Researcher of the Month
Nash and Pareto: New findings in game theory
Risks and returns, price equilibria, or systemic risk – if a situation involves multiple criteria or actors, concepts such as the Nash equilibrium and Pareto optimality can help in decision-making. So far, the Nash and Pareto concepts have been regarded as inherently different. Birgit Rudloff, deputy head of the Institute for Statistics and Mathematics at WU (Vienna University of Economics and Business), has now shown, however, that the two concepts coincide under certain conditions. If the Pareto optimization problem for any non-cooperative game is constructed in a specific way, the set of all Pareto optimal solutions coincides with the set of all Nash equilibria. This insight makes it possible to apply Pareto optimality methods and algorithms to Nash equilibria.
The Nash equilibrium, known from game theory, refers to a situation where no player can unilaterally change their strategy without being worse off afterwards. This concept plays an important role in determining prices and quantities in competitive markets. The Pareto optimality refers to a solution where two or more (often conflicting) criteria are considered at the same time – for example when optimizing a portfolio to maximize returns while at the same time minimizing the risks.
For decades, these two concepts have been regarded as separate. Together with Zachary Feinstein from the Stevens Institute of Technology, USA, Birgit Rudloff has shown that this is not necessarily the case. For any non-cooperative game, the Pareto optimization problem can be constructed in such a way that the set of all Pareto optimal solutions coincides with the set of all Nash equilibria. This applies not only to classic Nash games but also to generalized Nash games and even vector-valued games.
The advantage of this characterization is that it makes it possible to apply Pareto optimality methods and algorithms to Nash equilibria. Under certain conditions, algorithms can now be used to compute the set of all Nash equilibria. This is in contrast to the typical fixed point iteration approaches used so far, which can construct just a single Nash equilibrium.
About Birgit Rudloff
Birgit Rudloff is deputy head of the WU Institute for Statistics and Mathematics and spokeswoman of the PhD specialization in Economics and Business. A native of eastern Germany, she earned her PhD in financial mathematics at Martin Luther University Halle-Wittenberg. She went on to complete a one-year research stay at the Institute for Pure and Applied Mathematics (IMPA) in Rio de Janeiro and worked as a postdoctoral researcher at TU Wien. From 2006 to 2015, Birgit Rudloff was an assistant professor at Princeton University, Department of Operations Research and Financial Engineering. She joined WU in 2015.
In her research, Birgit Rudloff focuses especially on problems in financial mathematics that involve multiple criteria at the same time, for example the management of systemic risk in bank networks, the dynamic mean-risk portfolio optimization problem, and dynamic Nash equilibria. She is particularly interested in time consistency and successfully generalized the Bellman principle for dynamic programming to cover multivariate problems.
Birgit Rudloff’s research has been published in leading international journals, including Operations Research, Finance and Stochastics, Bernoulli, the SIAM Journal on Financial Mathematics, Mathematical Programming, and Mathematics of Operations Research. Her contributions have earned her several international awards.