research 203, 404–408. Bilbao, J.M., 2000. Cooperative Games on Combinatorial Structures. Kluwer Academic Publishers, Dordrecht. Brink, R. van den, 2011. On hierarchies and communication. Social Choice and Welfare, http://dx.doi.org/10.1007/s00355-011-0557-y (forthcoming). Brink, R. van den, van der Laan, G., Moes, N., 2012. Fair agreements for sharing international rivers with multiple springs and externalities. Journal of Environmental Economics and Management 63, 388–403. Brink, R. van den, van der Laan, G., Pruzhansky, V., 2011. Harsanyi power solutions for graph-restricted games. International Journal of Game Theory 40, 87–110. Brink, R. van den, van der Laan, G., Vasil’ev, V., 2007. Component efficient solutions in line-graph games with applications. Economic Theory 33, 349–364. Derks, J., Haller, H., Peters, H., 2000. The selectope for cooperative games. International Journal of Game Theory 29, 23–38. Grabisch, M., 2009. The core of games on ordered structures and graphs. 4OR 7, 207–238. Herings, P., van der Laan, G., Talman, D., 2008. The average tree solution for cycle free games. Games and Economic Behavior 62, 77–92. Herings, P., van der Laan, G., Talman, D., Yang, Z., 2010. The average tree solution for cooperative games with limited communication structure. Games and Economic Behavior 68, 626–633. Myerson, R., 1977. Graphs and cooperation in games. Mathematics of Operations Research 2, 225–229. Shapley, L.S., 1953. A value for n-person games. In: H.W., Kuhn, Tucker, A.W. (Eds.), Contributions to the Theory of Games, vol. II. Princeton University Press, pp. 307–317. Shapley, L.S., 1971. Cores of convex games. International Journal of Game Theory 1, 1–26. Tijs, S., Borm, P., Lohmann, E., Quant, M., 2011. An average lexicographic value for cooperative games. European Journal of Operational Research 213, 210–220. Vasil’ev, V., 1982. On a class of operators in a space of regular set functions. Optimizacija 28, 102–111 (in Russian).
4 The proof of this claim is available from the corresponding author.
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Weighted component fairness for forest games✩ Sylvain Béal a,∗ , Eric Rémila b , Philippe Solal c a
Université de Franche-Comté, CRESE, 30 Avenue de l’Observatoire, 25009 Besançon, France
b
Université de Lyon, LIP, UMR 5668 CNRS-ENS Lyon-Université Lyon 1, France
c
Université de Saint-Etienne, CNRS UMR 5824 GATE Lyon Saint-Etienne, France
article
info
Article history: Available online 30 March 2012
abstract We study the set of allocation rules generated by component efficiency and weighted component fairness, a generalization of component fairness introduced by Herings et al. (2008). Firstly, if the underlying TU-game is superadditive, this set coincides with the core of a graph-restricted game associated with the forest game. Secondly, among this set, only the random tree solutions (Béal et al., 2010) induce Harsanyi payoff vectors for the associated graph-restricted game. We then obtain a new characterization of the random tree solutions in terms of component efficiency and weighted component fairness. © 2012 Elsevier B.V. All rights reserved.
1. Introduction In this article we study TU-games where the cooperation possibilities are represented by the links of an undirected communication graph as in Myerson (1977). We assume that the communication graph is a forest, i.e. each component of the graph is a tree. A forest game is a pair consisting of a TU-game and a forest on the agent set. Herings et al. (2008) introduce the Average Tree solution for the class of forest games, which can be characterized by two axioms: component efficiency and component fairness. Component efficiency asserts that the members of a component ought to allocate to themselves the total worth available to them. When a link is severed in a component of a forest, this component breaks up into two new components that will be called the proper cones incident to the link. Component fairness asserts that deleting a link in a component yields for both resulting proper cones the same average change in payoff, where the average is taken over the players in each cone. Combining component efficiency with component fairness implies that the members of each proper cone receive the sum of two parts: the worth of their proper cone and a share of the surplus
✩ For helpful comments received, the authors want to thank an anonymous referee, Gerard van der Laan, Hans Peters and the participants at S.I.N.G. 5 in Amsterdam and at the conference ‘‘Bargaining, Evolution and Networks’’ in honor of Hans Haller. Financial support by the National Agency for Research (ANR) – research program ‘‘Models of Influence and Network Theory’’ (MINT) ANR.09.BLANC-0321.03 – and the ‘‘Mathématiques de la décision pour l’ingénierie physique et sociale’’ (MODMAD) project is gratefully acknowledged. ∗ Corresponding author. Tel.: +33 0 3 81 66; fax: +33 0 3 81 66. E-mail addresses: [email protected] (S. Béal), [email protected] (E. Rémila), [email protected] (P. Solal).
0165-4896/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.03.004
generated by the deletion of the link. This share is precisely the relative size of this proper cone. In a sense, one can consider these cones as two bargaining entities which agree upon the distribution of a surplus according to their relative sizes. Many other sharing systems can be considered. For instance, Brink et al. (2007) study line-graph games and their axiom of equal loss property implies that the two cones will get the same share of the surplus. In this article, we explore the consequences of this specific sharing system. As a further extension, we introduce a system of weights that determines the share of the surplus between the two proper cones incident to a link. Then, we keep component efficiency as an axiom, replace component fairness by the new axiom of weighted component fairness, and study three properties of the induced set of allocation rules. Firstly, we establish that for a fixed system of weights, component efficiency and weighted component fairness yield a unique allocation rule on the class of forest games. The expression of this rule reveals that it induces a payoff vector of the cone-restricted game associated with the forest game, which is defined as the restriction of the underlying TU-game on the set of cones of the forest. Secondly, we explore the relationships between the set of allocation rules generated by component efficiency and weighted component fairness, and the core of the cone-restricted games associated with the forest games. We provide a necessary and sufficient condition for the nonemptiness of the core of a conerestricted game and then show that it forms a polytope within each component. Then, we prove that a payoff vector belongs to the (nonempty) core of the cone-restricted game if and only if it is obtained by component efficiency and weighted component fairness for some system of weights. This result is encouraging because it points out that fairness and stability considerations are not always incompatible. As far as we know it is the first
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
