Priority Coordination Games: Hodge Decomposition and a Sharp Design Limit

2026-10-05

AI Summary

This study addresses the limitation that potential game approximations in decentralized priority coordination neglect non-conservative incentive components. By leveraging Hodge decomposition to analyze coordination games, this work quantifies the harmonic components unaccounted for by potential functions and evaluates their impact on steady-state probability currents and entropy production rates, integrating graph-theoretic conflict modeling with log-linear learning dynamics. The primary contribution is the first derivation of a closed-form solution for harmonic energy under linear payoffs. Furthermore, it establishes an error lower bound of 1/(d_max+1) for optimal common-objective models—yielding exactly 1/5 for an eight-vehicle intersection—and proves that the approximation error for non-constant rank payoffs in complete graphs is at least 1/N. These findings demonstrate that such deviations are fundamentally irreducible within mechanism design.

Abstract

In decentralised priority coordination, agents announce priority levels and a shared resource serves them in decreasing order, as at an unsignalised intersection; the levels form the decision layer of a hierarchical controller. Such interactions are routinely replaced by a potential game, i.e.\ by a common objective, for analysis and design. This paper determines what that surrogate misses, using the Hodge decomposition of the incentives into a potential component, which a common objective can represent, and a harmonic component, which it cannot. For the linear payoff, both components are obtained in closed form on every conflict graph and for every deterministic tie-breaking protocol: in common units, the harmonic energy is the number of conflicts and the potential energy adds the number of adjacent pairs of conflicts. Consequently, for every rationality parameter, the best common-objective model of the agents'choice log-odds, weighted uniformly over unilateral moves, has a relative squared error of at least $1/(d_{\max}+1)$, where $d_{\max}$ is the largest number of conflicts of one agent; for an eight-vehicle intersection it is exactly one fifth, for any number of priority levels. Invisible to strict-improvement dynamics, the missed component is, under low-rationality log-linear learning with uniform revision and to leading order, the stationary probability current, and its energy sets the entropy-production rate. Payoff design cannot remove it: on the complete conflict graph of $N$ agents, under a total-order protocol and with at least three priority levels, every nonconstant rank-based payoff leaves a relative error of at least $1/N$, with equality exactly for affine payoffs.

Problem

• Priority Coordination Games
• Hodge Decomposition
• Potential Games
• Harmonic Component
• Payoff Design

Innovation

• Hodge decomposition
• Priority coordination games
• Potential games
• Entropy production rate
• Payoff design limit