Rooted-tree solutions for tree games∗

Sylvain Beal † Eric Remila ‡ Philippe Solal §

July 10, 2009

Abstract

In this paper, we study cooperative games with limited cooperationpossibilities, represented by a tree on the set of agents. Agents in thegame can cooperate if they are connected in the tree. We introducenatural extensions of the average (rooted)-tree solution [see Herings, vander Laan, Talman, 2008]: the marginalist tree solutions and the randomtree solutions. We provide an axiomatic characterization of each of thesesets of solutions. By the way, we obtain a new characterization of theaverage tree solution.

Keywords: Average tree solution – Communication structure – Marginalcontributions – Random (order) values.

JEL Classification: C71.

∗We thank an associate editor and two referees for valuable suggestions and comments.For helpful comments received, the authors also want to thank participants of the gametheory seminar at Tilburg University, MDOD seminar at University Paris I and EUROXXIII conference. Financial support by the Complex Systems Institute of Lyon (IXXI)and the National Agency for Research (ANR) – research program “Models of Influenceand Network Theory” – is gratefully acknowledged. A previous version of this article hasbeen circulated under the title “Tree solutions for graph games”.

†GREDEG, University of Nice Sophia-Antipolis, France beals@gredeg.cnrs.fr‡LIP, E.N.S. Lyon, University of Lyon and IXXI, France eric.remila@ens-lyon.fr§CREUSET, University of Saint-Etienne, France philippe.solal@univ-st-etienne.fr

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1 Introduction

A situation in which a finite set of agents can obtain certain payoffs by coopera-tion can be described by a cooperative game with transferable utility, or simply aTU-game, being a pair consisting of a finite set of agents and a function on theset of coalitions of agents that assigns a worth to each coalition of agents. Thisworth represents the maximal wealth of transferable utilities that the membersof a coalition can obtain if they agree to cooperate. A payoff vector specifies foreach agent the payoff that this agent can expect when he or she participates to aTU-game. A (single-valued) solution for a class of TU-games is a mapping thatassigns to each element of this class a payoff vector. The best known solutionis the Shapley value, proposed by Shapley [1953]. At the Shapley value, everyagent receives the average of all his marginal contributions to each coalitionformed by his entrance according to an ordering of the agent set. Weber [1988]introduced natural extensions of the Shapley value, the so-called random (order)values. A random (order) value is defined as a probabilistic distribution overthe set of marginal contribution vectors. Weber [1988] provided an axiomaticcharacterization of the set of random values on the class of all TU-games with afixed set of agents.

In this paper we study TU-games with restrictions on cooperation possibili-ties. The first to model restricted cooperation by means of an undirected graphwas Myerson [1977]. He introduced graph games, which consist of a TU-gameand an undirected graph on the agent set. The links in the graph represent thebilateral communication possibilities between the agents. Agents can only coop-erate if they are connected. In this paper we assume that the communicationgraph is represented by a tree. The best-known solution for graph games is theMyerson value, a solution that assigns to each graph game the Shapley valueof a restricted game as defined in Myerson [1977]. Myerson characterized thissolution by (component) efficiency and an axiom of fairness.

Recently, Herings, van der Laan and Talman [2008] introduced a new solutionfor the class of tree games, the so-called average (rooted)-tree solution. Theaverage tree solution is the average of specific marginal contributions vectors,where each marginal contribution vector is defined according to an orientationof the tree. Quite naturally, Herings, van der Laan and Talman [2008] followedDemange [2004] and envisaged as the set of orientations the set of rooted trees.A rooted tree is an orientation of a tree such that all links are directed awayfrom a designated agent, called the root. Such a rooted tree describes how theagents can communicate with each other: two agents cannot communicate with

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each other if one is not a subordinate of the other. The marginal contributionof an agent in a rooted tree is equal to the worth of the coalition consisting ofthis agent and all her subordinates minus the sum of the worths of the coalitionsconsisting of any of her successors and all subordinates of this successor. Theaverage tree solution can be characterized by efficiency and an alternative axiomof fairness. On these points, see Herings, van der Laan and Talman [2008].