characterization of the core in terms of fairness, although it is obtained on the cone-restricted game and not on the game itself. Furthermore, we find that the extreme points of the core are computed through the set of all orientations of a forest. This result is not an extension of the characterization of the core of convex TU-games in terms of the marginal vectors associated with all permutations of the player set (see Shapley, 1971), even if the orientations of a forest are used in a similar fashion as the permutations of the players. Thirdly, we establish a link between the random tree solutions of a forest game and the Harsanyi solutions of the associated cone-restricted game. Harsanyi solutions have been studied by Derks et al. (2000) for games where cooperation possibilities are unrestricted. The average tree solution can be computed as the average of marginal contribution vectors defined according to all rooted trees of a forest, where a rooted tree consists in directing all links away from a selected agent, called the root. A random tree solution consists in a weighted average of these marginal contribution vectors, where the weights are given by a probability distribution over the set of contribution vectors. We show that among the component efficient and weighted component fair allocation rules, only the random tree solutions generate Harsanyi payoff vectors in the associated cone-restricted games. Since core allocations of the cone-restricted games with a nonempty core are generated by component efficiency and weighted component fairness, it amounts to saying that the search for Harsanyi solutions among core allocations boils down to selecting the set of random tree solutions. As a corollary, we provide an alternative characterization of the random tree solutions in terms of component efficiency and weighted component fairness for suitable systems of weights. While Béal et al. (2010) characterize the set of random tree solutions by axioms that are similar to the axioms used by Shapley (1953) for the characterization of the Shapley value for TU-games, this result is more appealing in the framework of graph games since the considered axioms do not rely exclusively on the characteristic function of the games. This article is organized as follows. Section 2 contains definitions and a preliminary result. The axiom of weighted component fairness is introduced in Section 3, in which we establish that, for a given system of weights, component efficiency and weighted component fairness yield a unique allocation rule on the class of forest games. In Section 4, we investigate the structure of the core of the cone-restricted game. Section 5 is devoted to the relationship between the Harsanyi solutions of the cone-restricted game and the random tree solutions. Section 6 concludes. 2. Preliminaries 2.1. TU-games For a finite set of agents N = {1, . . . , n}, each S ∈ 2N is a coalition of cardinality denoted by s. In many practical situations some coalitions may not be meaningful, because of the lack of communication possibilities, or certain institutional constraints. Thus we shall consider a subcollection Ω ⊆ 2N of feasible coalitions with the convention that ∅ ∈ Ω (see Bilbao, 2000 and Grabisch, 2009). A TU-game on N is a characteristic function v : Ω −→ R such that v(∅) = 0. For each S ∈ Ω , v(S ) is interpreted as the worth of S. By C (Ω ), we denote the real linear space of all TU-games v on Ω , which can be identified with the corresponding linear sub-space of R|Ω | of dimension |Ω | − 1. A payoff vector x ∈ Rn of v ∈ C (Ω ) is a vector giving a payoff xi ∈ R to any agent i ∈ N. For each S ∈ 2N , the notation xS stands for i∈S xi . An allocation rule on C (Ω ) is a map f that assigns to each v ∈ C (Ω ) a payoff vector f (v) ∈ Rn .
145
2.2. Graphs The restrictions on communication possibilities faced by the group of agents N can be represented by an undirected graph (N , L), where the set of nodes is the set N, and the set of links L is a subset of the set of unordered pairs of elements of N. For simplicity, we write ij for the link {i, j}. The set Li = {j ∈ N : ij ∈ L} is the set of links of i ∈ N. For each S ∈ 2N \ {∅}, L(S ) = {ij ∈ L : {i, j} ⊆ S } is the set of links between agents in coalition S. The graph (S , L(S )) is the subgraph of (N , L) induced by S. A sequence of distinct agents (i1 , . . . , ip ) is a path in (N , L) if ik ik+1 ∈ L for k = 1, . . . , p − 1. A graph is connected if there exists a path between any two agents, and it constitutes a tree if this path is unique for any two agents. A coalition S is connected if (S , L(S )) is connected. A coalition C ∈ 2N is a component of (N , L) if (C , L(C )) is connected and for each i ∈ N \ C , the subgraph (C ∪ {i}, L(C ∪ {i})) is not connected. The set of components of (N , L), denoted by N /L, forms a partition of N. A forest is a graph in which each component is a tree. The set of cones of a forest (N , L) consists of the set of components N /L, the set ∅, and for each C ∈ N /L and ij ∈ L(C ) the two components of (C , L(C ) \ {ij}). A cone strictly included in C is a proper cone of C . The unique agent of a nonempty proper cone K who has a link with the complement C \ K is called the head of K . Note that K ⊆ C is a cone if and only if the coalition C \ K is connected. In order to insist on the link ij, K(j,i) denotes the cone with head i and K(i,j) its complement with head j in the corresponding component. Let ∆L and ∆0L be the set of cones and the set of nonempty proper cones of (N , L) respectively. An orientation ◦ of a graph (N , L) is a directed graph (N , L◦ ) obtained from (N , L) by replacing each link ij by a directed link: either (i, j) or (j, i). Let ◦(ij) be the orientation of link ij ∈ L. 2.3. Solutions for graph games and cone-restricted games A graph game on N is a pair (v, L) such that v ∈ C (2N ) is a TU-game and (N , L) is a communication graph. In what follows, we only consider the set F of all forest games on N, i.e. the set of all pairs (v, L) where v ∈ C (2N ) and (N , L) is a forest. An allocation rule on F is a map f that assigns to each (v, L) ∈ F a payoff vector f (v, L) ∈ Rn . While for the class C (Ω ) it is explicitly assumed that only coalitions in Ω are feasible, a graph game (v, L) indicates that the entire set of coalitions 2N may be affected by (N , L). The payoff vector f (v, L) determines how the communication links affect both the formation of coalitions and the process of redistribution. Since the communication graph is meant to reduce the set of feasible coalitions, it is often convenient to express f (v, L) as a payoff vector g (v Ω ) on some class C (Ω ) where v Ω : Ω −→ R is the restriction of v to the set Ω , i.e. for all S ∈ Ω , v Ω (S ) = v(S ). The restricted game v Ω incorporates both the possible gains from cooperation as modeled by v and the restrictions on communication induced by (N , L) and reflected by Ω . The axioms that are considered can tell how the restrictions imposed by communication graph (N , L) shrinks the set of feasible coalitions from 2N to Ω . For instance, if one interprets the communication graph as the set of direct and indirect communication possibilities between the agents, then a minimal requirement is that Ω should be included in the set of connected coalitions. In the following, we will consider the set of cones of a forest as the set of feasible coalitions. For a forest (N , L), let C (∆L ) be the set of cone-restricted games. There are two justifications for considering the class C (∆L ). Firstly, the axioms that we are going to present on F will imply that the coalitions in ∆L are necessary and sufficient for computing a solution. Secondly, two complementary cones reflect the original idea of von Neumann and Morgenstern
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in which the set of players is split in two competing coalitions. An interpretation in terms of Nash bargaining will also be discussed later on in the article. On the class F , Herings et al. (2008) introduce a new allocation rule called the Average Tree solution, that can be characterized by component efficiency and component fairness. Component efficiency requires that the payoffs in a component add up to the worth of that component. Component efficiency. For each (v, L) ∈ F and each C ∈ N /L, it holds that: fC (v, L) = v(C ). Component fairness says that deleting a link between two agents yields for both resulting cones the same average change in payoffs. Component fairness therefore emphasizes that the gains associated with linking one cone to its complement in a component should be attributed to these two proper cones, rather than to the two agents whose link is deleted, and the gains should be proportional to the size of the cones. Component fairness. For each (v, L) ∈ F and each link ij ∈ L, it holds that: 1 fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) k(j,i)
=
1 fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}) .
