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Bi-capacities -- Part I: definition, Möbius transform and interaction
Michel Grabisch, Christophe Labreuche
TL;DR
The paper addresses how to generalize capacity-based decision models when criterion scales are bipolar. It develops the mathematical machinery of bi-capacities, including their structure, Möbius transform, derivatives, and game-theoretic indices, while remaining mainly theoretical. The resulting framework makes familiar fuzzy-measure notions available for bipolar settings.
Problem
Capacity-based models can be inadequate for bipolar scales, where scores range from negative through neutral to positive values and coalition importance and overall scoring must be generalized.
Method
The paper defines bi-capacities on ternary positive, negative, and neutral alternatives, then develops their lattice structure, Möbius transform, derivatives, Shapley value, and interaction index.
Results
The framework provides corresponding structural and game-theoretic notions for bi-capacities, including a basis of bi-unanimity games and interaction expressed through derivatives and the Möbius transform.
Takeaways & Limitations
Bi-capacities offer a more general bipolar framework that encompasses models such as the symmetric Choquet integral and Cumulative Prospect Theory.
Takeaways & Limitations
The treatment remains largely theoretical, and decomposable bi-capacities based on t-conorms or uninorms are not developed.
Abstract
from arXiv · showhide
Bi-capacities arise as a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours, encompassing models such as Cumulative Prospect Theory (CPT). The aim of this paper in two parts is to present the machinery behind bi-capacities, and thus remains on a rather theoretical level, although some parts are firmly rooted in decision theory, notably cooperative game theory. The present first part is devoted to the introduction of bi-capacities and the structure on which they are defined. We define the Möbius transform of bi-capacities, by just applying the well known theory of M\" obius functions as established by Rota to the particular case of bi-capacities. Then, we introduce derivatives of bi-capacities, by analogy with what was done for pseudo-Boolean functions (another view of capacities and set functions), and this is the key point to introduce the Shapley value and the interaction index for bi-capacities. This is done in a cooperative game theoretic perspective. In summary, all familiar notions used for fuzzy measures are available in this more general framework.
1 Introduction
Capacities model decision behavior effectively on unipolar scales but can be inadequate when criteria scores are bipolar. The paper introduces bi-capacities to represent positive, negative, and neutral criterion states more generally.
- Motivation: Bipolar scales represent affective scores from negative through neutral to positive values.Examples include bounded cardinal, unbounded cardinal, and ordinal scales.
- Problem: The problem is to generalize coalition importance and overall scoring from unipolar to bipolar criteria.
- Existing models: Symmetric positive and negative parts yield the symmetric Choquet integral, while separate capacities yield the Cumulative Prospect Theory model.
- Contribution: Bi-capacities assign values to ternary alternatives containing fully satisfied, fully unsatisfied, and neutral criteria.They use v(A, B), where A denotes totally satisfied criteria and B totally unsatisfied criteria.
- Paper scope: The first part develops the structure, Möbius transform, k-additivity, derivatives, Shapley value, and interaction index for bi-capacities.
2 Preliminaries
The preliminaries establish capacities, pseudo-Boolean functions, derivatives, and finite distributive lattices as the mathematical foundations for bi-capacities. Birkhoff’s theorem supplies the lattice decomposition framework used later.
- Capacities: A capacity is a monotone set function normalized by ν(∅)=0 and, when normalized, ν(N)=1.Its conjugate and additive forms are also introduced.
- Pseudo-Boolean functions: Capacities correspond to nonnegative monotonic pseudo-Boolean functions under the bijection between subsets and binary vectors.
- Derivatives: S-derivatives are defined recursively from finite differences, and capacity derivatives can therefore be indexed by subsets and evaluation points.
- Lattices: A finite distributive lattice has unique least upper bounds and greatest lower bounds, together with top and bottom elements.
- Lattice terminology: Down-sets contain every element below each of their members, with principal ideals providing the basic generated examples.
- Lattice decomposition: Birkhoff’s theorem states that every finite distributive lattice is isomorphic to the down-sets of its join-irreducible elements.The resulting irredundant decomposition is unique in a distributive lattice.
3 Bi-capacities
Bi-capacities are order-preserving functions on disjoint positive and negative criterion sets. The framework includes normalized, CPT-type, symmetric, asymmetric, additive, and decomposable variants.
- Definition: Q(N) consists of pairs (A, B) of disjoint subsets of N, representing positive and negative sides.
- Definition: A bi-capacity is a function on Q(N) that increases with its first argument and decreases with its second.
- Normalization: Normalization requires v(N, ∅)=1 and v(∅, N)=-1, implying nonnegative positive-only and nonpositive negative-only values.
- Special cases: CPT-type bi-capacities separate positive and negative parts through two normalized capacities, while equal capacities produce the symmetric case.
- Special cases: Additive bi-capacities are CPT-type models whose two component capacities are additive.
- Scope: More generally, decomposable bi-capacities can be constructed using t-conorms or uninorms with neutral element 0, but this topic is not developed here.
4 The structure of Q(N)
Q(N) is a ternary lattice of disjoint positive and negative subsets, with order determined by inclusion in the positive component and reverse inclusion in the negative component. Alternative representations expose different structural properties, including join-irreducible layers.
- Q(N) representation: Q(N) is equivalent to mappings from N to {−1, 0, 1}, so it contains 3^n elements.
- Q(N) lattice: The lattice order is (A, B) ⊑ (C, D) when A ⊆ C and B ⊇ D, with supremum (A ∪ C, B ∩ D) and infimum (A ∩ C, B ∪ D).
