Causal Cooperative Game (CCG)

• CCG is a framework that conceptualizes machine learning tasks as cooperative games where nodes and labels act as players whose causal contributions are assessed.
• It employs Shapley value attribution in graph settings to capture group-level effects, enhancing stability and reducing variance under adversarial and distributional shifts.
• In multi-label classification, CCG integrates causal invariance and counterfactual rewards to improve rare-label prediction and overall model interpretability.

The Causal Cooperative Game (CCG) conceptualizes machine learning tasks as cooperative games in which multiple components—such as nodes in a graph or groups of labels—jointly contribute to robust prediction through explicit modeling of their causal and interaction effects. Integrating principles from causal inference and cooperative game theory, CCG frameworks provide a formal foundation for capturing group-level influences and promoting stability, interpretability, and generalization, especially under distributional shifts, adversarial perturbations, and rare-event settings (Xue et al., 20 May 2025, Fan et al., 30 Nov 2025).

1. Formalization of the Causal Cooperative Game Principle

CCG models learning schemas—such as neighborhood sampling in GNNs or label prediction in multi-label classification (MLC)—as cooperative games, where “players” (nodes or label subgroups) form coalitions whose contributions to the prediction task are assessed through causal and game-theoretic principles.

• Graph-based setting: Each neighbor $v_i$ of a target node $v_r$ is treated as a player, forming coalitions $S \subseteq N(v_r)$ whose collective effect on $v_r$'s label $y_r$ is measured via causal-graph–based value functions. The cooperative sampling payoff $v(S)$ for a coalition is determined by its causal impact on $y_r$ (Xue et al., 20 May 2025).
• MLC setting: The label set $\mathcal{L}$ is partitioned into disjoint causal subgraphs, each modeled as a player $P_k$. The utility $U_k$ for player $P_k$ encourages accurate classification, invariance across environments, and robustness via explicit counterfactual reasoning (Fan et al., 30 Nov 2025).

2. Cooperative Causal Modeling and Value Attribution

The core technical innovation in CCG is to treat the estimation of causal effects as a coalition-value assignment problem. This leverages the Shapley value from cooperative game theory to assess not only the direct causal effect of an individual component but, crucially, its average group-level contribution.

Shapley-Value Attribution in Graphs:

For a node $v_i$ in the neighborhood $N(v_r)$, its cooperative causal influence is quantified as

$$\phi_i(v) = \sum_{S \subseteq N(v_r) \setminus \{v_i\}} \frac{|S|!\,\bigl(|N(v_r)| - |S| -1\bigr)!}{|N(v_r)|!} \bigl(v(S\cup\{v_i\}) - v(S)\bigr)$$

where $v(S)$ is the causal payoff of coalition $S$ (Xue et al., 20 May 2025).

Neural SEM for Label Interaction:

In MLC, Neural Structural Equation Models (NSEMs) are constructed to capture directed causal dependencies among labels, parameterizing for each label $\ell_i$

$$h_i = f_i(x, \{h_j:j \in \text{Pa}(\ell_i)\}; \theta),\quad \hat y_i = \sigma(h_i)$$

with parameters learned to reflect causal relations and to amplify rare-label interactions (Fan et al., 30 Nov 2025).

3. Algorithmic Realizations

3.1 CoCa-Sampling in GNNs

The CoCa-sampling algorithm in Cooperative Causal GraphSAGE (CoCa-GraphSAGE) iteratively evaluates each candidate neighbor’s Shapley-valued causal contribution by marginalizing over all coalitions of fixed size. The steps involve:

1. For each $v_i \in N(v_r)$, sum over all $(M-1)$-sized coalitions $S$ not containing $v_i$.
2. Estimate the marginal causal weight of $v_i$ in $S$, approximated via kernel-density approaches.
3. Aggregate into a discrete distribution over $N(v_r)$ based on accumulated Shapley-weighted contributions.
4. Sample a set $C \subseteq N(v_r)$ with probability proportional to the normalized cooperative scores (Xue et al., 20 May 2025).

3.2 CCG Optimization in Multi-Label Classification

The CCG framework for MLC integrates:

• Causal invariance loss: Enforced via contrastive loss and cross-environment prediction consistency to ensure that learned representations are robust to spurious, non-causal variations.
• Counterfactual curiosity reward: Penalizes the divergence in model outputs between true and counterfactual samples, using Jensen–Shannon divergence to focus learning on causally relevant features.
• Rare-label enhancement: Employs amplification factors in the loss for rare labels, dynamic label-reweighting, and a priority queue to up-weight underperforming classes periodically (Fan et al., 30 Nov 2025).

4. Empirical Results and Robustness

CoCa-GraphSAGE (Graph Representation Learning)

• Competitive clean accuracy compared to GraphSAGE, GCN, and GAT.
• Under feature and structure perturbations (Bernoulli XOR, Gaussian noise), outperforms individual-effect models by 8–15% (vs GraphSAGE) and by 4–10% (vs single-node Causal GraphSAGE).
• Substantially reduced prediction variance, indicating enhanced embedding stability under repeated trials (Xue et al., 20 May 2025).

CCG for Multi-Label Classification

• On rare-label prediction, removal of any core CCG component (SEM loss, counterfactual reward, invariance loss, rare-label weighting) results in 2–5% drop in Rare-F1, demonstrating the synergy and necessity of all facets.
• Under temporal OOD shifts, maintains Rare-F1 drops of ≈7.4%, versus ≈16.4% for non-CCG baselines.
• Qualitative causal graphs align with known domain structures, enhancing model interpretability (Fan et al., 30 Nov 2025).

5. Theoretical and Methodological Insights

CCG architectures establish that:

• Modeling group (coalition) effects alleviates confounding biases and elevates robustness, transcending limitations of methods that treat each component as an independent predictor.
• The Shapley-value–driven selection identifies maximally informative, noise-resistant subsets—whether nodes for aggregation in GNNs or label blocks for prediction in MLC.
• Incorporation of counterfactual perturbation and invariance learning provides principled defenses against spurious correlations and supports generalization.

A plausible implication is that CCG frameworks are intrinsically suited to domains where interaction effects, spurious correlations, or class imbalance play a major role.

6. Interpretability, Limitations, and Future Directions

• Interpretability is enhanced: CCGs yield explicit value decompositions (via Shapley values in GNNs or Neural SEM edges in MLC) which can be aligned with domain-knowledge causal structures.
• Computational demands increase due to coalition enumeration, but practical heuristics (sampling, pairwise decomposition) are deployed to ensure tractability.
• Open directions include broader integration with model-agnostic feature selection, extension to higher-order relational structures, and investigation into dynamic/online coalition modeling.

In summary, the Causal Cooperative Game paradigm unifies explicit causal discovery with cooperative interaction modeling, yielding robust, interpretable, and generalizable learning across graph and multi-label domains (Xue et al., 20 May 2025, Fan et al., 30 Nov 2025).