k(i,j)
(1)
The Average Tree solution can be expressed as the average of marginal contribution vectors, which require some definitions to be described. By a rooted tree tr on the subgraph (C , L(C )), we mean an orientation that arises from a component C of a forest (N , L) by selecting agent r ∈ C , called the root, and directing all links of L(C ) away from r. For a given tree (C , L(C )), each agent r ∈ C is the root of exactly one rooted tree tr in (C , L(C )). Note also that for any rooted tree tr on C , any agent k ∈ C \ {r }, there is exactly one directed link (j, k); agent j is the unique predecessor of k and k is a successor of j in tr . Denote by sr (j) the possibly empty set of successors of agent j in tr . An agent k is a subordinate of j in tr if there is a directed path from j to k. The set Sr (j) denotes the union of the set of all subordinates of j in tr and {j}. Pick any (v, L) ∈ F , any component C ∈ N /L, any root r ∈ C , and consider the marginal contribution vector mr (v, L) on Rc defined as:
∀i ∈ C ,
mri (v, L) = v(Sr (i)) −
v(Sr (j)).
(2)
j∈sr (i)
(v, L) of i ∈ C in tr is thus equal to the worth of the coalition consisting of agent i and all his subordinates in tr minus the sum of the worths of the coalitions consisting of any successor of i and all subordinates of this successor in tr . The Average Tree solution is the allocation rule AT on F which assigns to each (v, L) ∈ F the payoff vector in which agent i in a component C gets the average over r ∈ C of mri (v, L): The marginal contribution mri
∀C ∈ N /L, ∀i ∈ C ,
ATi (v, L) =
1 c r ∈C
mri (v, L).
(3)
For further developments on the Average Tree solution, see Herings et al. (2010) and Baron et al. (2011). Béal et al. (2010) introduce and characterize the set of Random Tree solutions on the class of forest games. An allocation rule on F is a Random Tree solution, denoted by RTq , if for each forest (N , L) and each component C ∈ N /L, there is a probability distribution qC = (qC (r ))r ∈C over the set of c rooted trees in (C , L(C )) such that:
∀C ∈ N /L, ∀i ∈ C ,
q
RTi (v, L) =
qC (r )mri (v, L).
r ∈C
Thus, AT is the Random Tree solution where qC (r ) = 1/c.
3. Weighted component fairness In this section, we introduce the axiom of weighted component fairness, which can be motivated by the following remark. Note that in (1), the weight of each proper cone K(j,i) ⊂ C , ij ∈ L(C ), can be measured by its relative size k(j,i) /c since (1) is equivalent to: k(i,j) fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) c k(j,i) fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}) . = c The ratio of the sizes k(i,j) /k(j,i) determines the ratio of the payoff variations. Theorem 1 will point out that k(j,i) /c can also be considered as the share of the surplus generated by the deletion of the link ij. Another interpretation in terms of Nash bargaining is provided below. We want to extend the axiom of component fairness by considering all possible weights for the proper cones. The induced solutions are also payoff vectors of the cone-restricted game. We will show that these payoff vectors have some interesting properties. Formally, a system of weights on the set of forests on N is a function α which assigns to each L the system of weights
α(L) = αK (L) : K ∈ ∆0L , αK (L) ∈ [0, 1], αK (L) + αC \K (L) = 1 , where C is the component containing cone K . Let A be the set of all systems of weights α . α -component fairness. For each α ∈ A, each (v, L) and each link ij ∈ L, it holds that:
αK(i,j) (L) fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) = αK(j,i) (L) fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}) . The axiom of component fairness as introduced in (1) corresponds to the special case where αK(i,j) (L) = k(i,j) /c, where c is the size of the component C and k(i,j) is the size of the proper cone K(i,j) ⊂ C . In case the payoff variations of the two cones and their weights are non null, α -component fairness can be rewritten as: fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) fK(i,j) (v, L) − fK(i,j) (v, L \ {ij})
=
αK(j,i) (L) αK(i,j) (L)
,
that is, the ratio of the payoff variations equals the ratio of weights. An alternative interpretation of the axiom can be formulated. Imagine that the coalitions K(j,i) and K(i,j) have to bargain on the consequences of creating the link ij. It seems natural to choose the stand-alone payoffs (fK(j,i) (v, L \ {ij}), fK(i,j) (v, L \ {ij})) obtained by the cones if they are not connected by the link ij as the disagreement point in this two-entity bargaining. Let us make the mild assumption that v ∆L is cone-modular, i.e. for each C ∈ N /L and each link ij ∈ L(C ), it holds that:
v(C ) ≥ v(K(j,i) ) + v(K(i,j) ). The set of feasible agreements is given by {(fK(j,i) (v, L), fK(i,j) (v, L)) ∈ R2 : fK(j,i) (v, L)+ fK(i,j) (v, L) ≤ u}, where u ∈ R is a constant reflecting the quantity of utility over which the coalitions K(j,i) and K(i,j) negotiate. For the purpose of the interpretation, the constant u is chosen sufficiently large to have the disagreement point in the interior of the set of feasible agreements. Furthermore, assume that the bargaining power of the coalitions during the negotiation is given by the weights αK(j,i) (L) and αK(i,j) (L). Then the asymmetric Nash bargaining solution of this bargaining problem is precisely given by our axiom of α -component fairness. In any (strict) conemodular game, the creation of the link ij connecting two cones K(j,i) and K(i,j) generates some surplus. By the axiom of component efficiency, the disagreement point (fK(j,i) (v, L \{ij}), fK(i,j) (v, L \{ij})) reduces to the pair of stand-alone worths (v(K(j,i) ), v(K(i,j) )), and the
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