- Q(N) lattice: Top and bottom are (N, ∅) and (∅, N), and bi-capacities are order-preserving mappings from Q(N) to R.
- Alternative operations: The alternative operations ⊔′ and ⊓′ do not form a lattice, as shown by a counterexample involving complementary pairs.
- Lattice properties: Q(N) is a distributive lattice formed from 2^n Boolean sub-lattices sharing (∅, ∅), but it is not complemented.
- Sub-lattices: The lattice contains intervals of type 2^k × 3^l, with the parameters determined by the changes between positive and negative components.
- Alternative representation: Q*(N) replaces each pair (A, B) by (A, B^c), making A the scores equal to 1 and B the scores equal to 0 or 1.
- Decomposition and layers: Join-irreducible elements define layers Q^[k](N), where layer k contains elements whose irredundant decomposition has k join-irreducibles.For Q(N), layer k contains elements with |B|=n−k.
5 M¨obius transform of bi-capacities
The paper extends Möbius-transform machinery from capacities to bi-capacities on Q(N), deriving the corresponding Möbius function and transform. It then characterizes special cases, including CPT-type, additive, and k-additive bi-capacities.
- Möbius function: The Möbius function on Q(N) is nonzero only when (A, A′) ⊑ (B, B′) and A′ ∩ B = ∅, with sign determined by |B\A| + |A′\B′|.Otherwise, the Möbius function is zero.
- Möbius function: The transform is derived by viewing Q(N) as a product poset and multiplying the component Möbius functions.The proof identifies Q(N) with 3^n and translates the resulting conditions into subset notation.
- Möbius transform: For a normalized bi-capacity, the transform satisfies m(∅, N) = −1, and it can be represented in matrix form using a transform matrix T(n).The matrix T(n) has a fractal structure analogous to the classical-capacity transform.
- Special cases: For CPT-type bi-capacities v(A, B) = ν1(A) − ν2(B), the transform has a restricted form, including m(∅, N) = −1 and zero coefficients for specified nonempty pairs.The result also yields expressions for symmetric and asymmetric bi-capacities.
- Special cases: For additive bi-capacities, the transform is nonzero only at join-irreducible elements and the bottom element, with coefficients determined by ν1 and ν2.These join-irreducibles correspond to positive and negative singleton components.
- k-additivity: A bi-capacity is k-additive when its Möbius transform vanishes on Q^[l](N) for l > k, equivalently when m(A, B) = 0 whenever |B| < n − k.This definition extends the familiar notion of k-additivity from capacities.
6 Derivatives of bi-capacities
The paper defines left and right derivatives for bi-capacities by translating ternary pseudo-Boolean differences to Q(N), then recursively generalizes them to mixed derivatives. Monotonicity ensures these derivatives are nonnegative.
- Definition of derivatives: Bi-capacities correspond to ternary pseudo-Boolean functions f: {−1, 0, 1}^n → R through f(1_S, −1_T) = v(S, T).This correspondence supplies the setting for defining derivatives by changes among ternary variable values.
- Definition of derivatives: The left derivative adds an element to the positive argument, while the right derivative removes an element from the negative argument.They are respectively Δ_i,∅v(S, T) = v(S ∪ i, T) − v(S, T) and Δ_∅,iv(S, T) = v(S, T \ i) − v(S, T).
- Order properties: When a bi-capacity is monotone, its left and right derivatives are nonnegative.This connects the derivative construction to the order properties of bi-capacities.
- Higher-order derivatives: Mixed derivatives are defined recursively by applying left and right difference operators to disjoint positive and negative element sets.The general derivative expands as an alternating sum over subsets of the selected elements.
- Higher-order derivatives: For example, a mixed derivative combines four bi-capacity evaluations with alternating signs.The displayed second-order expressions include Δ_i,jv and Δ_ij,∅v.
- Möbius representation: The derivative of a bi-capacity can be expressed through its Möbius transform, generalizing the corresponding result for classical capacities.The paper proves this relationship by induction over the selected positive and negative elements.
7 Shapley value and interaction index
The paper extends cooperative-game concepts to bi-cooperative games, where players may support a defender coalition, join a defeater coalition, or remain indifferent. It defines Shapley values and interaction indices for this bipolar setting and relates them to derivatives and Möbius transforms.
- Bi-cooperative games: Bi-cooperative games interpret v(S,T) as the worth of defender coalition S opposed by defeater coalition T, with remaining players indifferent.Ternary voting games provide one interpretation, with positive, negative, and abstaining voters.
- Bi-unanimity games: Bi-unanimity games form a basis for bi-capacities, allowing games to be represented through Möbius coefficients.Their Möbius transform is nonzero only at the centered pair, although some bi-unanimity games are not normalized bi-capacities.
- The Shapley value: The framework introduces Shapley values for both defender and defeater participation, extending the classical contribution concept to bipolar games.The resulting operator maps bi-cooperative games to 2n-dimensional vectors.
- The Shapley value: The Shapley-value construction is characterized through linearity, null-player, symmetry, and efficiency-related axioms, with symmetry and efficiency equivalent to the unanimity-game axiom under the stated conditions.The axioms ensure that player labels do not affect computation and that contributions are allocated consistently.
- The interaction index: The interaction index generalizes classical interaction through recursion formulas and can be expressed using derivatives or the Möbius transform.For k-additive capacities, interactions vanish above order k and equal the Möbius coefficient at order k.
- The interaction index: The paper derives interaction-index results for k-additive, CPT-type, asymmetric, and symmetric bi-capacities.These cases connect the bipolar interaction framework to familiar capacity structures and CPT representations.