A recursive solution concept for multichoice games

A recursive solution concept for multichoice games Fabien LANGE Michel GRABISCH Université Paris I - Panthéon-Sorbonne CERMSEM, 106-112 bd de l’Hôpital 75013 Paris, France email [email protected] email [email protected] June 17, 2007 Abstract. We propose a new axiomatization of the Shapley value for cooperative games, where symmetry and efficiency can be discarded and replaced with new natural axioms. From any game, an excluded-player game is built by discarding all coalitions that contain a fixed player. Then it is shown that the Shapley value is the unique value satisfying the linearity axiom, the nullity axiom, the excluded-null-player axiom, and the equity axiom. In the second part, by generalizing the above material, the Shapley value for multichoice games is worked out. Key words. Shapley value, multichoice games, equity, generalized nullity axiom. 1 Introduction The value or solution concept of a game is a key concept in cooperative game theory, since it defines a rational imputation given to the players if they join the grand coalition. In this respect, the Shapley value remains the best known solution concept [11], and it has been axiomatized by many authors in various ways (see especially Weber [13], or the survey by Monderer and Samet [8]). If the definition and axiomatization of the Shapley value is well established for classical cooperative TU-games, the situation is less clear when considering variants of classical TU-games, like multichoice games [7], games in partition function form [12], etc. In this paper, we focus on multichoice games, where players are allowed to have several (and totally ordered) levels of participation. Hence, a solution for multichoice games assigns a numerical value to each possible participation level and to each player. The original proposal of Hsiao and Raghavan [7] for the Shapley value has been, up to our knowledge, scarcely used due to its complexity. Another proposal is due to Faigle and Kern [5], and compared to the former one by Branzei et al. [3], and also by the authors [6]. The value proposed by Faigle and Kern, although elegant but still with a very high computational complexity, is more rooted in combinatorics than in game theory, and takes as a basis the 1 expression of the Shapley value using maximal chains in the lattice of coalitions. In [6], the authors proposed an alternative view having the same complexity than the usual Shapley value for classical TU-games. It turned out that our value was identical to the egalitarian value proposed by Peters and Zank [10], although with different axioms and some restrictions (namely, all players should have the same set of participation levels). Although close to the axiomatization proposed by Weber for classical TU-games, our axiomatization in [6] suffered from a complex symmetry axiom, hard to interpret, the fundamental problem there being that the classical notion of symmetry among players cannot hold since two different players may have a different set of participation levels (note that this difficulty was avoided by Peters and Zank, since they considered multichoice games with all players having the same set of participation levels). In this paper, we propose a new axiomatization for the so-called egalitarian value, which is based essentially on carriers and on a recursive scheme, and which does not make use of a symmetry axiom. In Section 3, we present the main ideas applied on classical TU-games, and we come up with a very simple and natural axiomatization using linearity, a nullity axiom which uses also carriers, and an equity axiom stating that the sharing should be uniform and efficient for the unanimity game based on the grand coalition (this is in fact a very weak version of the efficiency axiom). In Section 4, the same process is applied to multichoice games. An additional axiom (called decreased level axiom) is used, to take into account the case where a player does not participate at the highest level. In the sequel, N refers to the set of positive integers. In order to avoid a heavy notation, we will often omit braces for subsets, by writing i instead of {i} or 123 for {1, 2, 3}. Furthermore, cardinalities of subsets S, T, . . . will be denoted by the corresponding lower case letters s, t, . . . 