constant u is equal to the worth v(C ) of the component C resulting from the merge of the two cones. Strict cone-modularity ensures that v(C ) > v(K(j,i) ) + v(K(i,j) ). As a consequence, the combination of strict cone-modularity and the axioms of component efficiency and α -component fairness imply that the two cones K(j,i) and K(i,j) bargain over the set of efficient distributions of the (positive) surplus produced by the creation of link ij with disagreement point (v(K(j,i) ), v(K(i,j) )). It will be shown in the next section that this surplus division belongs to the core of the cone-restricted game provided that the underlying game is cone-modular. In that sense, the bargaining which is inherent in the axiom α -component fairness leads to stability in the cone-restricted game. We show in Theorem 1 that the combination of α -component fairness and component efficiency generates a unique solution. It is obtained as the solution of a system of linear equalities which we describe before stating Theorem 1. Pick any component C ∈ N /L of a forest (N , L) and consider the directed subgraph (C , L◦ (C )) obtained from the orientation ◦. Define a collection of real numbers {bK : K ∈ ∆L \ {∅}} such that:
∀K ∈ ∆0L ,
b C = b K + b C \K ,
(4)
where C is the component containing cone K . From (C , L◦ (C )) and {bK : K ∈ ∆L \ {∅}}, construct the following system of c linear equations with c unknowns:
xk = bC
xk = bK(j,i)
for each directed link (j, i) ∈ L◦ (C ),
(5)
where K(j,i) denotes the proper cone whose head is i and obtained by deleting the directed link (j, i). Because the directed subgraph (C , L◦ (C )) has exactly c − 1 directed links, this procedure selects c − 1 proper cones plus the cone C . The lemma below will be important in proving Theorem 1. Lemma 1. For each collection {bK : K ∈ ∆L \ {∅}} and each orientation ◦ of (C , L(C )), the system (5) admits exactly one solution. Proof. Pick any collection {bK : K ∈ ∆L \ {∅}} satisfying (4), any orientation ◦ of (C , L(C )) and any (j, i) ∈ L◦ (C ). Substituting xk k∈K (j,i)
= bK(j,i) in (5) by the complementary equation
k∈K(i,j)
xk = bC −
bK(j,i) yields an equivalent system. Thus, it suffices to prove that a system of c linear equations with equation k∈C xk = bC and, for each (j, i) ∈ L◦ , either equation xk = bK(j,i)
k∈K(j,i)
or
xk = bC − bK(j,i)
k∈K(i,j)
has a unique solution. To construct such a system, we consider an ordering of the elements of C such that σ (i) is the agent at position i ∈ C in the ordering. We define the ordering σ such that σ (1) is a leaf of the tree (C , L(C )) and for each i ∈ C , σ (i) is a leaf of the (sub)tree obtained by deleting the set {σ (1), . . . , σ (i − 1)}. Let {σ (i), σ (j)} ∈ L(C ) and assume without loss of generality that i < j. We have the following facts: σ (i) is a leaf of the tree obtained by deleting the set {σ (1), . . . , σ (i − 1)}; σ (i) is linked to σ (j) in this tree; for each k ∈ C such that σ (k) ∈ K(σ (j),σ (i)) , it holds that k ≤ i. By construction, for each i ∈ C there is a unique cone Kσ (i) such that σ (i) is the head of Kσ (i) and k ≤ i for each σ (k) ∈ Kσ (i) . In particular, Kσ (c ) = C . Now, order the unknowns according to σ and choose the equation associated with the cone Kσ (i) for each σ (i). We get the following linear system:
∀i ∈ C ,
ful to consider the linear system of the form (5) with constant terms bC = v(C ) and for each directed link (j, i), bK(j,i) = v(K(j,i) ). Such a system will be called a ◦-system associated with (v, L) in component C . Denote by x◦ (v, L) its unique solution. Note that each marginal contribution vector mr (v, L) ∈ Rc as defined in (2) is the unique solution of the tr -system associated with (v, L) in component C , where tr is the orientation inducing the directed tree tr rooted at q r ∈ C . Hence, a Random Tree payoff vector (RTi (v, L))i∈C is a convex combination of these solutions over the c orientations that induce the c rooted trees of (C , L(C )). We are now equipped to state the first result of this section. Theorem 1. For each system of weights α ∈ A on the set of forests on N, there is a unique allocation rule f α on F that satisfies component efficiency and α -component fairness. Moreover, for each (v, L) ∈ F it holds that f α (v, L) = g α (v ∆L ) where for each C ∈ N /L and each i ∈ C, giα (v ∆L ) = v(C ) −
v(K(i,j) )
j∈Li
−
αK(i,j) (L) v(C ) − v(K(j,i) ) − v(K(i,j) ) .
(6)
and
k∈K(j,i)
For each forest game (v, L) ∈ F and each directed subgraph (C , L◦ (C )), C ∈ N /L, obtained from the orientation ◦, it is also use-
j∈Li
k∈C
147
xσ (k) =
bKσ (i)
σ (k)∈Kσ (i)
which is lower triangular with each diagonal term equal to 1. Thus, it admits a unique solution.
Proof. Consider an allocation rule f satisfying component efficiency and α -component fairness for some α ∈ A. From Lemma 1, we only have to show that component efficient and α -component fairness generate a system of type (5). Pick any (v, L) ∈ F and any C ∈ N /L. Component efficiency implies that fC (v, L) = v(C ).
(7)
By component efficiency, we also have for each ij ∈ L(C ), fK(i,j) (v, L\ {ij}) = v(K(i,j) ), fK(j,i) (v, L \{ij}) = v(K(j,i) ), and fK(j,i) (v, L) = v(C )− fK(i,j) (v, L). Therefore, using the fact that αK(i,j) (L) + αK(j,i) (L) = 1, α -component fairness reduces to fK(i,j) (v, L) = v(K(i,j) ) + αK(i,j) (L) v(C ) − v(K(j,i) ) − v(K(i,j) ) . (8) For each component C of size c, there is one linear equation (7) and c − 1 linear equations of type (8). It follows that Eqs. (7) and (8) constitute a linear system of c equations with c unknowns. This is a system of the type (5) where the constant terms are: bC = v(C ), and
∀ij ∈ L(C ),
bK(i,j) = v(K(i,j) ) + αK(i,j) (L) v(C ) − v(K(j,i) ) − v(K(i,j) ) .
Clearly, these constant terms satisfy (4). By Lemma 1, this linear system has a unique solution (fiα (v, L))i∈C . Continuing in this fashion for each component C of (N , L), we conclude that component efficiency and α -component fairness yield a unique solution f α (v, L). Next, pick any i ∈ C . By component efficiency, we have: fiα (v, L) = v(C ) −
j∈Li
fKα(i,j) (v, L).
Using the expression of fKα (v, L) given in (8), we obtain: (i,j)
fiα (v, L) = v(C ) −
v(K(i,j) )
j∈Li
−
αK(i,j) (L) v(C ) − v(K(i,j) ) − v(K(j,i) ) .
j∈Li
The right-hand side of this expression is precisely the right-hand side of (6). Since only cones of (N , L) are used to compute f α (v, L), the result follows.
