ON THE COMPLEXITY OF COMPUTING THE MYERSON VALUE BY

ON THE COMPLEXITY OF COMPUTING THE MYERSON VALUE BY DIVIDENDS E. ALGABA, J.M. BILBAO, J.R. FERNÁNDEZ, N. JIMÉNEZ, AND J.J. LÓPEZ Matemática Aplicada II, Escuela Superior de Ingenieros Camino de los Descubrimientos, 41092 Sevilla, Spain http://www.esi2.us.es/~mbilbao/sevigame.htm

Abstract. 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 2N 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

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E. ALGABA, J.M . BILBAO, J.R. FERNÁNDEZ, N. JIM ÉNEZ, 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; N g :

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 2N 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 ½ 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 2N 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 1 , S 2 ; S 1 6= S 2 , be maximal feasible coalitions of S. If S 1 \ S 2 6= ;, then S 1 [ S 2 2 F since F is union stable and S 1 [ S 2 µ S. This contradicts the fact that S 1 and S 2 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 2 disjoint. 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 2N be a union stable system. The F-restricted game vF : 2N ! R; is de…ned by X v F (S) := v(T ): T 2CF (S)

A union stable structure is a triple (N; v; F) where (N; v) is a game and F µ 2N is a union stable system.

COM PLEXITY OF COMPUTING THE MYERSON VALUE

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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 ¡N 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 ¡N ; where LF : ¡N ! ¡N ; is de…ned by LF (v) = v F (see Bilbao of the unanimity games [3]). Then vF can be expressed as a linear combinationP corresponding to the feasible coalitions, that is, vF = T 2F dv F (T ) uT ; where F dvF (T ) is the dividend of T in the game v and dvF (;) = 0, [9] The linearity of the Shapley value implies that the Myerson value satis…es, for every i 2 N , X dvF (S) : ¹i (N; v; F) = jSj fS2F: i2Sg

Moreover, for every S 2 F; we have

v(S) = vF (S) =

X

dvF (T ):

fT 2F:T µSg

From this expression we obtain the following recursive algorithm: dvF (;) = 0; dvF (S) = v(S) ¡

X

dvF (T ):

fT 2F:T ½Sg

The description of the above dividend algorithm is as follows: ¢ ¡ Algorithm dividend N; vF dvF (;) à 0 2 for i from 1 to n 6 6 2 6 for j from 1 to S (i) 6 6 6 ³ ´ ³ ´ P 6 6 j j 6 6 dvF Si à v Si ¡ fT 2F:T ½S j g dvF (T ) i 6 4 6 6 end for 6 4 end for

where Sij is the j-th feasible coalition of size i and S (i) is the number of feasible coalitions of cardinal i: 2. Algorithm complexity The time complexity function f : Z+ ! Z+ of an algorithm A is the maximal number f (n) of iterations of a universal Turing machine makes before halting, taken over all inputs of size n: We say that an algorithm has space complexity at most f (n); if it can be computed by a Turing machine with space demand (cells and tapes) at most f(n): Let f and g be functions from Z+ to Z+ : We write f (n) = O(g (n)) ; in words f is of the order of g, if there are positive integers c and n0 such that f (n) · cg (n) for all n ¸ n0 : We write f (n) = ­ (g (n)) if the opposite happens, that

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E. ALGABA, J.M . BILBAO, J.R. FERNÁNDEZ, N. JIM ÉNEZ, AND J.J. LÓPEZ

is, g (n) = O(f (n)) : If f and g have exactly the same rate of growth, then we write f (n)¡ =¢ £ (g (n)) : For instance, if p (n) is a polynomial of degree d; then p (n) = £ nd : The above O­£-notation was proposed by Knuth [10]. For a more detailed exposition, see the book of Gács and Lovász [7]. We have the following result for spatial and temporal complexity of the dividend algorithm. Theorem 2.1. Let (N; v; F) be a union stable structure. To compute all dividends of the restricted game requires a space ­ (jFj) and a time O(3n ) : Proof. First of all, notice that it is su¢ces to calculate dividends of the feasible coalitions and that the¡ number of feasible coalitions of size i; denoted by S (i), ¢ satis…es that S (i) · ni for i = 1; : : : ; n: The execution time of the dividend algorithm satis…es t(dividend) = 1 + t(loop1) = 1 +

