Solution concepts for games with general coalitional structure
Gleb Koshevoy a,∗, Dolf Talman b
a Central Institute of Mathematics and Economics, Russian Academy of Sciences, Nakhimovskii prospect 47, 117418 Moscow, Russiab CentER, Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
h i g h l i g h t s
• Theory of marginal values is introduced for games with restrictions to form coalitions.• Theory is based on the notion of nested sets and complex of nested sets.• We study core stability of the solutions.• We introduce half-space supermodularity, which is weaker than convexity of a game.
a r t i c l e i n f o
Article history:Received 14 March 2013Received in revised form17 December 2013Accepted 19 December 2013Available online 27 December 2013
a b s t r a c t
We introduce a theory of marginal values and their core stability for cooperative games with transferableutility and arbitrary set systems representing the set of feasible coalitions. The theory is based on thenotion of strictly nested sets in a set system. For each maximal strictly nested set, we define a uniquemarginal contribution vector. Using these marginal contribution vectors several solutions concepts areintroduced.
The gravity center or GC-solution of a game is defined as the average of the marginal vectors overall maximal strictly nested sets. For union stable set systems, buildings sets, the GC-solution differs fromMyerson-type solutions. The half-space or HS-solution is defined as the average of the marginal vectorsover the class of so-called half-space nested sets and is appropriate for example when feasible coalitionsrepresent social networks. The normal tree orNT-solution is defined as the average of themarginal vectorsover all so-called NT-nested sets and is appropriate when feasibility of a coalition is based on bilateralcommunication between players. For graphical building sets, the NT-solution is equal to the average treesolution. We also study core stability of the solutions and show that the conditions under which the HS-and NT-solutions belong to the core are weaker than conditions under which the GC-solution is stable.
For amore general set system, there exists a uniqueminimal building set containing the set system, itsbuilding covering. As solutions for games on an arbitrary set system of feasible coalitions we propose totake the solutions for its building covering with respect to theM-extension of the characteristic functionof the game.
In the classical model of cooperative games with transferableutility it is assumed that any subset of agents can form a coalitionand obtain some worth which can be freely distributed amongstits members. The problem is how much payoff each player shouldreceive. Inmany situations of cooperation, there are restrictions forforming coalitions. One of the most well-known examples is theMyerson communication graph game (Myerson, 1977), in which
the coalitional structure on the players set is modeled by meansof a graph. The vertices of the graph are labeled by the players,and players which label nodes of an edge are able to communicate.A subset of agents can form a feasible coalition if the inducedsubgraph on the subset is connected. Another example is a gameon a poset; see Faigle and Kern (1992) and Derks and Gilles (1995).In such a game there is a partial order on the set of players, or ahierarchy, and a coalition of players is feasible if its members forman ideal (filter). An extension of games on posets is consideredin Bilbao and Edelmann (2000), where there is given an anti-exchange closure operator on the set of players, and a coalition isfeasible if it is closed with respect to the closure operator. In allthese models, marginal (contribution) vectors are defined, albeitusing different methods, and Shapley type of values are studied as
20 G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30
solutions. For a game with a general coalition structure one facestwo problems. One is to find an appropriate notion of a ‘‘marginalvector’’ and another one is how to weigh these values.
In this paper we consider cooperative games with transferableutility and general coalitional structure. The coalitional structureis given by a set system F , being a collection of subsets of a finiteset of n elements, denoted by [n] := {1, . . . , n}. Elements of [n]might be considered as economic agents, or players, and elementsofF are feasible coalitions that players are able to form.We alwaysassume that the grand coalition [n] is a feasible coalition. A gameon the set system F is a characteristic function v : F → Rassigning to each feasible coalition S ∈ F its worth v(S), being themaximum amount of payoff that coalition S can obtain by itself forits members. A solution is a mapping from the class of games on Fto the n-dimensional vector space Rn and assigns for any game inthe class a payoff to every player.
We first construct for a game with an arbitrary set systemthe M-extension of the game, by using its Möbius inversion, andthen discuss several solution concepts. This solves the problemof defining marginal vectors for a general coalition structure. Incase the set system is given by the sets of vertices of all connectedsubgraphs of a connected graph the M-extension of a gamecoincides with the Myerson-restriction of the game consideredin Myerson (1977). By using the M-extension of a game weintroduce its M-solution as the average of the marginal vectorsinduced by all permutations. For graph games the M-solution isequal to the Myerson value, introduced in Myerson (1977), and forgames on union stable set systems it is the Myerson value definedin Algaba et al. (2001).
To study the problem of weighing marginal vectors, weconsider, to start with, so-called building sets on the player set. Aset system on [n] is a building set if it contains all singletons, theset [n], and together with any two non-disjoint sets also the unionof these sets. In social terms, the union of intersecting networksthat are feasible is also a network that can be formed. Such set-systems closed under union have also been considered in Algabaet al. (2001) and Faigle et al. (2010). To a connected undirectedgraph defined on [n] is associated the graphical building set on[n] consisting of the vertex sets of all connected subgraphs of thegraph. The class of games on graph systems, introduced inMyerson(1977), is therefore a subclass of the class of games on building sets.For an arbitrary set system on [n] there exists a unique minimalbuilding set, called its building covering, containing the set system.
An important property of a building set on [n] is that anysubset of [n] has a unique maximal (with respect to set-inclusion)partition of elements of the building set. Due to this property,to any permutation of the elements of [n] a unique collection ofn elements of the building set is associated. To be more precise,let σ = (σ (1), . . . , σ (n)) be any permutation of the elementsof [n]. Then the set [n] \ {σ−1(n)} is uniquely partitioned intomaximal elements of the building set, say, S1, . . . , Sk. For each Sj,j = 1, . . . , k, denote by tj themaximal element of [n] in Sj such thatσ−1(tj) ∈ Sj, and next partition each Sj \ {σ−1(tj)}, and so on. Inthis way, we obtain a so-called strictly nested set, being a family ofelements of the building set, including [n] itself, such that any twoof its elements are either disjoint or one is a subset of the other,and, moreover, the union of any collection of subsets of disjointelements of the family is not an element of the building set. Nestedsets for buildings were introduced in algebraic geometry; see, forexample, De Concini and Procesi (2005), Feichtner and Sturmfels(2005) and Postnikov (2005). When a strictly nested set consistsof n different sets it is said to be maximal. For a building set F ,any maximal strictly nested set is a vertex of the nested complexof F . Every maximal strictly nested set describes a unique way inwhich the grand coalition [n] can be built from elements of onlythe building set by, starting with the empty set, letting players join
to one or more feasible coalitions to construct larger and largerfeasible coalitions until the grand coalition has been constructed.
In case the building set on [n] is the power set 2[n] there aren! maximal strictly nested sets and each maximal strictly nestedset is a chain that corresponds to a unique permutation. If foran arbitrary building set a maximal strictly nested set is not achain, it corresponds to a collection of permutations, being theset of all linear extensions of the partially ordered set that isinduced by the structure of the strictly nested set and thereforealso corresponds to a rooted tree. Given a gamewith characteristicfunction on a building set, for any maximal strictly nested set ofthe set system, we define a unique marginal vector in which everyplayer receives as payoff the amount he contributes inworthwhenjoining his subordinates in the tree. Since a vertex of a nestedcomplex corresponds to a rooted tree on [n] and all connectedideals of this tree are elements of the underlying building set,we can define the marginal vector by the same rule as is donein Demange (2004) for a hierarchy. We show that each marginalvector of a game defined in this way is equal to themarginal vectorof the M-extension of the game calculated with respect to any ofthe permutations being a linear extension of the poset induced bythe rooted tree.
For a cooperative game on a building set we introduce severalnew solution concepts which take into account the uniqueness ofmarginal vectors. The GC-solution (or gravity center solution) isdefined as the average of the marginal values over all maximalstrictly nested sets in the building set. The GC-value differs fromthe M-value and the Myerson value defined in Myerson (1977)and Algaba et al. (2001) because in the GC-solution every marginalvector is only counted once like in the Shapley value. Let usnotice that recently, in Faigle et al. (2010), a Shapley value wasdefined using the Monge algorithm. For union stable set-systemsthis solution is different from the Myerson value. For the class ofgames onbuilding sets, theGC-solution coincideswith this Shapleyvalue as is proven in the Appendix. However, the solutions differfor games on set systems which are non-building sets, and thecomplexity of computing the Shapley value in Faigle et al. (2010)on building sets is always O(n!) while the complexity for the GC-solution depends on the structure of the building set and rangesfrom O(n) to O(n!).