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Lemma 1 and Theorem 1 are also useful to construct new allocations rules. By Eq. (8) in Theorem 1, each allocation rule satisfying component efficiency and α -component fairness distributes to the members of the cone K(i,j) the sum of two parts. One part is the worth of K(i,j) and the second part is a share αK(i,j) (L) of the surplus v(C ) − (v(K(j,i) ) + v(K(i,j) )) generated by the creation/deletion of the link ij. For instance, the allocation rule AT distributes this surplus according to the relative size of the cone. Instead, consider the case where this surplus is equally distributed to the members of the two cones incident to the link ij. The associated system of weights, denoted by e, is such that eK (L) = 1/2 for each forest (N , L) and each proper cone K ∈ ∆0L . From this, we derive the following axiom of e-component fairness.1 e-component fairness. For each (v, L) ∈ F and each link ij ∈ L, it holds that: fK(j,i) (v, L) − fK(j,i) (v, L \ {ij}) = fK(i,j) (v, L) − fK(i,j) (v, L \ {ij}).
(9)
As for AT, the allocation rule f e is equal to the average of the solutions of a set of linear systems of the form (5). More precisely, we have the following result. Theorem 2. For each forest game (v, L) ∈ F , each component C ∈ N /L and each agent i ∈ C , the payoff fie (v, L) is equal to the average over the set of all orientations of the solutions of the ◦-systems associated with (v, L) in component C . Proof. Pick any (v, L) ∈ F , any C ∈ N /L. From (8), we have:
∀ij ∈ L(C ),
fKe(j,i) (v, L) = v(K(j,i) ) +
v(C ) − v(K(j,i) ) − v(K(i,j) ) 2
.
On the other hand, consider the solutions x◦ (v, L) = (x◦k (v, L))k∈C of the ◦-systems associated with (v, L) in component C . There are exactly 2c −1 such linear systems, one for each orientation ◦ of the subgraph (C , L(C )). Define the payoff vector (AOk (v, L))k∈C as the average of the solutions x◦ (v, L) over the set of all orientations:
∀k ∈ C ,
AOk (v, L) =
1
2c −1
◦
x◦k (v, L).
Proceeding in this way for each C ∈ N /L, we obtain a unique payoff vector AO(v, L) ∈ Rn . Pick any link ij ∈ L(C ) and compute the aggregate payoff vector AOK(j,i) (v, L). We get: AOK(j,i) (v, L) =
1 2c −1
+
v(K(j,i) )
◦:◦(ij)=(j,i)
v(C ) − v(K(i,j) ) .
Obviously, there are as much orientations such that ◦(ij) = (j, i) as orientations such that ◦(ij) = (i, j). Thus, we have:
1 c −2 c −2 2 v( K ) + 2 v( C ) − v( K ) ( j , i ) ( i , j ) 2c −1 = fKe(j,i) (v, L).
AOK(j,i) (v, L) =
(i,j)
AOK(j,i) (v, L)
= v(K(j,i) ) +
v(C ) − v(K(j,i) ) − v(K(j,i) )
= AOK(j,i) (v, L \ {ij}) +
2
v(C ) − AOK(j,i) (v, L \ {ij}) − AOK(i,j) (v, L \ {ij}) 2
,
and so
1 AOK(j,i) (v, L) − AOK(j,i) (v, L \ {ij}) 2 1 = AOK(i,j) (v, L) − AOK(i,j) (v, L \ {ij}) , 2 which is precisely the axiom of e-component fairness in (9). Therefore, AO satisfies component efficiency and e-component fairness on F . By Theorem 1, we conclude that f e = AO. In view of Theorem 2, the only difference between AT(v, L) and f e (v, L) is the subset of orientations from which the average of the solutions x◦ (v, L) is taken. Besides the fact that f e coincides with the Average Orientation solution AO on F , it possesses several interesting features. Firstly, it is worth noting that x◦ (x, L) can equivalently be obtained on F by component efficiency and α -component fairness for a system of weights α ∈ A such that αK(j,i) (L) = 0 if (j, i) ∈ L◦ (C ) and αK(j,i) (L) = 1 otherwise. Secondly, even if the Average Orientation solution is the average of 2c −1 payoff vectors for each component C , it can be obtained by much simpler computations for some special classes of graph games. As an example, for line-graph games, e-component fairness coincides with the equal loss property introduced by Brink et al. (2007) and the Average Orientation solution resumes to the average of only two payoff vectors, corresponding to the two rooted spanning trees where the two pending players of the line are the roots (see Theorem 3.3 in Brink et al., 2007). Thirdly, in case Ω = 2N , Shapley (1971) shows that the Shapley value is the center of gravity of the core – the average of its extreme points – of each supermodular game belonging to C (2N ). Tijs et al. (2011) show that the Alexia value coincides with the Shapley value in supermodular games. It is possible to extend this result to the class of games C (∆L ): e-component fairness and component efficiency on F selects the Alexia value in each cone-modular v ∆L associated with the forest game (v, L), and the Alexia value is the center of gravity of this cone-modular game.2 4. Fairness and the cone-restricted core
◦:◦(ij)=(i,j)
In the same way, we obtain AOK(i,j) (v, L) = fKe
each link ij ∈ L(C ), AO satisfies component efficiency, from which we also obtain:
(v, L). By compo-
nent efficiency of f e , we get: AOK(i,j) (v, L) + AOK(j,i) (v, L) = v(C ). Since this equality holds for each (v, L) ∈ F , each C ∈ N /L and
For a fixed L, Theorem 1 indicates that for each system of weights α , the map v −→ f α (v, L) determines an allocation rule g α : C (∆L ) −→ Rn . In this section, we study the properties of these solutions. A payoff vector x ∈ Rn is acceptable with respect to the set ∆L if xK ≥ v(K ) for each K ∈ ∆L . The core is the most well-known solution for TU-games. Given a cone-restricted game v ∆L , the core of v ∆L is the set of payoff vectors that are both acceptable and component efficient: Core(v ∆L ) = x ∈ Rn : ∀C ∈ N /L, xC = v(C ),
and ∀K ∈ ∆0L , xK ≥ v(K ) .
1 Another weighted extension of component fairness, called α -TIBS Fairness, is proposed in Brink et al. (2012) for the distribution of welfare in international river basins. Contrary to our system of weights, in the α -TIBS Fairness the weight of a cone is obtained by summing the weights of its members.