2 Mathematical background We begin by recalling necessary material on lattices (a good introduction on lattices can be found in [4]), in a finite setting. A lattice is a set L endowed with a partial order ≤ such that for any x, y ∈ L their least upper bound x ∨ y and greatest lower bound x ∧ y always exist. For finite lattices, the greatest element of L (denoted ⊤) and least element ⊥ always exist. x covers y (denoted x ≻ y) if x > y and there is no z such that x > z > y. A ranked lattice is a pair (L, r), where L is a lattice and the rank function r : L → N satisfies the property that r(y) = r(x) + 1 whenever y covers x in L. The lattice is distributive if ∨, ∧ obey distributivity. An element j ∈ L is join-irreducible if it cannot be expressed as a supremum of other elements. Equivalently j is join-irreducible if it covers only one element. The set of all join-irreducible elements of L is denoted J (L). An important property is that in a distributive lattice, any element x can be written as an irredundant supremum of join-irreducible elements in a unique way (this is called the minimal decomposition of x). We denote by η ∗ (x) the set of join-irreducible elements in the minimal decomposition of x, and we denote by η(x) the normal decomposition of x, defined as the set of join-irreducible elements smaller or equal to x, i.e., η(x) := {j ∈ J (L) | j ≤ x}. Let us rephrase differently the above result. We say that Q ⊆ L is a downset of L if x ∈ Q and y ≤ x imply y ∈ Q. For any subset P of L, we denote by 2 O(P ) the set of all downsets of P . Then, by Birkhoff’s theorem [2], the mapping η is an isomorphism of L onto O(J (L)). Given lattices (L1 , ≤1 ), . . . , (Ln , ≤n ), the product lattice L = L1 × · · · × Ln is endowed with the product order ≤ of ≤1 , . . . , ≤n in the usual sense. Elements of x can be written in their vector form (x1 , . . . , xn ). We use the notation (xS , y−S ) to indicate a vector z such Q that zi = xi if i ∈ S, and zi = yi otherwise. Similarly L−i denotes k6=i Lk if N \ i 6= ∅,
and the singleton set {()} otherwise. By this way, for any vector x, (), x simply denotes x. All join-irreducible elements of L are of the form (⊥1 , . . . , ⊥i−1 , ji , ⊥i+1 , . . . , ⊥n ), for some i and some join-irreducible element ji of Li . A vertex of L is any element whose components are either top or bottom. We denote Γ(L) the set of vertices of L. 3 A new axiomatization of the Shapley value for classical cooperative games In the whole paper, we consider an infinite denumerable set Ω, the universe of players. As usual, a game on Ω is a set function v : Ω → R such that v(∅) = 0, which assigns to each coalition S ⊆ Ω its worth v(S). We denote by 2Ω (power set of Ω) the set of coalitions. In this section, we focus on the particular case of classical cooperative games, that is to say, each player has the only choice to cooperate or not. A set N ⊆ Ω is said to be a carrier of a game v when for all S ⊆ Ω, v(S) = v(N ∩ S). Thus a game v with carrier N ⊆ Ω is completely defined by the knowledge of the coefficients {v(S)}S⊆N and the players outside N have no influence on the game since they do not contribute to any coalition. In this paper, we restrict our attention to finite games, that is to say, games that posses a finite carrier N with n elements. We denote by G(N ) the set of games with the finite carrier N . For the sake of clarity, and to avoid any ambiguity, the domain of v ∈ G(N ) will be restricted to the elements of 2N . G denotes the set of all finite games: G := {G(N ) | N ⊆ Ω, n ∈ N}. Identity games of G(N ) are particular games defined by ( 1 if T = S, ∀S ⊆ N \ {∅}, δS (T ) := 0 otherwise. A