Soft cooperation systems and games

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of General Systems , Vol. 47, Issue 3 on 2018, available at: https://doi.org/10.1080/03081079.2017.1413099 ARTICLE TEMPLATE Soft cooperation systems and games J.R. Fern´andez, I. Gallego, A. Jim´enez-Losada and M. Ord´o˜nez Dpto. Matem´atica Aplicada II. Escuela T´ecnica Superior de Ingenieros. Universidad de Sevilla. Spain ARTICLE HISTORY Compiled November 24, 2017 ABSTRACT A cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation. In order to obtain this allocation a characteristic function is known. It establishes the profit of each coalition of agents if this coalition decides to act alone. Originally players are considered symmetric and then the allocation only depends on the characteristic function, this paper is about cooperative games with an asymmetric set of agents. We introduced cooperative games with a soft set of agents which explains those parameters determining the asymmetry among them in the cooperation. Now the characteristic function is defined not over the coalitions but over the soft coalitions, namely the profit depends not only on the formed coalition but also on the attributes considered for the players in the coalition. The best known of the allocation rules for cooperative games is the Shapley value. We propose a Shapley kind solution for soft games. KEYWORDS game theory; soft sets; Shapley value; cooperation systems 1. Introduction In a cooperative situation for a finite set of agents (players) with a common utility we must look for a fair allocation of this profit among them. Nowadays cooperative games are being used in numerous fields of engineering and artificial intelligence in order to solve this kind of situations. A cooperative game, see Driessen (1988), is defined by a characteristic function which establishes the worth of each subset of players (coalition). Using this information, a payoff vector containing the chosen allocation of the profit is obtained. A value for cooperative games is a function determining a payoff vector for each cooperative game. The most known value is the Shapley value (Shapley 1953). But there are a lot of values more in the literature and, in order to decide which is the most appropriate for our situation each value is provided with a set of reasonable conditions (axioms). Originally, the characteristic function is the only information about the players that we use to formulate a value, and then players are considered symmetric in their actions and relations. But in real life cooperative situations are more complex. Any additional information about the players must change the formulation of the values. So, several works in recent year have developed different versions of cooperative games with CONTACT A. Jim´enez-Losada. Email: a[email protected] additional information about the players: coalition structures (Aumann and Dreze 1974), a priori unions (Owen 1977), communication structures (Myerson 1977), conference structures (Myerson 1980), permission structures (Gilles et al. 1992), fuzzy communication structures (Jim´enez-Losada et al. 2010), fuzzy authorization structures (Gallardo et al. 2015), proximity relations (Fern´andez et al. 2016), fuzzy settings (Borkotokey and Mesiar 2013) or fuzzy restrictions (Gallardo et al. 2017). Some of these studies evidenced the importance of the structures of sets in the analysis of the restrictions or modifications in the game. So several known set systems were used in this sense: convex geometries (Bilbao and Edelman 2000), antimatroids (Algaba et al. 2004) or matroids (Bilbao et al. 2001). Also, new particular partial ordered sets have been introduced