Sequential Decision Making

The project aims at a reconsideration of the theory of non-cooperative games, in particular the theory of extensive form games, without simplifying finiteness or discreteness assumptions. By redoing the mathematical foundations of game theory, with the goal of proving characterization results in each and every step, we hope to extend the current confines of game theory and break new ground that allows for further applications of game theory. Immediate tasks are to extend the proof of Kuhn`s theorem to the general framework and to provide a rigorous model for randomized strategies. Further, for most applications, a "down-discreteness" condition seems natural and appears to allow for a significant simplification of the relevant definitions, thereby providing a compact "canonical" definition of the primitives of game theory. In the next step we hope to explore the possible range of preference relations that can govern sequential decision-making in games.

Game theory, the mathematical theory of strategic interaction, is one of the great intellectual achievements of the twentieth century. It was shaped in the 1950ties and was then applied during the second half of the past century in a number of fields, including economics, political science, and biology. When game theory was originally formulated, a number of simplifying assumptions were made, in particular finiteness. Those soon proved to be an obstacle for applications. The purpose of this project is to reprove the fundamental results of game theory without these simplifying assumptions. Our work prior to the project has resulted in a characterization of when basic objects, like strategies, make sense. During this project we have extended this framework to one that is appropriate for applications. In particular, we have devised a general formalism for games that allows for infinite action spaces and an infinite horizon yet does not preclude any of the applications from the literature. In fact, we are able to nest all earlier attempts in that direction, resulting in the most general framework for (extensive form) game available up to date. The second main contribution discovered during this project concerns one of the key early results in game theory, the existence of a subgame perfect equilibrium for games of perfect information. While this was proved for the finite case in the 1950ties, our result provides a characterization for when this extends to the case of arbitrary cardinality, in particular to the infinite case.