Bilevel Optimization and Conjectural Equilibrium: Theoretical Results and Numerical Algorithms (An Invited Tutorial Paper)

We study a mixed oligopoly model, including a nonprivate firm that maximizes the convex combination of domestic social surplus and net profit, under the concepts of consistent conjectural variations (CCVE), Cournot–Nash, and perfect competition equilibria and analyze the behavior of these equilibria. We compare the results to develop an optimality criterion for a parameter, decided by the nonprivate firm, known as socialization level. To work with the CCVE, we need to define the concept of consistency; hence, we also consider a classic oligopoly, which only includes private firms that maximize net profit, and reformulated it as a non-cooperative game, named meta-game, where the players (the private firms) select a conjecture as their strategies; then, we prove that the Nash equilibrium for the meta-game implies the CCVE. After that, due to its similarities with the meta-game, we study the Tolls Optimization Problem (TOP). Taking advantage of the sensitivity analysis for convex quadratic optimization and the filled function method; then, we present an efficient algorithm to solve the TOP. Finally, we apply these results to a financial model in which a group of sectors buy, sell, or exchange a set of instruments; each sector faces some uncertainty when selecting its assets and liabilities to optimize its portfolio structure. The problem is modeled using the concepts of conjectural variations and meta-game and we present results concerning the CCVE.