Regret Bounds for Competitive Resource Allocation with Endogenous Costs
This provides a formal regret-theoretic justification for decentralized competitive allocation in modular systems, addressing a fundamental challenge distinct from partial observability.
The paper tackles the problem of online resource allocation with endogenous costs, where costs depend on interactions among modules, and shows that competitive allocation via multiplicative weights achieves O(sqrt(T log N)) regret, outperforming uniform (Omega(T)) and gated (O(T^{2/3})) methods.
We study online resource allocation among N interacting modules over T rounds. Unlike standard online optimization, costs are endogenous: they depend on the full allocation vector through an interaction matrix W encoding pairwise cooperation and competition. We analyze three paradigms: (I) uniform allocation (cost-ignorant), (II) gated allocation (cost-estimating), and (III) competitive allocation via multiplicative weights update with interaction feedback (cost-revealing). Our main results establish a strict separation under adversarial sequences with bounded variation: uniform incurs Omega(T) regret, gated achieves O(T^{2/3}), and competitive achieves O(sqrt(T log N)). The performance gap stems from competitive allocation's ability to exploit endogenous cost information revealed through interactions. We further show that W's topology governs a computation-regret tradeoff. Full interaction (|E|=O(N^2)) yields the tightest bound but highest per-step cost, while sparse topologies (|E|=O(N)) increase regret by at most O(sqrt(log N)) while reducing per-step cost from O(N^2) to O(N). Ring-structured topologies with both cooperative and competitive links - of which the five-element Wuxing topology is canonical - minimize the computation x regret product. These results provide the first formal regret-theoretic justification for decentralized competitive allocation in modular architectures and establish cost endogeneity as a fundamental challenge distinct from partial observability. Keywords: online learning, regret bounds, resource allocation, endogenous costs, interaction topology, multiplicative weights, modular systems, Wuxing topology