n X t(loop2) i=1

= 1+

n S(i) X X

t(assignment) = 1 +

i=1 j=1

· 1+

n S(i) X X i=1 j=1

0

@1 +

i

2X ¡1 k=1

1

1A = 1 +

n S(i) X X (1 + t(sum)) i=1 j=1

n S(i) X X (1 + 2i ¡ 1) i=1 j=1

n S(i) n n µ ¶ X X X X n i i i = 1+ 2 =1+ S (i) 2 · 1 + 2 i i=1 j=1 i=1 i=1 n µ ¶ X n i = 2 = 3n : i i=0

Therefore, dividend has a time O(3n ): On the other hand, if it is taken into account that the computation of the dividends is by an ascending process which requires to keep the dividends of each one of the feasible coalitions, it is obtained ¤ that the required space is ­ (jFj) : Theorem 2.2. Let (N; v; F) be a union stable structure, v a zero-normalized game and let p = jCF (N )j. Then we have: a) To compute the Myerson value for a¡ player ¢ i, by the dividend algorithm, requires a space ­ (jFM j) and a time O 3jMj ; where M 2 CF (N) and i 2 M: b) To compute the Myerson value for all by the dividend algorithm, ¡ players, ¢ maxfjMj:M2CF (N)g : requires a space ­ (jFj) and a time O p ¢ 3

Proof. a) If the coalition N is not feasible and v is zero-normalized, the calculation of the Myerson value, for player i; it can be done through the maximal feasible coalition of N which the player i belongs to. This is, if M 2 CF (N ) such that i 2 M, then (M; vM ; FM ) is a union stable structure, where vM is the restriction of v to M and FM = fF 2 F : F µ M g, and it holds ¹i (N; v; F) = ¹i (M; vM ; FM ) : b) For computing the Myerson value of all players, it would have to establish two stages. In the …rst one, using dynamic programming, the dividends of all feasible coalitions are determined, for each F-component of the coalition N by the following scheme: let CF (N ) = fM1 ; : : : ; Mp g, then

COM PLEXITY OF COMPUTING THE MYERSON VALUE

5

dvF (;) Ã 0 2 for j from 1 to p 6 ³ ´ 6 6 dividend Mj ; vF Mj 6 6 6 F (S); for all S 2 FMj g {To compute dvM 6 j o n 6 6 F F (S) = d (S) d v vM 6 j 4 end for

In the second stage, the Myerson value is determined when all dividends are known. The algorithm is as follows 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

for j from 1 to p 2

for i from 1 to jMj j 6 6 dvF (S) 6 j P 6 ¹i (N; v; F) Ã fS2F :i2Sg Mj Mj 6 jSj n ³ ´o 6 j 6 (N; v; F) = ¹ ; v ¹ M 6 j Mj ; FMj i i 4 end for end for

Therefore, t(Myerson) = f(n) + g(n), where f (n) and g(n) are the functions that respectively indicate the calculation time in the two established stages, i.e., the corresponding time to determinate the dividends and, once that dividends are known, the calculation time of the Myerson value for each one of the players. Indeed, ¢¢ ¡ ¡ F · p 3jMj ; f(n) = t(loop) · p t dividend M; vM

where M 2 CF (N ) and satis…es that jM j = maxfjMj j ; 1 · j · pg. In the same way, we have g(n) = t(loop1) = p t(loop2) · pjM jt(sum) · pjM j2jMj :

Therefore,

³ n o´ O(t(M yerson)) = O(f(n) + g(n)) = O max p 3jMj ; pjMj2jMj :

Finally, since

jM j2jMj = 0; jMj!1 3jMj lim

we¡ have©that the computation of¡ the Myerson value for all players requires a time ª¢ ¢ O max p3jMj ; pjM j2jMj = O p 3jMj : Since all dividends have to be obtained ¤ the required space is ­ (jFj). Next, we analyze the complexity for the computation of the Myerson value in the communication situation in which the cooperation graph has a speci…c structure.

