SHAP-Based Interpretability Techniques

• SHAP-based interpretability techniques are methods that use Shapley values to quantify feature contributions and allocate credit in collaborative settings.
• They leverage share-function formalism to enforce fairness and reciprocity by measuring agents' marginal improvements in utility.
• These techniques ensure stability in data exchange protocols with tractable algorithmic solutions within the CLS complexity framework.

SHAP-based interpretability techniques refer to the use of Shapley value and its related cooperative game-theoretic credit allocation functions for quantifying and attributing value, utility, or impact among participants (such as agents, features, or datasets) in collaborative or competitive settings. In data exchange economies and multi-agent learning systems, the Shapley value formalism enables rigorous analysis of fairness, reciprocity, and stability in terms of agents' marginal contributions to communal utility. This approach forms the heart of a broad family of attribution mechanisms, formalized as share-functions, that guarantee principled division of surplus and enable robust design of incentive-compatible protocols in distributed data and machine learning systems (Akrami et al., 2 Dec 2024).

1. Shapley Value and Share-Function Formalism

The core formal mechanism is the Shapley value, originally defined for set functions $f(T)$. For a set of players $N$ and a value function $f(S)$ over subsets $S \subseteq N$, the Shapley value for player $i$ is given by

$$\phi_i(S) = \sum_{T\subseteq S\setminus\{i\}} \frac{|T|! (|S|-|T|-1)!}{|S|!} [f(T\cup\{i\}) - f(T)].$$

This measures, in expectation over permutations, the marginal contribution of $i$ to coalition utility.

For continuous, vector-valued utility scenarios (e.g., data exchanges), this is generalized via share-functions. In the data exchange model, each agent $i$ derives utility $u_i(x_i)$ from fractional shares of others' data, and the share-function

$$\psi_{ij}(x_j) = \frac{1}{n} \sum_{S\subseteq N\setminus\{i\}} \binom{n-1}{|S|}^{-1} [ u_j(x_j[S \cup \{i\}]) - u_j(x_j[S]) ]$$

quantifies agent $i$'s marginal impact on $j$'s benefit, aggregating over all coalition orders (Akrami et al., 2 Dec 2024).

2. Fairness and Reciprocity Principles

A central interpretability application of the Shapley mechanism in data markets is the enforcement of fairness, operationalized as reciprocity: agents' received benefit should be proportional to what they contribute to others, as measured through the share-functions. Reciprocity is defined via surplus: $\Delta_i(x) = \sum_{j} \psi_{ij}(x_j) - u_i(x_i).$ An exchange is exactly reciprocal if $\Delta_i(x) = 0$ for all $i$. Approximate forms (e.g., $\delta$-reciprocity where $|\Delta_i(x)| \leq \delta$) are used in algorithmic settings. The Shapley share, by satisfying monotonicity, normalization ($\psi_{ij}(x_j) = 0$ if $x_{ij}=0$), and efficiency ($u_j(x_j) = \sum_{i} \psi_{ij}(x_j)$), guarantees rigorous interpretability of data value transfers among agents (Akrami et al., 2 Dec 2024).

3. Stability (Core-Stability) and the Exchange Graph

Beyond fairness, interpretability in such systems requires stability guarantees: no coalition of agents should have incentive to unilaterally deviate and achieve strictly better outcome among themselves. This is formalized as core stability ($\epsilon$-core-stability), prohibiting profitable group deviations beyond a threshold $\epsilon$: $$\not\exists S\subseteq N, \exists y\ \text{on }S : u_i(y_i) > u_i(x_i)+\epsilon \quad \forall i \in S.$$ A combinatorial certificate for stability is given via the exchange graph $G(x,\alpha)$, where an acyclic structure certifies $\epsilon$-core-stability for $\alpha \leq \alpha(\epsilon)$. The cycle condition is

$\prod_{(i\to j)\in C} (1 - x_{ij}) = 0$

for all cycles $C$ (in exact form), connecting graph-theoretic and algebraic perspectives on interpretability of agent interactions (Akrami et al., 2 Dec 2024).

4. Existence and Uniqueness of Fair, Stable Solutions

A key result is that under mild regularity conditions (monotonicity, continuity of $u_i$, and required properties of $\psi_{ij}$), there always exists an exchange $x^*$ that achieves exact reciprocity and core-stability. The existence is constructive: by variable transformation $z_{ij} = \log(1/(1-x_{ij}))$, the feasible set becomes a convex polytope, and a continuous map $g:Z \to Z$ (adjusting flows based on surplus differentials) is shown, via Brouwer's fixed point theorem, to admit a fixed point corresponding to the required exchange. This establishes that data value attribution via Shapley shares is not only interpretable but also achievable in practice for general continuous (and not necessarily submodular) utilities (Akrami et al., 2 Dec 2024).

5. Algorithmic Computation and CLS Complexity

The computation of approximately fair and stable SHAP-based allocations falls into the complexity class CLS (Continuous Local Search), the intersection of PLS (Polynomial Local Search) and PPAD (Polynomial Parity Arguments on Directed graphs):

• Local search: a greedy algorithm iteratively balances surpluses, using lexicographic potentials to guarantee progress. Each step shifts fractional sharing along the exchange graph, respecting stability/reciprocity constraints. Under $L$-Lipschitz and submodular $u_i$, convergence to $\epsilon$-reciprocal, $\epsilon$-core-stable allocations is polynomial in $n$, $1/\epsilon$, and $L$.
• Fixed point: by piecewise-linear approximation $\tilde{g}$, an explicit reduction to finding an approximate fixed point for the flow adjustment map, also in polynomial time.

The CLS classification demonstrates the tractable and robust nature of these interpretability allocations compared to general cooperative game-theoretic division problems (Akrami et al., 2 Dec 2024).

6. Practical Implications and Protocols in Data Exchange

The theoretical framework for SHAP-based interpretability directly informs protocol design in real-world data exchange platforms, such as the AWA (Academic & Well-being Analytics) Data Exchange. Protocol implementations use:

• Shapley-based share-functions for transparent contribution crediting.
• Surplus tracking and local search for maintaining reciprocity and balance.
• Monitoring of the exchange graph for stability against strategic coalitions.
• Automated convergence to a nearly-fair, nearly-stable configuration, with all agents' marginal contributions quantifiable and published.

This guarantees both interpretability and robustness in collaborative data analytics, ensuring incentive compatibility, transparency, and coalition-proof data sharing (Akrami et al., 2 Dec 2024).

7. Connections to Broader Interpretability and Attribution Literature

SHAP-based methods originate from cooperative game theory and have broad application in model interpretability (e.g., SHAP explanations for feature attributions in machine learning), fairness-aware learning, and economic mechanism design. In the context of data sharing and exchange economies, their use as share-functions enables unambiguous, mathematically-grounded decomposition of value, and fosters novel connections between economic stability, algorithmic game theory, and interpretable machine learning. A plausible implication is the expanding applicability of SHAP-style value division to multi-party data governance regimes and collaborative AI systems (Akrami et al., 2 Dec 2024).