An Axiomatic Approach to the Stieltjes Integral Representations of Cardinal-Probabilistic Interaction Indices∗ Katsushige FUJIMOTO Faculty of Economics, Fukushima University, 1 Kanayagawa, Fukushima 960-1296, Japan [email protected]

Abstract The Shapley, Banzhaf, and chaining interaction indices are included in the class of probabilistic cardinal interaction indices which are obtained as the expected marginal interaction. This paper provides a new axiomatic characterization of cardinal-probabilistic interaction indices and shows that any cardinal-probabilistic interaction index can be represented as the Stieltjes integral with respect to non-decreasing functions on [0,1]. Keywords:

marginal interaction, cardinal-

probabilistic interaction index.

1

Introduction

Non additive measures, (e.g., fuzzy measures, characteristic functions in game theory, belief and plausibility functions in evidence theory, etc...), have a high potential for representing interactions among elements of objects to be considered, especially in multi-criteria decision making, where they have been already successfully applied. The study of the notion of interaction among players/criteria/attributes is relatively recent in the framework of cooperative game theory. The first attempt is probably due to Owen [14, 1972] for superadditive games. More recent developments are due to Murofushi and Soneda [13, 1993], Roubens [15, 1996], Grabisch [5, 1996], and Marichal and Roubens [12, 1999] and led to the concepts of Shapley interaction index, Banzhaf interaction index and chaining interaction index. Grabisch and Roubens [8, 1999] proposed and axiomatized cardinal-probabilistic interaction indices which encompass the three existing interaction indices. In this paper, we show a new axiomatic characterization and new integral expressions of cardinalprobabilistic interaction indices. ∗ This research was partially supported by the Japan Society for the Promotion of Science, Japan, Grant-in-Aid for Encouragement of Young Scientists, 13780349, 2001-2002.

2

Preliminaries

Let U be an infinite set, the universe of players. A game on U is a set function v : 2U → R with v(∅) = 0. We interpret the members of U as players, the members of 2U as coalitions, and the number v(S) for any S ⊆ U as the worth or power of a coalition S. A set N ⊆ U is a support of v if, for each S ⊆ U , v(S) = v(S ∩ N ). A finite game is the game which has a finite support. We denote by G the vector space of all finite games on U , by G N the subspace of G consisting of games with a finite support N , moreover, by I the set of all operators I : G → G, and by I N the set of all operators I N : G N → G N . In order to avoid heavy notations, we will whenever possible omit braces for singletons, e.g., writing v(i), S ∪ i instead of v({i}), S ∪ {i}. Cardinality of sets S, T, . . . will be denoted whenever possible by corresponding lower cases s, t, . . ., otherwise by the standard notation |S|, |T |, . . . 2.1 Basic definitions on games Addition of two games v and w in G is defined by (v + w)(S) := v(S) + w(S) for each S; multiplication of the game v by the scalar α is defined by (αv)(S) := αv(S) for each S. A game v is monotone (1-monotone) if v(S) ≤ v(T ) whenever S ⊆ T ; furthermore, given an integer k ≥ 2, a game v is k-monotone [3] if v(

k


i=1

Si ) ≥

 I⊆{1,...,k} I=∅

(−1)|I|+1 v(



Si )

(1)

i∈I

for any Si ⊆ U , 1 ≤ i ≤ k. The game which is kmonotone for any integer k ≥ 1 is called ∞-monotone, e.g., fuzzy measures, convex games, and belief functions are 1-, 2-, and ∞-monotone, respectively. If the Si ’s⊆ U are distinct, the formula (1) is expressed k k as v( i=1 Si ) ≥ i=1 v(Si ). Therefore, we can interapret that k-monotonic games represent the situation in which synergy effects exist on the combination of at most k coalitions/players.

