Weighted-Selective-Split Interaction Mechanism

• Weighted-selective-split interaction mechanism is a formal framework that models heterogeneous interactions using weighted participation factors and decomposed substructures.
• It underpins weighted interaction indexes in cooperative games and enables efficient algorithms for weighted list-coloring in split graphs under local constraints.
• The mechanism is applied in voting, network reliability, and decision making, balancing computational tractability with realistic modeling of non-uniform coalition formation.

A weighted-selective-split interaction mechanism is a formal structure for quantifying and leveraging the interplay between components—typically agents or subsets—in systems where interactions are both weighted (reflecting non-uniform likelihood or importance) and split (decomposed into substructures with differentiated interaction rules or constraints). This construct appears prominently in two major areas: cooperative game theory, in the guise of weighted interaction indexes for coalition games (Marichal et al., 2010), and combinatorial optimization, as an underpinning for weighted list-coloring in split graphs (Bentz, 2017). In both domains, the mechanism addresses how local or global constraints arising from system heterogeneity are coordinated or reconciled by algorithms or analytic indexes, taking into account probabilistic or weighted participation factors.

1. Weighted Interaction Indexes in Cooperative Games

Weighted interaction indexes generalize classical game-theoretic interaction measures—most notably, the Banzhaf interaction index—by introducing a weighted (probabilistic) perspective on coalition formation. Each coalition $S$ is assigned a weight $w(S)$, reflecting the probability or importance of $S$'s formation. In the independent probability model, $$w(S) = \prod_{i \in S} p_i \prod_{i \notin S}(1-p_i)$$, where $p_i$ is the activation probability for player $i$. The classical Banzhaf index corresponds to the special case $p_i = 1/2$ for all $i$. Weighted interaction indexes thus allow for differential likelihoods of coalition formation, accommodating more realistic modeling of systems with heterogeneity or empirical interaction frequencies.

Key properties of these indexes include:

• Linearity: The mapping $f \mapsto I_{B,p}(f,S)$ is linear in the pseudo-Boolean function $f$.
• Expected Difference Interpretation: $I_{B,p}(f,S)$ equals the expected discrete $S$-derivative of $f$ under weight $w$.
• Probabilistic Normalization: Coefficients arising in expansions can be interpreted as (conditional) probabilities of coalitions.
• Symmetry Under Uniform Weights: When $w$ is symmetric, the index reduces to a symmetric, cardinal-probabilistic interaction index.

2. Mathematical Formulation and Explicit Characterizations

For a game $f: \{0,1\}^n \to \mathbb{R}$ and a degree $k$, the best weighted least squares approximation is the unique multilinear polynomial $f_k$ minimizing

$$E_w[(f(x) - g(x))^2] = \sum_{x\in\{0,1\}^n} w(x)(f(x) - g(x))^2$$

where $w(x)$ is the weight/probability of coalition $x$.

The coefficient $a_k^*(S)$, associated with monomial $x_S$, is defined as the weighted interaction index $I_{B,p}(f,S)$. An explicit combinatorial formula is

$$I_{B,p}(f, S) = \sum_{T \supseteq S} (-1)^{|T\backslash S|} \left[\prod_{i \in T\backslash S} p_i \right] \left[\prod_{i \in N \backslash T} (1-p_i)\right] f(T)$$

Alternatively, this may be written as a weighted inner product: $$I_{B,p}(f, S) = \sum_{x \in \{0,1\}^n} w(x)\, f(x)\, v_S(x)$$ where $v_S(x)$ is the orthonormalized basis element for $S$ with respect to $w$.

3. Connection to Probabilistic Interaction Indexes

Weighted interaction indexes are a subclass of the family of probabilistic interaction indexes, where each index is formalized as an expected value of the discrete $S$-derivative over a probability measure on coalitions. In this context,

$$I_{B,p}(f, S) = \sum_{T \subseteq N \backslash S} Pr(T|S^c)\, (\Delta_S f)(T)$$

with $Pr(T|S^c)$ as the conditional probability on remaining players, parameterized by the $p_i$. The standard Banzhaf and Shapley interaction indexes are recovered as average (center of mass) values over all probabilities $p$ or over uniform $p$, respectively: $I_B(f,S) = \int I_{B,p}(f, S) dp$ Thus, the weighted mechanism interpolates continuously between classical power indexes and general probabilistic participation models.

4. Interpretation and Practical Applications

The center of mass interpretation provides a unifying view of classical indexes as averages over all possible parameterizations of the coalition probabilities. The weighted-selective-split interaction mechanism enables direct modeling of systems where coalition probabilities are non-uniform and empirically estimable.

Key application domains include:

• Voting and Political Coalitions: Modeling non-uniform formation probabilities of parties/blocs.
• Reliability and Network Systems: Assigning failure or functioning probabilities to components or subnetworks.
• Multi-criteria Decision Making: Weighting coalitions according to observed or desired frequencies, allowing interaction measures to reflect actual importance or likelihood of simultaneous criterion satisfaction.

The framework also permits computation of normalized indexes, such as Pearson correlation coefficients or $R^2$ for model fit in approximating games, enhancing its utility in model assessment and comparison.

5. Weighted-Selective-Split Interaction in Combinatorial Optimization

In combinatorial optimization, the mechanism underpins algorithms for the weighted locally bounded list-coloring problem in split graphs (Bentz, 2017). In this context, the split structure (clique $K$ and independent set $S$) interacts with weighted constraints representing local demands.

Formally, the problem asks for a coloring $f: V \to \{1, \ldots, k\}$ respecting:

• List constraints $f(v) \in L(v)$,
• Proper coloring,
• Local weight bounds: For each part $V_i$ and color $j$,

$\sum_{v \in V_i, f(v)=j} w(v) = W_{ij}$

In split graphs, where $K$ is a clique and $S$ an independent set, the assignment of colors to $K$ (subject to unique colors) is small when $k=O(1)$. Once colors are assigned to $K$, the remaining problem on $S$ reduces to a matching/flow problem for distributed weight quotas, allowing a (pseudo)polynomial-time algorithm.

6. Complexity, Flexibility, and Limitations

The efficiency of the weighted-selective-split mechanism in split graphs depends crucially on bounded parameters (e.g., number of colors $k$). If $k$ or weights are unbounded, the problem is NP-complete. This dichotomy reveals that the mechanism's tractability is tied to the selective split into strongly constrained (clique) and weakly constrained (independent set) substructures, governed by local weights and quotas.

For weighted interaction indexes in games, a major modeling decision is the specification of weight vectors $p$ or distributions $w$. The sensitivity to these parameters means that empirical or justified selection directly impacts measured interactions, requiring caution in interpretation when probabilities are only estimated or are otherwise uncertain.

7. Synthesis and Comparative Perspective

The weighted-selective-split interaction mechanism generalizes classical, homogeneous approaches by introducing weights that encode heterogeneous likelihoods, importance, or constraints at the coalition or subgraph level. In game theory, this yields a smooth continuum between Banzhaf, Shapley, and fully probabilistic indexes; in combinatorial optimization on split graphs and related structures, it yields efficient algorithms when split-part constraints and weights are bounded. In both cases, the mechanism formalizes the interplay between strict (selective) and relaxed (split) constraints under weighted participation, characterizing a wide spectrum of real-world systems with a unified and extendable analytic formalism.