The HS-solution (or half-space solution) is defined as theaverage ofmarginal vectors over a specific class ofmaximal strictlynested sets, namely the class of so-called HS-nested sets. In anHS-nested set every player forms with his subordinates in thecorresponding tree a feasible coalition, and also the remainingplayers do. The HS-solution is therefore appropriate if the feasiblecoalitions concern (social) networks connecting different groups ofagents.
The NT-solution (normal tree solution) is defined as the averageof marginal payoff vectors over another specific class of maximalstrictly nested sets, namely the class of so-called NT-nested sets.In the corresponding tree of an NT-nested set every player islinked to his successor in the tree, so that the NT-solution is amore appropriate solution concept if for example the coalitionalstructure involves bilateral communication or relations betweenplayers. On graphical buildings the collection of NT-nested setscorresponds to the collection of spanning normal trees of thegraph, introduced in Diestel (2005).
For games with an arbitrary building set the HS- or the NT-solution may not exist. For graphical buildings, however, the set ofNT-nested sets is a nonempty subset of the set of HS-nested sets.Moreover, for games on graphical building sets the NT-solutioncoincides with the Average Tree solution introduced in Heringset al. (2010) for the class of graph games. In case the set systemis equal to the power set of all coalitions all four solutions coincideand are equal to the Shapley value.
G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30 21
For all three solutionswe study core stability. The core of a gameconsists of the payoff vectors that cannot be blocked by any feasiblecoalition. We prove that, given a building set B as coalitionalstructure, all marginal vectors and therefore also the GC-solutionand the M-value belong to the core if the game is both B-superadditive andB-supermodular. The latter condition is weakerthan supermodularity. We also introduce the notion of half-spaceB-supermodularity, which weakens B-supermodularity. For anHS-nested set in B the corresponding marginal vector belongsto the core if the game is half-space B-supermodular and B-superadditive. This is a generalization of a core stability result forgames on rooted tree in Demange (2004). For an NT-nested set inB the corresponding marginal vector belongs to the core if thegame is half-space B-supermodular and 2-superadditive. Whenthe game is totally positive, the GC-solution is the gravity centerof the core. This latter property does not hold for the M-value orthe Myerson values defined in Algaba et al. (2001) and Myerson(1977).
As stated above, for an arbitrary set systemF on [n] there existsa unique minimal building set B(F ), the building covering of F ,which contains F . For a characteristic function v : F → R, wetake the restriction of the M-extension of v with respect to B(F )to obtain a game vF on the building covering B(F ) of F . TheM-, GC-, HS-, and NT-solutions for this latter gamewe define as thecorresponding solutions for the original game v onF . Core stabilityof these solutions is provided by the corresponding conditionsfor the building covering. On the class of convex geometries wecompare our solutions with the solution proposed in Bilbao andEdelmann (2000).
This paper is organized as follows. In Section 2, for a givenarbitrary set system the concept of strictly nested set is introducedand for a function on it its M-extension is defined using theMöbius inverse. It is shown that to every strictly nested set therecorresponds a rooted tree. For a maximal strictly nested set, aunique marginal vector is defined. In Section 3 maximal strictlynested sets and the M-extension of games on building sets arestudied. In Section 4 the different solution concepts for gameson building sets are introduced. In Section 5 core stability of thesolutions is studied. In Section 6 solutions for games on arbitraryset systems are discussed. In the Appendix it is proven that, fora building set, the GC-solution coincides with the Shapley valuedefined in Faigle et al. (2010) using the Monge algorithm.
2. TheM-extension and strictly nested sets
Let [n] = {1, . . . , n} be a finite set for some integer n ≥ 2 andlet F ⊆ 2[n] be a given set system on [n]. We assume that both ∅
and [n] belong to F and that for any function v : F → R it holdsthat v(∅) = 0.
For a function f : 2[n]→ R, let µ : 2[n]
→ R be its Möbiusinversion, i.e., µ satisfies
f (T ) =
T ′⊆T
µ(T ′), T ∈ 2[n].
The Möbius inversion of f is given by
µ(T ) =
T ′⊆T
(−1)|T |−|T ′|f (T ), T ∈ 2[n].
Definition 2.1. Let v : F → R be a function, then theM-extensionvF
: 2[n]→ R of v is given by the following conditions:
(i) vF (S) = v(S) for every S ∈ F .(ii) For theMöbius inversion of vF ,µF , it holds thatµF (S) = 0
for every S ∈ F .
Theorem 2.2. For a function v : F → R its M-extension vF is welldefined.
Proof. Consider the system of linear equations:
v(S) =
T∈F |T⊆S
µ(T ), S ∈ F .
The matrix for this system corresponds after appropriate reorder-ing of columns and rows by set inclusion to a 0–1 upper-triangularsquare matrix with all ones on the diagonal. Therefore the systemhas a unique solution µ(S), S ∈ F . Define µF
: 2[n]→ R by
µF (S) = µ(S) if S ∈ F and µF (S) = 0 if S ∈ F . Then vF is theunique function for which µF is its Möbius inverse, i.e.,
vF (S) =
T∈F |T⊆S
µF (T ), S ∈ 2[n]. �
Wehave the following interesting property of theM-extension.Let F1 ⊆ F2 be two set systems; then for any function v : F1 → Rit holds that
vF1 = (vF1 |F2)F2 . (1)
In particular this property holds when F2 is equal to F1.In Faigle et al. (2010) another extension v of v : F → R is
proposed based on the so-called Monge algorithm. This extensionalso possesses the property that v(T ) = v(T ) for every T ∈ F ,but it differs from the M-extension. For example, the extensionin Faigle et al. (2010) is not defined for F := 2[n]
\ {[n]}, whilethe M-extension is defined and gives vF ([n]) =
n−1k=1(−1)k+1
i1<···<ikv([n] \ {i1, . . . , ik}).
Next we define strictly nested sets in an arbitrary set system.
Definition 2.3. A subset N of F is a strictly nested set if it satisfiesthe following conditions:
(G1) For any different S, T ∈ N it holds that either S ⊂ T or T ⊂ Sor S ∩ T = ∅.
(G2) For any collection of k, k ≥ 2, disjoint subsets T1, . . . , Tk inNit holds that
T ′
1 ∪ · · · ∪ T ′
k ∈ F
for any nonempty T ′
j ⊆ Tj, j = 1, . . . , k.(G3) [n] ∈ N .
Property (G1) is known under the names nested sets, laminar orhierarchy; see, for example, Frank and Tardos (1988). This propertysays that if two sets in a strictly nested set are not disjoint thenone of them must be a subset of the other one. Property (G2) isthe strength of the nested property; see, for example, De Conciniand Procesi (2005). It says that no union of subsets of disjointmembers of a strictly nested set can be feasible. Notice that anychain {N1,N2, . . . ,Nk−1, [n]} of length k, 1 ≤ k ≤ n, with N1 ⊂
N2 ⊂ · · · ⊂ Nk−1 ⊂ [n] and Nj ∈ F for j = 1, . . . , k − 1, isa strictly nested set, since (G2) is automatically fulfilled. Property(G3) says that the set [n] is a member of any strictly nested set. Inparticular, {[n]} is a strictly nested set.
Example 2.4. Consider the set system F consisting of all single-tons of [n] and [n] itself. Then any strictly nested set N consists of[n] and at most n − 1 singletons, whereas a chain consists of [n]and at most one singleton.
To any strictly nested set N in F there corresponds a rootedtree FN , whose vertex set is labeled by a partition of [n], definedas follows. Because of (G1) and (G3) for any strictly nested set Nand i ∈ [n] there is a unique minimal element in N , denoted byTN (i), containing i. Let the ordering≼
N on [n] be defined by i≼N jif TN (i) ⊆ TN (j) and consider the partition of [n] constituted
22 G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30
Fig. 1. The rooted trees for Example 2.6.
from sets being equivalent elements of [n] with respect to ≼N .
Consider the quotient-set [n]/ ∼N , that is an element of [n]/ ∼
N
corresponds to a set of equivalent elements of ∼N . The ordering
≼N induces a poset on [n]/ ∼
N . The Hasse-diagram of this poset isthe rooted tree FN . More precisely, consider all maximal elementsof N different from [n]. Let TN
1 , . . . , TNm be these sets. Because of
(G1) and (G2) these sets are disjoint and their union is not equal to[n]. Hence, the set [n] \ (TN
1 ∪ · · · ∪ TNm ) consists of equivalent
elements and is the root of the tree FN . The successors of theroot are formed by the roots in the subtrees corresponding to therestrictions of N to each of the sets TN
1 , . . . , TNm . The existence
of the tree follows by induction, since the restriction of a strictlynested set to any such a set is a strictly nested set with respect tothe restriction of F to that set.
