Annals of Operations Research 109, 331–342, 2002 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
Reliability Importance Measures of the Components in a System Based on Semivalues and Probabilistic Values JOSEP FREIXAS and M. ALBINA PUENTE {josep.freixas, M.Albina.Puente}@upc.es Department of Applied Mathematics III and Polytechnic School of Manresa, Polytechnic University of Catalonia, Spain
Abstract. The main contribution of this paper consists in providing different ways to value importance measures for components in a given reliability system or in an electronic circuit. The main tool used is a certain type of semivalues and probabilistic values. One of the results given here extends the indices given by Birnbaum [3] and Barlow and Proschan [2], which respectively coincide with the Banzhaf [1] and the Shapley and Shubik [15] indices so well-known in game theory. Keywords: simple games, reliability, semivalues, probabilistic values, reliability importance measures AMS subject classification: 91A12, 91A40, 91A80
1.
Introduction
Few structures in mathematics arise in more contexts and lend themselves to more diverse interpretations than do simple games. In this work we deal with material based on the concept of power index, extensively developed in simple games, applied to the reliability field or circuit theory, without undervaluation other fields like threshold logic and Boolean algebra. A survey showing the connections of simple games with other mathematical subjects and a bibliography is given in Hilliard [10], an excellent introduction is also provided in Shapley [14] and connections, both mathematical and historical, between the analysis of simple games and the study of reliability theory are covered in Ramamurthy [13]. A classical problem in reliability consists in controlling the functioning of a mechanism formed by several components each one of them working independently and having a running distribution for each period of time t. In order to get an efficient control of the system it is necessary to measure the relative importance that each component has in it. This importance depends on two factors: the allocation the component has in the system (first factor) and its proper reliability (second factor). In this paper we introduce different components’ measures that take into account both factors: we will assume the allocation of each component is known and, that in a given period of time, we have complete information about the reliability of the components. That is, we know the probabilistic model that each component follows.
332
FREIXAS AND PUENTE
Birnbaum [3] and Barlow and Proschan [2] considered the problem of a priori quantification of relative importance of components of reliability systems. They referred to this as the problem of measurement of structural importance of components. In fact, Birnbaum rediscovered the Banzhaf index and Barlow and Proschan rediscovered the Shapley–Shubik index. These two measures fit in our framework. The paper is made up of four sections. Section 2 is devoted to explain the main definitions that permit the pursuit of the work, summing up the basic analogies between game theory and reliability. In section 3 binomial semivalues and multi-binary probabilistic values are introduced, their calculus can be done computing a partial derivative of the multilinear extension of the game. These kind of solutions serve us to the following study of the relative importance of the components in a given system from a probabilistic point of view. We present four models to measure this importance. The results by Birnbaum and Barlow and Proschan are extended using binomial semivalues and multi-binary probabilistic values. Finally, in section 4, some hints for future research are indicated. 2.
Background
2.1. Cooperative and simple games Let N = {1, 2, . . . , n} be the set of players and 2N be the set of coalitions (subsets of N). A cooperative game on N is a pair (N, v), where v : 2N → R such that v(∅) = 0 is the characteristic function. The game v is monotonic if v(S) v(T ) whenever S ⊂ T . A player i ∈ N is a dummy in a game v if v(S ∪ {i}) = v(S) + v({i}) for all S ⊆ N − {i}. Games on N form a real vector space GN endowed with the usual linear operations, given by (u + v)(S) = u(S) + v(S) and (λv)(S) = λv(S) for all S ⊆ N, u, v ∈ GN and λ ∈ R. The unanimity game for T ⊆ N is the game uT such that uT (S) = 1 if and only if S ⊇ T , and 0 otherwise. We can also interpret 2N as the set of all vectors (x1 , x2 , . . . , xn ) whose components are either 0 or 