value on G(N ) is a function Φ : G(N ) × N → R that assigns to every player i in a game v ∈ G(N ) his prospect Φ(v, i) for playing the game. For instance, the Shapley value [11] for cooperative games ΦSh is defined by ∀v ∈ G(N ), ∀i ∈ N, ΦSh (v, i) := X s! (n − s − 1)! (v(S ∪ i) − v(S)). n! (1) S⊆N \i The axiomatization is well-known. ΦSh is the sole value given on G(N ) satisfying (see also Weber [13]): Linearity (L): for any i ∈ N, Φ(v, i) is linear w.r.t the variable v. 3 Player i ∈ N is said to be null for v if ∀S ⊆ N \ i, v(S ∪ i) = v(S). Nullity (N): for any game v ∈ G(N ) and any i ∈ N null for v, Φ(v, i) = 0. For any permutation σ on N , we denote v ◦ σ the game defined by v ◦ σ(S) := v(σ(S)), ∀S ∈ 2N . Symmetry (S): for any permutation σ on N , any game v ∈ G(N ) and any i ∈ N , Φ(v, σ(i)) = Φ(v ◦ σ, i). This means that Φ must not depend on the labelling of the players. P Efficiency (E): for any game v ∈ G(N ), i∈N Φ(v, i) = v(N ). That is to say, the values of the players must be shared in proportion of the overall resources v(N ). We now introduce a new axiomatization of the Shapley value for classical cooperative games. For any game v ∈ G(N ) and any coalition S ∈ 2N , we denote by v S ∈ G(S) the restricted game v to the power set of S. For i ∈ N , v −i denotes the restricted game v N \i . Let us consider the following axioms for values on G. Excluded-null-player (ENP): for any finite set N ⊆ Ω and any game v ∈ G(N ), if i ∈ N is null for v, ∀j ∈ N \ i, Φ(v, j) = Φ(v −i , j). This simply means that if a null player leaves the game, then other players should keep the same value in the associated restricted game. Note that this axiom completes in a certain sense the above axiom (N) since the former deals with null players whereas the latter addresses the others. Therefore, one can merge (N) and (ENP): Generalized nullity (GN): for any finite set N ⊆ Ω and any game v ∈ G(N ), if i ∈ N is null for v, ( Φ(v, i) = 0, Φ(v, j) = Φ(v −i , j), for any player j ∈ N \ i. ( 1, if S = N , We define the particular unanimity game of G(N ) by uN (S) := 0, otherwise. Equity (Eq): for any finite set N ⊆ Ω, for any player i ∈ N , Φ(uN , i) = 1 . n This natural axiom simply states that in the particular game where the grand coalition is the unique to produce a unitary worth (all others giving nothing), all players should share the same fraction of this unit. 4 Theorem 1 ΦSh is the sole value on G satisfying axioms (L), (GN) and (Eq). Note that since the result is given over G, axioms (L) and (N) should be adjusted in accordance with the arbitrariness of the choice of N . Actually, it is sufficient to specify for these axioms “for any finite set N ⊆ Ω, for any game v ∈ G(N )”. Proof: First, let us check that ΦSh satisfies the axioms, which is already known for (L) and (N), and obvious for (Eq). It remains to check the validity of (ENP). We denote s! (n − s − 1)! by αs1 (n) := the Shapley’s coefficient in (1). Let v ∈ G(N ) and i ∈ N a n! player null for v. Then for j ∈ N \ i, X ΦSh (v, j) = αs1 (n) (v(S ∪ j) − v(S)) S⊆N \j = X 1 [αs1 (n) + αs+1 (n)] (v(S ∪ j) − v(S)), S⊆N \ij since i is null for v. Now s! (n − s − 1)! (s + 1)! (n − s − 2)! + n! n! s! (n − s − 2)! [s + 1 + n − s − 1] = n! s! (n − s − 2)! = = αs1 (n − 1). (n − 1)! 1 αs1 (n) + αs+1 (n) = (2) Thus ΦSh (v, j) = X αs1 (n − 1) (v(S ∪ j) − v(S)) S⊆(N \i)\j = ΦSh (v −i , j). Conversely, let us remind an intermediary result from Weber [13], that gives the general form of values Φ on G(N ) × N , under the nullity axiom and the linearity axiom: X ∀v ∈ G(N ), ∀i ∈ N, Φ(v, i) = piS (n) (v(S ∪ i) − v(S)), (3) S⊆N \i where the piS (n)’s are some real numbers. We now show the result by mathematical induction on the cardinality of N . For N := {1}, (3) gives Φ(uN , 1) = p1∅ (1) uN (1) = p1∅ (1). Besides, from (Eq), Φ(uN , 1) = 1. Thus p1∅ (1) = 1 and (1) is satisfied for n = 1. Let us assume that (1) is true for games of G(N ), n fixed in N. We now consider any game v ∈ G(N ′ ) with |N ′ | = n + 1. We suppose that player i ∈ N ′ is null for v. Then, for any player j ∈ N ′ \ i, by (3), there are some real coefficients pjS := pjS (n + 1), S ⊆ N ′ \ j, such that X j Φ(v, j) = [pS + pjS∪i ] (v(S ∪ j) − v(S)), S⊆N ′ \ij 5 since i is null for v. Moreover, by (ENP), we also have X Φ(v, j) = Φ(v −i , j) = αs1 (n) (v(S ∪ j) − v(S)). S⊆N ′ \ij Since these formula are true for any game v ∈ G(N ′ ) with player i null for v, they are in particular true for the game δS∪j + δS∪ij , S ⊆ N ′ \ ij. By identification of the two above formulae, we can deduce that for all j ∈ N ′ , for all S ⊆ N ′ \ ij, pjS + pjS∪i = αs1 (n). (4) We now recursively compute the coefficients pjS ’s. Precisely, we show that ∀j ∈ N ′ , ∀S ( N ′ \ j, pjS = αs1 (n + 1). (5) Let j ∈ N . Considering the unanimity game uN ′ , we have Φ(uN ′ , j) = pjN ′ \j = 1 = αn1 (n + 1), n+1 where the second equality is due to (Eq). Thus (5) is shown for S = N ′ \ j, that is to say s = n. Assuming (5) is true ∀S ⊆ N ′ \ j such that s is a fixed cardinality in {1, . . . , n}, then by (4), for any i ∈ S 1 (n) − αs1 (n + 1) pjS\i = αs−1 1 = αs−1 (n + 1), where the second equality comes from (2). Consequently, (5) is also satisfied for any T ⊆ N ′ \ j such that t = s − 1, thanks to correct choices of S and i. Finally, the result is proved for any subcoalition of N ′ \ j.  An important remark is that this new axiomatization has the advantage of characterizing ΦSh for all games of G, and not only for the games of G(N ), where N is a fixed finite set. This is due to the recursive nature of the axiom (ENP). We present now another axiomatization of ΦSh , where the generalized nullity axiom is outlined in another way. Definition 1 Let v ∈ G(N ) be any finite game. We call support of v, denoted by S(v), the minimal carrier of v, that is, \ S(v) := {C ∈ 2N }. C is a carrier of v Actually, a carrier axiom has been introduced for the first time by Myerson [9], saying that, if C is a carrier for the game v, then the worth v(C) should be shared only among the members of the carrier. It is shown that this axiom is equivalent to the conjunction of the above axioms (N) and (E). With regard to our work, we focus our attention on the support of the game and give an axiom for players in accordance with their membership of the support of the game. If there is no ambiguity, we denote by v S the restricted game v S(v) . 6 Restricted-support games (RS): for any finite set N ⊆ Ω, any game v ∈ G(N ), and any player i ∈ N , ( Φ(v S, i) if i ∈ S(v), Φ(v, i) = 0 otherwise. Corollary 2 ΦSh is the sole value on G satisfying axioms (L), (RS) and (Eq). To show this result, we propose an alternative characterization of the support of a game: Lemma 3 Let v ∈ G(N ) be any game. Then S(v) is the set of players which are not null for v. Proof: It suffices to show that a player i ∈ N is not null for v iff she belongs to every carrier of v. Suppose that player i ∈ N is null for v and let C be any carrier of v. Then for any S ∈ 2N , v(S ∩ (C \ i)) = v((S \ i) ∩ C) = v(S \ i) = v(S). Thus C \ i is a carrier of v. Conversely, if i is not null for v, then ∃S ′ ∈ 2N s.t. v(S ′ ∪ i) 6= v(S ′ ). Then for any carrier C, considering the subcoalition S := S ′ ∪i, if i 6∈ C then v(S ∩C) = v((S ′ ∩C)∪(i∩C)) = v(S ′ ) which should also equals v(S) = v(S ′ ∪ i). This contradicts v(S ′ ∪ i) 6= v(S ′ ).  4 The Shapley value of multichoice games In previous section, the lattice representing actions of players was L := {0, 1}Ω , 0 (resp. 1) denoting absence (resp. presence) of a player. Now, for every player i belonging to a finite carrier of players N , it is assumed that she may act at a level of participation k ∈ Li to the game. Actually, Li := {0, 1, 2, . . . , ⊤i } is a linear lattice, where 0 means absence of participation and ⊤i represents the maximal participation to the game. Thus L = L1 × · · · × Ln is the set of all possible joint actions of players of N . We denote by L(N ) the set of all cartesian products of finite linear