to describe determined situations: augmenting and decreasing systems (Ord´o˜nez and Jim´enez-Losada 2017) or embedded coalition structures (Alonso-Meijide et al. 2017). In this paper we introduce a new model to study asymmetric problems in cooperative games. Molodtsov (1999) initiated the concept of soft set theory as a new mathematical tool. He also proposed the application of soft sets to game theory defining soft games but in normal form. During the last years a lot of papers about soft sets and operations have been published (Maji et al. 2003; Yang 2008; Ali et al. 2009; Ge and Yang 2011; Sezgin and Atag 2011; Zhu and Wen 2013). We propose an application of soft sets to describe coalitions with information about the distribution of the tasks among the agents. The asymmetry properties among the players in a cooperative game are the parameters to define a soft set of players. We consider then a characteristic function over the soft subsets of players, the soft coalitions. The meaning of this worth for a soft coalition is the profit obtained for the coalition depending on the parameters satisfied by the players in the coalition. The next section is dedicated to the preliminaries about cooperative games and soft sets. In section 3 we introduce soft coalitions and we analyze the structure of a soft coalition system for a game. Soft cooperative games are defined in section 4. Section 5 shows a version of Shapley value for soft cooperative games. Finally in section 6 we provide this Shapley solution with an axiomatization. 2. Preliminaries 2.1. Posets A finite poset (partial ordered set) is a pair (K, <) where Kis a finite set and <is a partial order relation over K. The bottoms of the poset are the minimal elements in the order and the tops are the maximal ones. Let x, y ∈Kwith x<yin the poset (K, <). The interval [x, y] is [x, y] = {z∈K:x≤z≤y}.(1) We say ycovers xif x<yand [x, y] = {x, y}. The Hasse diagram of the poset is the graph whose vertices are the elements in K, whose links are the cover relations (we have a link between two elements if one of them covers the other one) and if x<y then yis drawn above x. A (maximal saturated) chain in the poset is a sequence of elements {x1, . . . , xp}in Ksuch that x1is a bottom, xpis a top and xkcovers xk−1 for all k= 2, . . . , p. Namely, a chain is a line in the Hasse diagram from a bottom to a top. A poset is named graded if all the chains have the same number of elements. 2 The power set of Kis the family of subsets of Kand it is denoted by 2K. The poset (2K,⊂) using the inclusion of sets as order is a boolean algebra. 2.2. Cooperative games We follow Driessen (1988) for this section. A cooperative game with transferable utility (game) is a pair (N, v) where Nis a finite set and v: 2N→Ris a real function over the power set of Nwith v(∅) = 0. The empty set is actually a degenerate idea of coalition and it is included only as a technical instrument. The elements of N={1,2, ..., n}are called players, the subsets S⊆Ncoalitions and v(S) is the worth of S. We suppose fixed the set of players Nin all the paper and then we identify the game (N, v) with the characteristic function v. The family of games with Nas set of players is denoted as GN. Let v∈ GN. Game vis superadditive if for two disjoint coalitions S, T ⊆N, S∩T=∅,v(S∪T)≥v(S) + v(T). If we interpret the worths in the characteristic function vas benefits then the above condition implies an incentive to cooperate. The marginal contribution of player iin vfor coalition Snot containing iis the number v(S∪{i})−v(S). A player i∈Nis a null player in vif all her marginal contributions are zero, i.e v(S∪{i}) = v(S) for all S(N\{i}. Two players i, j ∈Nare symmetric in vif their marginal contributions to coalitions not containing both are equal, i.e v(S∪{i}) = v(S∪{j}) for all S⊆N\{i, j}. Apayoff vector for a game v∈ GNis any x∈RNwhere, for each player i∈N, number xirepresents the payment of iowing to his cooperation possibilities. Given xa payoff vector and Sa coalition, we use the notation x(S) = Pi∈Sxi. An imputation for the game vis an efficient payoff vector satisfying the individual rationality principle, x(N) = v(N) and xi≥v({i}) for all player i∈N. A value for cooperative games over Nis a function ψ:GN→RNwhich assigns to each game v∈ GNa payoff vector ψ(v)∈RN. The element to measure the contribution of a player iin a coalition S with i /∈Sfor a game vis the marginal contribution given by v(S∪{i})−v(S). The Shapley value (Shapley 1953) is for each game v∈ GNand any player i∈N shi(v) = X {S⊆N:i/∈S} γN S[v(S∪{i})−v(S)] ,(2) where γN S=(n−|S|−1)!(|S|)! n!(3) is the probability of getting the marginal contribution between Sand S∪{i}in the boolean algebra 2N, namely the quotient between the number of chains containing this link and the total number of chains. If v∈ GNis superadditive then sh(v) is an imputation. Formula (2) is the most known of the Shapley value in the sense that it permits to reduce the calculations only to the marginal contributions. This value is the only one over GNsatisfying the following axioms. (S1) Efficiency. If v∈ GNit holds Pi∈Nshi(v) = v(N). (S2) Additivity. It is an additive function, that is if v1, v2∈ GNwe have the equality sh(v1+v2) = sh(v1) + sh(v2). (S3) Null player axiom. If i∈Nis a null player for the game vthen shi(v) = 0. (S4) Equal treatment axiom. If i, j are symmetric for vthen shi(v) = shj(v). 3 2.3. Soft sets We follow Maji et al. (2003) for this subsection. Consider two non-empty finite sets, one of them Uof elements and the another one Eof parameters to be issued to the elements in U. Asoft set over Uis a pair (F, A) where A⊆E, and F:A→2U. For each e∈A the set F(e) is named the e-approximation of the soft set. Soft sets are defined in Molodtsov (1999). The support of (F, A) is AF={e∈A:F(e)6=∅}.(4) If A⊆Ethen we use (∅, A) as the relative null soft set to Anamely ∅(e) = ∅for every e∈A. The relative whole soft set to Ais (U, A) where U(e) = Ufor each e∈A. Now we consider two different soft sets (F, A) and (G, B). The first soft set is a soft subset of the second one, (F, A)⊆(G, B), if A⊆Band F(e)⊆G(e) for all e∈A. If B⊆Athen (F, B) is the soft subset of (F, A) given by the restriction of Fto B, called the restriction of (F, A) to B. The (extended) union of both of the soft sets is defined as (F, A)∪(G, B)=(H, A ∪B)) with H(e) =    F(e)∪G(e),if e∈A∩B F(e),if e∈A\B G(e),if e∈B\A. The (restricted) intersection of both soft sets when A∩B6=∅is (F, A)∩(G, B) = (H, A ∩B) with H(e) = F(e)∩G(e) for all e∈A∩B. There exist another different concepts for union and also for intersection, the restricted union and the extended intersection. This paper can be developed doing the same things but using these other definitions. Although the definition of soft set (F, A) seems to suppose that A6=∅, we also consider a technical soft set for the situation without parameters denoted as (∅,∅). This new soft set is a soft subset of any soft set in U. Moreover, using (∅,∅) the intersection can be introduced also for softs sets with disjoint sets of parameters. Let (F, A),(G, B) be soft sets, we define (F, A)∩(G, B) = (∅,∅) if A∩B=∅. 