2 The proof of this claim is available from the corresponding author.
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
149
Note that Core(v ∆L ) is a convex polyhedron. A payoff vector x ∈ Rn is an extreme point of the core if there do not exist distinct payoff vectors z and y in Core(v ∆L ) and a ∈]0, 1[ such that x = az + (1 − a)y. The result of this section establishes that a game v ∆L ∈ C (∆L ) has a nonempty core if and only if it is cone-modular. Moreover, the core of a cone-modular game v ∆L coincides with the set of all allocations g α (v ∆L ) obtained by component efficiency and α -component fairness for all possible α ∈ A on F . In this sense, the core of a cone modular game is fair. Finally, we prove that the core of a cone-modular game is a polytope, and that its profile is constituted by the allocations x◦ (v, L) associated with all the possible orientations ◦ of (N , L). As a consequence, combining Theorems 2 and 3 yields that the Average Orientation solution is the barycenter of the core of any cone-modular game.
and x◦K
Theorem 3. A game v ∆L ∈ C (∆L ) has a nonempty core if and only if it is cone-modular. Moreover, for each cone-modular game v ∆L ∈ C (∆L ), the core is a polytope given by:
To complete the proof of the claim, let x be an extreme point of Core(v ∆L ). We show that at least one of the following equalities holds for each link ij ∈ L : xK(j,i) = v(K(j,i) ) or xK(i,j) = v(K(i,j) ). By contradiction, suppose there exists at least one link ij ∈ L such that both equalities are violated. Consider any such link ij ∈ L, choose any ϵ ∈ R and construct the payoff vector xϵ as follows: xϵi = xi + ϵ, xϵj = xj − ϵ and xϵk = xk for each k ∈ N \ {i, j}. For |ϵ|
Core v ∆L = x ∈ Rn : x = g α (v ∆L ), α ∈ A . Proof. Pick any game v ∆L ∈ C (∆L ). It is obvious that Core v ∆L = ∅ if v ∆L is not cone-modular. So, assume that v ∆L is cone-modular. Since the core is component decomposable, i.e. the payoff of each player i ∈ N on any core allocation is completely determined within the component to which he belongs, it is sufficient to focus the analysis on the case where (N , L) has a single component. So, assume without loss of generality that N is the unique component of (N , L). We proceed in four steps. (a) We show that Core v ∆L is a polytope. The core is a convex polyhedron, so it remains to verify that it constitutes a bounded set. Consider any i ∈ N and any payoff vector x ∈ Core v ∆L . To show: there exist ai ∈ R and bi ∈ R such that ai ≤ xi ≤ bi . For each j ∈ Li , we have: xKji ≥ v(Kji ). Let li be the cardinality of Li . Summing over the neighbors of agent i, we get:
xKji ≥
j∈Li
v(Kji ),
j∈Li
v(Kji ) −
sufficiently small, it easy to see that xϵ and x−ϵ belong to Core(v ∆L ) and x = (xϵ + x−ϵ )/2, which contradicts the premise that x is an extreme point of Core(v ∆L ). Thus the claim is true. It immediately follows that we can orientate all links in such a way that for each directed link (j, i) of L◦ , xK(j,i) = v(K(j,i) ). By definition and Lemma 1, x = x◦ (v, L). Therefore, the profile of Core(v ∆L ) coincides with the set in (11). (c) We claim that:
Core v ∆L ⊆ x ∈ Rn : x = g α (v ∆L ), α ∈ A . For each pair α 1 and α 2 in A, and each a ∈ [0, 1], we have aα 1 + (1 − a)α 2 ∈ A and 1 2 1 2 ag α (v ∆L ) + (1 − a)g α (v ∆L ) = g aα +(1−a)α (v ∆L ),
x ∈ Rn : x = g α (v ∆L ), α ∈ A
j∈Li
v(Kji ) − (li − 1)xN \{i} .
(12)
is a convex set of payoff vectors. On the other hand, we see that each element x◦ (v, L) belonging to the profile of Core v ∆L is obtained on F by component efficiency and α -component fairness, where α ∈ A is such that αK(j,i) (L) = 0 if (j, i) ∈ L◦ (C ) and
αK(ji) (L) = 1 otherwise. So, x◦ (v, L) = g α (v ∆L ) for such a system of weights α ∈ A. Since Core v ∆L is a convex hull of its profile by
Finally, by efficiency of x we obtain the following lower bound for xi :
(v, L). If (j, i) is an ele(j,i) (v, L) = v(K(j,i) ), which contradicts the fact (j,i) that x ∈ Core(v ∆L ). Thus, (i, j) is necessarily an element of L◦ and x◦K (v, L) = v(K(i,j) ). By component efficiency, we have xK(i,j) > (i,j) x◦K (v, L) = v(K(i,j) ). It follows that yK(i,j) < x◦K (v, L) = v(K(i,j) ), (i,j) (i,j) which contradicts the fact that y ∈ Core(v ∆L ). Therefore, the solution x◦ (v, L) is an extreme point of Core(v ∆L ). ment of L◦ , then x◦K
xKji \{i} .
j∈Li
xi ≥
(j,i)
without loss of generality that xK(j,i) < x◦K
where the equality follows from Eq. (6) in Theorem 1. This means that
Because any agent in N \ {i} belongs to all but one of the cones with head i, we can rewrite the previous inequality as follows: li xi ≥
(v, L) = v(N ) − v(K(j,i) ) ≥ v(K(i,j) ) by component effi-
or equivalently,
j∈Li
li xi ≥
(i,j)
ciency of x◦ (v, L) and cone-modularity of v ∆L . Assume, by way of contradiction, that x◦ (v, L) is not an extreme point of Core(v ∆L ). Then, we can choose x and y in Core(v ∆L ) such that x ̸= x◦ (v, L) and x◦ (v, L) = (x + y)/2. There necessarily exists a cone K(j,i) for which xK(j,i) ̸= x◦K (v, L). One can assume
v(Kji ) − (li − 1)v(N ).
(10)
j∈Li
The upper bound for xi is obtained by replacing in the equation of efficiencythe payoff xk of any agent k ∈ N \ {i} by (10). Conclude that Core v ∆L is a polytope and thus is defined as the convex hull of its extreme points. (b) We determine the profile of Core v ∆L , i.e. the collection of all extreme points of Core v ∆L . We claim that the profile of
∆ Core v L is given by: x ∈ Rn : x = x◦ (v, L) for some orientation ◦ .
(11)
n Pick any orientation ◦ of (N , L). The solution x◦ (v, ∆L) ∈ R of the L ◦-system associated with (v, L) belongs to Core v . To see this, note that by construction, the solution x◦ (v, L) is component efficient, and for each (j, i) ∈ L◦ , it holds that x◦K (v, L) = v(K(j,i) ), (j,i)
point (a), the claim follows by convexity of the set (12). (d) It remains to show the reverse inclusion:
x ∈ Rn : x = g α (v ∆L ), α ∈ A ⊆ Core v ∆L .
Pick any payoff vector g α (v ∆L ). By Eq. (8) in Theorem 1, we know that:
∀K ∈ ∆0L ,
gKα (v ∆L ) = v(K ) + αK (L) v(N ) − v(N \ K ) − v(K ) .
By cone-modularity of v ∆L , v(N ) − v(N \ K ) − v(K ) ≥ 0, and so gKα (v ∆L ) ≥ v(K ), which means that g α (v ∆L ) is acceptable for all proper cones of (N , L). By component efficiency of g α (v ∆L ), we conclude that g α (v ∆L ) isacceptable for all cones of (N , L). Thus we get g α (v ∆L ) ∈ Core v ∆L . Even if Theorem 3 states that the core of the restricted game coincides with the set of component efficient and α -component fair payoff vectors, it remains to examine whether some payoff vectors in the core are more plausible than others. The requirement than an allocation is obtained through a Harsanyi solution is one way to do that. The next section studies such a core selection.