In the spirit of what Weber [1988] did for the Shapley value, we build on thework of Herings, van der Laan and Talman [2008] by relaxing the assumptionthat each rooted tree has the same impact in the computation of the solution.Two new sets of solutions are proposed for the class of tree games: the set ofmarginalist tree solutions and the set of random tree solutions. A marginalist treesolution is defined as a linear combination of the marginal contribution vectorsand a random tree solution is defined as a probability distribution over the set ofall marginal contribution vectors. We provide an axiomatic characterization ofeach of these sets of solutions. In particular, we identify a subset of connectedcoalitions of a tree, called the set of cones, that is necessary and sufficient tocompute a marginalist tree solution. The axiom of cone equivalence reflects thisfact and states that the distribution of payoffs must be equal in two tree gamesin which any cone realizes the same worth. The set of cones of a tree consistsof the grand coalition, the empty coalition and, for each link of the tree, the twocomponents obtained after deleting the link. The other axioms also rely on theset of cones. Efficiency is replaced by proper cone efficiency, and a version ofnull agent is defined with respect to the set of cones. Together with linearity,these axioms characterize the set of marginalist tree solutions on the class oftree games with a fixed tree. It turns out that the set of marginal tree solutionsis a linear space whose dimension is equal to the size of the agent set. If wesubstitute proper cone efficiency by the standard axiom of efficiency and add anaxiom of positivity, then we obtain the set of random tree solutions. Positivityindicates that the solution assigns to each agent a positive payoff in the Diractree game on the grand coalition. As a corollary of these results, an alternativecharacterization of the average tree solution for tree games is offered.

The rest of the paper is organized as follows. Section 2 is devoted to defi-nitions. Section 3 contains the characterizations of the marginalist and randomtree solutions. Section 4 concludes.

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2 Preliminaries

Consider a finite set of agents N of size n, who face restrictions on communi-cation. Each subset S of N is called a coalition. The bilateral communicationpossibilities between the agents are represented by an undirected graph on N ,denoted by (N, L), where the set of nodes coincides with the set of agents N ,and the set of links L is a subset of the set of unordered pairs of elements ofN . For each agent i ∈ N , the set Li = {j ∈ N : {i, j} ∈ L} denotes theneighborhood of i in (N, L). The degree of an agent i ∈ N in (N, L), denotedby di, is the number of elements of Li. For each nonempty coalition S of N ,L(S) = {{i, j} ∈ L : i ∈ S, j ∈ S} is the set of links between agents in S. Notethat L(N) = L. The graph (S, L(S)) is the subgraph of (N, L) induced by S.A sequence of distinct agents (i1, i2, . . . , ip) is a path in (N, L) if {iq, iq+1} ∈ Lfor q = 1, . . . , p − 1. Two agents i and j are connected in (N, L) if i = j orthere exists a path from i to j. A graph (N, L) is connected if any two agentsin N are connected. A tree is a minimally connected graph (N, L) in the sensethat if a link is removed from L, (N, L) ceases to be connected. Equivalently,a tree is a connected graph such that only one path connects any two agents.Note that a tree on N has exactly n − 1 links. A coalition S is connected in(N, L) if (S, L(S)) is a connected graph. The empty coalition ∅ is trivially con-nected. Denote by C(L) the set of connected coalitions in (N, L). A coalition Sof N is a component of a graph (N, L) if the subgraph (S, L(S)) is maximallyconnected, i.e. if the subgraph (S, L(S)) is connected and for each i ∈ N\S,the subgraph (S ∪ {i}, L(S ∪ {i})) is not connected. Note that the collectionof components of (N, L) forms a partition of N . The concept of component isdefined similarly for each subgraph (S, L(S)).