2.2 Basic definitions on coalitions A coalition P (= ∅) is said to be a partnership[11] in a game v if v(R ∪ T ) = v(R) for any T  P and any R ⊆ U \ P . Therefore, any proper subset T of a partnership P is worthless and generates no new effect by forming any coalition U ⊇ T unless U ⊇ P , –that is, the partnership P behaves like a single hypothetical player [P ]. Clearly, any single player is a partnership. A player i is said to be a dummy in a game v if v(S ∪ i) = v(S) + v(i) for any S ⊆ U \ i. A coalition P is said to be a dummy partnership in a game v if P is a partnership and v(S ∪ P ) = v(S) + v(P ) for any S ⊆ U \ P. 2.3 Representations of games There exist several equivalent representations of a finite game v ∈ G N . The first representation is to give for any coalition S the number v(S). The second one is to observe that v can be expressed  in a unique way as: v(S) = T ⊆S m(v)(T ) ∀S ⊆ N . In game theory, the real coefficient {m(v)(T )}T ⊆N are called the dividends[10] of the coalitions in a game v. In combinatorics, m(v) viewed as a set function on N is called the M¨ obius transform of v, which is given by  m(v)(S) = T ⊆S (−1)s−t v(T ) for S ⊆ N . The third one is to regard as a pseudo-Boolean function (i.e., a function f : {0, 1}n → R). For a coalition S ⊆ N , let eS be the characteristic vector (i.e., the vector of {0, 1}n whose i-th component is 1 if and only if i ∈ S). Geometrically, the characteristic vectors are the 2n vertices of the hypercube [0, 1]n . Any game v ∈ G N can be viewed as a pseudoBoolean function. The correspondence is obtained    by f (x) = T ⊆N v(T ) i∈T xi i∈T (1 − xi ) for any x ∈ {0, 1}n , and v(S) = f (eS ) for all S ⊆ N , (see Hammer and Holzman[9]). The multi-linear extension of the pseudo-Boolean function f corresponding to v is obtained uniquely as the multi-linear function g : [0, 1]n → R which interpolate f at 2n vertices of [0, 1]n defined by g(x) =    n T ⊆N v(T ) i∈T xi i∈T (1 − xi ) for any x ∈ [0, 1] , (see Owen [14]). Given S = {i1 , . . . , is } ⊆ N , the S-th derivative of g is denoted as ∆S g(x) and defined by ∂ s g(x) ∆S g(x) := ∂xi1 · · · ∂xis

3

n

for x ∈ [0, 1] .

Representations of Interactions

The concept of interaction index is fundamental for it enables to represent the interaction phenomena modeled by a game on a set of players. The expression

interaction phenomena refers to complementarity or redundancy effects among players in coalitions resulting from the non-additivity of the underlying game. 3.1 k-Monotonicity and marginal interaction Let δi v(T ∪ i) be the marginal contribution of a player i to a coalition T ⊆ U \ i in a game v defined by δi v(T ∪ i) := v(T ∪ i) − v(S). Murofushi and Soneda [13] proposed a marginal interaction between two players i and j in the presence of players in T ⊆ U \ {i, j}, which was defined by ∆Iij (T ) := [v(T ∪ {i, j}) − v(T ∪ i)] − [v(T ∪ j) − v(T )], as an extension of the marginal contribution, –that is, i’s contribution to j’s marginal contribution to T ∪ {i, j}. This formula can also be expressed as ∆Iij (T ) = δi (δj v(T ∪ {i, j})). Later, Grabisch and Roubens [7] generalized this marginal interaction between two players to the marginal interaction among players in any finite coalition S in the presence of players in T ⊆ U \ S naturally as follows: δS v(S ∪ T ) := δi (δS\i v(S ∪ T )), where i ∈ S. It is easy to show by induction on s that  δS v(T ∪ S) = m(v)(K ∪ S) K⊆T

for any finite S ⊆ U and any T ⊆ U \ S. On the other hand, if g is the multi-linear extension of the pseudo-Boolean function corresponding to v ∈ G N , then δS v(S ∪ T ) = ∆S g(eS∪T ) for any S ⊆ N and any T ⊆ N \ S. –that is, the marginal interaction can be also represented as the S-derivative of g (see Grabisch, Marichal, and Roubens [6]). The following proposition shows that the marginal interaction represents synergy effects as positive values. It is easy to see from Proposition 4 in [2]. Proposition 1 A game v ∈ G is k-monotone iff δS v(T ∪ S) ≥ 0 for any T ⊆ U \ S and for any S ⊆ U such that s ≤ k. 3.2 An axiomatic characterization of the family of probabilistic interaction indices Consider the following axioms imposed on I ∈ I: • Additivity axiom (A) : I is an additive operator on G. • k-Positivity axiom (Pk ) : For any positive integer k and k-monotonic game v ∈ G, I(v)(S) ≥ 0 for any S ⊆ U such that s ≤ k. • Dummy Player axiom (D) : If i is a dummy player in a game v ∈ G, then (i) I(v)(i) = v(i), (ii) I(v)(S) = 0 for any S  i.

• Dummy Partnership axiom (DP) : If P is a dummy partnership in a game v ∈ G, then (i) I(v)(P ) = v(P ), (ii) I(v)(T ) = 0 for any T  P .



• Symmetry axiom (S) : For any v ∈ G and any permutation π on U (i.e., π is a bijection from U to U ), I(v)(S) = I(πv)(π(S)), where πv(π(S)) := v(S) and π(S) := {π(i)|i ∈ S}. • N -Symmetry axiom (SN ) : For any v N ∈ G N and any N -preserving permutation π

N

N

on U (i.e., π (N ) =

N ), I N (v N )(S) = I N (π N v N )(π N (S)).