lattices over N , and by L, the union of all these ones for every finite set N : n Y L(N ) := { Li | ⊤1 , . . . , ⊤n ∈ N}; L := {L(N ) | N ⊆ Ω, n ∈ N}. i=1 Note that it shall be useful for the sequel to introduce the following binary relation over L defined for all L ∈ L(N ), L′ ∈ L(N ′ ), by ( n = n′ , ′ LRL iff (⊤′1 , . . . , ⊤′n ) is a permutation of (⊤1 , . . . , ⊤n ). This relation is obviously an equivalence relation. We denote by L the quotient set L/R. 7 Thus, it turns out that L is isomorphic to the set of the partitions of positive integers, where a partition of a positive integer m is a finite nonincreasing sequence1 of positive Pn integers (λ1 , . . . , λn ) such that i=1 λi = m (see [1]). The λi ’s, corresponding to the maximal levels of participation of players, are called the parts of the associated partition. With a slight abuse of notation, we may assimilate L to the set of partitions of positive integers. For any λ := (λ1 , . . . , λn ) ∈ L, |λ| is defined by the sum of the λi ’s, i.e., the unique integer whose partition is given by λ. Also, let us endow L with the following order. For all λ := (λ1 , . . . , λn ) ∈ L, λ′ := (λ′1 , . . . , λ′n′ ) ∈ L, ( n′ ≤ n, . λ′ ≤ λ iff ∀i ∈ {1, . . . , n′ }, λ′i ≤ λi For instance, we have (2, 1, 1) ≤ (4, 3, 2, 1). Note that λ := (1) is the bottom of (L, ≤). Proposition 4 (L, ≤) is a ranked lattice, whose rank function is given by r(λ) = |λ|, ∀λ ∈ L. Proof: We show that supremum and infimum of (L, ≤) are respectively given by (s) ′ ′ (λ1 , . . . , λn ) ∨ (λ′1 , . . . , λ′n′ ) = (λ1 , . . . , λ(s) n ) := (λ1 ∨ λ1 , . . . , λn′ ∨ λn′ , λn′ +1 , . . . , λn ), (i) (i) (λ1 , . . . , λn ) ∧ (λ′1 , . . . , λ′n′ ) = (λ1 , . . . , λn′ ) := (λ1 ∧ λ′1 , . . . , λn′ ∧ λ′n′ ), where it is assumed without loss of generality that n ≥ n′ . Indeed, we easily check that (s) (s) (S) (S) (i) (i) (λ1 , . . . , λn′ ) ≤ (λ1 , . . . , λn ), (λ′1 , . . . , λ′n′ ) ≤ (λ1 , . . . , λn ). Besides, if (λ1 , . . . , λm ) is (S) (s) another partition greater than (λ1 , . . . , λn ) and (λ′1 , . . . , λ′n′ ), then m ≥ n and λj ≥ λj (s) (s) for all j = 1, . . . , n. This proves the unicity of (λ1 , . . . , λn ). Argument is the same for the infimum. Now let us define r over L by r(λ) := |λ|, and suppose that λ and λ′ satisfy λ′ ≻ λ (λ′ covers λ, see Section 2). Thus either n = n′ , and ∃!j ∈ {1, . . . , n} such that λ′j = λj + 1, or n′ = n + 1, with λ′j = λj for all j = 1, . . . , n and λn′ = 1. In both cases, we obtain r(λ′ ) = |λ′ | = |λ| + 1 = r(λ) + 1.  For L ∈ L, G(L) denotes the set of functions defined on L which vanish at ⊥ := (0, . . . , 0): this corresponds to multichoice games as introduced by Hsiao and Raghavan [7], where each player has a set of possible ordered actions. For the sake of commodity, we will assimilate any element L of L with its representative element in L. In this way, for any λ := (λ1 , . . . , λn ) ∈ L, v ∈ G(λ) means that v is any game with n players such that their maximal participations are given up to the order of players by λ1 , . . . , λn . We denote by G M the set of all multichoice games, that is to say, G M := {G(L) | L ∈ L}. The set J (L) of join-irreducible elements of L is {(0−i , ki ) | i ∈ N, k ∈ Li \ {0}}, using our notation for compound vectors (see Section 2); hence each join-irreducible element 1 In the sequel, elements of L are assumed to be given under this form. 8 (0−i , ki ), which we will often denote by ki if no ambiguity occurs, corresponds to a single player playing at a given level. Thus a value on G(L) is a function Φ : G(L) × J (L) → R that assigns to every player i playing at the level k in a game v ∈ G(L) his prospect Φ(v, ki ). Our aim is to define the Shapley value Φ(v, ki ) for each join-irreducible element ki . Our approach will take here a similar way, such as the axiomatization given for classical cooperative games. Note that an axiomatization of the Shapley value for multichoice games has already been done in [6] and [10]. The computed formula