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By (6), g α is a linear allocation rule on C (∆L ) so that for each
5. Harsanyi solutions In this section, we consider the class of Harsanyi solutions proposed by Vasil’ev (1982) and applied recently by Brink et al. (2012, 2011) to graph-restricted games.3 Each v ∈ C (Ω ) can be expressed in a unique way as:
v=
av (T )uT ,
T ∈Ω
where av (T ) ∈ R is called the Harsanyi dividend of T in v and where {uT : Ω −→ R : T ∈ Ω \ {∅}} is the collection of unanimity games given by: uT (S ) = 1 if T ⊆ S and uT (S ) = 0 otherwise. A sharing system on Ω is a system p = (pS )S ∈Ω \{∅} , where pS is an s-dimensional vector to each player i ∈ N a share assigning S pSi ∈ [0, 1] such that i∈S pi = 1. For a sharing system p, the corresponding Harsanyi solution f p induces the Harsanyi payoffs defined as:
∀v ∈ C (Ω ), ∀i ∈ N ,
p
fi (v) =
pSi av (S ).
α
We prove that an allocation rule g , α ∈ A, on C (∆L ) induced by the allocation rule f α (·, L), is a Harsanyi solution if and only if it is obtained from component efficiency and α -component fairness for some ‘‘credible’’ system of weights α . Imagine that each agent i ∈ N sums the shares of the surpluses collected through his links. His payoff is increasing is these shares αK(j,i) since (6) is decreasing in the complementary shares αK(i,j) . The system of weights α is ‘‘credible’’ if the computed total share is less than or equal to one unit. For such a system, this means that if an agent has a large number of links, then he must divide this unit to specify his share for numerous surpluses. At the same time, an agent with a unique link can enjoy the whole surplus created by his unique link. Thus, his sole neighbor will not benefit from this surplus, but can get some surpluses through his other links. In other words, the search for Harsanyi solutions boils down to selecting systems of weights that possess interesting properties. For instance, this result enables to show that an allocation rule g α is a Harsanyi solution if and only if it is induced by a Random Tree solution. As a corollary, we provide a new characterization of the Random Tree solutions on F in terms of component efficiency and α -component fairness. Let H be the set of Harsanyi solutions on C (∆L ). Note that H is a convex set. Theorem 4. For a given L, let g α , α ∈ A, be an allocation rule on C (∆L ) induced by the allocation rule f α (·, L). The following assertions are equivalent. 1. The allocation rule g α belongs to H . 2. The system of weights α is such that for each i ∈ N, it holds that
αK(i,j) (L) ≤ 1.
(13)
j∈Li
3. There is q such that g α (v ∆L ) = RTq (v, L) for each v ∈ C (2N ). Proof. (1) ⇐⇒ (2). Pick any α ∈ A and define the system p = (pK )K ∈∆L \{∅} as follows: for each nonempty cone K ∈ ∆L , pK is a k-dimensional real vector assigning to each i ∈ K the value pKi = 1 −
αK(i,j) (L).
K ∈∆L \{∅}
Consider a nonempty cone K ⊆ C , C ∈ N /L. Observe that for each i ∈ C \ K and each link ij ∈ L(C ), either K(i,j) ⊇ K or K(j,i) ⊇ K , and there is exactly one link ij ∈ L(C ) such that K(i,j) ⊇ K . From this observation and from (6) in Theorem 1, we immediately get: ∆
giα (uK L ) =
pKi 0
if i ∈ K if i ∈ N \ K .
In case C = K , we see from the expression of pC that every agent i ∈ C gets precisely the payoff expressed in (13): ∆
giα (uC L ) = 1 −
αK(i,j) (L).
j∈Li
Therefore, we can rewrite g α (v ∆L ) as follows:
S ∈Ω : i∈S
v ∆L ∈ C (∆L ), we have: ∆ av ∆L (K )g α (uK L ). g α (v ∆L ) =
(14)
j∈Li ∩K
Note that the system p depends only on the forest (N , L).
3 The set of all Harsanyi solutions is also called the selectope (see Derks et al., 2000).
giα (v ∆L ) =
∀i ∈ N ,
pKi av ∆L (K ).
K ∈∆L :
i∈K
By component efficiency of g α , we obtain:
i∈C
∆
giα (uK L ) =
i∈K
∆
giα (uK L ) =
pKi = 1.
i∈K
Thus, p is a sharing system if and only if for each nonempty cone K and each i ∈ K , pKi ∈ [0, 1], i.e. if and only if (13) holds. We conclude that g α is a Harsanyi solution on C (∆L ) if and only if (13) holds. (1) ⇐⇒ (3). To show: for any v ∆L ∈ C (∆L ), g α (v ∆L ) is a Harsanyi payoff vector if and only if g α (v ∆L ) is a Random Tree payoff vector. The set of payoff vectors g α (v ∆L ), α ∈ A, is a convex set by Theorem 3. The profile of this convex set, given by (11), is generated by the following systems of weights. For L and each component C ∈ N /L, there is an orientation ◦ of (C , L(C )) such that for each orientated link (j, i), αK(i,j) (L) = 1, and so αK(j,i) (L) = 0. Using component efficiency and α -component fairness on F , the induced payoff vectors g α (v ∆L ) are such that giα (v ∆L ) = x◦i (v, L), i ∈ C , C ∈ N /L. If an orientation ◦ is not a rooted tree, some agent has at least two predecessors in the induced orientated subgraph and so (13) is violated. In case ◦ is a rooted tree, each agent belonging to this rooted tree has at most one predecessor and (13) is satisfied. Thus, the systems of weights constructed in this manner satisfy (13) on L if and only if the orientations ◦ on the components of L are rooted trees. In such a case, by (6) in Theorem 1, giα (v ∆L ) = mri (v, L), i ∈ C , C ∈ N /L. By the preceding point, we know that assertion 2 is equivalent to assertion 1. Therefore, we conclude that among the elements of the profile (11), only the marginal contribution vectors induce Harsanyi payoffs vectors, where the sharing system p is constructed as in (14) from the vector of weights α(L) defined above. Because a convex combination of Harsanyi payoff vectors is still a Harsanyi payoff vector, the set of Random tree payoff vectors – the convex hull of the marginal vectors – belongs to the set of Harsanyi payoff vectors. It remains to show that the set of Harsanyi payoff vectors of the form g α (v ∆L ), α ∈ A, is included in the set of Random tree payoff vectors. It suffices to show that the marginal contribution vectors constitute the profile of the convex set of these Harsanyi payoff vectors. Suppose, by way of contradiction, that this assertion is false. Thus, there exists a Harsanyi payoff vector g α (v ∆L ) in this profile such that for at least one link ij ∈ L we have αK(j,i) and αK(i,j) in ]0, 1[. Let (i1 , i2 , . . . , iq ) be a maximal path in L such that for any link ik ik+1 , k ∈ {1, 2, . . . , q − 1}, on this path, αK(i ,i ) (L) and k+1 k
S. Béal et al. / Mathematical Social Sciences 64 (2012) 144–151
αK(ik ,ik+1 ) (L) in ]0, 1[. Since the path is maximal, αK(i1 ,j) ∈ {0, 1} for each j ∈ Li1 \ {i2 } and αK(iq ,j) ∈ {0, 1} for each j ∈ Liq \ {iq−1 }. Let ε ∈ R. For |ε| sufficiently small, define the system of weights α ε which differs from α only on L, and where – for each link ik ik+1 , k ∈ {1, 2, . . . , q − 1}, on the path,
αKε(i
k+1 ,ik )
(L) = αK(ik+1 ,ik ) (L) + ε and
αKε(i
k ,ik+1 )
(L) = αK(ik ,ik+1 ) (L) − ε;
– for any other link ij ∈ L, αKε
αK(i,j) (L).
