Bi-capacities -- Part II: the Choquet integral

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 second part focuses on the definition of Choquet integral. We give several expressions of it, including an expression w.r.t. the M"obius transform. This permits to express the Choquet integral for 2-additive bi-capacities w.r.t. the interaction index.

💡 Research Summary

The paper introduces bi‑capacities, a natural extension of classical capacities (fuzzy measures) designed for decision‑making contexts where the underlying evaluation scale is bipolar, i.e., it simultaneously captures gains and losses. After recalling the limitations of ordinary capacities—namely their inability to represent asymmetric attitudes toward gains and losses—the authors define a bi‑capacity μ as a set‑function μ:2^N → ℝ^2, with components μ⁺ (gain side) and μ⁻ (loss side). Both components satisfy monotonicity (μ⁺ is non‑decreasing, μ⁻ is non‑increasing with respect to set inclusion) and normalization (μ⁺(∅)=0, μ⁺(N)=1, μ⁻(∅)=0, μ⁻(N)=1). This formulation subsumes ordinary capacities as the special case μ⁻≡0 and aligns with the structure of Cumulative Prospect Theory (CPT), where separate weighting functions are used for gains and losses.

The core contribution is the definition and analysis of the Choquet integral with respect to a bi‑capacity. For a real‑valued function f:N→ℝ, the integral is defined as I_μ(f)=I_{μ⁺}(f)−I_{μ⁻}(f), i.e., the difference between the standard Choquet integrals of the gain and loss components. To handle this construct analytically, the authors employ the Möbius transform ζ of μ, which uniquely decomposes μ into elementary contributions: μ(A)=∑{B⊆A} ζ(B). Using ζ, the Choquet integral admits a linear representation
 I_μ(f)=∑{S⊆N} ζ(S)·Δ_f(S),
where Δ_f(S) denotes the set‑difference operator applied to f. This expression mirrors the classic Möbius‑based formula for ordinary capacities but now captures bipolar contributions.

A particular focus is placed on 2‑additive bi‑capacities, i.e., those for which ζ(S)=0 whenever |S|>2. In this case the transform reduces to first‑order coefficients a_i=ζ({i}) and second‑order interaction indices I_{ij}=ζ({i,j}). The Choquet integral simplifies to a tractable polynomial:
 I_μ(f)=∑{i} a_i f(i)+∑{i<j} I_{ij} min{f(i),f(j)}.
Because a_i and I_{ij} are themselves bipolar (each has a gain and a loss part), the formula simultaneously accounts for individual importance and pairwise synergy or antagonism on both sides of the scale. This result provides a practical computational tool for models that require only up to pairwise interactions, a common assumption in many applied settings.

The paper further connects the theoretical framework to established decision‑theoretic models. By mapping CPT’s value function v and weighting functions w⁺, w⁻ onto μ⁺ and μ⁻, the authors show that the CPT expected utility is exactly the Choquet integral I_μ(v). This demonstrates that bi‑capacities can reproduce the full CPT representation while offering a more general algebraic structure. In cooperative game theory, the Möbius coefficients of a bi‑capacity correspond to generalized Shapley values and Banzhaf indices that distinguish contributions to gains and to losses, thereby extending classic fairness concepts to bipolar environments.

Finally, the authors discuss limitations and future directions. The current treatment is static; extending bi‑capacities to dynamic or stochastic settings, handling higher‑order (k‑additive with k>2) interactions, and developing efficient algorithms for large‑scale problems remain open challenges. Nonetheless, the paper establishes a solid mathematical foundation for bipolar decision analysis, provides explicit formulas for 2‑additive cases, and opens avenues for integrating bi‑capacities into behavioral economics, multicriteria decision making, and cooperative game theory.