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Bayesian Recovery for Probabilistic Coalition Structures

Angshul Majumdar

TL;DR

This work casts Probabilistic Coalition Structure Generation (PCSG) as a sparse recovery problem $y = A w^{\star} + \varepsilon$ and analyzes three sparse estimation paradigms under highly coherent, near-duplicate coalition designs. It shows that $\ell_1$-based methods and Orthogonal Matching Pursuit fail to recover the true coalition support in this regime due to violation of the irrepresentable condition and indistinguishable residual correlations, respectively. In contrast, Sparse Bayesian Learning with a Gaussian-Gamma hierarchy achieves support consistency, exploiting a concave sparsity penalty that suppresses spurious near-duplicates and concentrates mass on true coalitions with high probability as the sample size grows. The results establish a rigorous separation between convex, greedy, and Bayesian sparse approaches for PCSG, with Bayesian methods offering robust recovery under extreme column coherence and uncertainty. This has implications for coalition formation under uncertainty and motivates further empirical and theoretical exploration of Bayesian sparse methods in cooperative game contexts.

Abstract

Probabilistic Coalition Structure Generation (PCSG) is NP-hard and can be recast as an $l_0$-type sparse recovery problem by representing coalition structures as sparse coefficient vectors over a coalition-incidence design. A natural question is whether standard sparse methods, such as $l_1$ relaxations and greedy pursuits, can reliably recover the optimal coalition structure in this setting. We show that the answer is negative in a PCSG-inspired regime where overlapping coalitions generate highly coherent, near-duplicate columns: the irrepresentable condition fails for the design, and $k$-step Orthogonal Matching Pursuit (OMP) exhibits a nonvanishing probability of irreversible mis-selection. In contrast, we prove that Sparse Bayesian Learning (SBL) with a Gaussian-Gamma hierarchy is support consistent under the same structural assumptions. The concave sparsity penalty induced by SBL suppresses spurious near-duplicates and recovers the true coalition support with probability tending to one. This establishes a rigorous separation between convex, greedy, and Bayesian sparse approaches for PCSG.

Bayesian Recovery for Probabilistic Coalition Structures

TL;DR

This work casts Probabilistic Coalition Structure Generation (PCSG) as a sparse recovery problem and analyzes three sparse estimation paradigms under highly coherent, near-duplicate coalition designs. It shows that -based methods and Orthogonal Matching Pursuit fail to recover the true coalition support in this regime due to violation of the irrepresentable condition and indistinguishable residual correlations, respectively. In contrast, Sparse Bayesian Learning with a Gaussian-Gamma hierarchy achieves support consistency, exploiting a concave sparsity penalty that suppresses spurious near-duplicates and concentrates mass on true coalitions with high probability as the sample size grows. The results establish a rigorous separation between convex, greedy, and Bayesian sparse approaches for PCSG, with Bayesian methods offering robust recovery under extreme column coherence and uncertainty. This has implications for coalition formation under uncertainty and motivates further empirical and theoretical exploration of Bayesian sparse methods in cooperative game contexts.

Abstract

Probabilistic Coalition Structure Generation (PCSG) is NP-hard and can be recast as an -type sparse recovery problem by representing coalition structures as sparse coefficient vectors over a coalition-incidence design. A natural question is whether standard sparse methods, such as relaxations and greedy pursuits, can reliably recover the optimal coalition structure in this setting. We show that the answer is negative in a PCSG-inspired regime where overlapping coalitions generate highly coherent, near-duplicate columns: the irrepresentable condition fails for the design, and -step Orthogonal Matching Pursuit (OMP) exhibits a nonvanishing probability of irreversible mis-selection. In contrast, we prove that Sparse Bayesian Learning (SBL) with a Gaussian-Gamma hierarchy is support consistent under the same structural assumptions. The concave sparsity penalty induced by SBL suppresses spurious near-duplicates and recovers the true coalition support with probability tending to one. This establishes a rigorous separation between convex, greedy, and Bayesian sparse approaches for PCSG.
Paper Structure (9 sections, 3 theorems, 24 equations)

This paper contains 9 sections, 3 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. PCSG-inspired sparse regression model
  3. Failure of $\ell_{1}$ and OMP in PCSG designs
  4. $\ell_{1}$ fails: irrepresentable condition violation
  5. OMP fails: nonvanishing mis–selection probability
  6. Sparse Bayesian Learning: support consistency in PCSG designs
  7. Main result
  8. Proof of Theorem \ref{['thm:sbl']}
  9. Conclusion

Key Result

Proposition 3.1

Assume the near–duplicate structure of Section sec:model and $\rho_{\mathrm{in}}\to 1$ as $m\to\infty$, while $\rho_{\mathrm{out}}<1$ stays fixed. Then, for all sufficiently large $m$, condition eq:irrep fails; in particular, for some constant $c>0$ independent of $m$.

Theorems & Definitions (5)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1