3. Soft coalition systems. We consider a fixed finite set of players Nin a cooperative situation and also a universal set of parameters Ewhich define how the players are organized to carry out the cooperation: issues, actions or tasks aimed at getting a common profit. If we take a soft set of N, (F0, A0), we are determining the parameters needed for the cooperation, A0, and which members of Nare able to make it, F0. If e /∈A0then players have decided not to take into account parameter e, for instance if eis a determined task then it is not made. If e∈A0and F(e) = S6=∅then this parameter is associated to coalition S, so if eis a task we understand that this task is made by the players in S. Finally if e∈A0but F(e) = ∅then we suppose that eis associated to an external element, the task is made by other figures beyond N. The external elements are not players because they are out of the game in the sense that their payoffs are fixed. Definition 3.1. A soft set of players is any soft set (F0, A0) over Nwith parameters in Esuch that A06=∅and there is at least one e∈A0with F0(e)6=∅. 4 If a soft set of players is formed, players get the profit of the cooperation and they pay the externalities. Now they have to allocate the rest of the profit among them. Coalitions are the instruments of the players to bargain their payoffs. But the externalities defined in the soft set of players are not negotiable. So the idea of coalition is not exactly any soft subset of the soft set of players as we could expect1, it must be a soft subset of the soft set of players but also must accept the externalities already decided. Definition 3.2. Let (F0, A0) be a soft set of players over Nwith parameters in E. The soft set (F, A) is a soft coalition of (F0, A0) if it is a soft subset of (F0, A0) and A0\(A0)F0⊆A\AF. The family of soft coalitions of (F0, A0) is denoted as SC(F0, A0). Hence, a soft coalition of a soft set of players (F0, A0) is a soft subset of (F0, A0) assuming the externalities of (F0, A0). The simplification of the soft cooperation to a crisp one is done in the following sense. The crisp coalition of (F, A)∈SC(F0, A0) is SA F=[ e∈A F(e).(5) We take particularly S∅ ∅=∅. Observe that we always suppose SA0 F06=∅. The concept of soft subset of (F, A) assumes that for any parameter e∈A\AFthe restriction (F, A\{e}) is a soft subset. But in our situation we think that they are not comparable. So we introduce a particular partial order relation among the soft coalitions. Definition 3.3. Let (F0, A0) be a soft set of players. A soft coalition (F, A)∈ SC(F0, A0) is smaller than another (G, B)∈SC(F0, A0) if (F, A)∈SC(G, B). This relation is denoted as (F, A)v(G, B). The family of coalitions with the inclusion relation is a boolean algebra in the classical situation but not in our version. Suppose N={1,2,3}and E={e1, e2, e3}. We consider the soft set of players (F0, A0) given by A0=Eand F0(e1) = {1,2}, F0(e2) = {2,3},F0(e3) = ∅. So parameter e3is satisfied by an externality (for instance (∅,∅) is not a soft coalition in this case). The Hasse diagram of the poset of soft coalitions SC(F0, A0) with the relation in Definition 3.3 is shown in Figure 1. Each triple in the picture represents a soft coalition (F, A) with the image of the parameters by Fordered. We use a star to indicate that the parameter is not considered. For instance (∗,{2,3},∅) represents (F, {e2, e3}) with F(e2) = {2,3}and F(e3) = ∅. The reader can see on the picture that if we do not modify the concept of soft subset to soft coalition, it will be impossible to calculate some marginal contributions. For instance, ({2},{3},∅) only covers one soft subset without player 3, ({2},∅,∅), but there are two soft coalitions covered by it, ({2},∅,∅) and ({2},∗,∅). Using the concept of soft subset we cannot calculate the marginal contribution of player 3 in front of ({2},∗,∅). Now player 3 can measure his contribution to ({2},{3},∅) taking into account both of the options for player 2 into the soft coalition, or parameter e1is not assigned or it is assigned to an externality, ({2},∅,∅) and ({2},∗,∅). 