Safe Pricing Mechanisms for Distributed Resource Allocation with Bandit Feedback

In societal-scale infrastructures, such as electric grids or transportation networks, pricing mechanisms are often used as a way to shape users' demand in order to lower operating costs and improve reliability. Existing approaches to pricing design for safety-critical networks often require that users are queried beforehand to negotiate prices, which has proven to be challenging to implement in the real-world. To offer a more practical alternative, we develop learning-based pricing mechanisms that require no input from the users. These pricing mechanisms aim to maximize the utility of the users' consumption by gradually estimating the users' price response over a span of $T$ time steps (e.g., days) while ensuring that the infrastructure network's safety constraints that limit the users' demand are satisfied at all time steps. We propose two different algorithms for the two different scenarios when, 1) the utility function is chosen by the central coordinator to achieve a social objective and 2) the utility function is defined by the price response under the assumption that the users are self-interested agents. We prove that both algorithms enjoy $\tilde{O} (T^{2/3})$ regret with high probability. We then apply these algorithms to demand response pricing for the smart grid and numerically demonstrate their effectiveness.