The Fixed Server Locality Gap of Count Load Assignment Games

RESUMO

Marginal-contribution pricing makes the social objective an exact potential, but does not ensure that every stable assignment is efficient. We study indivisible clients with heterogeneous workloads, arbitrary nonnegative assignment costs, and private eligibility menus over a fixed number of shared servers and optional local execution. Each server's social cost is its occupancy multiplied by its aggregate workload and a server-specific coefficient. For every fixed server cap $M$, we prove that the price of anarchy over this class is $\Theta_M(\kappa^{1-2^{-M}})$ as the workload-ratio cap $\kappa$ grows. The upper bound applies to every equilibrium and every feasible comparison, without an acyclic-comparison or strongly-connected-component restriction. Its proof combines a common-multiplier certificate, a source-weighted residual inequality and an ordered moment recurrence. A matching chain family has a realization by mobile unmanned aerial vehicle (UAV) clients and fixed ground servers, with positive altitude, positive-width coverage and an explicitly paid wireless baseline. We prove exact waypoint elimination and give an input-computable component refinement. Scheduling and congestion identities delimit which established bounds transfer. Exact enumeration of 2,800 synthetic games identifies inefficient equilibria, while 270 primary-parameterized synthetic instance-budget records evaluate incremental latency relative to a feasible multistart reference. Tightness concerns the heterogeneity exponent for fixed $M$, not matching leading constants or measured frequency of the worst-case configurations.