It is assumed that only connected coalitions of (N, L) are able to cooperate.If a coalition S belongs to C(L), then its members can fully coordinate their ac-tions. The worth obtained by cooperation between connected agents is describedby the function v : C(L) −→ R such that v(∅) = 0. The function v definesa graph game with transferable utility on (N, L). We take the communicationgraph (N, L) to be fixed, and we assume that (N, L) is a tree. Therefore, weconsider the vector space (over the field R) CN,L of all tree games v on (N, L).For each coalition S ∈ C(L)\{∅}, define the unanimity tree game uS on C(L)as uS(T ) = 1 if T ⊇ S, and uS(T ) = 0 otherwise. Define also the Dirac treegame 1S on C(L) as 1S(T ) = 1 if T = S, and 1S(T ) = 0 otherwise. It iswell-known that the set of unanimity tree games and the set of Dirac tree gamesform two bases for CN,L.

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A payoff vector x ∈ Rn for a tree game v ∈ CN,L is an n-dimensional vectorgiving a payoff xi ∈ R to each agent i ∈ N . A payoff vector x ∈ Rn is efficientif it distributes the worth v(N) among the agents, i.e. if

∑i∈N xi = v(N). A

(single-valued) solution on CN,L is a function f that assigns to every v ∈ CN,L apayoff vector f(v) ∈ Rn. Such a solution f represents a method for measuringthe value of playing a particular role in a tree game.

In order to calibrate the importance of each agent in the different connectedcoalitions, we define specific contribution vectors as in Herings, van der Laanand Talman [2008]. To describe these contribution vectors we first give somedefinitions concerning rooted trees. By a rooted tree ti, we mean a (directed)graph that arises from a tree (N, L) by selecting agent i ∈ N , called the root,and directing all links away from the root. Each agent i ∈ N is the root ofexactly one rooted tree ti on (N, L). Note also that for any rooted tree ti on(N, L), any agent k ∈ N\{i}, there is exactly one directed link (j, k); agent jis the unique predecessor of k and k is a successor of j in ti. Denote by si(j)the possibly empty set of successors of agent j ∈ N in ti. An agent k is asubordinate of j in ti if there is a directed path from j to k, i.e. if there is asequence of distinct agents (i1, i2, . . . , ip) such that i1 = j, ip = k and for eachq = 1, 2, . . . , p− 1, iq+1 ∈ si(iq). The set Si(j) denotes the union of the set ofall subordinates of j in ti and {j}. So, we have si(j) ⊆ Si(j)\{j}. A rootedtree reflects the idea that two agents incident to a communication link do nothave equal access or control to that link.

Pick any v ∈ CN,L, any rooted tree ti, and consider the marginal contributionvector mi(v) on Rn defined as

mij(v) = v(Si(j))−

∑k∈si(j)

v(Si(k)) (1)

for every j ∈ N . The marginal contribution mij(v) of j ∈ N in ti is thus equal

to the worth of the coalition consisting of agent j and all his subordinates in timinus the sum of the worths of the coalitions consisting of any successor of j andall subordinates of this successor in ti. Note that for each tree game v ∈ CN,L,each rooted tree ti and each agent j ∈ N , we have∑

k∈Si(j)

mik(v) = v(Si(j)). (2)

A solution f is called a marginalist tree solution on CN,L if for each v ∈ CN,L,f(v) is a linear combination of the marginal contribution vectors mi(v), i.e. if

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there exists a vector α ∈ Rn such that

f(v) =∑i∈N

αimi(v). (3)

The solution f is a random tree solution if α in (3) is a probability distributionon N , i.e. if αi ≥ 0 and

∑i∈N αi = 1. Herings, van der Laan and Talman

[2008] introduced and characterized the average tree solution, denoted by AT.This random tree solution assigns to every game v ∈ CN,L the average of allmarginal contribution vectors mi(v), i.e.

