InfluenceNet: AI Models for Banzhaf and Shapley Value Prediction

TL;DR

Power indices quantify agent influence in cooperative games but exact computation scales poorly with coalition size, exhibiting exponential complexity $O(2^m)$ and $O(m!)$ for Banzhaf and Shapley-Shubik indices. The paper demonstrates that a simple neural network, inspired by Marginal Contribution Networks, can accurately approximate both indices for large $n$ ($n \ge 10$) while dramatically reducing computation time. It provides a comprehensive workflow—dataset generation via three strategies, Monte Carlo labeling, feedforward NN training with a three-layer architecture, and graph-based analyses—to benchmark speed and accuracy against existing tools. This approach broadens the practical applicability of power-index analysis to large-scale voting systems and complex multi-agent environments, enabling rapid assessment of influence dynamics in contexts such as corporate governance, distributed networks, and political decision-making.

Abstract

Power indices are essential in assessing the contribution and influence of individual agents in multi-agent systems, providing crucial insights into collaborative dynamics and decision-making processes. While invaluable, traditional computational methods for exact or estimated power indices values require significant time and computational constraints, especially for large $(n\ge10)$ coalitions. These constraints have historically limited researchers' ability to analyse complex multi-agent interactions comprehensively. To address this limitation, we introduce a novel Neural Networks-based approach that efficiently estimates power indices for voting games, demonstrating comparable and often superiour performance to existing tools in terms of both speed and accuracy. This method not only addresses existing computational bottlenecks, but also enables rapid analysis of large coalitions, opening new avenues for multi-agent system research by overcoming previous computational limitations and providing researchers with a more accessible, scalable analytical tool.This increased efficiency will allow for the analysis of more complex and realistic multi-agent scenarios.

Figures (11)

• Figure 1: Models performance on uniform random datasets Mean Absolute Error (MAE) across different dataset configurations The x-axis represents the sparsity threshold (p), and the y-axis shows the MAE values. Subplots show variations across: (A-C) uniform rule values with 20 rules and 10, 20, and 50 agents respectively; (D-F) low-variance Gaussian distribution (mean=5) with 20 rules across 10, 20, and 50 agents; (G-I) high-variance Gaussian distribution (mean=15) with 20 rules for 10 and 20 agents. Each line within subplots represents a model trained with a different p value while maintaining consistent coalition parameters.
• Figure 2: Impact of rule value distribution on model performance using Mean Absolute Error (MAE). Comparison across (A) uniform rule value, (B) low-variance Gaussian rule value, and (C) high-variance Gaussian rule value distributions. All configurations maintain consistent number of rules and agents. The x-axis represents the sparsity threshold (p), and the y-axis shows MAE values.
• Figure 3: Cross-model comparison of Mean Absolute Error (MAE) for different numbers of agents. Results show model performance on datasets with (A) 10 agents, (B) 20 agents, and (C) 50 agents, while maintaining consistent rule values and number of rules across all configurations. The x-axis represents the sparsity threshold (p), and the y-axis shows MAE values.
• Figure 4: Impact of number of agents distribution on model performance using Mean Absolute Error (MAE). Comparison across (A) n=10, (B) n=20 (C) n=50 agents. All configurations maintain consistent number of rules and rule value, but are unpadded and so have different dimensions. The x-axis represents the datasets sparsity threshold (p), and the y-axis shows MAE values.
• Figure 5: Correlation comparison of coalition Banzhaf distribution and the coalitions as a graph statistics. All figures were based on coalitions with p threshold of 0.1, n=10 agents and uniform rule value. Notice the similar strength and reverse sign comparing the mean Banzhaf values and the 'req' and 'ban' average degree in figure (A) and (B), respectively.