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E. ALGABA, J.M . BILBAO, J.R. FERNÁNDEZ, N. JIM ÉNEZ, AND J.J. LÓPEZ

Let (N; v; G), where G is a path 1 ¡ n with n vertices and (N; v) a game with transferable utility. If the vertices are enumerated from left to right, the set F of feasible coalitions is given by F = f[i; j] : 1 · i · j · ng [ f;g;

where [i; j] = fi; i + 1; : : : ; j ¡ 1; jg: Notice that except the empty set, the feasible coalitions can be arranged as entries on a matrix A 2 Mn (R); upper triangular, such that the unitary coalitions are kept on the diagonal. On the row k; k = 1; : : : ; n, the coalitions of the set f[k; j] : k · j · ng are arranged in an orderly way. The player k belongs to all coalitions of the submatrix (aij ); for i = 1; : : : ; k and j = k; : : : ; n. Hence, the total number of coalitions which contain player k is Fk = k(n ¡ k + 1): It can also be observed that on the i-th superdiagonal would be all feasible coalitions with i players. On this, there is a number of elements equal to (n ¡ i + 1) and, therefore, the number of feasible coalitions with i players is given by S(i) = n ¡ i + 1: Moreover, the total number of nonempty feasible coalitions is n n X X n2 + n : S(i) = (n ¡ i + 1) = jFj = 2 i=1 i=1 Example 2.1. Let us consider the path 1 ¡ n with n = 5 vertices. Then the collection of feasible coalitions is F = f[i; j] : 1 · i · j · 5g [ f;g:

The nonempty coalitions can be kept as entries of the following upper triangular matrix: 0 1 1 12 123 1234 12345 B 2 23 234 2345 C B C B 3 34 345 C B C @ 4 45 A 5 Note that the number of feasible coalitions with i players is

S(i) = 5 ¡ i + 1 = 6 ¡ i; for i = 1; : : : ; 5:

In the same way, the number of coalitions which a player k belongs to is given by Fk = k(5 ¡ k + 1) = k(6 ¡ k) for k = 1; : : : ; 5: Theorem 2.3. Let (N; v; G) be a communication situation, where G is a path 1¡n. To compute all dividends of the restricted game requires a space ­(n2 ) and a time £(n4 ): Proof. For computing the dividends is used the dividend algorithm as above. Now then, in this particular case, the number of feasible coalitions of size i is given by S(i) = n ¡ i + 1; and the total number of nonempty feasible coalitions contained in a feasible coalition T with jT j = i is given by i X i2 + i : (i ¡ k + 1) = 2

k=1

Note that if the coalition T is excluded and the empty coalition is included, then the¡ total¢ number of feasible coalitions strictly contained in T is given by C (i) = i2 + i /2 :

COM PLEXITY OF COMPUTING THE MYERSON VALUE

7

Therefore, the execution time of the algorithm is t(dividend) = 1 + t(loop1) = 1 +

n X t(loop2) i=1

n S(i) n S(i) X X X X = 1+ t(assignment) = 1 + (1 + t(sum)) i=1 j=1

= 1+

n S(i) X X i=1 j=1

0

@1 +

C(i)

X

k=1

1

1A = 1 +

i=1 j=1

n S(i) X X

(1 + C(i))

i=1 j=1

¶ n µ X i2 + i 1+ S(i) 2 i=1 ¶ n µ 2 X i +i+2 = 1+ (n ¡ i + 1) 2 i=1

= 1+

= 1+ =

n ¢ 1X ¡ 2 i + i + 2 (n ¡ i + 1) 2 i=1

n4 + 6n3 + 23n2 + 18n + 24 : 24

Thus, the time t(dividend) = £(n4 ): The recursive process requires to keep the dividends ¡ 2 ¢ corresponding to all the feasible nonempty coalitions whose number is n + n /2 . Therefore, the required space is ­(n2 ): ¤

From Myerson value expression in terms of the dividends of the restricted game, the following result is deduced, using dynamic programming.