(j,i)
(L) = αK(j,i) (L), and so αKε(i,j) (L) =
As g α (v ∆L ) is a Harsanyi vector payoff, α(L) satisfies (13) since assertion 1 is equivalent to assertion 2, and the sharing system p ε is constructed as in (14). Thus, for |ε| sufficiently small, g α (v ∆L ) is also a Harsanyi payoff vector since
j∈Li
αKε(j,i ) (L) = 1
αKε(j,i ) (L) =
q
j∈Liq
αK(j,i1 ) (L) + ε ≤ 1,
j∈Li
1
1
αK(j,iq ) (L) − ε ≤ 1
j∈Liq
j∈Li
ε
αK(j,i) (L) =
Cones also form an interesting family of feasible coalitions. The set of cones is in fact an union stable system as introduced by Algaba et al. (2001), i.e. a set of coalitions such that the union of any two elements in the set with a nonempty intersection belongs to the set. Furthermore, the set of cones satisfies two properties which are in the same spirit as the properties used by Brink (forthcoming) in order to characterize the sets of feasible coalitions arising from communication graphs and hierarchies. A collection of feasible coalitions P are called accessible from a feasible coalition S if P is a partition of S \ {i} for some i ∈ N. Such an agent i is called a starting point of S. Player j is a called successor of i, denoted by i ◃ j, if i is a starting point of some S and j is a starting point of some coalition accessible from S. The set of cones satisfies path property and accessibility. The path property is an adaptation of the 2-path property used in Brink (forthcoming) and requires the uniqueness of a ‘‘◃-path’’ between any two players. Accessibility requires that for any player of a feasible coalition, there exists at most one successor of this player in the complementary coalition. The set of cones is the unique set of feasible coalitions that satisfies the path property and accessibility.4 References
and for each other link ij ∈ L,
151
αK(j,i) (L) ≤ 1.
j∈Li ε
It follows that the convex combination g α (v ∆L ) + g α (v ∆L ) /2 is also a Harsanyi payoff vector for |ε| sufficiently small, which contradicts the initial assumption. Thus, the marginal contribution vectors constitute the profile of the Harsanyi payoff vectors of the form g α (v ∆L ), α ∈ A.
−ε
From Theorems 1 and 4, we derive a new characterization of the Random Tree solutions. Theorem 5. An allocation rule on F is a Random Tree solution if and only if it satisfies component efficiency and α -component fairness for some α satisfying (13) for each L. 6. Conclusion Let us motivate further the restriction to the cone-restricted core and the use of cones of a forest. It is true that the cone-restricted core of a forest game can be much larger than the usual core of the Myerson-restricted game. Nevertheless, it satisfies desirable properties. Firstly, efficiency of each element of the cone-restricted core is trivially met. Secondly, part (a) of proof of Theorem 3 shows that the payoff of each player in the cone-restricted core is bounded in spite of the fact that several singleton coalitions are not feasible. The latter property is not guaranteed for other restricted cores (see Grabisch, 2009 for a review of this literature). Thirdly, it can be proved that any payoff vector in the cone-restricted core satisfies the null player property introduced in Béal et al. (2010).
Algaba, E., Bilbao, J.M., Borm, P., López, J.J., 2001. The Myerson value for union stable structures. Mathematical Methods of Operations Research 54, 359–371. Baron, R., Béal, S., Rémila, E., Solal, P., 2011. Average tree solutions and the distribution of Harsanyi dividends. International Journal of Game Theory 40, 331–349. Béal, , Rémila, E., Solal, P., 2010. Rooted-tree solutions for tree games. European Journal of Operational Research 203, 404–408. Bilbao, J.M., 2000. Cooperative Games on Combinatorial Structures. Kluwer Academic Publishers, Dordrecht. Brink, R. van den, 2011. On hierarchies and communication. Social Choice and Welfare, http://dx.doi.org/10.1007/s00355-011-0557-y (forthcoming). Brink, R. van den, van der Laan, G., Moes, N., 2012. Fair agreements for sharing international rivers with multiple springs and externalities. Journal of Environmental Economics and Management 63, 388–403. Brink, R. van den, van der Laan, G., Pruzhansky, V., 2011. Harsanyi power solutions for graph-restricted games. International Journal of Game Theory 40, 87–110. Brink, R. van den, van der Laan, G., Vasil’ev, V., 2007. Component efficient solutions in line-graph games with applications. Economic Theory 33, 349–364. Derks, J., Haller, H., Peters, H., 2000. The selectope for cooperative games. International Journal of Game Theory 29, 23–38. Grabisch, M., 2009. The core of games on ordered structures and graphs. 4OR 7, 207–238. Herings, P., van der Laan, G., Talman, D., 2008. The average tree solution for cycle free games. Games and Economic Behavior 62, 77–92. Herings, P., van der Laan, G., Talman, D., Yang, Z., 2010. The average tree solution for cooperative games with limited communication structure. Games and Economic Behavior 68, 626–633. Myerson, R., 1977. Graphs and cooperation in games. Mathematics of Operations Research 2, 225–229. Shapley, L.S., 1953. A value for n-person games. In: H.W., Kuhn, Tucker, A.W. (Eds.), Contributions to the Theory of Games, vol. II. Princeton University Press, pp. 307–317. Shapley, L.S., 1971. Cores of convex games. International Journal of Game Theory 1, 1–26. Tijs, S., Borm, P., Lohmann, E., Quant, M., 2011. An average lexicographic value for cooperative games. European Journal of Operational Research 213, 210–220. Vasil’ev, V., 1982. On a class of operators in a space of regular set functions. Optimizacija 28, 102–111 (in Russian).
4 The proof of this claim is available from the corresponding author.