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 ≳ K log m, and hence yields welfare-optimal structures. The second scheme uses an ℓ 1 -penalised estimator; under a restricted eigenvalue condition, we derive ℓ 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.
Coalition structure generation (CSG) is a central optimisation problem in cooperative game theory and multi-agent systems: given a set of agents N = {1, . . . , n} and a characteristic function v : 2 N → R, the goal is to partition N into disjoint coalitions so as to maximise the total social welfare C∈P v(C) over all coalition structures P. The search space has size given by the Bell numbers, which grow super-exponentially with n, and even in the simplest transferable utility settings, CSG is NP-hard and demands sophisticated algorithmic machinery [1,2]. Over the last two decades, an extensive line of work has developed dynamic programming, branch-and-bound, and anytime algorithms for exact or approximate CSG, with a strong emphasis on worst-case complexity and pruning of the coalition structure search space [3,4,5,6,7,2]. More recent work in Algorithmica and related venues has further explored structural and complexity aspects of coalition structures [8,9], including preferences over partitions and refined efficiency concepts.
In parallel, the multi-agent systems and JAAMAS communities have developed a rich literature on coalition formation under uncertainty, where coalition values are not known a priori but must be learned or inferred from data. One influential line of work adopts a Bayesian reinforcement learning viewpoint: agents repeatedly form coalitions, observe noisy payoffs, and update beliefs about the types or capabilities of their potential partners, leading to new concepts such as the Bayesian core and sequentially optimal coalition formation policies [10,11]. These models explicitly acknowledge that, in many applications, coalition values are only revealed through episodic interaction, and they provide principled ways to balance exploration and exploitation in repeated coalition formation. More generally, surveys and case studies in cooperative and multi-objective MAS highlight coalition formation as a key mechanism for distributed decision making and resource allocation, with applications ranging from energy systems to communication networks and robotics [12,13].
The work on probabilistic CSG (PCSG) provides a complementary formalism in which the source of uncertainty lies in stochastic agent attendance rather than in noisy payoffs. In the PCSG model of [14], each agent is active in a given instance with a specified probability, and the goal is to find a coalition structure maximising expected social welfare under this probabilistic attendance model. Subsequent work has designed approximation algorithms and studied the complexity of PCSG in more detail, including two approximation schemes for PCSG and related variants in the JAAMAS literature [15,16]. These contributions firmly place PCSG alongside classical CSG as a fundamental combinatorial optimisation problem, but they focus primarily on the algorithmics of computing good coalition structures within the probabilistic attendance model itself.
In many real-world settings, however, the uncertainty is neither purely combinatorial (random attendance) nor purely epistemic in the sense of unknown but fixed coalition values. Instead, system designers often observe episodic data of the form (X t , Y t ), t = 1, . . . , T, where X t ∈ {0, 1} m encodes which of m candidate coalitions (or coalition features) are active in episode t, and Y t ∈ R is a noisy scalar reward generated by the underlying characteristic function. Examples include online advertising and sponsored search, where different subsets of advertisers are shown in each round; logistics and transport, where subsets of carriers collaborate on shared loads; and robotic or sensing teams, where only a few coalition templates are actually instantiated in each episode. In these domains, one rarely observes full values v(C) for all C ⊆ N ; instead, one observes a sequence of noisy aggregate payoffs and must both learn about the underlying cooperative structure and exploit it to compute high-welfare coalition structures. This motivates a probabilistic CSG framework that is explicitly data-driven and statistical in nature. From a modelling perspective, one natural way to exploit structure in episodic observations is to posit that only a small number of coalitions have non-negligible marginal contributions to welfare. Formally, if we index a collection of m candidate coalitions by j ∈ [m] and write θ ⋆ j for the (unknown) contribution of coalition j, then the assumption that only K ≪ m of these are truly relevant corresponds to a sparse parameter vector θ ⋆ ∈ R m . The observed episodic payoffs can then be modelled as
with sub-Gaussian noise ε t , exactly in the spirit of high-dimensional sparse linear models familiar from compressive sensing and statistical learning. In this view, probabilistic CSG becomes a two-level problem: (1) statistically, we must recover the support and magnitude of θ ⋆ from T episodes, ideally using T that scales only logarithmically with m; and (2) combinatorially,
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