Engineering Social Optimality via Utility Shaping in Non-Cooperative Games under Incomplete Information and Imperfect Monitoring

Abstract: In this paper, we study decentralized decision-making where agents optimize private objectives under incomplete information and imperfect public monitoring, in a non-cooperative setting. By shaping utilities-embedding shadow prices or Karush-Kuhn-Tucker(KKT)-aligned penalties-we make the stage game an exact-potential game whose unique equilibrium equals the (possibly constrained) social optimum. We characterize the Bayesian equilibrium as a stochastic variational inequality; strong monotonicity follows from a single-inflection compressed/stretched-exponential response combined with convex pricing. We give tracking bounds for damped-gradient and best-response-with-hysteresis updates under a noisy public index, and corresponding steady-state error. The framework accommodates discrete and continuous action sets and composes with slower discrete assignment. Deployable rules include: embed prices/penalties; publish a single public index; tune steps, damping, and dual rates for contraction. Computational experiments cover (i) a multi-tier supply chain and (ii) a non-cooperative agentic-AI compute market of bidding bots. Relative to price-only baselines, utility shaping attains near-centralized welfare, eliminates steady-state constraint/capacity violations when feasible, and accelerates convergence; with quantization, discrete equilibria track continuous ones within the mesh. The blueprint is portable to demand response, cloud/edge scheduling, and transportation pricing and biosecurity/agriculture. Overall, utility shaping plus a public index implements the constrained social optimum with stable equilibria under noise and drift-an operations-research-friendly alternative to heavy messaging or full mechanism design.