1In the classical cooperative game theory all the subsets of the set of players are coalitions. 5 {1,2},{2,3},Ø {2},{2,3},Ø {1},{2,3},Ø {1,2},{3},Ø {1,2},{2},Ø Ø,{2,3},Ø ★,{2,3},Ø {2},{3},Ø {2},{2},Ø {1,2},Ø,Ø {1,2},★,Ø {1},{2},Ø {1},{3},Ø Ø,★,Ø Ø,Ø,Ø ★,★,Ø ★,Ø,Ø Ø,{2},Ø ★,{2},Ø {2},Ø,Ø {2},★,Ø Ø,{3},Ø ★,{3},Ø {1},Ø,Ø {1},★,Ø Figure 1. Soft coalition system. If (F0.A0) is a soft set of players over Nwith parameters Ethen its assignment set is M(F0, A0) = {(i, e) : e∈A0, i ∈F0(e)}.(6) Associated to a given subset of assignments K⊆M(F0, A0) we define a particular soft set (FK, BK) with BK={e∈E:∃i∈Nwith (i, e)∈K}and, (7) FK(e) = {i∈N: (i, e)∈K}∀e∈BK.(8) Proposition 3.4. Let (F0, A0)be a soft set of players. The poset (SC(F0, A0),@) satisfies the following properties. (a) The top is (F0, A0)and the set of bottoms is Φ(F0, A0) = {(∅, A) : A0\(A0)F0⊆A⊆A0}. (b) For each (∅, A)∈Φ(F0, A0)the interval [(∅, A),(F0, A0)] is equivalent to the boolean algebra 2M(F0,A0). (c) If K⊆M(F0, A0)then (FK, BK)∪(∅, A)∈[(∅, A),(F0, A0)] with (∅, A)∈ Φ(F0, A0)is the only one in the interval identified with K. (d) It is a (Pe∈A0|F0(e)|)-graded poset and its number of chains is 2|(A0)F0|Pe∈A0|F0(e)|!. (e) For any (F, A)∈SC(F0, A0), the number of chains2from the bottom to (F, A) 2The number of chains in the subposet SC(F, A). 6 is 2|AF|Pe∈A|F(e)|!and the number of chains3from (F, A)to the top is  X e∈A|F0(e)\F(e)|+X e∈A0\A|F0(e)| !. Proof. All the statements can be proved jointly. It is evident since definition of soft coalition that (F0, A0) is the top of the structure. Let (∅, A)∈Φ(F0, A0). Suppose (G, B)∈SC(F0, A0) with (G, B)v(∅, A). We have (G, B)∈SC(∅, A). As (G, B)⊆ (∅, A) then B⊆Aand G(e) = ∅for all e∈B. As also A⊆Bthen (G, B)=(∅, A). Consider a soft coalition in the bottom, (∅, A)∈Φ(F0, A0). If (G, B)∈ [(∅, A),(F0, A0)] then K(G,B)={(i, e) : e∈B, i ∈G(e)} ⊆ M(F0, A0). Moreover, if (G0, B0)6= (G, B) satisfies K(G,B)=K(G0,B0)then there exists e∈B\B0(or e∈B0\B) such that G(e) = ∅(or G0(e) = ∅). But as (∅, A)⊆(G0, B0) then e /∈Aand as (∅, A)v(G, B) then e∈A. Now we take K⊆M. We can define the soft coalition (FK, BK)∪(∅, A)∈[(∅, A),(F0, A0)] where BK={e∈E:∃i∈Nwith (i, e)∈K} and for every e∈BK,FK(e) = {i∈N: (i, e)∈K}. So 2M(F0,A0)is identified one to one with the interval [(∅, A),(F0, A0)] for all (∅, A)∈Φ(F0, A0). The other statements follow from the above identification. We have the cardinalities |Φ(F0, A0)|= 2|(A0)F0| and |M(F0, A0)|=Pe∈A0|F0(e)|. As each chain ends in only one bottom then the number of chains is |M(F0, A0)|!|Φ(F0, A0)|= X e∈A0 |F0(e)|!! 2|(A0)F0|. Obviously if we consider the subposet from the bottom to (F, A) then we get SC(F, A) and the number of chains follows from the before reasoning using that Φ(F, A) = {(∅, B) : A\AF⊆B⊆A}. Finally, observe that we can identify the interval [(F, A),(F0, A0)] with the boolean algebra 2Mwhere M={(i, e) : e∈A, i ∈F0(e)\F(e)}∪{(i, e) : e∈A0\A, i ∈F0(e)}. 4. Soft games. We introduce in this section the concept of cooperative soft game. Definition 4.1. A soft game over Nis a triple (F0, A0, v) where (F0, A0) is a soft set of players and v:SC(F0, A0)→Rwith v(∅, A) = 0 if (∅, A)∈Φ(F0, A0). The set of soft games over Nis denoted as SGN. The worth of a soft coalition (F, A) in a soft game is understood as the profit obtained by the involved players SA Ftaking into account that parameters in Aare 3The number of chains in the subposet [(F, A),(F0, A0)]. 