Theorem 2.4. Let (N; v; G) be a communication situation, where G is a¡path ¢ 1¡n. 2 ­ n To compute the Myerson value by dividend algorithm requires a space and a ¡ ¢ time £ n4 :

Proof. First of all, all dividends are calculated, using the dividend algorithm. Next, for computing the Myerson value of player k; it is necessary to consider that this player belongs to Fk = k(n ¡ k + 1) feasible coalitions. Since 1 Fk = (n + 1)2 ¡ 4

µ

1 (n + 1) ¡ k 2

¶2

;

we obtain, for any value of k, 1 n · Fk · (n + 1)2 : 4 Then, we have that ¡ the ¢ required time to evaluate the sum in the expression of the Myerson value is O n2 , for each player. Therefore, the complexity for computing the Myerson value is determined ¡ ¢ by the complexity for computing the dividends. So, the required time is £ n4 : The required space ­(n2 ) is an immediately con¤ sequence of Theorem 2.3.

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E. ALGABA, J.M . BILBAO, J.R. FERNÁNDEZ, N. JIM ÉNEZ, AND J.J. LÓPEZ

3. Restricted games by convex geometries A convex geometry is a set system F µ 2N which satis…es the following properties: (C1) ; 2 F, (C2) If A 2 F and B 2 F, then A \ B 2 F; (C3) If A 2 F and A 6= N; there exists i 2 N n A such that A [ i 2 F.

The elements of a convex geometry are called convex sets. Given a convex set S 2 F, an element i 2 S is an extreme point of S if S n i 2 F. The set of extreme points of S is denoted by ex(S) and S ¡ = S n ex(S): If F is a convex geometry and S 2 F, then the interval [S ¡ ; S] = fT 2 F : S ¡ µ T µ Sg is a Boolean algebra in the lattice (F; [; \) (for a survey see Edelman and Jamison [5]). If F is a convex geometry such that fig 2 F, for all i 2 N; and it is also a [-stable system, then F is called a partition convex geometry. The interest in the study of convex geometries comes from the search of combinatorial structures that generalize the obtained results with other structures of cooperation used in theory of games. As we have indicated in the preceding section, the dividends of the restricted game are determinant for computing the Myerson value. For any set system F, we can obtain the dividends with the following formula X (¡1)jSj¡jT j vF (T ): dvF (S) = T µS

However, we are interested in studying conditions in which the Myerson value can be calculated in terms of the dividends in the game (N; v); and if it is possible, in terms of its characteristic function. The next theorem (see Bilbao [3]) generalizes the result obtained by Owen [13] who studies the computation of the dividends of the restricted game by a communication situation (N; v; G) where G is a tree. Theorem 3.1. Let (N; v; F), where F µ 2N is a partition convex geometry and (N; v) is a game. If S 2 F, then X (¡1)jSj¡jT j v(T ): dvF (S) = fT :Snex(S)µT µSg

Proof. The collection fuT : T 2 F; T 6= ;g is a basis of the space of the restricted games by F. Thus, for every S 2 F, X dvF (T ): v(S) = vF (S) = fT 2F:T µSg

If we consider the functions v; dvF : F ! R; and we apply the Möbius inversion function [14] to the partially ordered set (F; µ), the dividends of vF can be expressed in terms of the values of the game v. Indeed, for all S 2 F, we have X X dvF (T ) () dvF (S) = v(T )¹(T; S): v(S) = fT 2F:T µSg

fT 2F:T µSg

The Möbius function in the case of a convex geometry [5, Theorem 4.3] is given by ½ (¡1)jSj¡jT j ; if S n T µ ex(S) ¹(T; S) = 0; otherwise,