AT(v) =1

n

∑i∈N

mi(v). (4)

3 Axiomatic characterizations

In this section, we provide axiomatic characterizations of the set of marginalisttree solutions and of the set random tree solutions on the class of tree gamesCN,L. The axiomatic development of random tree solutions will allow us topresent an alternative characterization of the average tree solution. In particular,we show that the set of marginalist tree solutions defines a linear space ofdimension n, the size of the agent set. These results rely on a null agentaxiom, a linearity axiom, a positivity axiom, two efficiency axioms, an axiomof communication ability, and an axiom of cone equivalence. This last axiomstates that connected coalitions which do not induce a cone (see below for adefinition) in the underlying tree are useless to determine the allocation betweenagents over CN,L. The null agent axiom is adapted from the well-known nullagent axiom used to characterize the Shapley value [Shapley, 1953] in gameswithout communication restrictions. The axiom of communication ability istaken from van den Brink, van der Laan and Pruzhansky [2007]. The otheraxioms are standard axioms in cooperative game theory.

Efficiency For each v ∈ CN,L, it holds that∑

i∈N fi(v) = v(N).

Linearity For each v ∈ CN,L, each w ∈ CN,L and each a ∈ R, it holds thatf(v + w) = f(v) + f(w) and f(av) = af(v).

Positivity For the Dirac tree game 1N , fi(1N) ≥ 0 for each i ∈ N .

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A tree game v ∈ CN,L is called point unanimous if there is a ∈ R such thatv = a1N . Thus, in point unanimous games the worth of coalition is zero if itdoes not contain all the (connected) agents.

Communication ability If v ∈ CN,L is point unanimous, then there existsc ∈ R such that fi(v) = c for each i ∈ N .

Before introducing the other axioms, some definitions are in order. Let (N, L)be a tree. The set of cones of (N, L) consists of N , ∅ and for each {i, j} ∈ Lthe two components that are obtained after deletion of link {i, j}. Every coneexcept N is called a proper cone. The unique agent of a nonempty proper coneK who has a link with the complement N\K is called the head of the cone andis denoted by h(K). Thus the tree (N, L) contains 2(n−1)+2 = 2n cones, andeach agent i ∈ N is the head of exactly di nonempty proper cones. Denote by∆N,L the set of cones in (N, L). Observe that K ∈ C(L) belongs to ∆N,L if andonly if N\K ∈ C(L). A proper cone T ∈ ∆N,L is a successor of a (nonempty)proper cone K ∈ ∆N,L if either T ⊆ K and {h(T ), h(K)} ∈ L or K = {i} forsome i ∈ N and T = ∅. The set of successors of a proper cone K is denoted bys(K), where s(∅) = ∅.

Two tree games v and w of CN,L are cone equivalent if for each K ∈ ∆N,L,it holds that v(K) = w(K).

Cone equivalence For each pair of cone equivalent tree games v and w ofCN,L, it holds that f(v) = f(w).

Given a tree game v ∈ CN,L, agent i ∈ N is a null agent in v if for eachproper cone K ∈ ∆N,L such that h(K) = i, it holds that v(K) =

∑T∈s(K) v(T )

andv(N) =

∑{K∈∆N,L:h(K)∈Li,i∈N\K}

v(K).

Null agent For each v ∈ CN,L and each null agent i ∈ N , it holds thatfi(v) = 0.

Proper cone efficiency For each nonempty proper cone K ∈ ∆N,L, it holdsthat

∑i∈N fi(1K) = 0.

Note that if a solution f satisfies efficiency, then it satisfies proper cone

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efficiency. In the proofs of the two following propositions we assume that theelements of ∆N,L are indexed by integers k ∈ {0, 1, 2, . . . , 2n− 1} in such a waythat K0 = ∅ and K2n−1 = N . We also denote by [p] the first p positive integers,i.e. the set {1, 2, . . . , p}.