Example 2.5. Consider the set system F = {{1}, {2}, {3}, {4},{1, 2}, {1, 3}, {1, 2, 3, 4}}. Then N = {{1}, [n]} is a strictly nestedset with rooted tree FN having {2, 3, 4} as root and {1} as itssuccessor. The collection N ′
= {{1}, {2}, [n]} is not a strictlynested set because {1, 2} belongs to F . The collection N ′′
= {{1},{1, 2}, [n]} is a strictly nested setwith rooted tree FN ′′
having {3, 4}as root, {2} as the successor of the root and {1} as the successorof {2}.
A strictly nested set N is maximal if it contains n differentnonempty sets. Notice that an arbitrary set system may not havea maximal strictly nested set. For example, for n = 4, the setsystem {{1}, {4}, {1, 4}, {2, 3}, {1, 2, 3}, {2, 3, 4}, [4]}has nomax-imal strictly nested sets.1 In Section 3 we discuss set systems thatalways contain maximal strictly nested sets. To every maximalstrictly nested set N in F there corresponds a rooted tree FN withvertex set [n]. In such a case, the ordering ≼
N has no multipleequivalent elements, and therefore ([n], ≼N ) is a poset.
Example 2.6. Consider again set system F = {{1}, {2}, {3}, {4},{1, 2}, {1, 3}, {1, 2, 3, 4}}. ThenF has fourmaximal strictly nestedsets, {{1}, {4}, {1, 2}, {1, 2, 3, 4}}, {{2}, {4}, {1, 2}, {1, 2, 3, 4}},{{1}, {4}, {1, 3}, {1, 2, 3, 4}}, {{3}, {4}, {1, 3}, {1, 2, 3, 4}}. We de-pict the corresponding rooted trees in Fig. 1. Notice that none ofthe four maximal strictly nested sets is a chain.
The rooted tree of any maximal strictly nested set in a setsystem determines a marginal vector, as in Demange (2004) for ahierarchy. If v is a game on set system F , then at marginal vectormv(N ) of v with respect to any maximal strictly nested set Nplayer i ∈ [n] receives as payoff what this player contributes inworthwhen he joins his subordinates in the corresponding tree FN
to form feasible coalition TN (i). For i ∈ [n], let SN (i) denote the setof successors of player i in the tree FN corresponding to maximalstrictly nested set N .
1 If a strictly nested set contains the singleton set {1}, then it cannot contain {2, 3}or {2, 3, 4}. Thus it contains either {1, 4} or {1, 2, 3}, or none of them. In both cases,it consists of at most three sets. Similarly, if a strictly nested set contains {4}, then italso consists of atmost three sets. Finally, if a strictly nested set contains {2, 3}, thenit cannot contain {1} or {4}, or {1, 4}. Thus it contains either {1, 2, 3} or {2, 3, 4}, andagain it consists of at most three sets.
Definition 2.7. Let v : F → R be a function. For a maximalstrictly nested set N in F , the marginal vector of v with respectto N is given by
mvi (N ) = v(TN (i)) −
j∈SN (i)
v(TN (j)), i ∈ [n].
For a maximal strictly nested set N in F , denote by SN the setof permutations on [n]which are linear extensions (total orderingson [n]) of the poset ([n], ≼N ). In this way to a set system F isassociated the set of permutations
SF:= ∪N SN ,
where the union is taken over the set of all maximal strictly nestedsets in F .
Nowwe show how this set of permutations is related to theM-extension. For this the following notions are of use. Denote by Snthe set of all permutations on [n].
Definition 2.8. Let f : 2[n]→ R be a function. For a permutation
σ ∈ Sn, the marginal vector mf (σ ) of f with respect to σ is givenby
mfσ−1(i)
(σ ) = f ({σ−1(1), . . . , σ−1(i)})
− f ({σ−1(1), . . . , σ−1(i − 1)}), i ∈ [n].
Remark 1. For the set system 2[n], any maximal strictly nested settakes the form of a chain Nσ = {{σ−1(1), . . . , σ−1(i)} | i ∈ [n]}for some permutation σ ∈ Sn. The marginal vector of a functionv : 2[n]
→ R with respect to the maximal strictly nested set Nσ isnothing but the marginal vector mv(σ ). However, for an arbitraryset system F , a chain Nσ may not be a maximal strictly nested setin F , even not when σ ∈ SF .
Remark 2. The marginal vector mv(N ) of a function v : F → Rwith respect to a strictly nested set N in F can also be interpretedas follows. Consider the Möbius inversion µN of the M-extensionof the restriction v|N of v to N . Then we get
mvi (N ) = µN (TN (i)), i ∈ [n].
We have the following theorem.
Theorem 2.9. Let v : F → R be a function and let vF be its M-extension. Then, for every maximal strictly nested set N inF and anypermutation σ ∈ SN it holds that
mvF(σ ) = mv(N ).
For the proof of Theorem 2.9 we need the following lemma.
Lemma 2.10. Let v : F → R and let T1 and T2 be two disjointsubsets of 2[n] such that for any nonempty T ′
1 ⊆ T1 and T ′
2 ⊆ T2 itholds that T ′
1 ∪ T ′
2 ∈ F . Then
vF (T1 ∪ T2) = vF (T1) + vF (T2).
G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30 23
Proof. The proof follows because for the Möbius inversion µF ofvF it holds that µF (T ′
1 ∪ T ′
2) = 0 for any T ′
1 ⊆ T1 and T ′
2 ⊆ T2. �
Proof of Theorem 2.9. Take any maximal strictly nested set N inF and permutation σ ∈ SN . Let i ∈ [n] and let S := {σ−1(1), . . . ,σ−1(i)} and j = σ−1(i). Since σ ∈ SN , we have that TN (j) is asubset of S. BecauseN is amaximal strictly nested set, for any non-empty subset B ⊆ TN (j) and any nonempty subset B′
⊆ S \ TN (j)it holds that B ∩ B′
= ∅ and B ∪ B′∈ F . Hence, by Lemma 2.10,
vF (S) = vF (TN (j)) + vF (S \ TN (j))= v(TN (j)) + vF (S \ TN (j)).
The latter equality holds because TN (j) ∈ F . Similarly, it holdsthat
vF (S \ {j}) = vF (TN (j) \ {j}) + vF (S \ TN (j))
=
k∈SN (j)
v(TN (k)) + vF (S \ TN (j)).
Therefore, we have
mvj (N ) = vF (S) − vF (S \ {j}) = mvF
σ−1(i)(σ ). �
Corollary 2.11. Let N1 and N2 be two different maximal strictlynested sets in F ; then
SN1 ∩ SN2 = ∅.
Proof. Since the union of N1 and N2 is not a strictly nested set,there exists a function v : F → R satisfying mv(N1) = mv(N2).From Theorem 2.9 it follows that SN1 ∩ SN2 = ∅. �
From this result it follows that the set of permutations SF ispartitioned into SN over all maximal strictly nested sets N in F .
3. Building sets
In Algaba et al. (2001) the concept of partition system isintroduced as a combinatorial abstraction of connected subgraphsof a graph. Partition systems are equivalent to the following notionof building sets introduced in Postnikov (2005).
Definition 3.1. A set system B on [n] is a building set if it satisfiesthe following conditions:
(B1) For any S, T ∈ B such that S ∩ T = ∅ it holds that S ∪ T ∈ B.(B2) B contains all singletons {i}, i ∈ [n].
We assume in this paper that any building set on [n] alsocontains [n]. A natural example of a building set is the collectionof vertex sets of all connected subgraphs of a connected graph.
Example 3.2. Let G = (V (G), E(G)) be a connected graph withvertex set V (G) = [n] and edge set E(G) ⊆ {{i, j} ⊆ [n] | i = j}.Then the set system consisting of the vertex sets of all connectedsubgraphs of G forms a building set, called the graphical buildingB(G) of G.
Postnikov, see Postnikov (2005), defines B-nested sets for abuilding set B as follows.
Definition 3.3. A subset N of a building set B on [n] is a B-nestedset if it satisfies the following conditions:
(N1) For any different S, T ∈ N , it holds that either S ⊂ T orT ⊂ S or S ∩ T = ∅.
(N2) For any collection of k, k ≥ 2, disjoint subsets T1, . . . , Tk inNit holds that T1 ∪ · · · ∪ Tk ∈ B.
