From Necklaces to Coalitions: Fair and Self-Interested Distribution of Coalition Value Calculations
Authors: Terry R. Payne, Luke Riley
Organizations: School of Computer Science and Informatics, University of Liverpool, Liverpool, UK. · Quant, London, UK.
Abstract
A key challenge in distributed coalition formation within characteristic function games is determining how to allocate the calculation of coalition values across a set of agents. The number of possible coalitions grows exponentially with the number of agents, and existing distributed approaches may produce uneven or redundant allocations, or assign coalitions to agents that are not themselves members. In this article, we present the \emph{Necklace-based Distributed Coalition Algorithm} (N-DCA), a communication-free algorithm in which each agent independently determines its own coalition value calculation allocation using only its identifier and the total number of agents. The approach builds on the notion of Increment Arrays (IAs), for which we develop a complete mathematical framework: equivalence classes under circular shifts, periodic IAs, and a rotated designation scheme with formal load-balance guarantees (tight bounds). We establish a bijection between canonical representative IAs and two-colour combinatorial necklaces, enabling the use of efficient necklace generation algorithms to enumerate allocations in constant amortised time. N-DCA is, to the best of our knowledge, the only distributed coalition value calculation algorithm for unrestricted characteristic function games to provably satisfy five desirable properties: no inter-agent communication, equitable allocation, no redundancy, balanced load, and self-interest. An empirical evaluation against DCVC (Rahwan and Jennings 2007) demonstrates that, although DCVC is faster by a constant factor, this difference becomes negligible under realistic characteristic-function evaluation costs, while N-DCA offers advantages in working memory, scalability, and the self-interest guarantee.
This paper studies coalition formation as a decentralized dynamical process driven by unilateral exit-and-join decisions. Agents evaluate local moves using the Aumann-Dreze value, so payoffs are computed within the agent's current coalition rather than through a globally negotiated coalition structure. The resulting model links cooperative payoff allocation with noncooperative best-response behavior: a terminal partition is precisely a coalition structure with no admissible, individually profitable exit-and-join deviation. We establish equilibrium characterizations, identify conditions under which the dynamics admit scalar Lyapunov or exact-potential representations, and analyze how switching and acceptance costs shape local stability. Numerical experiments test finite-time stabilization, cost sensitivity, and a special convex-game benchmark.
Additively separable hedonic games (ASHGs) are a prominent model of coalition formation where agents' preferences are derived from their individual valuations of peers. While social welfare maximization in ASHGs has traditionally focused mostly on utilitarian welfare, Nash welfare -- a well-established metric in economics which balances fairness with efficiency and offers scale invariance -- has been entirely overlooked. In this paper, we initiate the study of Nash welfare in ASHGs. We point out desirable properties fulfilled by partitions with high Nash welfare. This includes guaranteed contractual Nash stability in symmetric games, even for any approximation of Nash welfare. This is particularly appealing since, as for other welfare notions, Nash welfare turns out to be NP-hard to maximize, even for the ASHG subclass of symmetric aversion to enemies games (AEGs). A main focus of our study is on approximation algorithms for the Nash welfare objective. We present packing-based algorithms with approximation ratios for well-established subclasses of ASHGs: n−1 for AEGs and 2n for appreciation of friends games. This is complemented by a strict inapproximability result showing it is NP-hard to approximate Nash welfare within a factor of 1.0000759 in general ASHGs. Further, we investigate the restricted settings with an upper bound on the coalition size or number of coalitions, and draw the boundary between the cases admitting efficient algorithms and those yielding NP-hardness: bounding the allowed size or number of coalitions by 2 admits polynomial-time solvability, whereas bounds of 3 or more yield NP-hardness or unbounded inapproximability.
Traditional distributed consensus protocols classify nodes as either honest-but-faulty or actively malicious (Byzantine). However, in organizational structures, departmental agents rarely fit this binary. Instead, they exhibit bounded rationality and self-interested preferences while operating under asymmetric information. This paper presents the Organizational Consensus Algorithm (OCA), a mechanism design framework tailored for internal negotiation and decision coordination. OCA models inter-departmental conflict as an incomplete information dynamic game, integrating internal token staking, an exception-triggered challenge mechanism, and confidence-weighted consensus rules. Rather than enforcing instantaneous total ordering, OCA leverages a retrospective penalty system driven by delayed verifiable outcomes to deter structural bias and reduce exhaustive coordination overhead. A Python simulation prototype was developed to evaluate OCA. Across independent trials with varying organizational scales, OCA reports lower coordination overhead, higher informative reporting rates, and bounded welfare loss in noisy environments. Crucially, these results remain conditional on the stated simulation model and do not by themselves establish a general truthful equilibrium.