Bayesian Learning with Adaptive Load Allocation Strategies

We study a Bayesian learning dynamics induced by agents who repeatedly allocate loads on a set of resources based on their belief of an unknown parameter that affects the cost distributions of resources. In each step, belief update is performed according to Bayes' rule using the agents' current load and a realization of costs on resources that they utilized. Then, agents choose a new load using an adaptive strategy update rule that accounts for their preferred allocation based on the updated belief. We prove that beliefs and loads generated by this learning dynamics converge almost surely. The convergent belief accurately estimates cost distributions of resources that are utilized by the convergent load. We establish conditions on the initial load and strategy updates under which the cost estimation is accurate on all resources. These results apply to Bayesian learning in congestion games with unknown latency functions. Particularly, we provide conditions under which the load converges to an equilibrium or socially optimal load with complete information of cost parameter. We also design an adaptive tolling mechanism that eventually induces the socially optimal outcome.