(N3) [n] ∈ N .
By definition, any strictly nested set in B is a B-nested set. Theconverse is also true.
Lemma 3.4. Let B be a building set on [n] and let N be a B-nestedset. Then N is a strictly nested set in B .
Proof. For simplicity suppose that k = 2 and for disjoint T1, T2 inN there exist nonempty T ′
1 ⊆ T1 and T ′
2 ⊆ T2 such that T ′
1∪T ′
2 ∈ B.Because of property (B1) it holds that T1∪T ′
2 ∈ B and again by (B1)T1 ∪ T2 ∈ B, which contradicts (N2). �
From the lemma it follows that for a building set B the set ofB-nested sets coincides with the set of strictly nested sets in B.
Theorem 3.5. Let B be a building set on [n]. Then for any permuta-tion σ ∈ Sn there exists a maximal strictly nested set N in B suchthat σ ∈ SN .
Proof. Let σ ∈ Sn be a permutation. We construct a maximalstrictly nested set N such that σ ∈ SN as follows. For any k ∈ [n],let N(k) be a maximal element of B that contains σ−1(k) andis a subset of {σ−1(1), . . . , σ−1(k)}. Such a set exists, due to thedefinition of a building set and since {σ−1(k)} ∈ B. Notice thatN(1) = {σ−1(1)} and N(n) = [n]. Then we obtain a collection Nof n sets containing also [n]. We still have to check property (N2).Suppose (N2) is not valid, then there exists disjointN(k1) andN(k2)such that N(k1)∪N(k2) ∈ B. Let k2 > k1, then N(k1)∪N(k2) ∈ Band N(k1) ∪ N(k2) ⊆ {σ−1(1), . . . , σ−1(k2)}, which contradictsmaximality of N(k2). �
From this theorem and the previous lemma it follows thatany building set contains maximal strictly nested sets. Anotherinteresting property of a building set is that any T ∈ B is includedin some maximal strictly nested set.
Lemma 3.6. Let B be a building set on [n] and T ∈ B . Then thereexists a maximal strictly nested set in B which contains T among itselements.
Proof. The restriction of B to T is a building set, denoted by B|T .By the previous theorem, there exists a maximal B|T -nested set inB|T . Let us extend this set to a maximal B-nested set N on [n].Pick an element i ∈ [n] \ T and choose a maximal element of Bthat is a subset of T ∪ {i} and contains i. By adding this set to N ,N becomes a B|T∪{i}-nested set in B|T∪{i}. In the next step, pick anelement i′ in [n]\(T ∪{i}) and choose amaximal element ofB thatis a subset of T ∪{i}∪{i′} and contains i′, and add this set toN , andso on. In the last step [n] will be added, since [n] ∈ B, and we endup with a maximal B-nested set, which is then a maximal strictlynested set in B. �
For building sets theM-extension takes an easy form.
Lemma 3.7. Let B be a building set on [n] and let v : B → R be afunction; then the M-extension vB of v is given by
vB(T ) =
kj∈1
v(Tj), T ∈ 2[n],
where {T1, . . . , Tk} is the unique partition of T ⊆ [n] into maximalelements of B .
Proof. For any set T ⊂ [n], there exists a uniquemaximal partitionof T in sets from B as follows by induction. Take any t ∈ T andconsider a maximal set of B which contains t and is contained inT . Such a set exists and denote this set by T1. Then T1 togetherwith the partition of T \ T1, which exists by induction, form thedesiredpartition of T . Because of property (B1), any subset S, S ⊂ T ,having a non-empty intersection with more than one element ofthe partition does not belong to B, and hence µB(S) = 0. Thisimplies the proposition. �
24 G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30
Because of this lemma and Theorem 2.9 we may construct amaximal strictly nested set corresponding to any permutationσ ∈ Sn as follows. To every element {σ−1(1), . . . , σ−1(k)},k = 1, . . . , n, of the chain Nσ , we associate the unique partitioninto maximal elements of B; then the collection of all such setsconstitutes the same maximal strictly nested set as constructed inthe proof of Theorem 3.5.
In Section 6 we consider arbitrary set systems. For an arbitraryset system F on [n], there exists a minimum building set B(F )that contains F , called the building covering of F . Since theintersection of two buildings sets is also a building set, B(F ) isuniquely defined. For the building set B(F ), any strictly nestedset in F is a B(F )-nested set.
Proposition 3.8. Let F be a set system and let B(F ) be the buildingcovering of F . Then every strictly nested set in F is a B(F )-nestedset.
Proof. LetN be a strictly nested set inF . Let T1, . . . , Tk be disjointelements in N . Then we have to check that the union T1 ∪ · · · ∪
Tk cannot take the form A1 ∪ · · · ∪ At , where A1, . . . , At is anon-disjoint family in F . Suppose not, then such a non-disjointfamily A1, . . . , At exists. Then there exists at least one element ofthis family that has a non-empty intersection with at least twoelements of the family T1, . . . , Tk. This contradicts with property(G2). �
4. Solution concepts for games on building sets
Let B be a building set system on [n] and let v : B → Rbe a function. We consider a cooperative game with transferableutilities as follows. We take B to be the set of feasible coalitionson the set of n players and, for T ∈ B, v(T ) the worth of feasiblecoalition T . Denote by V(B) the set of cooperative games on B. Asolution is a mapping from V(B) to Rn.
Based on Definitions 2.8 and 2.7, the following two solutionconcepts come naturally.
Definition 4.1. For a building set B on [n] and for a game v ∈
V(B), the following two solutions are given:
• The M-solution is the average of the marginal vectors mvB(σ )
over the set of all permutations σ ∈ Sn.• TheGC- (or gravity center) solution is the average of themarginal
vectorsmv(N ) over all maximal strictly nested sets N in B.
For a game v on a building set B, we consider its M-extensionvB and take for the M-solution the Shapley value for this M-extension. For union stable set systems the M-solution coincideswith theMyerson-type value defined inAlgaba et al. (2001), and forgraphical building sets it is the Myerson value defined in Myerson(1977).
The GC-solution takes the average of all marginal vectors withrespect to the maximal strictly nested set. Denote by n(B) thenumber of maximal strictly nested sets in building set B, and, fora maximal nested set N in B, denote by p(N ) the cardinality ofthe corresponding set SN of permutations. Then the GC-solutionis equal to
GC(v) =1
n(B)
N
mv(N )
and according to Theorem 2.9 theM-solution is equal to
M(v) =1n!
N
p(N )mv(N ),
where both summations are over all maximal strictly nested setsin B.
In general, for a building set B, the function p is not a constant,and, therefore, the GC-solution differs from the M-solution. TheM-solution is a weighted average, with weights depending on thestructure of the building set, and theGC-solution is an un-weightedaverage of all different marginal vectors.
Recently, in Faigle et al. (2010), the Shapley value was definedfor arbitrary set systems using the Monge algorithm. In theAppendix we prove that on the class of games on building setsthis value coincides with the GC-solution. For games on other setsystems these two solutions may differ.
The tree corresponding to a maximal strictly nested set N ina building set B describes how the grand coalition [n] can beformed by letting players join to feasible coalitions to form largerfeasible coalitions, starting with the empty set. For a maximalstrictly nested set N and i ∈ [n], let SN (i) be the set of successorsof i in the tree FN i.e., j ∈ SN (i) if TN (j) is a maximal element ofN in TN (i) \ {i}. Then player i forms the larger feasible coalitionTN (i) in N by joining simultaneously to all feasible coalitionsTN (j), j ∈ SN (i), that were formed by his successors. These lattersub-coalitions form a maximal partition of the set TN (i) \ {i}of subordinates of i (property G1) and satisfy that their unionis not feasible (property G2), i.e., these coalitions are not ableto cooperate without player i. For the formation of the feasiblecoalition TN (i), player i receives as payoff what he contributes inworth, described in the corresponding marginal vector. One of theplayers, the root of the tree, forms the grand coalition [n], which isalso feasible (property G3), by joining simultaneously to all feasiblecoalitions formed by its successors. The collection of maximalstrictly nested sets in F describes all different possibilities inwhich the grand coalition can be formed in this way.
For the GC-solution it is implicitly assumed that all maximalstrictly nested sets in a building set are admissible and every playertherefore receives his marginal contribution. In every maximalstrictly nested set a player forms together with his subordinatesin the corresponding tree a feasible coalition. When for examplethe coalitional structure is based on domination relations betweenplayers it seems appropriate to consider allmaximal strictly nestedsets in the set system for which a coalition is feasible if it containsall the players that are being (strictly) dominated by a player in thecoalition, e.g., see Faigle and Kern (1992).
