Sequential Decision Making, Part II

The project aims at generalizing the classical framework for non-cooperative games in extensive form. In particular, the classical results in game theory the existence and uniqueness of outcomes induced by strategies, the existence of subgame perfect equilibria for perfect information games, the equivalence of extensive form games that have the same semi-reduced normal form, or the equivalence of mixed and behavioral strategies under perfect recall are investigated without any finiteness assumptions. Preferences for players that do not necessarily conform to the expected utility axioms will be considered from the perspective of sequential consistency. Furthermore, the measure theoretic framework required for solution concepts like Perfect Bayesian Equilibrium or Sequential Equilibrium will be extended so as to allow for infinite action spaces and horizons. Generalizing the definitions of choice systems or information sets will allow for novel insights into certain aspects of absent-mindedness and connect this research to behavioral economics. We aim at providing characterizations in each step, so as to identify necessary on top of sufficient conditions.

The development of interactive decision theory, also known as game theory, began during the middle of the last century. In the meantime game theory has become the dominant methodology in the social sciences, in particular in economics. The necessities of economic theory, however, have generated applications that transcended the traditional framework of finite game theory. Hence, this project has tackled the task of putting game theory on new foundations that are appropriate for modern economic theory. This concerns in particular the generalization of the key classical results without resorting to the finiteness assumptions, which were originally adopted for simplicity.In the now completed, second part of this research project we have proposed a general theory of dynamic games, also known as extensive form games. This theory encompasses nor only representations that can do without any cardinality assumptions, but also a generalization of one of the fundamental building blocks of solution theory: The existence of subgame perfect equilibrium for games of perfect information. More precisely, we offer a characterization of the existence of equilibria, because the conditions that we propose are not only sufficient for existence but also necessary. Since these conditions apply to the topology on the universal set of plays, this result defines the framework in which game theory can be operative. While pursuing this we have also found a number of new results that are relevant for applications. For instance, we were able to characterize when the one-shot deviation principle holds true. Another example is a novel characterization of the concept of perfect recall, which we can now define purely in terms of the primitives of the game, instead via derived objects like strategies.