1. Thus 2N = {0, 1}N . Finally, every permutation θ of N induces a linear automorphism of GN given by (θv)(S) = v(θ −1 S) for all S ⊆ N and all v. A value on N is a map : GN → R n , that assigns to every game v a vector [v] with components i [v] for all i ∈ N. Following Weber [16], is a semivalue if it satifies the following properties: (i) linearity: [v + v ] = [v] + [v ] and [λv] = λ[v] for all v, v , and λ; (ii) symmetry: θi [θv] = i [v] for all θ, i and v; (iii) positivity: if v is monotonic, then [v] 0; (iv) dummy player: if i is a dummy in game v, then i [v] = v({i}). For an equivalent definition of semivalue and the following characterization of semivalues by means of weighting coefficients, which will be intensively used here, see
RELIABILITY IMPORTANCE MEASURES
333
Dubey et al. [8]. Set n =| N |. Then: n−1 n−1 = 1 and pk 0 for every k, the expression (a) for every {pk }n−1 k=0 pk k k=0 such that ps v S ∪ {i} − v(S) , ∀i ∈ N, ∀v ∈ GN s = |S| , i [v] = S⊆N−{i}
defines a semivalue on GN ; (b) conversely, every semivalue can be obtained in this way; (c) the correspondence given by {pk } → is bijective. So, the payoff that a semivalue allocates to any player in any game is a weighted sum of his/her marginal contributions in the game. Semivalues provide a wide family of solutions including the Banzhaf value and having the Shapley value as the only efficient member. In Weber [16], probabilistic values are defined from linearity, positivity and the dummy player property. Semivalues are characterized among the set of all probabilistic values by symmetry and, finally, the Shapley value is the semivalue characterized by efficiency. The expression (see Weber [16]) pSi v S ∪ {i} − v(S) , (2.1) ϕi [v] = S⊆N−{i}
where for any player i ∈ N, {pSi : S ⊆ N − {i}} is a collection of constants satisfying i i S⊆N−{i} pS = 1 and pS 0, defines a probabilistic value on GN . Let i view his participation in a game as consisting merely of joining some coalition S, and then receiving as a reward his marginal contribution to the coalition. If pSi is the probability that he/she joins coalition S, then ϕi [v] is simply his expected payoff from the game. Owen [11] introduced the multilinear extension (abbreviated MLE) of a cooperative game (N, v) defined by xi (1 − xi ) v(S) (2.2) f (x1 , x2 , . . . , xn ) = S⊆N
i∈S
i ∈S /
for 0 xi 1, i = 1, . . . , n, and he obtained the Shapley value and the Banzhaf value [12]:
1 1 ∂f 1 ∂f ,..., . (t, . . . , t) dt, βi [v] = i [v] = ∂xi 2 2 0 ∂xi A game v on a given player set N is simple if v(S) = 0 or v(S) = 1 for all S ⊆ N. We will suppose that v is not equal to zero. In this case, the game is completely determined by its set of winning coalitions W = {S ⊆ N: v(S) = 1}. The set SN of all monotonic simple games on N becomes a lattice under the standard composition laws given by
(v ∧ v )(S) = min v(S), v (S) and (v ∨ v )(S) = max v(S), v (S) .
334
FREIXAS AND PUENTE
A power index on N is a map : SN → R n , that assigns to every monotonic simple game v a vector [v] with components i [v] for all i ∈ N. We shall say that a power index satisfies the transfer property if [v ∨ v ] = [v] + [v ] − [v ∧ v ]
for all v, v ∈ SN .
This property was introduced by Dubey [7] to replace additivity in his characterization of the Shapley–Shubik [15] power index, and used by Dubey and Shapley [9] to axiomatize the Banzhaf–Coleman power index. Carreras et al. [6] axiomatize semivalues for monotonic simple games replacing linearity by the transfer property and provide their definition in terms of weighting coefficients. 2.2. Structure function. The reliability function A Boolean function of n variables is a function on {0, 1}n taking values in {0, 1}. We shall now see that Boolean functions arise in a natural way in reliability theory. A system or structure is assumed to consist of n components and, without loss of generality, we use N = {1, 2, . . . , n} to denote the set of components. We consider the state of the system at a fixed moment of time. The state of the system is assumed to depend only on the states of the components. We shall distinguish between only two states – a functioning state and a failing state. This dichotomy applies to the structure as well as to each component. To indicate the state of the ith component, we assign a binary indicator variable xi to component i and define 1 xi = 0
if component i is functioning, if component i is failing.
Similarly the binary variable y indicates the state of the structure, that is, 1 y= 0
if the structure is functioning, if the structure is failing.
