Computing all solutions of linear generalized Nash equilibrium problems

In this paper we consider linear generalized Nash equilibrium problems, i.e., the cost and the constraint functions of all players in a game are assumed to be linear. Exploiting duality theory, we design an algorithm that is able to compute the entire solution set of these problems and that terminates after finite time. We present numerical results on some academic examples as well as some economic market models to show effectiveness of our algorithm in small dimensions.

A generalized Nash equilibrium approach for optimal control problems of autonomous cars

We consider optimal control problems with ordinary differential equations that are coupled by shared, possibly nonconvex, constraints. For these problems, we use the generalized Nash equilibrium approach and provide a reformulation of normalized Nash equilibria as solutions to a single optimal control problem. By this reformulation, we are able to prove existence, and in some settings, exploiting convexity properties, we also get a limited number or even uniqueness of the normalized Nash equilibria. Then, we use our approach to discuss traffic scenarios with several autonomous vehicles, whose dynamics is described through differential equations, and the avoidance of collisions couples the optimal control problems of the vehicles. For the solution to the discretized problems, we prove strong convergence of the states and weak convergence of the controls. Finally, using existing optimal control software, we show that the generalized Nash equilibrium approach leads to reasonable results for a crossing scenario with different vehicle models.

How to Select a Solution in Generalized Nash Equilibrium Problems

We propose a new solution concept for generalized Nash equilibrium prob- lems. This concept leads, under suitable assumptions, to unique solutions, which are generalized Nash equilibria and the result of a mathematical procedure modeling the process of finding a compromise. We first compute the favorite strategy for each player, if he could dictate the game, and use the best response on the others’ favorite strategies as starting point. Then, we perform a tracing procedure, where we solve parametrized generalized Nash equilibrium problems, in which the players reduce the weight on the best possible and increase the weight on the current strategies of the others. Finally, we definethelimitingpoints of this tracingprocedureas solutions. Under our assumptions, the new concept selects one reasonable out of typically infinitely many generalized Nash equilibria.

An algorithm for equilibrium selection in generalized Nash equilibrium problems

Recently a new solution concept for generalized Nash equilibrium problems was pub- lished by the author. This concept selects a reasonable equilibrium out of the typically infinitely many. The idea is to model the process of finding a compromise by solving parametrized generalized Nash equilibrium problems. In this paper we propose an algorithmic realization of the concept. The model produces a solution path, which is under suitable assumptions unique. The algorithm is a homotopy method that tries to follow this path. We use s emismooth Newton steps as corrector steps in our algorithm in order to approximately solve the generalized Nash equilibrium problems for each given parameter. If we have a unique solution path, we need three additional theo- retical assumptions: a stationarity result for the merit function, a coercivity condition for the constraints, and an extended Mangasarian–Fromowitz constraint qualification. Then we can prove convergence of our semismooth tracing algorithm to the unique equilibrium to be selected. We also present convincing numerical results on a test library of problems from literature. The algorithm also performs well on a number of problems that do not satisfy all the theoretical assumptions.