In case a building set reflects (social) connectivity of playersit might be desirable to restrict the admissible set of maximalstrictly nested sets and require that for every player the playersthat are not his subordinates in the corresponding tree form afeasible coalition. In this way the set of remaining players is ableto transmit information or to connect through this player to hissubordinates. For this ability the player receives as payoff hismarginal contribution in worth.When in amaximal strictly nestedset there is at least one player that is not able to connect the otherplayers in this way, the maximal strictly nested set will not beadmissible. For a set system F on [n] an element S ∈ F is a calledhalf-space in F if [n] \ S ∈ F .
Definition 4.2. For a set systemF on [n] amaximal strictly nestedset N is a half-space nested set (HS-nested set) if for every i ∈ [n]with SN (i) = ∅, the set TN (i) is a half-space in F .
Since for a maximal strictly nested set N every TN (i), i ∈ [n],belongs toF , we have thatN is anHS-nested set if for every i ∈ [n]the complement to a non-singleton TN (i) also belongs to F . Thismeans that in the corresponding tree FN , for every node it holdsthat after contracting the node and all its subordinates to its uniquepredecessor, the resulting set of nodes belongs to F . This restrictsthe collection of maximal strictly nested sets.
G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30 25
Example 4.3. Given a connected graph G on [n], a rooted treeis an HS-tree on G if it corresponds to an HS-nested set in thegraphical buildingB(G). Let us describe how to construct HS-treesby induction on the number of vertices of the graph. Suppose, forall graphs with fewer than n nodes, HS-trees are listed. Let G =
(V (G), E(G)) be a graph with vertex set V (G) = [n]. Pick a vertexi ∈ V (G) and let G1, . . . ,Gm be the components of the subgraph ofG on the vertex set [n] \ {i}. For each component Gj pick an HS-treeFj with respect to the building set B(Gj), j = 1, . . . ,m, and let rj bethe root of Fj. Since, for any j1 = j2, Gj1 and Gj2 are not connected,every B(Gj)-nested set is a B(G)-nested set. Now, we connect allnodes rj, j = 1, . . . ,m, to node i and obtain a tree F on [n]with rooti and as its successors r1, . . . , rm. This tree is an HS-tree if and onlyif for each j = 1, . . . ,m one of the following conditions is satisfied:
• {i, rj} ∈ E(G);• {i, rj} ∈ E(G) and for every component K of the subgraph of G
on the vertex set V (Gj) \ {rj} there exists a node w ∈ K suchthat {i, w} ∈ E(G);
• {i, rj} ∈ E(G) and there exists a singleton component {w} of thesubgraph of G on the vertex set V (Gj) \ {rj} such that {i, w} ∈
E(G).
A building set may also reflect the possibilities of communica-tion between players. Players communicate bilaterally with eachother and coalitions of players are feasible if they contain enoughopportunities for the players to communicate with each other, notnecessarily being the sets of connected vertices of a connectedgraph. In this case a maximal strictly nested set can only be admis-sible if every player is able to communicate and therefore formsa feasible coalition with each of his successors in the correspond-ing tree. For this ability to communicate a player receives as payoffhismarginal contribution inworth.When at least one communica-tion link is missing in the corresponding tree, the maximal strictlynested set cannot be formed and is not admissible.
Definition 4.4. For a set systemF on [n] amaximal strictly nestedset N is an NT-nested set if for every i ∈ [n] and j ∈ SN (i) it holdsthat {i, j} is an element of F .
A maximal strictly nested set N is an NT-nested set if everyplayer is able to cooperate with each of his successors in thecorresponding tree FN . This restricts the collection of maximalstrictly nested sets.
Example 4.5. Let B(G) be a graphical building for a connectedgraph G on vertex set [n]. Then there is a bijection betweenthe collection of maximal NT-nested sets in B(G) and the set ofspanning normal trees on G (see Diestel (2005), p. 15). The latterforms a subset of rooted spanning trees of G. However, not everyrooted spanning tree of a graph G corresponds to an NT-tree. Weproceed to constructNT-trees by induction on the cardinality of thevertex set. Suppose for all graphsG on [k], k < n, the correspondingNT-trees have been constructed. Let G = (V (G), E(G)) be a graphwith vertex set V (G) = [n]. Pick a vertex i ∈ V (G); let G1, . . . ,Gmbe the components of the subgraph of G on the vertex set [n] \ {i}.For each j = 1, . . . ,m there is an edge {i, rj} ∈ E(G) with rj ∈
V (Gj). Let Fj be an NT-tree in Gj with root rj, j = 1, . . . ,m. Then anNT-tree in G with root i is obtained by connecting each Fj to i byedge (i, rj), j = 1, . . . ,m.
For a building set every NT-nested set is an HS-nested set. Thenext example gives the collection of maximal strictly nested sets,NT-nested sets, andHS-nested sets for a graphical building setwiththe graph being a line-tree on [n].
Example 4.6. Let An = ([n], E) with E = {{i, i + 1} | i =
1, . . . , n−1} be a line-tree on [n]. Then there is a bijection betweenmaximal B(An)-nested sets and rooted binary trees on [n]. Recallthat, for a rooted binary tree, every node has either exactly twosubordinates or none. We proceed by induction on n. Suppose thatfor k < n such a bijection between B(Ak)-nested trees and planebinary trees on [k] exists. Note thatB(Ak) consists of intervals, thatis sets of the form {a, a + 1, . . . , a + b} with a, a + b ∈ [k]. Picka vertex i ∈ [n]. In any B(An)-nested set, there are exactly twosuccessors sets of i, since these subsets are intervals and because ofcondition (N2), {1, . . . , i−1} and {i+1, . . . , n}. Then anyB(Ai−1)-nested set in {1, . . . , i−1} is a binary tree aswell as is anyB(An−i)-nested set in {i + 1, . . . , n}. This provides the required bijection.The number of such trees is the Catalan number Cn :=
(2n)!(n+1)!n! . HS-
nested sets inB(Ak) are constructed by the above construction andonly the first and third items are possible. In Fig. 2we depict an HS-tree.There are
ni=1 F(i + 1)F(n − i + 2) HS-trees, where F(k), k =
1, 2, . . . , n+ 1, denotes the Fibonacci sequence for n+ 1. Any NT-tree takes the form: the root r has as successors vertices r − 1 andr + 1, and vertex i, i = r , has as successor vertex i − 1 if i < r andvertex i + 1 if i > r . In total there are n NT-trees, because for eachi ∈ [n] there is one NT-tree having i as root.
For games onbuilding setswe introduce the following twomoresolution concepts.
Definition 4.7. Let a building set B on [n] be given. For a gamev ∈ V(B) the following solutions are given:
• The HS- (or half-space) solution is the average of the marginalvectorsmv(N ) over all HS-nested sets N in B.
• The NT- (or normal tree) solution is the average of the marginalvectorsmv(N ) over all NT-nested sets N in B.
The GC-solution is just the average of all different marginalvectors, while the Myerson value is a weighted average ofthese vectors with the weights determined by the number ofpermutations that correspond to the maximal strictly nested sets.The HS-solution is a completely new solution concept and takesthe average of a specific set of marginal vectors, whereas theNT-solution takes the average of at least for games on graphicalbuilding sets an even more specific set of marginal vectors. Forgraphical building sets the NT-solution coincides with the averagetree solution introduced in Herings et al. (2010).
Example 4.8. Let G = ([n], E) be a circular graph with n = 4 andE = {{1, 2}, {2, 3}, {3, 4}, {1, 4}}. Then the Myerson value is theaverage of (4! =)24 marginal vectors, four of them showing uptwice, the GC-solution is the average of all 20 different marginalvectors, the HS-solution is the average of 16 of those marginalvectors, and the NT-solution is the average of 8 of those lattermarginal vectors.
Example 4.9. Let G = ([n], E) be a graph with n = 4 and E =
{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {3, 4}}. Then the Myerson value is theaverage of 24 marginal vectors, two of them showing up twice,the GC-solution is the average of all 22 different marginal vectors,the HS-solution is the average of 18 of those marginal vectors,and the NT-solution is the average of 14 of those latter marginalvectors. There are five rooted trees corresponding to maximalstrictly nested sets that have node 1 (or node 3) as root and sixtrees that have node 2 (or node 4) as root, there are five HS-treesthat have node 1 as root and four HS-trees that have node 2 as root,and there are three spanning normal trees that have node 1 as rootand four spanning normal trees that have node 2 as root.
