Interval Shapley Rule in Continuous Allocation

• Interval Shapley Rule is a framework that extends classical Shapley values to continuous settings by integrating agent contributions over the interval [0,1].
• It replaces discrete summations with integrals and averages marginal contributions over all orderings, bridging cooperative game theory with continuous uncertainty.
• The method employs axiomatic extensions such as Homogeneous Increments Sharing to provide unique, balanced indices for cost-sharing and power allocation in cooperative environments.

The Interval Shapley Rule generalizes the classical Shapley value framework to settings in which agents’ choices, inputs, or contributions are not discrete (e.g., binary or multilevel), but rather span an interval—typically the real interval [0,1]. This extension, studied in power index theory, economic and engineering allocation, and cooperative game theory, enables principled measurement and allocation of influence, power, or cost in systems where uncertainty, aggregative behavior, or continuous policy alternatives predominate. The rule appears in various formalizations, including as an axiomatization of generalized Shapley indices for interval simple games, a fairness rule within continuous two-tier voting models, and an equitable interval cost-sharing method under uncertain cooperative environments.

1. Mathematical Formulation and Generalization

The Interval Shapley Rule arises when classical (discrete) coalition games are extended to games or aggregation problems where decision variables are real-valued or lie in a continuum.

In the interval decision setting, a power or allocation index is defined analogously to the classical Shapley value, replacing summations over coalitions with integrals or expectation over the interval domain. For a monotone aggregation function $v\colon [0,1]^n\to [0,1]$, the generalized Interval Shapley value for agent $i$ can be stated as:

$$\Psi_i(v) = \frac{1}{n!} \sum_{\pi\in S_n} \int_{[0,1]^n} \left\{ [v(x_{\pi_{<i}},1_{\pi_{\geq i}}) - v(x_{\pi_{<i}},0_{\pi_{\geq i}})] - [v(x_{\pi_{\leq i}},1_{\pi_{>i}}) - v(x_{\pi_{\leq i}},0_{\pi_{>i}})] \right\}\, dx$$

This captures the reduction in aggregate outcome uncertainty when agent $i$’s value is revealed, averaged over all orderings of the agents.

For cooperative cost allocation under interval uncertainty, the interval Shapley value is typically given in interval form:

$$\text{Sh}_j(N, \dots) := [\varphi_j(N, \underline{w}),\ \varphi_j(N, \overline{w})],$$

where $\underline{w}$ and $\overline{w}$ denote the lower and upper border games associated with interval parameters, and $\varphi_j$ is the classical Shapley formula evaluated at these extremes (Gonçalves-Dosantos et al., 29 Jul 2025).

2. Axiomatic Characterization

A fundamental result is that the axioms characterizing the classical Shapley–Shubik index—Efficiency, Anonymity, and Null Player—are insufficient to obtain a unique extension to interval decisions (Kurz et al., 2019). The interval setting requires the additional axiom of Homogeneous Increments Sharing (HIS):

• HIS: If the worth of a coalition $S$ is uniformly increased over some domain, then this increase is shared among members of $S$ by weights depending only on $|S|$, $n$, and $i\in S$.

With this axiom and a discretization condition (ensuring indices converge under regular interval game approximations), a unique power index $\Psi$ emerges, generalizing the Shapley–Shubik index to interval games. This index is the natural limit of the (j,k) simple games under step-function embeddings, and thus recovers the classical index where appropriate.

3. Comparison with Classical and Square Root Rules

The Interval Shapley Rule contrasts notably with the Penrose square root rule, long established for binary collective decision models under independent votes (Kurz et al., 2016):

• Binary Model (Penrose): In two-tier voting, fair representation requires top-tier power indices (e.g., Banzhaf) proportional to $\sqrt{n_i}$, due to the combinatorial swing probabilities under iid voting.
• Interval Model: With single-peaked preferences on a real interval and positive intra-constituency correlation, the pivotality (power) in the upper tier converges to the Shapley value, and fairness demands $\varphi_i\propto n_i$—that is, a linear relationship. Here, the top-level voting weights are chosen such that the Shapley values align proportionally with constituency sizes.

This distinction is crucial in applications such as weighted assemblies, economic allocation where continuous choices and correlated environments appear.