The assumption that the state of the system is completely determined by the states of its components implies the existence of a Boolean function f : {0, 1}n → {0, 1} such that y = f (x) where x = (x1 , . . . , xn ). We will suppose that f is not equal to zero. In the terminology of reliability theory, the function f is called a structure function. A structure function f such that f (0) = 0 is analogous to the characteristic function of a simple game used in game theory. Let f be a structure on N. We say f is monotonic if x, y ∈ {0, 1}n and x y imply f (x) f (y). A monotonic structure f is called semicoherent. Notice the parallelism between a semicoherent structure such that f (0) = 0 and a monotonic simple game. From now on, we suppose that at a given instant of time t, component i has the qi = probability p i of being in the functioning state and the complementary probability
RELIABILITY IMPORTANCE MEASURES
335
Table 1 Game theory
Reliability
player coalition winning coalition losing coalition simple game (v) monotonic simple game (v) multilinear extension (f ) Shapley–Shubik index ([v]) Banzhaf index (β[v])
component characteristic vector true vector false vector structure function (f ), f (0) = 0 semicoherent structure (f ), f (0) = 0 reliability function (fˆ) Barlow–Proschan index ([f ]) Birnbaum index (β[f ])
1−p i of being in the failing state. Let Xi (t) be the random variable representing the state of the component i, that is, 1 if the component i is in functioning state, Xi (t) = 0 if the component i is in failing state, and also let X(t) = (X1 (t), X2 (t), . . . , Xn (t)). To simplify the notation we remove “t” and write Xi (t) as Xi and X(t) as X. We also assume that the components are (stochastically) independent, that is, the random variables X1 , X2 , . . . , Xn are independently distributed. The random variable f (X) represents the state of the system at the given instant of time t, that is, 1 if the system is in functioning state, f (X) = 0 if the system is in failing state. The reliability function of a structure f on N with independent components is the function fˆ : [0, 1]n → [0, 1] defined by
ˆ p j (1 − p j ) , f (p) = Prob f (X) = 1 = S∈W
j ∈S
j ∈S /
2 , . . . , p n ) and the collection W is induced by the structure function f . where p = ( p1 , p In game theory, fˆ is referred to as Owen’s MLE of the characteristic function of a game. In table 1 we summarize the main analogies between game theory and reliability. 3.
Relative importance of components in a system
In this section we focus on the study of the relative importance of components in a system. We confine our analysis to the case where the components and hence the system cannot be repaired. In attempting to achieve high reliability of a system, a basic problem is that of evaluating the relative importance of components. There are two possible approaches for the quantification of relative importance (power) of components (players) –
336
FREIXAS AND PUENTE
the axiomatic and the probabilistic approaches. In an axiomatic approach, the reasonable properties that a measure of importance is expected to satisfy are first stated as axioms. Then an attempt is made to find the measures that satisfy the axioms, if existing at all. In the probabilistic approach, a stochastic description of the given reliability structure (voting game) is first visualized. The relative importance of a component (player) is then taken as the probability that functioning or failure of a component (a vote of yes or no for a player) makes a difference in the functioning or failure of the system (in the acceptance or rejection of the proposal). In this context we raise the question: What is the probability that the system works with the component i in position “ON” and does not work with component i in position “OFF”? The same question is encountered in game theory: what is the probability that the vote of player i will affect the outcome of the vote on a bill? In other words, what is the probability that a bill will pass if i votes for it, but fail if i votes against it? To answer the first question we need to specify a probability model for p k , k = 1, . . . , n, where n is the number of components and 0 p k 1 is the probability that component k works. There are many ways in which this could be done, but at a level of abstraction suitable for power indices at least four possible assumptions have a claim to naturalness. Deterministic homogeneity assumption: All the components have the same (known) probability of functioning state. That is, p k = p for all k, where 0 p 1 is the (known) probability that the components work. Deterministic independent assumption: The probability of every component being in the functioning state is known in a fixed interval of time. That is, p k = p k for all k, where 0 p k 1 is the (known) probability that the k-th component works. General homogeneity assumption: A number p is chosen from the uniform distribution on I = [a, b] ⊆ [0, 1] and p k = p for all k. General independence assumption: Each p k is chosen independently from the uniform distribution on Ik = [ak , bk ] ⊆ [0, 1]. Notice that the first and the third case, where the components have the same probability of functioning, may be suitable for systems of physically identical components. However, the second and the fourth case, where every component has a different probability of functioning, are suitable to study systems of physically different components. In the two first cases the functioning probabilities of the components are known in a fixed interval of time. In the third and the fourth case we have information about the interval that contains the functioning probabilities of components. Under the first two assumptions we are going to see how the “binomial semivalues” and the “multi-binary probabilistic values”, we are going to introduce, answer the initial question. The two following propositions apply to cooperative games, whereas in the main theorem we use their restrictions to semicoherent structures (monotonic simple games), understanding that a game v has f as MLE and a semicoherent structure function f has fˆ as reliability function.