26 G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30
Fig. 2. An HS-tree in the case of a line-tree.
The next example shows that in non-graphical building sets HS-and NT-nested sets may not exist.
Example 4.10. Let the building set B on [n] consist of all single-tons and the set [n]. This building set has nmaximal strictly nestedsets, each consisting of [n] and all singletons but one. Each of thesemaximal nested sets is neither an NT-nested nor an HS-nested set.The GC-solution is equal to
GCj(v) = v({j}) +1n(v([n]) −
i∈[n]
v({i)}), j ∈ [n].
5. Core stability
In this section we present conditions under which the solutionsdefined in the previous section are elements of the core. Our maingoal is to establish such conditions for building sets. In Section 6we consider arbitrary set systems.
Definition 5.1. Let v be a game on the set system F on [n]; thenthe core C(v) is given by
C(v) = {x ∈ Rn| x([n]) = v([n]), x(S) ≥ v(S), S ∈ F }.
Obviously it holds that C(vF ) ⊆ C(v), where vF is the M-extension of v. For a building set B and game v ∈ V(B) it followsfrom Lemma 3.7 that C(v) = C(vB).
Let B be a building set on [n].
Definition 5.2. A function f : B → R is B-supermodular if
f (A) + f (B) ≤ f (A ∪ B) + f B(A ∩ B)
for any A, B ∈ F such that A ∩ B = ∅.
Definition 5.3. A function f : B → R is B-superadditive if for anydisjoint family T1, . . . , Tk in B such that T1 ∪ · · · ∪ Tk ∈ B it holdsthat
f (T1 ∪ · · · ∪ Tk) ≥ f (T1) + · · · + f (Tk).
Definition 5.4. A function f : B → R is 2-superadditive if for anydisjoint pair of sets T1 and T2 in B such that T1 ∪ T2 ∈ B it holdsthat
f (T1 ∪ T2) ≥ f (T1) + f (T2).
When B = 2[n], the conjunction of B-supermodularity and 2-superadditivity is equivalent to the usual supermodularity condi-tion that f (A)+ f (B) ≤ f (A∪B)+ f (A∩B) for all A, B ⊆ [n]. Noticethat in this case B-superadditivity follows from 2-superadditivity,which is not the case in general.
Theorem 5.5. Let v ∈ V(B) be a B-supermodular and B-super-additive game on a building set B . Then the core C(v) contains theGC-solution and is equal to the convex hull of the marginal vectorsmN (v) over all maximal strictly nested sets N in B .
To any strictly nested set N in a set system F a simplicial coneK(N ) is associated spanned by the vectors ξT , T ∈ N , in Rn, whereξT ,i = 1 if i ∈ T and zero otherwise, and the two vectors ±ξ[n].We denote by Σ(F ) the collection of all simplicial cones inducedby F .
Proposition 5.6. For a building set B it holds that Σ(B) is asimplicial cone complex, that is for any maximal strictly nested setsN1 and N2 in B it holds that K(N1) ∩ K(N2) = K(N1 ∩ N2).
Proof. We have to prove that K(N1 ∩ N2) ⊇ K(N1) ∩ K(N2). Thisfollows from the fact that a cone K(N ) does not contain vectors ξTwith T ∈ B and T ∈ N . If this is not the case and such a vectorξT exists, then ξT is equal to the sum of some ξN1 , . . . , ξNk withN1, . . . ,Nk disjoint elements of N . Then N1 ∪ · · · ∪ Nk = T ∈ B,which contradicts property (N2) ofmaximal strictly nested set. �
Because of Theorem 3.5, ∪N K(N ) = Rn, i.e., Σ(B) is a fullsimplicial fan (Fulton, 1993).
Proof of Theorem 5.5. A function f : B → R is Σ(B)-super-modular if the extension of f by affinity on each cone K(N ), Nbeing a maximal strictly nested set in B, yields a concave func-tion. Because of Theorem 3.5, each cone K(N ) is the union of cones
G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30 27
corresponding to permutations, which implies that, for a Σ(B)-supermodular function f , the M-extension f B is a submodularfunction on 2[n], and therefore due to Theorem2.9 and to Edmond’stheorem in Edmonds (1970) (see also Shapley (1953)), C(f B) andtherefore also C(f ) are equal to the convex hull of the marginalvectorsmf (N ) over all maximal strictly nested sets N in B.
We still have to check that v is Σ(B)-supermodular. One-dimensional cones of the fan Σ(B) correspond to strictly nestedsets {A, [n]}, A ∈ B. Let N be a maximal strictly nested set in B.Pick an i ∈ [n] and consider the unique element u(i) ∈ [n] suchthat i ∈ SN (u(i)). Denote by T (u(i)) the (unique) maximal set of Bwhich contains u(i), does not contain i, and is contained in T (u(i)).Then (N \{TN (i)})∪{T (u(i))} is also amaximal strictly nested set.The cones in Σ(B) corresponding to this set and N are adjacentfull-dimensional cones, and any pair of adjacent full-dimensionalcones in Σ(B) is of such a form for suitable N and i. The lin-ear relation between the spanning vectors of the one-dimensionalcones corresponding to the strictly nested sets {TN (i), [n]} and{T (u(i)), [n]} on the one hand and the spanning vectors of the com-mon facet on the other hand is of the form
ξTN (i) + ξT (u(i)) = ξTN (u(i)) −
j∈U
ξTN (j) +
k∈M
ξTN (k),
where j ∈ U if TN (j) is a maximal element of N in TN (u(i)) \
(TN (i) ∪ T (u(i))), and k ∈ M if TN (k) is a maximal element of N
in TN (i) ∩ T (u(i)). Because of B-supermodularity we have
v(TN (i)) + v(T (u(i)))
≤ v(TN (i) ∪ T (u(i))) + vB(TN (i) ∩ T (u(i))). (2)
Then B-superadditivity implies
v(TN (i) ∪ T (u(i))) +
j∈U
v(TN (j)) ≤ v(TN (u(i))), (3)
and due to property (N2) of B-nested sets we have
vB(TN (i) ∩ T (u(i))) =
k∈M
v(TN (k)).
Summing up the inequalities (2) and (3) and the last equality, weget
v(TN (i)) + v(T (u(i))) +
j∈U
v(TN (j))
≤ v(TN (u(i))) +
k∈M
v(TN (k)),
which implies that v is Σ(B)-supermodular. �
As a consequence of the proof of this theorem we obtain thefollowing result proven by Postnikov (2005, Proposition 7.5). LetR++ = {x ∈ R | x > 0} and for T ∈ 2[n] let ∆T := {x ∈ R[n]
+ |i xi = 1, xk = 0 for k ∈ T } denote a face of the unit simplex.
Proposition 5.7. Given a building set B , for any function µ : B →
R++ the normal fan to the polytopeT∈B
µ(T )∆T ,
is the simplicial fan Σ(B).
Proof. Given a functionµ : B → R++, define the game f µ: B →
R+ by
f µ(S) =
T∈B|T⊂S
µ(T ), S ∈ B.
Due to Danilov and Koshevoy’s theorem in Danilov and Koshevoy(2000) the core of the game f µ is a Minkowski sum of simplices,
C(f µ) =
T∈B
µ(T )∆T .
It is easy to see that, becauseµ(T ) > 0 for any T ∈ B, for the func-tion f µ theB-supermodularity andB-superadditivity inequalitiesare strict. Hence, the inequalities (2) and (3) are also strict and thisimplies that the normal fan to C(f µ) coincides with the simplicialfan Σ(B). �
From this proposition it follows that the GC-solution is the gravitycenter of the core C(v) in case the game v is totally positive. Recallthat a function f : B → R is said to be totally positive if the linearsystemT∈B|T⊆S
µ(T ) = f (S), S ∈ B,
has a positive solution.If a building set B contains at least one maximal HS-nested set
or maximal NT-nested set, we can ensure that the marginal vectorcorresponding to such a maximal strictly nested set belongs to thecore under weaker conditions than B-supermodularity.
Definition 5.8. A function f : B → R is half-space B-super-modular if, for any S, T ∈ B such that S ∩ T = ∅ and at leastone of the sets S or T is a half-space in B, it holds that
f (S) + f (T ) ≤ f (S ∪ T ) + f B(S ∩ T ). (4)
For B = 2[n] half-space 2[n]-supermodularity coincides with2[n]-supermodularity, since any subset of [n] is a half-space in 2[n].For other buildings setsB half-spaceB-supermodularity isweakerthan B-supermodularity, since we do not require validity (4) fortwo non-half-spaces.