is the same. However, the former uses a symmetry axiom which is not really natural, whereas the latter is less intuitive and requires more material. Another important difference in [10] is that the extended Shapley value is only given for multichoice games where the number of possible actions is the same for all players. Moreover, both are not given in a simple recursive way on the whole set G M . Let us give first the following axioms generalizing the ones given for classical games. Linearity (LM ): for any L ∈ L, for all join-irreducible ki ∈ J (L), Φ(v, ki ) is linear on the set of games G(L), which directly implies X Φ(v, ki ) = pkxi v(x), with pkxi ∈ R. x∈L For some k ∈ Li , k 6= 0, player i is said to be k-null (or simply ki is null ) for v ∈ G(L) if v(x, ki ) = v(x, (k − 1)i ), ∀x ∈ L−i . If ⊤i is null for v and ⊤i = 1, player i is simply said to be null for v. Nullity (NM ): for any L ∈ L, for any game v ∈ G(L), for any player i which is k-null for v, Φ(v, ki ) = 0. For some i ∈ N , and v ∈ G(L), if ⊤i 6= 1, we define by v −⊤i the restriction of v to the product L−i × (Li \ ⊤i ). As usual, v −i denotes the restriction of v to the product L−i . Excluded-null-player (ENPM ): for any L ∈ L, for any game v ∈ G(L), for any player i ∈ N such that ⊤i = 1, if i is null for v, ∀j ∈ N \ i, Φ(v, ⊤j ) = Φ(v −i , ⊤j ). Decreased-level (DLM ): for any L ∈ L, for any game v ∈ G(L), for any player i ∈ N such that ⊤i 6= 1, if ⊤i is null for v, (i) ∀k ∈ Li \ {0, ⊤i }, (ii) ∀j ∈ N \ i, Φ(v, ki ) = Φ(v −⊤i , ki ). Φ(v, ⊤j ) = Φ(v −⊤i , ⊤j ). Likewise the previous section, (NM ), (ENPM ) and (DLM ) may be merged in the following axiom: 9 Generalized nullity (GNM ): for any L ∈ L, for any game v ∈ G(L), for any player i which is k-null for v,  (  0 if l = k    Φ(v, li ) = Φ(v −⊤i , l ) otherwise for any level l ∈ {1, . . . , k}, i ( −i  Φ(v , ⊤j ) if ⊤i = 1    Φ(v, ⊤j ) = Φ(v −⊤i , ⊤ ) otherwise if k = ⊤i and for any player j ∈ N \ i. j For any L(∈ L, we define the particular unanimity game of G(L) by 1, if x = ⊤, u⊤ (x) := 0, otherwise. Equity (EqM ): for any L ∈ L, for any player i ∈ N , Φ(u⊤ , ⊤i ) = 1 . n Theorem 5 Under axioms (LM ), (GNM ), and (EqM ), Φ is given on G M by: Φ(v, ki ) = X x∈Γ(L−i ) h(x)! (n − h(x) − 1)! [v(x, ki ) − v(x, (k − 1)i )], n! (6) for any finite set N ⊆ Ω, ∀L ∈ L(N ), ∀v ∈ G(L), ∀ki ∈ J (L), where h(x) := |{j ∈ N \ i | xj = ⊤j }|. Proof: It is clear that the above formula satisfies (LM ) and (NM ). Conversely, we check that Φ satisfies axioms (ENPM ) and (DLM ), (EqM ) being easy to verify. Let v ∈ G(L) and i ∈ N a player null for v (Li = {0, 1}). Note that the classical Shapley’s h(x)! (n − h(x) − 1)! 1 coefficients appear in the formula, under the form αh(x) (n) = . Then n! for j ∈ N \ i, X 1 Φ(v, ⊤j ) = αh(x) (n) (v(x, ⊤j ) − v(x, ⊤j − 1)) x∈Γ(L−j ) = X 1 1 [αh(x) (n) + αh(x)+1 (n)] (v(x, 0i , ⊤j ) − v(x, 0i , ⊤j − 1)), x∈Γ(L−ij ) since i is null for v and ⊤i = 1. Then, by refering to (2), we have 1 1 1 αh(x) (n) + αh(x)+1 (n) = αh(x) (n − 1). Thus Φ(v, ⊤j ) = X x∈Γ((L−i )−j )
1 αh(x) (n − 1) v(x, ⊤j ) − v(x, ⊤j − 1) = Φ(v −i , ⊤j ), 10 and (ENPM ) is satisfied. If ⊤1 6= 1 and ⊤i is null for v, (DLM )-(ii) is obtained in the same way. Besides, (DLM )-(i) is also easy to check since ki < ⊤i and the set of indices Γ(L−i ) under the Sigma symbol of (6) does not depend on Li . We now show that the formula is uniquely determined by the axioms. Under (LM ) and (NM ), Φ is given by X Φ(v, ki ) = (7) pkxi (L) [v(x, ki ) − v(x, (k − 1)i )], x∈L−i for any finite set N ⊆ Ω, ∀L ∈ L(N ), ∀v ∈ G(L), ∀ki ∈ J (L), with pkxi (L) ∈ R. Indeed, assuming ki is null for v, X Φ(v, ki ) = pkxi (L) v(x) x∈L = X h x∈L−i = X i ki i (L) v(x, 0 ) + · · · + p (L) v(x, ⊤ ) pk(x,0 i i (x,⊤i ) i) i i (L)] v(x, ki ) + [pk(x,(k−1) (L) + pk(x,k i) i) X X i (L) v(x, li ). pk(x,l i) (8) x∈L−i l6=k−1,k x∈L−i Consider v ′ ∈ G(L−i ) and extend it to G(L): ( v ′ (x), v(x, li ) = 0, if l = k − 1, k otherwise. Then ki is null for v, hence (8) applies