Title: A Bilevel Optimization Method for an Exact Solution to Equilibrium Problems with Binary Variables
Speaker: Sauleh Siddiqui, Dept. of Environmental Science, Dept. of Math & Stat, American University; Research Fellow at the German Economic Research Institute (DIW Berlin).
Date: Tuesday February 9, 2021, 2:30 PM
Abstract: Bilevel optimization problems are used to model decisions that are constrained by other decisions, such as setting policies in energy markets, making long-term infrastructure decisions, and optimizing hyperparameters in machine learning models. We provide an introduction to bilevel optimization and propose a new method to find exact Nash equilibria in games with binary decision variables. We include compensation payments and incentive-compatibility constraints from non-cooperative game theory directly into an optimization framework in lieu of using first-order conditions of a linearization, or relaxation of integrality conditions. The reformulation offers a new approach to obtain and interpret dual variables to binary constraints using the benefit or loss from deviation rather than marginal relaxations. The method endogenizes the trade-off between overall (societal) efficiency and compensation payments necessary to align incentives of individual players. We provide existence results and conditions under which this problem can be solved as a mixed-binary linear program. We apply the solution approach to a stylized nodal power-market equilibrium problem with binary on-off decisions. This illustrative example shows that our approach yields an exact solution to the binary Nash game with compensation. We compare different implementations of actual market rules within our model, in particular constraints ensuring non-negative profits (no-loss rule) and restrictions on the compensation payments to non-dispatched generators. We discuss the resulting equilibria in terms of overall welfare, efficiency, and allocational equity.