(A) says that interaction indices should be decomposable additively whenever games are decomposable additively. (Pk ) says that synergy effects should be represented as positive interactions. (DP) says that (i) the effects of dummy partnerships should be their own worth, and that (ii) there should be no interaction between a dummy partnership and its outside. (S) or (SN ) says that the names of the players should play no role in determining the interaction indices. Proposition 2 If I ∈ I satisfies (A) and (Pk ), then I is a linear operator. The following lemma is immediately obtained by applying Proposition 2 to Propositions 3 and 5 in [7] and (P9) in [8]. Lemma 3.1 Let N ⊆ U be a finite set. I ∈ I N satisfies (A), (Pk ), and (D) iff, for any S ⊆ N , there exists a family of non-negative real constants {pN S (T )}T ⊆N \S  satisfying T ⊆N \S pN (T ) = 1 such that S I N (v N )(S) =



a probabilistic interaction index I ∈ I is defined by, for any S ⊆ N and any v ∈ G,  pN (4) I(v)(S) = S (T )δS v(T ∪ S)

N pN S (T )δS v (S ∪ T ).

(2)

T ⊆N \S

Moreover, I ∈ I N satisfies (A), (Pk ), (D), and (SN ) iff, for any S ⊆ N , there exists a family of non-negative real constants {pns (t)}t=0,...,n−s satisfying n−s n−s n ps (t) = 1 such that t=0 t  I N (v N )(S) = pns (t)δS v N (S ∪ T ). (3) T ⊆N \S n Here, pN S (T ) and ps (t) in Lemma 3.1 can be viewed as probability distributions on N \ S. Therefore, interaction indices expressed by (2) and/or (3) are represented as the expected values of marginal interactions over all coalitions on N \ S. Now, we define the notion of probabilistic interaction index and cardinal-probabilistic interaction index:

T ⊆N \S

N subject to T ⊆N \S pN S (T ) = 1 and pS (T ) ≥ 0 for any T ⊆ N \ S; a cardinal-probabilistic interaction index I ∈ I is defined by, for any S ⊆ N and any v ∈ G,  I(v)(S) = pns (t)δS v(T ∪ S) (5) T ⊆N \S

n−s   n ps (t) = 1 and pns (t) ≥ 0 for any subject to t=0 n−s t t = 0, . . . , n − s, where N is a finite support of v. Theorem 1 I ∈ I is a cardinal-probabilistic interaction index iff I satisfies (A), (Pk ), (D), and (S). Besides, if I ∈ I is a cardinal-probabilistic interaction index given in the form of (5), then for every non-negative integer s ≤ n there uniquely exists a nondecreasing function Fs : [0, 1] → [0, 1] such that 1 xt (1 − t)n−s−t dFs (x). (6) pns (t) = 0

Moreover, this I ∈ I can be also expressed by using Sderivative of the multi-linear extension g of the pseudoBoolean function corresponding to v ∈ G as follows: 1 I(v)(S) = ∆S g(x, . . . , x) dFs (x). (7) 0

Here, equations (5) and (6) are interpreted as follows (following Dubey, et.al.[4]): chose x in [0, 1] at random in accordance with Fs , and construct a random coalition T by letting each player other than players in S with probability x, independently of the other players (i.e., (6) represents the probability that T or T ∪ S is constructed). Then, I(v)(S) is the expected marginal interaction among players of S over all coalitions on N \ S.

3.3 Typical probabilistic interaction indices In this subsection, we introduce three typical cardinal-probabilistic interaction indices, the Shapley [5] , Banzhaf [7] , and the chaining interaction indices [12] , and two transforms which also can be regarded as cardinal-probabilistic interaction indices, the M¨ obius and co-M¨ obius transforms. Definition 1 [5, 7, 12] Let v ∈ G N . The Shapley, Banzhaf, and chaining interaction indices for a coalition S ⊆ N with respect to v are denoted by ISh , IB , Ich (∈ I N ) and defined by ISh (v)(S) :=

 T ⊆N\S

(n − t − s)!t! δS v(T ∪ S), (n − s + 1)!