4. Applications in Cost Allocation and Uncertainty

Interval Shapley rules are deployed in cooperative cost-sharing with parameter uncertainty, notably in interval inventory games (Gonçalves-Dosantos et al., 29 Jul 2025). Demands or order frequencies for each agent are modeled as intervals, leading to interval cost functions for coalitions:

$w^I(S) = 2a\sqrt{\sum_{i\in S} m_i^2}$

Here, $m_i$ is agent $i$'s order frequency interval, and $a$ is a fixed joint ordering cost. The interval Shapley value averages the marginal cost contributions of each agent across all coalition formation orderings for both the best-case and worst-case (using the lower and upper endpoints). This yields allocations as intervals, reflecting uncertainty, and ensures a Balanced Cost (BC) property, uniquely characterizing this rule for monotonic interval cost games.

A case paper involving Spanish airport perfume inventories illustrates that the interval Shapley rule results in fairer, more precise, and stable cost intervals than non-cooperative or simplistic allocations, even as computational complexity grows with the number of agents.

5. Influence, Power, and Importance in Interval Systems

The interval setting extends power indices to continuous aggregation regimes, relevant in economics, engineering (e.g., sensor networks), and security analysis (Kurz, 2018). The pivotality concept is reframed in terms of average uncertainty reduction when an agent’s value is revealed:

• For general aggregation functions $f\colon [0,1]^n\to [0,1]$, the Shapley-type importance measure is:

$$v_i(f) = \frac{1}{n!} \sum_{T \in S_n} \int_{[0,1]^n} \left[f(x_{T_{<i}}, 1, x_{T_{>i}}) - f(x_{T_{<i}}, 0, x_{T_{>i}})\right]\,dx$$

This framework is adaptable to inference about agent (or variable) importance in ML explainability and attribution contexts, provided the aggregation map is monotonic and continuous on the interval.

6. Fairness, Sharp Thresholds, and Relation to Banzhaf Values

A core insight is that uniform smallness of interval Shapley values across agents implies that threshold phenomena—e.g., abrupt transition from one collective state to another—become sharp (Kalai et al., 9 Feb 2025). For monotone Boolean functions, the length of the $\varepsilon$-threshold interval, where the function switches from mostly 0 to mostly 1, is bounded by $O(1/\log 1/t)$ with $t$ the maximal Shapley value:

$$p_{1-\varepsilon}(f) - p_{\varepsilon}(f) \leq C\frac{\log(1/\varepsilon)}{\log(1/t)}$$

Similarly, in the balanced function case, small Banzhaf values (individual influences at $p=1/2$) imply small maximal Shapley values. This demonstrates the power indices' relevance for analyzing and designing robust, balanced cooperative systems.

7. Computational Considerations and Limitations

Interval Shapley computation inherits, and typically exceeds, the computational complexity of the discrete case due to the combinatorial growth in the number of coalition orderings and, in the interval case, the necessity to evaluate bounds for all relevant permutations.

• For practical deployments (e.g., cost allocation in large consortiums), sampling and approximation methods are required for scalability.
• The rule’s meaningfulness depends on certain monotonicity and consistency conditions within the interval game; failure to satisfy these may render the interval operations ill-defined.
• The interpretability and informativeness of allocations decrease as parameter intervals widen, reflecting intrinsic data uncertainty rather than computation per se.

Summary Table: Central Formulas and Properties

Context	Mathematical Expression	Key Property
Generalized Shapley value (interval)	$$\displaystyle \Psi_i(v) = \frac{1}{n!} \sum_{\pi\in S_n} \int [\Delta v(\cdot)] dx$$	Integrates pivotality over interval input space
Interval inventory cost allocation	$$\displaystyle \text{Sh}_j = [\varphi_j(N, \underline{w}),\, \varphi_j(N, \overline{w})]$$	Allocates cost interval per agent
Linear Shapley Rule (two-tier voting)	$$\displaystyle \varphi(q; w_1,\dots,w_m) \propto (n_1,\dots,n_m)$$	Proportional fairness for correlated voter blocks
Threshold interval bound (Boolean case)	$$\displaystyle p_{1-\varepsilon}(f) - p_{\varepsilon}(f) \leq C\frac{\log(1/\varepsilon)}{\log(1/t)}$$	Sharpness linked to max Shapley value

This synthesis outlines the Interval Shapley Rule’s central role in modern extensions of cooperative game theory and power index analysis, encompassing technical foundations, axiomatization, fairness criteria under continuous decision spaces, and robust allocation mechanisms under uncertainty.