COM PLEXITY OF COMPUTING THE MYERSON VALUE

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and for T µ S, we have that S n T µ ex(S) if and only if S n ex(S) µ T: Therefore, dvF (S) =

X

(¡1)jSj¡jT j v(T ):

fT :Snex(S)µT µSg

2 Note that if F = 2N , then all elements of every coalition are extreme points. That is, for every S 2 2N ; we have that S n ex(S) = ;; and as vF = v, for any S 2 2N ; it holds, dv (S) = dvF (S) =

X

(¡1)jSj¡jT j v(T );

T µS

and it is the expression of the dividends. Thus, the Myerson value of a restricted game by a partition convex geometry is given by

¹i (N; v; F) =

X

fS2F:i2Sg

2 1 4 jSj

X

3

(¡1)jSj¡jT j v(T )5 ;

fT :Snex(S)µT µSg

for all i 2 N: The last result establishes a new algorithm which is called dividend2 that will permit to compute the dividends of the restricted game. The description of the algorithm is as follow. Algorithm dividend2 dvF (;) à 0 2 For i from 1 to n 6 6 2 6 For j from 1 to S(i) 6 6 6 ³ ´ j P 6 6 j 6 6 dvF Si à fT :S j nex(S j )µT µS j g (¡1)jSi j¡jT j v(T ) i i i 6 4 6 6 end for 6 4 end for

¯ ¯ ¯ ¯ where Sij is the j-th coalition such that ¯Sij ¯ = i and S (i) is the number of feasible coalitions of cardinal i:

We make use of the dividend2 algorithm for computing the Myerson value in restricted games by partition convex geometries. The dividends are calculated by means of an ascending procedure which requires a previous storage of the extreme points of each feasible coalition in a table.

Theorem 3.2. Let (N; v; F) be a union stable structure, where (N; F) is a partition convex geometry and (N; v) is a game. To compute the dividends in the restricted ¢game, with the dividend2 algorithm, requires a space ­ (jFj) and a time ¡ O 2D jFj ; where D = max fjex(S)j : S 2 Fg :

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Proof. Because of description of the dividend2 algorithm, we deduce that the execution time is n X t(loop2) t(dividend2) = 1 + t(loop1) = 1 + i=1

= 1+

n S(i) X X i=1 j=1

= 1+

S(i) n X X t(assignment) = 1 + t(sum) i=1 j=1

n S(i) X X i=1 j=1

= 1 + 2D+1

j

2 ¢ 2jex(Si )j · 1 +

n S(i) X X i=1 j=1

n S(i) X X i=1 j=1

2 ¢ 2D

1 < 1 + 2D+1 jFj :

¡ ¢ Therefore, t(dividend2) = O 2D jFj : Furthermore, if we suppose that the dividends are calculated by an ascending procedure, then we obtain that the space ¤ complexity is ­ (jFj). Theorem 3.3. Let (N; v; F) be a union stable structure, where (N; F) is a partition convex geometry and (N; v) is a game: ¡To compute Myerson ª¢ value by div© idend2 requires a space ­(jFj) and a time O max n jFj ; 2D jFj ; where D = max fjex(S)j : S 2 Fg : Proof. First of all, we calculate all dividends ¡ of the ¢ feasible coalitions with the dividend2 algorithm. It requires a time O 2D jFj . For computing the Myerson value for the player i; X dvF (S) ; ¹i (N; v; G) = jSj fS2F:i2Sg

we have that each player belongs to Fi coalitions where Fi < jFj. The required time to evaluate the sum for every player ¡ is©O(jFj) : Thus,ª¢to compute the Myerson ¤ value for n players requires a time O max n jFj ; 2D jFj : Finally, we analyze the complexity in the family of all connected coalitions of the path 1 ¡ n; which is a partition convex geometry. The purpose of studying, again, this family of feasible coalitions is to make clear the importance of using the formula of the above theorem, to obtain a signi…cant reduction in the computation time of the dividends and the computation time of the Myerson value. Notice that the number of extreme players of each feasible coalition S is given by ½ 2; if jSj ¸ 2 jex (S)j = 1; if jSj = 1: Theorem 3.4. Let (N; v; G) be a communication situation, where G ¡ is¢a path 1¡n. To compute all dividends in the restricted game requires a space ­ n2 and a time £(n2 ): Proof. For computing the dividends, we use the dividend2 algorithm and only consider the feasible coalitions. The number of coalitions with i players is given by