Theorem 5.9. Let v : B → R be a half-space B-supermodular andB-superadditive game on building set B . Then, for any maximal HS-nested set N in B , the marginal vector mv(N ) belongs to the coreC(v).
Proof. Let N be a maximal HS-nested set in B. We have to provethat for every Q ∈ B it holds thatj∈Q
mvj (N ) ≥ v(Q ). (5)
For a set Q ∈ N denote by SN (Q ) the set of successors of Q inthe tree FN , i.e., i ∈ SN (Q ) if i ∈ Q and i ∈ SN (j) for some j ∈ Q .We proceed by induction on the number of components ofQ in thetree FN . When Q is connected in FN , (5) takes the form
v(Q ∪ (∪j∈SN (Q ) TN (j))) ≥ v(Q ) +
j∈SN (Q )
v(TN (j)). (6)
This inequality holds because of B-superadditivity of v.Suppose (5) holds for any Q ∈ B having at most l components
in FN . Consider any Q ∈ B having l + 1 components in FN .Denote by Q0, Q1, . . . ,Ql these components. One can easily checkthat inequality (5) is equivalent to
lk=0
v(Qk ∪ (∪j∈SN (Qk) TN (j))) ≥ v(Q ) +
lk=0
j∈SN (Qk)
v(TN (j)). (7)
Because of property (G2) of strictly nested sets, we can orderQ0,Q1, . . . ,Ql such that, for any k = 0, Qk belongs to TN (j) for somej ∈ SN (Q0), and all such TN (j) cannot be singletons.
28 G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30
Pick j ∈ SN (Q0) such that TN (j) ∩ Q = ∅. Denote byQ j1, . . . ,Q
jt the components of Q ∩ TN (j). Due to half-space B-
jt are disjoint subsets of sets of the NT-nested set
N , there exists a partition {P1, . . . , Ps} of the set [t], such that
vB(Q j1 ∪ · · · ∪ Q j
t ) =
sh=1
v(∪u∈Ph Qju),
where, for each h = 1, . . . , s, the union ∪u∈Ph Qju belongs to B.
Because of the induction, inequality (7) holds for each set∪u∈Ph Q
ju, h = 1, . . . , s, that is
u∈Ph
v(Q ju ∪ (∪k∈SN (Q j
u)TN (k)))
≥ v(∪u∈Ph Qju) +
u∈Ph
k∈SN (Q j
u)
v(TN (k)). (9)
Summing up these inequalities over h = 1, . . . , s together withinequality (8), we get
v(Q ∪ TN (j)) +
tu=1
v(Q ju ∪ (∪h∈SN (Q j
u)TN (h)))
≥ v(Q ) + v(TN (j)) +
tu=1
h∈SN (Q j
u)
v(TN (h)).
Continue this procedure by adding every TN (j), j ∈ SN (Q0), andwe get inequality (7), which proves the theorem. �
As a consequence of this theorem,we obtain that if theHS-solutionexists, which is always the case for graphical building sets, thenit belongs to the core under half-space B-supermodularity andB-superadditivity of the game. Furthermore, from the proof ofTheorem 5.9 the next proposition immediately follows.
Proposition 5.10. Let B be a building set and let N be an HS-nestedset such that every Q ∈ B induces a subtree on FN , and let v : B →
R be a B-superadditive function. Then the marginal vector mN (v)belongs to the core C(v).
If the graph G is a tree on [n], then for the graphical build-ing B(G) every NT-nested set satisfies the assumption of Proposi-tion 5.10. Therefore, for a superadditive function v : B(G) → R itholds that the NT-solution belongs to the core. Such a specificationof Proposition 5.10 was proven in Demange (2004); see also Her-ings et al. (2010).
For the case when the building set B contains an NT-nestedset, which is always the case for graphical building sets, we canweaken the B-superadditivity requirement in Theorem 5.9 to 2-superadditivity.
Theorem 5.11. Let v : B → R be a half-space B-supermodularand 2-superadditive game on building set B . Then, for any maximalNT-nested set N inB , the marginal vector mv(N ) belongs to the coreC(v).
Proof. We have to prove that for every Q ∈ B it holds thatj∈Q
mvj (N ) ≥ v(Q ). (10)
Because B is a building set it holds for every i ∈ [n] that thecomplement of TN (i) belongs to B and therefore TN (i) is a half-space.
Weagain proceed by induction on the number of components ofQ in the tree FN . First consider the case when Q is one componentin FN . For such Q we have to establish inequality (10). For anyK ⊂ SN (Q ) and k′
∈ SN (Q )\K , the setsQ∪(∪k∈K TN (k))∪TN (k′),Q ∪ (∪k∈K TN (k)) and TN (k′) belong to B and the last set is ahalf-space in B. Therefore, by half-space B-supermodularity of vit holds that
v(Q ∪ (∪k∈K TN (k)) ∪ TN (k′))
≥ v(Q ∪ (∪k∈K TN (k))) + v(TN (k′)). (11)
By removing FN (j), j ∈ SN (Q ), one by one from Q ∪ (∪j∈SN (Q )
TN (j)) and applying (11), we obtain (10).Now, by repeating the proof of the previous theorem for the
case when the intersection Q contains more than one componentin the tree FN we obtain the validity of inequality (10). �
From this theorem it follows that under half-space B-super-modularity and 2-superadditivity of the game the NT -solutionbelongs to the core.
We remark that for a connected graph G, B(G)-superadditivityis equivalent to 2-superadditivity, because according to Zelevinsky(2006, Proposition 7.3) a building set B is a graphical buildingif and only if for any I, J1, . . . , Jk ∈ B such that I ∪ J1 ∪ · · · ∪
Jk ∈ B it holds that there exists i satisfying I ∪ Ji ∈ B. Due tothis characterization, for a graphical building the assumptions inTheorem 5.9 boil down to the assumptions in Theorem 5.11.
6. Solutions for arbitrary set systems
For arbitrary set systems we have the following analogs of theabove defined solutions and core stability theorems.
Let F be a set system on [n] and let V(F ) be the set of gamesonF .We consider the building coveringB(F ) and the (restricted)M-extension function vF
: B(F ) → R.
Definition 6.1. For a game v ∈ V(F ) the following solutions aregiven:
• The GC-solution is the average of the marginal vectors mvF(N )
over all maximal strictly nested sets N in B(F ).• The HS-solution is the average of the marginal vectors mvF
(N )over all HS-nested sets N in B(F ).
• The NT-solution is the average of the marginal vectors mvF(N )
over all NT-nested sets N in B(F ).
First we state the core stability results and then give someexamples.
Theorem 6.2. Let v ∈ V(F ) be a game on F .
• For a maximal strictly nested set N in B(F ), the marginal vectormvF
(N ) belongs to the core C(v) if the M-extension vF of v isB(F )-supermodular and B(F )-superadditive.
• For a maximal HS-nested set N in B(F ), the marginal vectormvF
(N ) belongs to the core C(v) if the M-extension vF of v ishalf-space B(F )-supermodular and B(F )-superadditive.
• For a maximal NT-nested set N in B(F ), the marginal vectormvF
(N ) belongs to the core C(v) if the M-extension vF of v ishalf-space B(F )-supermodular and 2-superadditive.
Proof. Wepresent a proof of the second item. FromProposition 3.8and Theorem 5.9 it holds that the marginal vector mv(N ) isa vertex of the core C(vB(F )) because according to (1) the M-extension of the restriction vF
|B(F ) is equal to the M-extensionvF .
Because C(vB(F )) ⊂ C(v), it holds that mv(N ) ∈ C(v). For apoint x in the core C(v) it holds that v(TN (i)) ≤
j∈TN (i) xj for all
G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30 29
i ∈ [n]. We have n independent inequalities because for any twoof the sets TN (i), i ∈ [n], it holds that either one is a subset of theother or the two sets do not intersect. Because of this mv(N ) is avertex of C(v). �
Wemay also define a generalization of the Myerson value for agame on an arbitrary set system.
Definition 6.3. For a game v ∈ V(F) the M-solution, denotedby M(v), is the average of the marginal vectors mvF
(σ ) over allpermutations σ ∈ Sn.
We have the following result.
Theorem 6.4. Suppose that the M-extension vF of a game v ∈
V(F) is B(F )-supermodular and B(F )-superadditive. Then the M-solution M(v) belongs to the core C(v).
As example consider the case that F = {[n]}. This set systemhas no maximal strictly nested set. Its building covering is equal to{[n], {1}, . . . , {n}}. The M-extension of v([n]) is a function whichequals zero on all proper subsets of [n]. The GC-solution coincideswith the M-solution and is equal to the egalitarian solution
GCj(v) =1nv([n]), j ∈ [n].
