Coalition formation is usually studied as a static problem: given a set of agents and their values, find the stable partition. But coalitions in practice form and dissolve dynamically — agents join groups, discover they're being treated unfairly, leave, and regroup.
Zhu and Han model this as a control system. Splits occur when an agent's Shapley value in a coalition is negative — they contribute more than they receive. Merges occur when combining two groups would produce surplus gains for all members. The dynamics track two quantities simultaneously: fairness gaps (how far each agent is from their Shapley value) and system surplus (how much total value remains uncaptured by the current partition).
A vector Lyapunov function combines both measures. The discrete-time LaSalle invariance principle guarantees convergence to what they call Shapley-Fair Merge-Stable partitions — states where no agent has a fairness deficit and no pair of coalitions would benefit from merging. Convergence is finite-time, not just asymptotic.
The structural point: fairness and efficiency are usually in tension in mechanism design — optimizing one degrades the other. Here, they're coupled through the dynamics rather than traded off in the objective. The split mechanism enforces fairness (removing exploited agents), the merge mechanism recovers efficiency (capturing unclaimed surplus), and the Lyapunov function ensures both improve monotonically. The resolution comes from treating coalition formation as a process with a direction, not an optimization with a constraint.