and reduces to: X i i Φ(v, ki ) = [pk(x,(k−1) (L) + pk(x,k (L)] v ′ (x) = 0, i) i) x∈L−i i (L) = by axiom (NM ). Since this is true for all v ′ ∈ G(L−i ), this implies pk(x,(k−1) i) ki −p(x,ki ) (L), ∀x ∈ L−i . Introducing this in (8) yields: Φ(v, ki ) = 0 = X X i (L) v(x, li ). pk(x,l i) x∈L−i l6=k−1,k i Since this must hold for any game v for which ki is null, we deduce that pk(x,l (L) = 0, i) k i (L), the result is proven. ∀l 6= k − 1, k. Letting pkxi (L) := p(x,k i) As for the previous theorem, we compute coefficients of (7) by a basic transfinite induction, which is an extension of mathematical induction on sets endowed with a wellfounded relation. A binary relation R is wellfounded on a set E if every nonempty subset of E has an R-minimal element; that is, for every nonempty subset X of E, there is an element m of X such that for every element x of X, the pair (x, m) is not in R. Considering the strict order < associated to ≤, it is easy to see that < is wellfounded on L. Thus, the inductive step rests on showing the formula over G(λ) if it is true for games defined over all predecessors of λ in (L, ≤). Consequently, if the formula is also satisfied 11 
on G (1) , then the induction hypothesis applies and the result is satisfied for any game of G M . The case λ = (1) corresponds to classical cooperative games with one player, which has been verified in the proof of Theorem 1. Indeed, in this case, J (L) has only one element one can denote by 11 (which is also one of the only two elements of L), for which (7) under (EqM ) writes Φ(v, 11 ) = v(11 ). Besides, (6) writes Φ(v, 11 ) = X h(x)! (n − h(x) − 1)! [v(x, 11 ) − v(x, 01 )] n! x∈{()} = v(11 ). For any λ := (λ1 , . . . , λn ) ∈ L \ {(1)}, let us assume that (6) holds for all games of G(λ′ ) such that λ′ ≺ λ. We now show that under (ENPM ), (DLM ) and (EqM ), the unicity of all coefficients in (7) is given for any game v ∈ G(λ). This being done, as it has been checked that (6) satisfies the axioms, the result will be proved. Let N be any set of players of cardinality n, and L be any linear lattice such that maximum levels ⊤1 , . . . , ⊤n , in any order, are given by λ. • We show the unicity of the Φ(v, ki )’s, for any player i ∈ N such that ⊤i 6= 1, and any level k < ⊤i . Indeed, in this case, if ⊤i is null for any v ∈ G(L), by (DLM )-(i), X pkxi (L′ ) [v(x, ki ) − v(x, (k − 1)i )], Φ(v, ki ) = Φ(v −⊤i , ki ) = x∈L′−i where L′ := L−i ×(Li \⊤i ). Since associated partition of L′ is one of the predecessors of λ, thus all pkxi (L′ ) are known by assumption. For any x ∈ L \ {⊥}, let us denote δx the identity game defined by ( 1 if y = x, ∀y ∈ L, δx (y) := 0 otherwise. Consequently, by identification of coefficients for the ( above formula and the one δ(x,ki ) if k < ⊤i − 1, straightforwardly given by (7) for the ⊤i -null game v := δ(x,⊤i −1) + δ(x,⊤i ) if k = ⊤i − 1, ki ki ′ we obtain px (L) = px (L ), for all x ∈ L−i . • Now, let i ∈ N be any player and j ∈ N \ i such that ⊤j 6= 1. Then for any game v ∈ G(L) for which ⊤j is null, X i Φ(v, ⊤i ) = p⊤ x (L) [v(x, ⊤i ) − v(x, ⊤i − 1)] x∈L−i = X i p⊤ x (L) [v(x, ⊤i ) − v(x, ⊤i − 1)] (9) x∈L−i xj 6=⊤j ,⊤j −1 + X y∈L−ij
⊤i i (L) [v(y, ⊤j − 1, ⊤i ) − v(y, ⊤j − 1, ⊤i − 1)]. (L) + p p⊤ (y,⊤j ) (y,⊤j −1) 12 Besides, by (DLM )-(ii), Φ(v, ⊤i ) = Φ(v −⊤j , ⊤i ) X ′ i = p⊤ x (L ) [v(x, ⊤i ) − v(x, ⊤i − 1)] (10) x∈L′−i xj 6=⊤j −1 + X ′ i p⊤ (y,⊤j −1) (L ) [v(y, ⊤j − 1, ⊤i ) − v(y, ⊤j − 1, ⊤i − 1)], y∈L′−ij where L′ := L−j × (Lj \ ⊤j ). Consequently, by identification of coefficients for (9) ⊤i ′ i and (10) in the game v := δ(x,⊤i ) which is ⊤j -null, we obtain p⊤ x (L) = px (L ) for all x ∈ L−i such that xj 6= ⊤j , ⊤j − 1. Thus, we have proved the unicity of coefficients i p⊤ x (L) for all i ∈ N , and for all x ∈ L−i such that ∃j ∈ N \ i, xj 6= ⊤j , ⊤j − 1. i • Lastly, it remains to show the unicity of the p⊤ x (L)’s, where i ∈ N and x ∈ L−i such that ∀j ∈ N \ i, xj ∈ {⊤j , ⊤j − 1}. This in view, we consider the partition  Ci,m i∈N ;0≤m≤n−1 of