7 satisfied and after paying the externalities. Therefore it is logical to take v= 0 in Φ(F0, A0). A soft game over Nfixes first the soft set of players formed and also the coalition into N(its crisp coalition) decided by the players. This coalition has a determined configuration with regard to the parameters in order to get a determined profit, including if they consider to satisfy several parameters by external actions. And second, the game establishes a characteristic function over the soft coalitions of this soft set of players. So, the problem in a soft game is how to allocate the worth of the soft set of players, and the worths of other soft coalitions are only the instrument to get the allocation vector. Usually games are considered superadditive, then we introduce this concept for soft games. Definition 4.2. Let (F0, A0, v)∈ SGNbe a soft game. The game is superadditive if for all (F, A),(G, B)∈SC(F0, A0) with (F, A)∩(G, B)∈Φ(F0, A0) it holds v(F, A) + v(G, B)≤v((F, A)∪(G, B)) . A game (F0, A0, v)∈ SGNdetermines a worth for each soft coalition (F, A) including those with parameters out of the support (with A\AF6=∅). In some situations these distinctions between not being in the support or being empty cannot be interesting. We introduce the following concept. Definition 4.3. Let (F0, A0) be a soft set of players. A soft coalition (F, A) is full if A=AF∪(A0\(A0)F0). SCfull(F0, A0) represents the family of full soft coalitions. A soft game (F0, A0, v) is full if v(F, A) = v(F, AF∪(A0\(A0)F0)) for all (F, A)∈SC(F0, A0). The family of full games is denoted as FSGN. If the soft game is full4then the only interesting soft coalitions are the full ones. Hence, we can consider the game restricted to the full coalitions. The concept of full can be extended to soft sets in general, a soft set (F, A) is full if AF=A. Observe that full soft set coincides with the concept of soft set studied in Zhu and Wen (2013). Finally we can see classical cooperative games as a special family of soft games. Definition 4.4. A soft game (F0, A0, v)∈ SGNis crisp if v(F, A) = v(G, B) for all pair of soft coalitions (F, A),(G, B)∈SC(F0, A0) with SA F=SB G. Each crisp soft game (F0, A0, v)∈ SGNis identified with the classical game w(F0,A0,v)∈ GSA0 F0defined as w(F0,A0,v)(S) = v(F, A) for any (F, A)∈SC(F0, A0) with SA F=S(we can always find one). Hence we can consider the family of crisp soft games as the set of all the classical games defined over coalitions in N, we use GN ∗=[ {S⊆N:S6=∅} GS.(9) Of course GN ∗⊆ FSGN. Let i∈Nbe a player. We define the i-soft set relative to A⊆Eas (Fi, A) with 4Full soft sets and therefore full soft games can be introduced in another equivalence way. Let (F0, A0) be a soft set of players. A soft coalition (F, A) is spanning if A=A0. A soft game (F0, A0, v) is spanning if v(F, A) = v((F, A)∪(∅, A0)) for all (F, A)∈SC(F0, A0). It is easy to test that (F0, A0, v) is full if and only if it is spanning. 8 Soft Null Player Axiom. Suppose iis an e-null player for the soft game (F0, A0, v). We will prove that for any A0\(A0)F0⊆A⊆A0the pair (i, e) is a null player for the game wA (F0,A0,v). Let K⊆M(F0, A0) be a coalition with (i, e)/∈K. Observe that (FK∪{(i,e)}, BK∪{(i,e)})=(FK, BK)∪(Fi,{e}) since Proposition 3.4. Hence we get wA (F0,A0,v)(K∪{(i, e)}) = v(FK∪{(i,e)}, BK∪{(i,e)})∪(∅, A) =v((FK, BK)∪(∅, A)∪(Fi,{e})) = v((FK, BK)∪(∅, A)) =wA (F0,A0,v)(K). As the Shapley value satisfies the null player axiom (S3) sh(i,e)(wA (F0,A0,v)) = 0 for all every set A0\(A0)F0⊆A⊆A0. Proposition 5.2 implies that She i(F0, A0, v) = 0. Soft equal treatment axiom. Consider iin ereplaceable by jin e0for the soft game (F0, A0, v). It is easy to see that (i, e),(j, e0) are symmetric for wA (F0,A0,v)because if K⊆M(F0, A0)\{(i, e),(j, e0)}then, following an above