For this example the HS- and NT-solutions do not exist.Let us consider the case of convex geometries, set systems
considered in Bilbao and Edelmann (2000).
Definition 6.5. A set system F on [n] is a convex geometry if Fis stable under intersection and the following shelling propertyholds. For any T ∈ F , including T = ∅, it holds that there existsi ∈ [n] \ T such that T ∪ {i} is an element of F .
To a convex geometry F is associated a collection Bas(F ) oflinear orders, a subset of Sn; see Danilov and Koshevoy (2005).These permutations correspond to all maximal chains in F . Bilbaoand Edelmann (2000) define the Shapley value for a convexgeometry F by the B-solution, B(v), given by
B(v) :=1
|Bas(F )|
σ∈Bas(F )
mv(Nσ ).
Example 6.6. Let G be a tree; then the graphical building B(G) isa convex geometry. In this case the four solutions GC-, HS-, NT-,andM-solutions all differ from the B-solution. Remark that for theline-tree An, there are only two permutations, the identical and thereverse to the identical, which define HS-trees.
This example shows that different view points on the samecoalitional structure yield different on solutions.
In contrast to Theorem 3.5 for building sets, in an arbitrary sys-tem the union of permutations that correspond to the maximallystrictly nested sets may not coincide with the set of all permuta-tions. For example, this is the case for convex geometries which donot contain all singletons.
Example 6.7. Let ([n], ≺) be a poset; then the set of ideals form aconvex geometry. Recall that a subset I ⊂ [n] is an ideal if for anyi ∈ I and j ≺ i it holds that j ∈ I . Denote by I(≺) the set of allideals. This set is stable under union and intersection. Therefore,the building covering of it is equal to I(≺) ∪ {{1}, . . . , {n}}, that iswe have to add allmissing singletons. Given a function v : I(≺) →
R, the B-solution is the average ofmv(Nσ ) over all permutations σbeing linear extensions of≺. This solution is different from theGC-,HS-, NT-, andM-solutions.
Example 6.8. In the previous example let n = 3 and 3 ≺ 1and 3 ≺ 2. Then the set of ideals consists of the sets {3},{1, 3}, {2, 3} and {1, 2, 3}. This is a convex geometry with buildingcovering the set of ideals plus the singletons {1} and {2}. Thisbuilding cover is also the graphical building set correspondingto a line-tree with node 3 connected to both 1 and 2. TheM-extension of a game v is obtained by setting vF ({1}) =
vF ({2}) = 0. Then the B-solution is ( 12 (v(123) − v(23) + v(13)
− v(3)), 12 (v(123) + v(23) − v(13) − v(3)), v(3)), and the NT-
solution and the HS-solution are both equal to ( 13 (v({1, 2, 3}) −
v({2, 3})), 13 (v({1, 2, 3})−v({1, 3})), 1
3 (v({1, 2, 3})+v({1, 3})+
v({3}))). The M-solution is ( 16 (2v({1, 2, 3}) − 2v({2, 3}) +
Next is an example of another important class of convexgeometries.
Example 6.9. Let X = {x1, . . . , xn} be a set in Rk. Then define theconvex geometry F as follows, a set A is in F if the convex hullof the points xa, a ∈ A, contains no xb, b ∈ A, i.e., co({xa | a ∈
A})∩X = {xa | a ∈ A}. Consider for example the points x1 = (0, 0),x2 = (1, 0), x3 = (2, 0), and x4 = (1, 1). The correspondingconvex geometry F consists of all subsets of {1, 2, 3, 4} except{1, 3} and {1, 3, 4}. The building covering is equal to F ∪ {1, 3, 4}.For a game v, its M-extension has to be specified at {1, 3} and{1, 3, 4}: vF ({1, 3}) = v({1}) + v({3}) and vF ({1, 3, 4}) =
v({1, 4}) + v({3, 4}) − v({4}). The B-solution is different from theother solutions.
Acknowledgments
We thank the editor and referees for useful comments andsuggestions. This research has been done while the first authorwas visiting CentER, Tilburg University, on a fellowship of theNetherlands Organization of Scientific Research (NWO).
Appendix
Here we prove that, for building sets, the GC-solution coincideswith the Shapley value defined in Faigle et al. (2010), shortly theFGH-Shapley value. Let us note that the GC-solution and the FGH-Shapley value might be different for non-building sets and thatfor building sets the complexity of the FGH-Shapley value is O(n!)while the complexity of the GC-solution depends on the structureof the building set and ranges from O(n) to O(n!). We brieflyrecall the Monge algorithm and the FGH-Shapley value; for detailssee Faigle et al. (2010).
Let F be a building set. Take a linear extension of the set-inclusion partial ordering ofF on 2[n], and letµF be the restrictionof this linear extension to F . Notice that we consider a reverseranking of the sets of F compared to Faigle et al. (2010). Let σ =
(i1, . . . , in) be any permutation, being a linear ordering of [n]. Forthe input σ , the Monge algorithm produces a list of elements of F ,and an ordering on [n].
At the first step we get the set S1 := [n] and element iS1 := i1.At the second step, we choose a subset of S1 \ {i1}, which is
maximal with respect to µF . Let S2 be such a subset of [n] \ {i1}. Atthe same step, we pick a minimal element of S2 with respect to theordering σ . Let iS2 be such an element.
At the next step, we choose a subset of [n] \ {i1, iS2} which ismaximal with respect to µF . Let S3 be such a subset. Simultane-ously, we pick aminimal element of S3 with respect to the orderingσ . Let iS3 be such an element.
30 G. Koshevoy, D. Talman / Mathematical Social Sciences 68 (2014) 19–30
At the next step, we choose amaximal subset in [n]\{i1, iS2 , iS3}with respect to µF and a minimal element in such a subset withrespect to σ , and so on until we get the empty set.
As a result we get a collection of n sets S1, S2, . . . , Sn and apermutation σ ′
= (iS1 , iS2 , . . . , iSn). These data define the Mongemarginal vector by the rule:miSk
(σ ) := v(Sk)−
Sk′⊂Sk\{iSk }v(Sk′),
k = 1, . . . , n.Denote by Π the set of different permutations (rankings) being
the outputs of the Monge algorithm. Then the FGH-Shapley valueis defined as the average of the Monge marginal vectors for allpermutations in Π .
Theorem A.1. Let B be a building set. Then the GC-solution on Bcoincides with the FGH-Shapley value on B .
Proof. We show that an output of theMonge algorithm is a strictlymaximal nested set in B and that the set of permutations Π
coincides with the set of linear extensions of the poset induced bythe trees corresponding to all maximal strictly nested sets in B.
We run in parallel the Monge algorithm and construct a tree Ton [n].
In the first step of the Monge algorithm we get a set S1 := [n]and element i1 = iS1 . As root of the tree T we take iS1 .
Consider amaximal partition of S1\{iS1} constituted from sets ofB. Because B is a building set, such a partition is unique. Becauseof this, a maximal (with respect to µB) element of B which is asubset of [n] \ {iS1} is an element of the partition P([n] \ {iS1}). LetS2 be such a set, and let iS2 ∈ S2 be the minimal element of S2 withrespect to ordering σ .
We take (iS1 , iS2) as edge of T and S2 as the set of vertices of thesubtree of T with root iS2 .
The same line of arguments shows that in the next step of theMonge algorithm the set S3 is either an element of the partitionP(S2 \ {iS2}) or of the partition P(([n] \ {iS1}) \ S2). In the first case,we take (iS2 , iS3) as edge of T and S3 as the set of vertices of thesubtree with root iS3 . In the second case, we take (iS1 , iS3) as edgeof T and S3 as the set of vertices of the subtree with root iS3 .
Analogously, for each step, it holds true for k = 3, . . . , n thatSk is an element of the unique maximal partition of Sk′ for somek′ < k. This defines edge (iSk′ , iSk) of T . Because µB is an ordering,after step n, T is indeed a tree. Moreover, for any k ∈ [n], every setof the partition P(Sk′ \Sk) gives rise to a subtree of T . Therefore, the
collection of sets being the output of the Monge algorithm for σ isa maximal strictly nested set.
Let N be a maximal strictly nested set and let T be thecorresponding tree. Then for any permutation π from SN theoutput of theMonge algorithmwill produce the collectionN and aranking τ whichmay differ from π . It is easy to show that τ ∈ SN .For any other π ′ from SN , we get the collection N and the sameranking τ . �
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