Let A = (apq)p∈[n],q∈[m] be an n × m matrix with real coefficients. Let A•qand Ap• denote its qth column and its pth row respectively. For two n × mmatrices A and B, denote by < A |B > the inner product of A and B definedas:

< A |B >=∑p∈[n]

∑q∈[m]

apqbpq.

For any m ∈ N, denote by 0m the m-dimensional null vector.Consider a solution f on CN,L satisfying cone equivalence and linearity. Such

a solution can be represented by a n × (2n − 1) matrix Af defined as follows:for any p ∈ [n] and any q ∈ [2n− 1],

Afpq = fp(1Kq).

So, for any v ∈ CN,L, the payoff vector obtained under f can be expressed asfollows:

f(v) =∑

q∈[2n−1]

Af•qv(Kq).

Clearly, the mapping f 7−→ Af is a linear space isomorphism, and the linearspace of all n × (2n − 1) real matrices has dimension n(2n − 1). Denote thislinear space by L(R2n−1, Rn). From these observations, we get the followingresult.

Proposition 3.1 The set of solutions f on CN,L satisfying linearity and coneequivalence coincides with the linear space L(R2n−1, Rn) of dimension n(2n−1).

Adding proper cone efficiency and null agent to linearity and cone equivalence,we derive the following intermediary result.

Proposition 3.2 The set of solutions f on CN,L satisfying linearity, coneequivalence, proper cone efficiency and null agent is a linear space of dimen-sion at most n.

Proof. Denote by E the set of solutions f on CN,L satisfying linearity, coneequivalence, proper cone efficiency and null agent. Note that if f and g belong

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to E, then f + g ∈ E and af ∈ E for each a ∈ R. It follows that E constitutesa linear space over R. By Proposition 3.1, E ⊆ L(R2n−1, Rn). The basic idea ofthe proof is to use the axioms so as to construct a linear subspace of L(R2n−1, Rn)of dimension n(2n− 2) and orthogonal to E. This will prove that E is a linearsubspace of L(R2n−1, Rn) of dimension at most n(2n− 1)− n(2n− 2) = n.

For each nonempty cone K ∈ ∆N,L, each agent i ∈ N , construct the n ×(2n− 1) matrix AK,i as follows: for each p ∈ [n] and each q ∈ [2n− 1],

aK,ipq =

{uK(Kq) if p = i0 otherwise.

Let uK be the line vector of R2n−1 defined as uKq = uK(Kq) for each q ∈

[2n − 1]. We see that AK,ii• = uK and AK,i

p• = 02n−1 for each p ∈ [n], p 6= i.

Notice that AK,ii• does not depend on the agent i. We have:

< Af |AK,i >=∑

q∈[2n−1]

afiqa

K,iiq =

∑q∈[2n−1]

fi(1Kq)uK(Kq).

Using linearity and the fact that

uK =∑

q∈[2n−1]:Kq⊇K

1Kq ,

we get:

< Af |AK,i >=∑

q∈[2n−1]:Kq⊇K

fi(1Kq) = fi

( ∑q∈[2n−1]:

Kq⊇K

1Kq

)= fi(uK).

If i 6= h(K), then i is a null agent in uK and so < Af |AK,i >= 0. For eachagent i, there are exactly 2n− 2− di such proper cones.

Next, for each k ∈ [2n − 2], define the n × (2n − 1) matrix Ak as follows:for each p ∈ [n] and each q ∈ [2n− 1],

akpq =

{1 if q = k0 otherwise.

By proper cone efficiency, we immediately get for each k ∈ [2n− 2]:

< Af |Ak >=∑p∈[n]

fp(1Kk) = 0.