RELIABILITY IMPORTANCE MEASURES
337
Proposition 3.1. Let 0 p j 1 and q j = 1 − pj , j = 1, . . . , n. Then the coefficients pSi = j ∈S p j j =i, q j , for S ⊆ N − {i}, define a probabilistic value, which will be j ∈S /
called multibinary probabilistic value ϕ i . Proof. We have to prove that pSi 0, for S ⊆ N − {i} and S⊆N−{i} pSi = 1. The first part is immediate from the definition. For the second part, if n = 1 we have to take p∅1 = 1. For n > 1 we are going to prove that i pS = pj qj = (p j + q j ) = 1. S⊆N−{i}
j ∈S
S⊆N−{i}
j =i
j ∈S / j =i
We will proceed by induction on the number of players n. If n = 2, for i = j we have i i i p∅i = q j and p{j } = p j , then p∅ + p{j } = q j + p j = 1. Assume that the property is true for a certain integer n − 1, we shall see that it is also true for n. Let l = i. pSi = pj qj S⊆N−{i}
S⊆N−{i}
=
j ∈S
S⊆N−{i,l}
j ∈S / j =i
pj
j ∈S
qj
(pl + q l ) =
(pj + q j ) = 1.
j =i
j ∈S / j =i, j =l
In particular, if p j = p wherein 0 p 1 for each j , the coefficients ps = p s (1 − p)n−1−s for s = 0, 1, . . . , n − 1, define a semivalue p , which will be called binomial semivalue. Proposition 3.2 extends Carreras and Freixas’ calculus (see [4, pp. 150–151]) for binomial semivalues. Proposition 3.2.Let ϕ be a multi-binary probabilistic value defined by coefficients pSi = j ∈S p j j =i q j , for S ⊆ N − {i}, where q j = 1 − p j and f is the MLE of j∈ /S
a cooperative game v, then ϕ i [v] = Proof.
∂f (p 1 , p 2 , . . . , p n ) ∂xi
for all i in N.
The ith partial derivative of the multilinear extension f given in (2.2), is ∂f (x1 , x2 , . . . , xn ) = xj (1 − xj ) v S ∪ {i} − v(S) . ∂xi S⊆N−{i} j ∈S j =i j ∈S /
Note now that the Weber characterization of probabilistic values, given in (2.1), implies that ∂f ϕ i [v] = pj (1 − pj ) v(S ∪ {i}) − v(S) = (p1 , p 2 , . . . , p n ). ∂x i S⊆N−{i} j ∈S j =i j ∈S /
338
FREIXAS AND PUENTE
As a consequence any binomial semivalue, p [v], may be calculated as where f is the MLE of a cooperative game v, for each i ∈ N. The following theorem answers the question formulated at the beginning under the four assumptions. ∂f (p, p, . . . , p), ∂xi
Theorem 3.3. Let MiD.H , MiD.I , MiG.H and MiG.I be respectively the relative importance measures under the deterministic homogeneity assumption, the deterministic independence assumption, the general homogeneity assumption and the general independence assumption for component i ∈ N. Then, (1) The answer to the question under the deterministic homogeneity assumption is given p by component i s binomial semivalue MiD.H = i (f ). (2) The answer to the question under the deterministic independent assumption is given by component i s multi-binary probabilistic value MiD.I = ϕ i (f ). (3) The answer to the question under the general homogeneity assumption is given by component i s
b ˆ ∂f G.H = (p, p, . . . , p) dp. Mi i a ∂p (4) The answer to the question under the general independence assumption is given by component i s n an + bn ∂ fˆ a1 + b1 G.I ,..., . (bj − aj ) Mi = ∂p i 2 2 j =1 Proof. The probability that the system works with component i in state “ON” and does not work with component i in state “OFF” is ∂ fˆ p j (1 − p j ) = ( p1 , p 2 , . . . , p n ). ∂ p i j ∈S S∪{i}∈W, S ∈ /W S⊆N−{i}
j =i j∈ /S
2 , . . . , p n does not involve p i . Notice that this polynomial in p 1 , p (1) Under the deterministic homogeneity assumption the functioning probability of every component is known and equal to p. Then the probability that the system works with component i in state “ON” and does not work with component i in ˆ state “OFF” is ∂∂pfi (p, p, . . . , p) which, by proposition 3.2, coincides with the binop mial semivalue i (f ). (2) Under the deterministic independent assumption the functioning probability of every component is known and equal to p k , k = 1, . . . , n, hence, the probability that the system works with component i in state “ON” and does not work with component i ˆ in state “OFF” is ∂∂pfi (p1 , p 2 , . . . , p n ) and, by proposition 3.2, coincides with the multi-binary probabilistic value ϕ i (f ).
