Sparse Probabilistic Coalition Structure Generation: Bayesian Greedy Pursuit and $\ell_1$ Relaxations
Angshul Majumdar
TL;DR
This work treats coalition structure generation as a data-driven, probabilistic problem where coalition values are learned from episodic observations. It introduces a sparse linear model for episode payoffs and presents two complementary estimation pipelines: Bayesian Greedy Coalition Pursuit (BGCP), a greedy OMP-like method with high-probability exact support recovery under coherence conditions, and an $\ell_1$-regularised estimator (Lasso) with non-asymptotic error and support bounds under a restricted eigenvalue condition. The authors prove end-to-end welfare guarantees, showing that BGCP achieves welfare-optimal coalition structures with high probability in the sparse regime, while the $\ell_1$ pipeline yields quantitative welfare gaps that shrink as $\sqrt{(\log m)/T}$. They also compare these sparse approaches to baselines (episodic plug-in and dense least-squares), identifying regimes where sparsity-based methods outperform and where classical dense methods may be more appropriate, with careful attention to design conditions, sample complexity $T = O(K\log m)$, and computational trade-offs. The results bridge high-dimensional statistics with cooperative game-theoretic CSG, offering principled, scalable strategies for learning and exploiting sparse coalition structure in uncertain environments. Practical impact lies in efficiently discovering welfare-enhancing coalitions from limited episodic data, enabling scalable planning in multi-agent systems under uncertainty.
Abstract
We study coalition structure generation (CSG) when coalition values are not given but must be learned from episodic observations. We model each episode as a sparse linear regression problem, where the realised payoff \(Y_t\) is a noisy linear combination of a small number of coalition contributions. This yields a probabilistic CSG framework in which the planner first estimates a sparse value function from \(T\) episodes, then runs a CSG solver on the inferred coalition set. We analyse two estimation schemes. The first, Bayesian Greedy Coalition Pursuit (BGCP), is a greedy procedure that mimics orthogonal matching pursuit. Under a coherence condition and a minimum signal assumption, BGCP recovers the true set of profitable coalitions with high probability once \(T \gtrsim K \log m\), and hence yields welfare-optimal structures. The second scheme uses an \(\ell_1\)-penalised estimator; under a restricted eigenvalue condition, we derive \(\ell_1\) and prediction error bounds and translate them into welfare gap guarantees. We compare both methods to probabilistic baselines and identify regimes where sparse probabilistic CSG is superior, as well as dense regimes where classical least-squares approaches are competitive.