reasoning, wA (F0,A0,v)(K∪{(i, e)}) = v((FK, AK)∪(Fi,{e})∪(∅, A)) =v(FK, AK)∪(Fj,{e0})∪(∅, A) =wA (F0,A0,v)(K∪{(j, e0)}), for all A0\(A0)F0⊆A⊆A0. As the Shapley value satisfies the equal treatment axiom (S4) we have by Proposition 5.2 She i(F0, A0, v) = 1 2|(A0)F0|X A0\(A0)F0⊆A⊆A0 sh(i,e)(wA (F0,A0,v)) =1 2|(A0)F0|X A0\(A0)F0⊆A⊆A0 sh(j,e0)(wA (F0,A0,v)) = She0 j(F0, A0, v). Additivity. We take wA (F0,A0,v1), wA (F0,A0,v2)for each A0\(A0)F0⊆A⊆A0. Also we consider wA (F0,A0,v1+v2). It holds wA (F0,A0,v1+v2)=wA (F0,A0,v1)+wA (F0,A0,v2), and then the axiom follows from the additivity of the Shapley value (S2), Sh(F0, A0, v1) + Sh(F0, A0, v2) = 1 2(A0)F0X A0\(A0)F0⊆A⊆A0 sh(wA (F0,A0,v1)) + wA (F0,A0,v2) =1 2(A0)F0X A0\(A0)F0⊆A⊆A0 sh(wA (F0,A0,v1+v2)) =Sh(F0, A0, v1+v2). Now we suppose Ya soft payoff function satisfying the four axioms. Let (F0, A0) be a soft set of players. For each (G, B)∈SC(F0, A0) we introduce the functions δ(G,B), 15 ξ(G,B)as δ(G,B)(F, A) = 1,if (G, B)=(F, A) 0,otherwise, and ξ(G,B)(F, A) = 1,if (G, B)v(F, A) 0,otherwise for all (F, A)∈SC(F0, A0). Of course, we have ξ(G,B)=P(G,B)v(F,A)δ(F,A)and the M¨obius inversion formula (Stanley 1986) applied to the poset SC(F0, A0) implies that there exist numbers µ((G, B),(F, A)) (and it is possible to calculate them) satisfying δ(G,B)=X (G,B)v(F,A) µ((G, B),(F, A))ξ(F,A). For each (G, B)∈SC(F0, A0)\Φ(F0, A0) we can define the soft game (F0, A0, δ(G,B)), (F0, A0, ξ(G,B)). If (F0, A0, v)∈ SGNthen we find a number r(G,B)for every (G, B)∈ SC(F0, A0)\Φ(F0, A0) such that v=X (G,B)∈SC(F0,A0)\Φ(F0,A0) v(G, B)δ(G,B)=X (G,B)∈SC(F0,A0)\Φ(F0,A0) r(G,B)ξ(G,B). Function Ysatisfies additivity then Y(F0, A0, v) = X (G,B)∈SC(F0,A0)\Φ(F0,A0) Y(r(G,B)ξ(G,B)). Observe that if e∈A0and i∈F0(e)\G(e) or e /∈Bthen iis e-null player for r(G,B)ξ(G,B), thus Ye i(F0, A0, r(G,B)ξ(G,B)) = 0 by the soft null player axiom. If e, e0∈B and i∈G(e), j∈G(e0) then player iin eis replaceable by jin e0for r(G,B)ξ(G,B), because for each (F, A)∈SC(F0, A0) with i /∈F(e) and j /∈F(e0) r(G,B)ξ(G,B)((F, A)∪(Fi,{e})) = 0 = r(G,B)ξ(G,B)((F, A)∪({j},{e0})). The soft equal treatment axiom implies that Ye i(F0, A0, r(G,B)ξ(G,B)) = Ye0 j(F0, A0, r(G,B)ξ(G,B)) = J. Finally if we apply soft efficiency we obtain X e∈A0X i∈F0(e) Ye i(F0, A0, r(G,B)ξ(G,B)) = X e∈BX i∈G(e) Ye i(F0, A0, r(G,B)ξ(G,B)) =J X e∈B|G(e)|! =r(G,B)ξ(G,B)(F0, A0) = r(G,B). Hence there is a unique worth for J. The above axiomatization implies another one for the Shapley crisp function. Now we think of a crisp payoff function Y:SGN→RN. Suppose (F0, A0, v)∈ SGN. 16 Efficiency is the same than (S1) of the Shapley value, X i∈N Yi(F0, A0, v) = v(F0, A0). A player iis a crisp null player in the above soft game if iis an e-null player for all e∈A0with i∈F0(e). Similarly, players i, j ∈Nare crisp symmetric if for all e, e0 with i∈F0(e) and j∈F0(e) we have that ien eis replaceable by jen e0. Theorem 6.2. The Shapley crisp payoff function Sh is the only crisp payoff function satisfying efficiency, crisp null player axiom, crisp equal treatment axiom and additivity. 7. Conclusions The paper introduced cooperative games with soft cooperation which permits to study new situations of asymmetric players in games. An analysis of the soft coalition structure was done. We also studied a soft Shapley value for these games. This paper pretends to be only a first introduction of the soft set theory in cooperative games. We think that there is an immense task to accomplish: crisp values, the core, stability, convexity, other solution concepts... References E. Algaba, J.M. Bilbao, R. van den Brink, A. Jim´enez-Losada 2004. “Cooperative games on antimatroids”. Discrete Mathematics 282: 1-15. M.I. Ali, F. Meng, X. Liu, W.K. Minc, M. Shabir 2009. “On some new operations in soft set theory”. Computers and Mathematics with Applications 57: 1547-1553. J.M. Alonso-Meijide, M. ´ Alvarez-Mozos, G.Fiestras-Janeiro, A. Jim´enez-Losada 2017.