(Non-cooperative) Game Theory studies strategic conflicts. As such, it can model many fundamental situations of economic life, and beyond. Well-known examples of such situations include auctions, elections, bargaining, sports competitions, competition between firms, lobbying, political debate, and even international conflict. Almost all members of the Center for Mathematical Economics work on game theory in one form or another. More specifically our main interests lie in the following subfields of game theory.

Strategic decisions of firms often have to be made under uncertainty and moreover imply long-ranging consequences, for example if they are hardly reversible due to physical or legal frictions. In this project we address these aspects with new methods. On the one hand we apply the most recent findings concerning model uncertainty ("Knightian Ambiguity") to such decisions. On the other hand we develop a theory of dynamic oligopoly games with irreversible capacity choices in continuous time. For such games, there exists no satisfying game theory, since a mathematically consistent game theoretic equilibrium concept (which allows for feedback) has not been successfully established so far. We use a new method for solving singular control problems, which has been (co-)developed by the principle investigator, to establish such a game theory. We will analyse in particular oligopoly games under Knightian uncertainty. For instance, we want to clarify the question whether the value of the option to wait disappears already with two firms. A first paper on the topic appears in Theory and Decision (Frank Riedel and Linda Sass). This research is jointly funded by the DFG and ANR.

Many economic problems are characterized by the interplay of strategic interaction and intertemporal effects of actions. Differential games are a very suitable tool to analyze behavior in such situations, and have been heavily used in areas like Industrial Organization and Resource Economics. Our research focuses on investment behavior of firms in dynamic oligopolies with product innovation. In particular we study how oligopolists in established markets should react to the anticipated and actual extension of the relevant product range due to product innovation, and how the effort to develop new products is affected by the position of a firm on the established market (see also the DFG project 'Dynamic Oligopolistic Competition between Innovating Firms')

We are also interested in approaches to game theory that question the usual rationality assumptions but still allow to build a meaningful theory (instead of the frequent ad hoc theories that were en vogue in the past). Evolutionary game theory is such an approach. Research at the IMW in this area has centered on the following topics and questions. Evolutionary Dynamics in Complex Models. Frank Riedel and Jörg Oechssler generalized the theory that was mainly confined to finite games that far to infinite games (Economic Theory 2001, Journal of Economic Theory 2002). They introduced the concept of evolutionarily robust strategy, and the use of the weak topology. In later work with the biologist Ross Cressman and the mathematician Josef Hofbauer, they derived differential equations for the study of such dynamics (Journal of Theoretical Biology 2006). They are surprisingly similar to the adaptive dynamics studied frequently in biology. In further work, they went on to study learning dynamics that are closer to human behavior, the so-called Brown–von Neumann–Nash dynamics. (Games and Economic Behavior 2009)

Does learning/evolution lead to behavior in the long-run that resembles that of highly rational players? Examples of work on this topic include Kuzmics (2004, Games and Economic Behavior); Kuzmics, (2011, Games and Economic Behavior); Balkenborg, Hofbauer, and Kuzmics (2013, Theoretical Economics).

If nature could freely choose individuals’ preferences, what preferences would she choose? Examples of work on this topic includes Herold and Kuzmics (2009, Games and Economic Behavior).

Strategic decisions of firms often have to be made under uncertainty and moreover imply long-ranging consequences, for example if they are hardly reversible due to physical or legal frictions. In this project we address these aspects with new methods. On the one hand we apply the most recent findings concerning model uncertainty ("Knightian Ambiguity") to such decisions. On the other hand we develop a theory of dynamic oligopoly games with irreversible capacity choices in continuous time. For such games, there exists no satisfying game theory, since a mathematically consistent game theoretic equilibrium concept (which allows for feedback) has not been successfully established so far. We use a new method for solving singular control problems, which has been (co-)developed by the principal investigator, to establish such a game theory. We will analyse in particular oligopoly games under Knightian uncertainty. For instance, we want to clarify the question whether the value of the option to wait disappears already with two firms. Publications include Irreversible Investment in Oligopoly by Jan-Henrik Steg (Finance and Stochastics 2010), Generalized Kuhnâ??Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources by Giorgio Ferrari, Jan-Henrik Steg, and Frank Riedel, SIAM Journal on Control and Optimization 2014. This research is funded by the DFG, Ri-1128-4-1.

Inspired by the seminal book by Schelling (1960), "The strategy of conflict", the study of focal points is concerned with the question as to how (highly rational) players could actually play a Nash equilibrium if there are many equilibria to choose from? This requires a high degree of coordination in players’ beliefs. Under what conditions is it plausible that playersâ?? beliefs are thus coordinated? What are the expected focal points in different situations? Examples of work on this topic includes Alos-Ferrer and Kuzmics (2013, Journal of Economic Theory).

Repeated games have a special structure: one â??stage gameâ? is played repeatedly at various points in time by the same set of individuals. Much of the current literature on this subject is concerned with the study of all equilibria in such games when monitoring is less than perfect (meaning players observe past action choices of opponents only imperfectly). At the IMW we have focused on slightly different questions. Among these is the question of focal points in repeated games (see also the paragraph on Focal Points) investigated by Kuzmics, Palfrey, and Rogers (2012, IMW working paper 468).

Network structures play an important role in many domains of social and economic interaction and recent years have seen an explosion of empirical and theoretical research aiming at a better understanding of factors generating certain types of networks and of the implications of certain types of networks on the behavior of agents in the network. Several researchers at the IMW have contributed and continue to do so to various aspects of this research field using analytical and numerical methods. In particular, it is studied which kind of networks are stable in a static (i.e. pairwise stable, pairwise Nash stable) and a dynamic (i.e. stochastically stable) sense for different general classes of individualsâ?? utility functions as well as for specific industrial organization models, like oligopolies with R&D. Furthermore, the effects of social networks on the inter-generational transmission of cultural traits or on the increase of wage inequality in the labor market has been studied.