Reward-Rate Congestion Games and Replicator--Dinkelbach Dynamics

Abstract

Reward rate is a key performance criterion in cyber-physical and robotic systems where time, workload, and coordination costs are limiting resources. We introduce reward-rate congestion games, where agents seek to maximize reward per unit execution time. The direct reward-rate game is generally not an exact potential game. We develop a Dinkelbach-based framework in which, for every fixed Dinkelbach parameter, the transformed game is an exact potential game. This yields a potential-level Dinkelbach iteration that terminates finitely at the optimal potential reward rate when the inner potential maximization problem is solved globally. We also provide a sufficient condition under which an equilibrium of the transformed game is an equilibrium of the original reward-rate game. To optimize aggregate performance, we introduce marginal externality corrections that make the corrected potential coincide with the Dinkelbach-transformed social reward-rate objective, thereby enabling optimization of the social reward rate. Finally, we develop a continuous-time replicator--Dinkelbach dynamics for reward-rate population games coupling fast replicator dynamics with a slow reward-rate update. We establish convergence of the fixed-parameter replicator dynamics, global asymptotic and local exponential stability of the reduced Dinkelbach dynamics, and local exponential stability of the coupled system for sufficiently slow Dinkelbach updates. The framework is illustrated on a continuous task-allocation problem.