these indices, where Ci,m denotes the set of elements of L−i whose m coordinates xj are ⊤j − 1 and the others are ⊤j . For any i ∈ N , we i show the unicity of the p⊤ x (L)’s by induction on m. For x ∈ Ci,0 , that is to say, M i x = ⊤−i := (⊤1 , . . . , ⊤i−1 , ⊤i+1 , . . . , ⊤n ), p⊤ x (L) is given by (Eq ): i Φ(u⊤ , ⊤i ) = p⊤ ⊤−i (L) = 1 . n i Let us suppose that all p⊤ x (L)’s are given for all elements of Ci,m , where m is fixed i in {0, . . . , n − 2}. We show the unicity of the p⊤ x (L)’s for x ∈ Ci,m+1 . Indeed, one can associate any x ∈(Ci,m+1 to any j0 ∈ N \ i such that xj0 = ⊤j0 − 1. We ⊤j0 if j = j0 , define x′ ∈ Ci,m by x′j := . Now, two situations may arise: either xj otherwise ⊤j0 6= 1 or ⊤j0 = 1. In the first case, we refer to (9) and (10) with j := j0 : by identification of the coefficients for the ⊤j -null game v := δ(x,⊤i ) + δ(x′ ,⊤i ) , we obtain ⊤i ⊤i ⊤i ⊤i ⊤i ′ ′ ⊤i i p⊤ x (L) + px′ (L) = px (L ), i.e. px (L) = px (L ) − px′ (L), where px′ (L) is given ′ i by hypothesis in the current induction, and p⊤ x (L ) is given by hypothesis in the backward transfinite induction. Finally, if ⊤j0 = 1, for any game v ∈ G(L) for which j0 is null, X i Φ(v, ⊤i ) = p⊤ x (L) [v(x, ⊤i ) − v(x, ⊤i − 1)] x∈L−i = X y∈L−ij0
⊤i i (L) [v(y, 0j0 , ⊤i ) − v(y, 0j0 , ⊤i − 1)]. (11) (L) + p p⊤ (y,1j ) (y,0j ) 0 0 Besides, by (ENPM ), Φ(v, ⊤i ) = Φ(v −j0 , ⊤i ) X ′ i = p⊤ y (L ) [v(y, ⊤i ) − v(y, ⊤i − 1)], y∈L′−i 13 (12) where L′ := L−j0 . Consequently, by identification of coefficients for (11) and (12) for ⊤i ⊤i ′ i the game v := δ(x,⊤i ) + δ(x′ ,⊤i ) which is j0 -null, we obtain p⊤ x (L) + px′ (L) = px (L ), ⊤i ⊤i ′ i i.e. p⊤ x (L) = px (L )−px′ (L). Note that even if m+1 choices of j0 are possible, one cannot guarantee the existence of such an index such that ⊤j0 = 1 for all i ∈ N , or such that ⊤j0 6= 1 for all i ∈ N . As a consequence, axioms (DLM )-(ii) and (ENPM ) are both necessary. i This ends the proof of the current inductive step: ∀i ∈ N , all p⊤ x (L)’s are given for any x ∈ L−i such that ∀j ∈ N \ i, xj ∈ {⊤j , ⊤j − 1}. Consequently, for all linear lattice L associated to λ, ∀ki ∈ J (L), ∀x ∈ L−i , all pkxi (L)’s are given, which also completes the inductive step of the transfinite induction.  References [1] G.E. Andrews, The theory of partitions, Addison-Wesley, 1976. [2] G. Birkhoff, Lattice theory, 3d ed., American Mathematical Society, 1967. [3] R. Branzei, D. Dimitrov, and S. Tijs, Models in cooperative game theory: crisp, fuzzy and multichoice games, Springer Verlag, to appear. [4] B.A. Davey and H.A. Priestley, Introduction to lattices and orders, Cambridge University Press, 1990. [5] U. Faigle and W. Kern, The Shapley value for cooperative games under precedence constraints, International Journal of Game Theory 21 (1992), 249–266. [6] M. Grabisch and F. Lange, Games on lattices, multichoice games and the Shapley value: a new approach, Mathematical Methods of Operations Research 65 (2007), 153–167. [7] C.R. Hsiao and T.E.S. Raghavan, Shapley value for multichoice cooperative games, I, Games and Economic Behavior 5 (1993), 240–256. [8] D. Monderer and D. Samet, Variations on the shapley value, Handbook of Game Theory No III (Ed. R.J. Aumann and S. Hart, eds.), Elsevier Science, 2002. [9] R.B. Myerson, Values of games in partition function form, Int. J. of Game Theory 6 (1977), 23–31. [10] H. Peters and H. Zank, The egalitarian solution for multi-choice games, Annals of Operations Research 137 (2005), 399–409. [11] L.S. Shapley, A value for n-person games, Contributions to the Theory of Games, Vol. II (H.W. Kuhn and A.W. Tucker, eds.), Annals of Mathematics Studies, no. 28, Princeton University Press, 1953, pp. 307–317. 14 [12] R.M. Thrall and W.F. Lucas, N-person games in partition function form, Naval Research Logistics Quarterly 10 (1963), 281–293. [13] R.J. Weber, Probabilistic values for games, The Shapley Value. Essays in Honor of Lloyd S. Shapley (A.E. Roth, ed.), Cambridge University Press, 1988, pp. 101–119. 15