“Some structural properties of a lattice of embedded coalitions”. International Journal of General Systems 46(2): 123-143. R.J. Aumann, J.H. Dreze J.H. 1974. “Cooperative games with coalition structures”. International Journal of Game Theory 3: 217-237. J.M. Bilbao, P.H. Edelman 2000. “The Shapley value on convex geometries”. Discrete Applied Mathematics 103: 33-40. J.M. Bilbao, T.S.H. Driessen, A. Jim´enez-Losada, E. Lebr´on 2001. “The Shapley value for games on matroids: the static model”. Mathematical Methods of Operations Research 53(2): 333-348. S. Borkotokey, R. Mesiar 2013. “The Shapley value of cooperative games under fuzzy settings: a survey”. International Journal of General Systems 43(1): 75-95. T.S.H. Driessen 1988. Cooperative Games, Solutions and Applications. Theory and Decision Library Series C, Kluwer Academic Publishers, Dordrecht, The Netherlands. J.R. Fern´andez, I. Gallego, A. Jim´enez-Losada, M. Ord´o˜nez 2016. “Cooperation among agents with a proximity relation”. European Journal of Operational Research 250: 555-565. J.M. Gallardo, N. Jim´enez, A. Jim´enez-Losada, E. Lebr´on 2015. “Games with fuzzy authorization structure: a Shapley value”. Fuzzy Sets and Systems 272: 115-125. J.M. Gallardo, N. Jim´enez, A. Jim´enez-Losada 2017. “Fuzzy restrictions and an application to cooperative games with restricted cooperation”. International Journal of General Systems 46(7): 772-790. 17 X. Ge, S. Yang 2011. “Investigations on some operations of soft sets”. World Academy of Science, Engineering and Technology 51: 1112-1115. R.P. Gilles, G. Owen, R. van den Brink 1992. “Games with permission structures: the conjunctive approach”. International Journal of Game Theory 20: 277-293. M. Grabisch, F. Lange 2007. “Games on lattices, multichoice games and the Shapley value: a new approach”. Mathematical Methods of Operations Research 65: 153-167. A. Jim´enez-Losada, J.R. Fern´andez, M. Ord´o˜nez, M. Grabisch (2010). “Games on fuzzy communication structures with Choquet players”. European Journal of Operational Research 207: 836-847. P.K. Maji, R. Biswas, A.R. Roy 2003. “Soft set theory”. Computers and Mathematics with Applications 45: 555-562. D. Molodtsov 1999. “Soft set theory, first results”. Computers and Mathematics with Applications 37: 19-31. R.B. Myerson 1977. “Graphs and cooperation in games”. Mathematics of Operations Research 2: 225-229. R.B. Myerson (1980). “Conference structures and fair allocation rules”. International Journal of Game Theory 9: 169-182. M. Ord´o˜nez, A. Jim´enez-Losada 2017. “Duality on combinatorial structures: an application to cooperative games. International Journal of General Systems 46(8): 839-852. G. Owen 1977. “Values of games with a priori unions”. Mathematical Economics and Game Theory. Lecture Notes in Economics and Mathematical Systems 141: 76-88. A. Sezgin, A.O. Atag 2011. “On operations of soft sets”. Computers and Mathematics with Applications 61: 1457-1467. L.S. Shapley 1953. “A value for n-person games”. In: Kuhn H.W., Tucker A.W. (eds.), Contributions to the theory of games II, Princeton University Press, New Jersey, pp. 307-317. R.P. Stanley 1986. Enumerative Combinatorics Volume I. Wadsworth and Brooks/Cole Advanced Books and Software. Monterrey, California. C.F. Yang 2008. “A note on soft set theory”. Computers and Mathematics with Applications 56: 1899-1900. P. Zhu, Q. Wen 2013. “Operations on soft sets revisited”. Journal of Applied Mathematics 1-7. 18