Equilibria, variational models and applications

FA-03: Equilibria, variational models and applications
Stream: Variational inequalities, Nash games, game theory, multilevel and dynamic optimization
Room: Nash
Chair(s): Mauro Passacantando

Multi-Leader-Follower Potential Games
Steffensen Sonja
We discuss a particular class of Nash games, where the participants of the game (the players) are devided into to groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996. We develop necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results.

A Decomposition Algorithm for Nash Equilibria in Intersection Management
Andreas Britzelmeier, Axel Dreves
We consider autonomous vehicles that communicate in order to find optimal and collision free driving strategies. Resulting in coupled optimal control problems with nonconvex shared constraints and we consider a generalized Nash equilibrium reformulation of the problem. To handle the nonconvexity, we employ a partial penalty approach and reformulate the problem as a generalized potential game. We propose a decomposition algorithm with penalty selection to avoid a priori hierarchies. Using dynamic programming, we prove convergence of our algorithm. Providing numerical and experimental results.

Lagrange multipliers for a non-constant gradient constraint problem
SOFIA GIUFFRE’
Aim of the talk is to show the existence of Lagrange multipliers associated with linear variational inequalities with a non-constant gradient constraint over convex domains. In particular, first we show the equivalence between the non-constant gradient constrained problem and a suitable obstacle problem, where the obstacle solves a Hamilton-Jacobi equation in the viscosity sense. Then, we obtain the existence of Lagrange multipliers associated to the problem. The results have been obtained, using the strong duality theory.

Radial solutions to a nonconstant gradient constraint problem
Attilio Marcian√≤, SOFIA GIUFFRE’
Our work deals of radial solutions to a nonconstant gradient constraint problem in a ball of R^n and the Lagrange multipliers associated with the problem. The problem is formulated by means of a variational inequality and under a suitable assumption we obtain, for n = 2, a necessary and sufficient condition, characterizing the free boundary. In particular, this condition allow us to define the explicit solution and the Lagrange multiplier. Finally, some examples illustrate the results.

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