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This means that 2n − 2 new matrices are orthogonal to Af . To sum up, weget

∑i∈N(2n − 2 − di) + 2n − 2 = n(2n − 2) matrices orthogonal to Af . It

remains to show that these matrices are linearly independent. So, pick any linearcombination of these matrices which is equal to the null matrix:∑

i∈N

∑K∈∆N,L:h(K) 6=i

αK,iAK,i +

∑k∈[2n−2]

αkAk = On×(2n−1), (5)

where On×(2n−1) represents the null n×(2n−1) matrix. Consider the n×(2n−1)matrix B =

∑k∈[2n−2] αkA

k. By construction, for each p ∈ [n], we have bpk =

αk for each k ∈ [2n − 2] and bp(2n−1) = 0. Hence, the n rows Bp•, p ∈ [n],are identical. Let γ = (α1, α2, . . . , α2n−2, 0) be the (2n− 1)-dimensional vectorcorresponding to each of these rows.

Now consider the first row AK,i1• of each AK,i, i ∈ N . By construction

AK,11• = uK and AK,i

1• = 02n−1 otherwise. Thus, we have:∑i∈N

∑K∈∆N,L:h(K) 6=i

αK,iAK,i1• =

∑K∈∆N,L:h(K) 6=1

αK,1uK .

Therefore, the restriction of (5) to the sums on the first rows of AK,i and Ak

yields: ∑K∈∆N,L:h(K) 6=1

αK,1uK + γ = 0(2n−1).

This means that the vector γ is a linear combination of unanimity games uK suchthat h(K) 6= 1. Continuing in this fashion for each agent i ∈ N , we obtain thatγ is a linear combination of unanimity games uK where h(K) 6= i. Equivalently,for each i ∈ N , γ is expressed as a linear combination of unanimity games uK

with a null coefficient for each proper cone K such that h(K) = i. It followsthat γ = 0(2n−1), which means that αk = 0 for each k ∈ [2n− 2]. Because thevectors uK are linearly independent,∑

K∈∆N,L:h(K) 6=i

αK,iuK = 0(2n−1)

for each i ∈ N implies that αK,i = 0 for each i ∈ N and each proper coneK such that h(K) 6= i. Therefore, the set of n(2n − 2) matrices AK,i and Ak

is linearly independent and generates a linear subspace of dimension n(2n − 2)orthogonal to E. It follows that E has dimension at most n. �

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The following theorem establishes that the vector space E coincides with theset of marginalist tree solutions.

Theorem 3.1 The set of marginalist tree solutions CN,L coincides with theset of solutions on CN,L satisfying linearity, cone equivalence, proper coneefficiency and null agent. Moreover, for each marginalist tree solution f , wehave

f =∑i∈N

fi(1N)mi.

Proof. Since each marginalist tree solution f is a marginal contribution vectoror a linear combination of marginal contribution vectors, it follows that such fsatisfies linearity on CN,L. That any marginalist tree solution f satisfies coneequivalence and null agent follows from the following argument. For any j ∈ Nand any rooted tree ti of a tree (N, L), Si(j) ∈ ∆N,L. Conversely, for anyK ∈ ∆N,L with head h(K) = j and obtained by deleting the link {i, j} from L,we have K = Si(j). By definition of the marginal vectors mi in (1), we concludethat only cones are used to compute a marginalist tree solution f , and so coneequivalence is satisfied by any such f on CN,L. From these observations, we alsoconclude that any marginalist tree solution f satisfies null agent on CN,L. Toshow that any marginalist tree solution f satisfies proper cone efficiency on CN,L,it suffices to note that for each nonempty proper cone K ∈ ∆N,L, 1K(N) = 0and then to use (2). We have shown that the set of marginalist tree solutionson CN,L belongs to the linear space E.

Because the set of marginalist tree solutions on CN,L is generated by theset of all linear combinations of the n marginal vectors mi, it constitutes alinear subspace of E. Proposition 3.2 establishes that E has dimension at mostn. Thus, to show that the linear subspace spanned by the marginal vectorscoincides with E, it suffices to verify that the set of marginal vectors are linearlyindependent. So, pick any linear combination f =

∑i∈N αim

i equal to the nullvector. In particular, f(1N) is null and so

∑i∈N αim

ij(1N) = 0 for any j ∈ N .