IB (v)(S) :=

T ⊆N\S



Ich (v)(S) :=

T ⊆N\S

1 δS v(T ∪ S), 2n−s

[3] G.Choquet, “Theory of capacities”, Fourier, vol.5, pp.131-295, 1953.

s(n − t − s)!(t + s − 1)! δS v(T ∪ S), n!

respectively. Both the Shapley and the chaining interaction indices for any {i} ⊆ N coincide with the Shapley power index[16] , the Banzhaf interaction index with the Banzhaf power index[1] , and the Shapley interaction index for any {i, j} ⊆ N with the Murofushi and Soneda’s interaction index defined in [13]. Clearly, the Shapley and Banzhaf interaction indices, ISh and IB , are cardinal-probabilistic interaction indices. On the other hand, the chaining interaction index Ich , the M¨ obius transform m, and the co-M¨ obius transform  m∗ , defined by m∗ (v)(S) = T ⊇N \S (−1)n−t v(T ), can be expressed in the form of (5) with pn s (t)

=

   1 0   s·(n−t−s)!(t+s−1)! n!

if s = 0 and t = 0, if s = 0 and t =  0, otherwise,

pn s (t) = 1 if t = 0 (i.e., T = ∅ ) and 0 otherwise, pn s (t) = 1 if t = n − s (i.e., T = N \ S ) and 0 otherwise,

respectively. Therefore, the chaining interaction index, the M¨ obius and co-M¨ obius transforms also can be regarded as cardinal-probabilistic interaction indices. Remark 1 The Shapley, Banzhaf, and chaining interaction indices, the M¨obius and co-M¨ obius transforms can be expressed as (5) and (6), and/or (7), with the non-decreasing function Fs (x) : [0, 1] → [0, 1] in the following table, where 1A is the characteristic function of A ⊆ [0, 1], i.e., 1A (x) = 1 if x ∈ A and 0 otherwise.

Fs (x) =

ISh x

IB 1[ 1 ,1] 2

Ich x · 1]0,1] s

m 1]0,1]

m∗ 1{1}

Acknowledgments The Author is grateful to Ivan Kojadinovic, Jean-Luc Marichal, and Toshiaki Murofushi for their valuable advises and comments.

References [1] J.F. Banzhaf. “Weighted voting does not work: A mathematical analysis”, Rutgers Law Review, vol.19,pp.317–343, 1965. [2] A.Chateauneuf and J-Y.Jaffray, “Some Characterization of Lower Probabilities and Other Monotone Capacities through the Use of M¨ obius Inversion”, Mathematical Social Sciences, Vol.17, pp.263-283, 1989.

Ann. Inst.

[4] P.Dubey, A.Neyman, and R.J.Weber, “Value Theory without Efficiency”, Mathematics of Operations Research, Vol.6, No.1,1981. [5] M.Grabisch, “k-order additive fuzzy measures,” Proc. of 6th International Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems(IPMU), pp.1345-1350, Granada, Spain, 1996. [6] M. Grabisch,J.-L.Marichal, and M.Roubens, “Equivalent representations of set functions”, Mathematics of Operations Research, Vol.25, pp.157-178, 2000. [7] M.Grabisch and M.Roubens, “An Axiomatic approach to the Concept of Interaction among Players in Cooperative Games”, International Journal of game theory, Vol.28, pp.547-565, 1999. [8] M.Grabisch and M.Roubens, “Probabilistic Interactions among players of cooperative game”, In: M.J.Machina and B.Munier (eds.), Beliefs, Interactions and Preferences, Kluwer Academic, 1999. [9] P.L.Hammer and R.Holzman, “Approximation of pseudo-Boolean functions; Applications to game theory”, ZOR—Methods and Models of Operations Research, vol.36, pp.3-21, 1992. [10] J.C.Harsanyi, “A bargaining model for cooperative n-person games”, In: A.W.Tucker and Luce (eds.), Contributions to the Theory of games, pp.325-335, Princeton Univ. Press., 1959. [11] E.Kalai and D.Samet, “Weighted Shapley values”, In: A.E.Roth(Ed.), The Shapley Value, Cambridge University Press, 1988. [12] J.-L. Marichal and M. Roubens, “The chaining interaction index among players in cooperative games”, In: N. Meskens and M. Roubens (eds.), Advances in Decision Analysis, pp. 69-85, Kluwer Acad. Publ., Dordrecht, 1999 [13] T.Murofushi and S.Soneda, “Techniques for reading fuzzy measure (III): interaction index,” Proc. of 9th Fuzzy System Symposium, Sapporo, Japan, pp.693696, 1993. In Japanese. [14] G. Owen. “Multilinear extension of games”, Management Sciences, vol.18, pp.64–79, 1972. [15] M. Roubens. “Interaction between criteria and definition of weights in mcda problems”. In 44th meeting of the European working group “ Multicriteria Aid for Decisions”, Brussels, Belgium, October 1996. [16] L.S.Shapley, “A value for n-person games”, H.W.Kuhn and A.W.Tucker (eds.), Contributions to the Theory of Games, II, Prinston Univ. Press, pp.307-317, 1953. [17] R.J.Weber, “Probabilistic values for games”, In: A.E.Roth(Ed.), The Shapley Value, Cambridge University Press, 1988.