COM PLEXITY OF COMPUTING THE MYERSON VALUE

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S(i) = n ¡ i + 1: Thus, the execution time of the algorithm is given by n X t(dividend2) = 1 + t(loop1) = 1 + t(loop2) i=1

S(i) n S(i) n X X X X = 1+ t(assignment) = 1 + t(sum) i=1 j=1

i=1 j=1

S(1)

= 1+

X

t(sum) +

j=1

X j=1

2

t(sum)

i=2 j=1

S(1)

= 1+

S(i) n X X

2 ¢ 21 +

= 1 + 4n :

n S(i) X X 2 ¢ 22 i=2 j=1

¡ ¢ 2 ): Since jFj = n2 + n /2 ; Theorem 3.2 implies Therefore, t(dividend2) =¡£(n ¢ ¤ that the required space is ­ n2 . Theorem 3.5. Let (N; v; G) be a communication situation, where G ¡ is¢a path 1¡n. To compute the Myerson value using dividend2 requires a space ­ n2 and a time O(n3 ):

Proof. ¡If G is ¢the path 1 ¡ n; then the family F of feasible coalitions satis…es that jFj = n2 + n /2 and D = max fjex (S)j : S 2 Fg = 2: Theorem that ¡ ©3.3 implies ª¢ to D max n jFj ; 2 jFj = n compute the Myerson value for players requires a time O ¡ ¢ ¡ ¢ ¤ O n3 and a space ­ n2 . References

[1] Algaba, E., J.M. Bilbao, P. Borm and J.J. López (1999): The Myerson value for union stable systems, Research Memorandum FEW 773, Tilburg University, The Netherlands. [2] Algaba, E., J.M. Bilbao, P. Borm and J.J. López (2000): The position value for union stable systems, Math. Meth. Oper. Res. 52 (2), 221-236. [3] Bilbao, J.M. (1998): Values and potential of games with cooperation structure, Int. J. Game Theory 27, 131–145. [4] Borm, P., Owen, G. and S.H. Tijs (1992): On the position value for communication situations, SIAM J. Disc. Math. 5, 305–320. [5] Edelman, P.H. and R.E. Jamison (1985): The theory of convex geometries, Geom. Dedicata 19, 247–270. [6] Edelman P.H. (1997): A note on voting, Math. Social Sciences 34, 37–50. [7] Gács, P. and L. Lovász (1999): Complexity of Algorithms, Lecture Notes, Yale University, available at http://www.esi2.us.es/~mbilbao/pd¢les/complex.pdf [8] Gilles, R.P., Owen, G. and R van den Brink (1992): Games with permission structures: the conjunctive approach, Int. J. Game Theory 20, 277–293. [9] Harsanyi, J.C. (1963): A simpli…ed bargaining model for the n-person cooperative game, Int. Economic Review 4, 194–220. [10] Knuth, D.E. (1976): Big omicron and big omega and big theta, ACM SIGACT News 8, 18–24. [11] Myerson, R.B. (1977): Graphs and cooperation in games, Math. Oper. Res. 2, 225–229. [12] Nouweland, A. van den, P. Borm and S.H. Tijs (1992): Allocation rules for hypergraph communication situations, Int. J. Game Theory 20, 255–268. [13] Owen, G. (1986): Values of graph-restricted games, SIAM J. Algebraic and Discrete Methods 7, 210–220. [14] Stanley, R.P. (1986): Enumerative Combinatorics, Vol. I, Wadsworth, Monterey, California.