Because mij(1N) = 1 if j = i and mi

j(1N) = 0 if j 6= i, we get αi = 0 for eachi ∈ N , the desired result. Finally, if f is a marginalist tree solution, then it iseasy to see from the above observations that fi(1N) = αi for each i ∈ N . �

If we substitute proper cone efficiency by efficiency and add positivity, we getthe following result:

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Theorem 3.2 The set of random tree solutions on CN,L coincides with theset of solutions on CN,L satisfying linearity, cone equivalence, efficiency, pos-itivity and null agent.

Proof. Note that a random tree solution belongs to E and satisfies efficiencyand positivity since

∑i∈N αi = 1 and αi ≥ 0, i ∈ N . Conversely, if a solution

f satisfies, linearity, cone equivalence, efficiency, positivity and null agent, thenit belongs to E by Theorem 3.1. Because f ∈ E and satisfies positivity andefficiency, we get

fi(1N) = αi ≥ 0 for each i ∈ N and∑i∈N

fi(1N) =∑i∈N

αi = 1.

Therefore, f is a random tree solution. This completes the proof of Theorem3.2. �

The next corollary follows immediately from Theorem 3.2.

Corollary 3.1 The average tree solution AT given by (4) is the unique so-lution on CN,L satisfying linearity, cone equivalence, efficiency, null agent andcommunication ability.

Recently, Mishra and Talman [2009] provided a new characterization of theaverage tree solution, which is comparable to the characterization contained inCorollary 3.1. Both characterizations use efficiency and linearity. Mishra andTalman [2009] considered a more demanding definition of null agent that relieson all marginal contributions of the Myerson restricted game. They dropped coneequivalence, and added a property of independence in unanimity games. Thisproperty requires that in two unanimity games, one corresponding to a connectedcoalition, say S, and the other corresponding to coalition S ∪ {j}, where agentj has a neighbor in S, the payoff of each agent i ∈ S\Lj must be the same inboth unanimity games. The interpretation is as follows. An agent i ∈ S canbe thought to represent an agent j 6∈ S in the unanimity game correspondingto S if the unique path from i to j does not contain any other agents from S.Agent i represents himself. Independence in unanimity games states that if thenumber of agents an agent represents in two unanimity games does not change,then he gets the same payoff in both games. This property is useful inasmuchas Herings, van der Laan and Talman [2008] showed that in unanimity games,the payoff allocated to an agent under the average tree solution is the fractionof the total number of agents he represents. Hence, the average tree solutionclearly satisfies independence in unanimity games.

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4 Conclusion

In this paper marginalist tree solutions are proposed for the class of tree games.These solutions generalize the solution introduced by Herings, van der Laan andTalman [2008]. We provide an axiomatic characterization of this set of solutionsand show that it constitutes a linear space whose dimension is equal to thenumber of agents. We also introduce and characterize the set of random treesolutions, a subset of the set of marginalist tree solutions. By the way, we obtaina new characterization of the solution introduced by Herings, van der Laan andTalman [2008]. As in Herings, van der Laan and Talman [2008], it appears thatthe connected components induced by a bridge – the (nonempty) proper conesof the underlying tree – play a central role in the analysis.

Several other extensions are left for future works. In particular, numerousrecent papers consider extensions of the Shapley value and random values in theframework of multi-choice games [see for instance Grabisch and Lange, 2007, andBranzei, Tijs and Zarzuelo, 2009]. Therefore defining and studying the averagetree solution and random tree solutions in this context would be a challengingissue.

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[9] Weber, R.J., 1988. Probabilistic values for games. In: Roth A. (ed.). TheShapley value: Essays in honor of Lloyd S. Shapley. Cambridge UniversityPress, 101–120.

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