On the Complexity of Computing the Myerson Value by Dividends

An algorithm describes a sequence of operations for solving a computational problem. There are often several algorithms for solving a problem and we are interested in analyzing the computational resources required to calculate the Myerson value. The complexity of a problem is the order of computational resources which are necessary and su¢cient to solve the problem. The algorithmic complexity is the cost of a particular algorithm. We say that a problem has polynomial complexity if its computational complexity is a polynomial in the measure of input size. We introduce new algorithmic procedures for computing the Myerson value by using dividends and we show that there exist problems with polynomial algorithm complexity. 1. Union stable systems We study cooperative games in which the cooperation among the players is partial. Several models of partial cooperation have been proposed, among which are those derived from communication situations as introduced by Myerson [11] and analyzed by Owen [13], and Borm, van den Nouweland and Tijs [4] [12]. We will give special attention to the union stable systems and we will study the complexity of the algorithm that, by means of the Harsanyi dividends [9], allows us to compute the Shapley value of the restricted game, i.e., the Myerson value. Some results on the complexity of computing the Myerson value will be provided in section 2. In section 3, we consider convex geometries introduced by Edelman and Jamison [5]. This concept gives rise to a special type of union stable structure and generalizes those communication situations in which the graph that models the bilateral relations among players is a tree. Algaba, Bilbao, Borm and López [1, 2] consider a partial cooperation model based on the so-called union stable systems, which is a generalization of the communication situations. Throughout this paper N denotes a ...nite set, and we use F μ 2 to denote the set system (N;F). De...nition 1.1. A set system F is called union stable if for all A;B 2 F such that A \B 6= ; it is satis...ed that A [B 2 F. Example 1.1. A communication situation is a triple (N; v;E), where (N; v) is a game and G = (N;E) is a graph. It is easy to see that the collection F = fS μ N : (S;E(S)) is a connected subgraph of Gg ; is a union stable system. E-mail address: [email protected]. 1 2 E. ALGABA, J.M. BILBAO, J.R. FERNÁNDEZ, N. JIMÉNEZ, AND J.J. LÓPEZ Example 1.2. Permission structures were de...ned by Gilles, Owen and van den Brink [8]. They assume that players who participate in a cooperative game restricted by a hierarchical organization in which there are players that need permission from certain other players before they are allowed to cooperate. Algaba et al. [1] showed that if the family A of subsets of N is derived from an disjunctive or conjunctive approach of an acyclic permission structure then A is a union stable system. Notice that a union stable system can not always be modelled by a communication graph. Let us consider N = f1; 2; 3; 4g and the collection F = f;; f1g; f2g; f3g; f4g; f1; 2; 3g; f2; 3; 4g; Ng : This set system is union stable, but does not coincide with the connected subgraph family of any graph. Example 1.3. Let N = f1; 2; : : : ; ng and consider the collection Fn of all the connected coalitions of the path 1¡ n, that is, Fn = f[i; j] : 1 · i · j · ng [ f;g ; where [i; j] = fi; i+ 1; : : : ; j ¡ 1; jg. Then Fn is a union stable system which corresponds to a voting situation in a unidimensional policy order (see Edelman [6]). De...nition 1.2. Consider F μ 2 and let S μ N . A set T μ S is called a Fcomponent of S if it is satis...ed that T 2 F and there exists no T 0 2 F such that T 1⁄2 T 0 μ S. The F-components of S are the maximal coalitions that belong to F and are contained in S. We denote by CF(S) the set of the F-components of S. Observe that the set CF(S) may be the empty set. Proposition 1.1. The set system F μ 2 is union stable if and only if for any S μ N such that CF(S) 6= ;, the F-components of S form a partition of a subset of S. Proof. Let F be a union stable system. Let S, S; S 6= S, be maximal feasible coalitions of S. If S \ S 6= ;, then S [ S 2 F since F is union stable and S [ S μ S. This contradicts the fact that S and S are F-components of S. Conversely, assume for any S such that CF(S) 6= ;, that its F-components form a partition of a subset of S. Suppose that F is not union stable, then there are A;B 2 F ; with A \B 6= ; and A [B = 2 F . Hence, there must be an F-component C1 2 CF(A[B), with A μ C1 and an F-component C2 2 CF(A[B), with B μ C2 such that C1 6= C2. This contradicts the fact that the F-components of A [B are disjoint. 2 Notice that if F is a union stable system such that fig 2 F for all i 2 N , then the F-components of S form a partition of S: De...nition 1.3. Let (N; v) be a game and let F μ 2 be a union stable system. The F-restricted game vF : 2 ! R; is de...ned by vF (S) := X T2CF(S) v(T ): A union stable structure is a triple (N; v;F) where (N; v) is a game and F μ 2 is a union stable system. COMPLEXITY OF COMPUTING THE MYERSON VALUE 3 De...nition 1.4. The Myerson value of a union stable structure (N;v;F) is given by the vector 1 (N; v;F) := © ¡N; vF¢, where © is the Shapley value: By ¡ we denote the set of all games (N; v): Given a union stable structure (N; v;F), the set of the unanimity games fuT : T 2 F ; T 6= ;g is a basis of the vector space LF ¡ ¡ ¢ ; where LF : ¡ ! ¡ ; is de...ned by LF(v) = vF (see Bilbao [3]). Then vF can be expressed as a linear combination of the unanimity games corresponding to the feasible coalitions, that is, vF = P T2F dvF (T )uT ; where dvF (T ) is the dividend of T in the game vF and dvF (;) = 0, [9] The linearity of the Shapley value implies that the Myerson value satis...es, for every i 2 N , 1i (N; v;F) = X fS2F : i2Sg dvF (S) jSj : Moreover, for every S 2 F ; we have v(S) = vF(S) = X fT2F:TμSg dvF (T ): From this expression we obtain the following recursive algorithm: dvF (;) = 0; dvF (S) = v(S)¡ X fT2F:T1⁄2Sg dvF (T ): The description of the above dividend algorithm is as follows: Algorithm dividend ¡ N; vF ¢ dvF (;)Ã