RELIABILITY IMPORTANCE MEASURES
339
(3) Under the general homogeneity assumption component i s importance is
b
b ˆ ∂f (p, p, . . . , p) dp = p s (1 − p)n−s−1 dp. i a ∂p a S∪{i}∈W, S ∈W / S⊆N−{i}
(4) Under the general independence assumption component i s importance is
bn
bi
b1 ˆ ∂f ... ... ( p1 , p 2 , . . . , p n ) d p1 · · · d pi · · · d pn . i an ai a1 ∂ p Developing the partial derivative of component i we obtain: bj bj p j d pj (1 − p j ) d pj (bi − ai ) S∪{i}∈W, S ∈ / W j ∈S S⊆N−{i}
= (bi − ai )
S∪{i}∈W, S ∈ /W S⊆N−{i}
aj
j =i j ∈S /
aj
(bj + aj ) aj + bj (bj − aj ) (bj − aj ) 1 − 2 2 j ∈S j =i j ∈S /
n an + bn ∂ fˆ a1 + b1 ,..., . (bj − aj ) = ∂ p 2 2 i j =1
Notice that under the general independence assumption if Ik = [0, 1] for all k = 1, . . . , n and under the deterministic homogeneity assumption with p = 1/2, the answer to initial question is given by the Banzhaf index (Birnbaum). In this case we obtain the known result: 1 ∂ fˆ 1 1 , ,..., . βi (f ) = ∂p i 2 2 2 However, under the general homogeneity assumption, if I = [0, 1] the answer is given by the Shapley–Shubik index (Barlow–Proschan) and we obtain the known result
1 ˆ ∂f (p, p, . . . , p) dp. i (f ) = i 0 ∂p We see that the Barlow–Proschan index i (f ) is the probability that component i caused system failure under the assumption that the lives of the components are independent, uniform and identically distributed random variables. 4.
Concluding remarks
We choose the model to be applied on theorem 3.3, according to the information on reliability’s components is deterministic or not and, at the same time, homogeneous (the same probability for each component) or not. This type of information can be more often
340
FREIXAS AND PUENTE
applied in systems or circuits than in situations of voting which depend on many aspects, like psychological reasons, which are extremely difficult to assess and, therefore, to implement just in a single number or in an interval. Of course, further models might be developed (for instance, considering a nonuniform distribution). The incorporation of the appropriate probabilistic model helps to get a high precision in computing the importance that each component has in the system. Concerning game theory, we have seen in the preceding section how multi-binary probabilistic values or binomial semivalues appear as importance measures of the components. Clearly, when considering other models, other values currently used in that field may appear. To illustrate this, let us consider an additional assumption that corresponds to a discrete distribution associated to a fixed number of values. General discrete homogeneity assumption: A number is chosen from the set {p1 , . . . , pk }, where 0 pi 1, with probability λi and
k i=1
λi = 1.
