DFG-Project A3G: Algorithms for solving mathematical equilibrium problems

In a joint DFG project with Professor Martin Schmidt from the University of Trier, a team led by mathematician Professor Tobias Harks from the University of Passau is researching the theoretical and algorithmic understanding of equilibrium problems, such as those that occur in the fields of transport and electricity.

In game theory, the Nash equilibrium describes a situation in which no player can increase their benefit by unilaterally changing their strategy. The concept can be applied to many everyday situations, for example in transportation. Here, the benefit of the players (negative travel time) depends on the routing decisions of other road users.

In a joint DFG project with the University of Trier, a team at the University of Passau led by Professor Tobias Harks, holder of the Chair of Mathematical Optimization, is researching so-called aggregative games with integer constraints, which are characterized by the fact that the utility function of a player, in this case the car driver, depends on his own strategy and an aggregate of the strategies of the other players and the individual strategy spaces can be discrete. The aim of the DFG project entitled "Aggregative mixed-integer equilibrium problems: existence, approximation and algorithm", A3G for short, is to develop a theoretical and algorithmic understanding of such equilibrium problems. In addition to the transport sector, the project also has applications in electricity and in transportation and communication networks.

"We are investigating complex equilibrium problems in which strategy spaces have discrete aspects," explains Professor Harks. "We are looking for mathematical structures to understand the existence of equilibria and then develop algorithms that we can use to calculate equilibria."

As one possible application, for example, the project deals with routing and transport protocols in communication networks in which data packets are sent through a network. Each individual user wants to achieve the highest possible bandwidth in order to ensure the best possible connection. All users of the communication network want this good connection equally, but there is only a certain total bandwidth. The project addresses the question of how such balances can be described and how they can be characterized and calculated.

The research team is tackling this problem with the help of various mathematical theories. Among other things, the theories of discrete convexity and algorithmic optimization methods are used.

In addition to Prof. Dr. Harks, his research assistant Julian Schwarz is also researching the topic at the University of Passau. Martin Schmidt, Professor of Nonlinear Optimization at the University of Trier, is involved in the project externally. The project will start in September 2024 and will be funded by the German Research Foundation (DFG) for a period of three years.

This text was machine-translated from German.