According to the following proposition we will see that semivalues – not necessarily binomial semivalues – may be used. Proposition 4.1. Every semivalue on GN , , may be expressed as nj=1 λj pj where n = |N|, each pj is a binomial semivalue, pi = pj if i = j and nj=1 λj = 1. Proof. Let us consider n distinct binomial semivalues p1 , . . . , pn , pi = pj , for i = j , each of them is defined by the weighting coefficients given by pk,i = pik (1 − pi )n−1−k for i = 1, . . . , n and for k = 0, . . . , n − 1. These semivalues are linearly independent. Indeed, if λ1 p1 + · · · + λn pn = 0, then we successively apply this linear combination to the games uN , uN−{n} , uN−{n,n−1} , . . . , u{1} and get, by always looking at player 1’s component λ p n−1 + λ2 p2n−1 + · · · + λn pnn−1 = 0, 1 1 λ p n−2 + λ2 p2n−2 + · · · + λn pnn−2 = 0, 1 1 .. .. .. .. .. . . . . . λ1 p1 + λ2 p2 + · · · + λn pn = 0, λ + λ + · · · + λ = 0. 1 2 n Its (Vandermonde) determinant, i>j (−1)n (pi − pj ), is not 0 because we are assuming that pi = pj for i = j . Thus, the system has the solution λ1 = λ2 = · · · = λn = 0. As a consequence of the fact that the subspace spanned by all semivalues within the space of linear maps from GN to Rn has dimension n we conclude that p1 , . . . , pn is a basis of the subspace spanned by all semivalues and, hence, every semivalue can be uniquely
RELIABILITY IMPORTANCE MEASURES
341
written as a linear combination = λ1 p1 + λ2 p2 + · · · + λn pn . Finally, as player i is a dummy in the game u{i} we use the dummy player property and obtain 1 = i [u{i} ] =
n j =1
p
λj i j [u{i} ] =
n
λj .
j =1
So, the answer to the main question proposed in section 3 under the general discrete homogeneity assumption brings us to consider a certain semivalue MiG.D.H =
n j =1
λj
∂ fˆ (pj , pj , . . . , pj ) = i (f ). ∂p i
A tool to be considered in using a specific measure M for a semicoherent structure f , is the ratio between components i and j , defined as Mi (f )/Mj (f ), which is useful to compare the relative importance measure between two arbitrary components. The study made by Carreras and Freixas (see section 3 in [5]) on monotonicity conditions concerning two players in the same game by semivalues applied to game theory suggests results on the variability of ratios. A further analysis of their study for semivalues and its development when M is a particular measure suggests a future research. Acknowledgments Research partially supported by Grant BMF 2000-0968 of the Science and Technology Spanish Ministry. The authors wish to thank two anonymous referees for their helpful comments. References [1] J.F. Banzhaf, Weighted voting doesn’t work: A mathematical analysis, Rutges Law Review 19 (1965) 317–343. [2] R.E. Barlow and F. Proschan, Importance of system components and fault tree event, Stochastic Processes and Their Applications 3 (1975) 153–172. [3] Z.W. Birnbaum, On the importance of different components in a multicomponent system, in: Multivariate Analysis II, ed. P.R. Krishnaiah (Academic Press, New York, 1969). [4] F. Carreras and J. Freixas, Some theoretical reasons for using (regular) semivalues, in: Logic, Game Theory and Social Choice, ed. H. de Swart (Tilburg University Press, 1999) pp. 140–154. [5] F. Carreras and J. Freixas, A note on regular semivalues, International Game Theory Review 2 (2000) 345–352. [6] F. Carreras, J. Freixas and M.A. Puente, Semivalues as power indices, European Journal of Operation Research (2001) forthcoming. [7] P. Dubey, On the uniqueness of the Shapley value, International Journal of Game Theory 4 (1975) 131–139. [8] P. Dubey, A. Neyman and R. Weber, Value theory without efficiency, Mathematics of Operations Research 6 (1981) 122–128. [9] P. Dubey and L.S. Shapley, Mathematical properties of the Banzhaf power index, Mathematics of Operations Research 4 (1979) 99–131.
342
FREIXAS AND PUENTE
[10] M.R. Hilliard, Weighted voting theory and applications, Ph.D. thesis, School of Operation Research and Industrial Engineering, Cornell University, Ithaca (1983). [11] G. Owen, Multilinear extensions of games, Management Science 18 (1972) 64–79. [12] G. Owen, Multilinear extensions and the Banzhaf value, Naval Research Logistics Quarterly 22 (1975) 741–750. [13] K.G. Ramamurthy, Coherent Structures and Simple Games (Kluwer Academic, Dordrecht, 1990). [14] L.S. Shapley, Simple games: An outline of the descriptive theory, Behavioral Science 7 (1962) 59–66. [15] L.S. Shapley and M. Shubik, A method for evaluating the distribution of power in a committee system, American Political Science Review 48 (1954) 787–792. [16] R. Weber, Probabilistic values for games, in: The Shapley Value, ed. A. Roth (Cambridge University Press, 1988) pp. 101–119.