NAAMSI · Non-Additive Axiomatic Models of Strategic Interaction

Non-Additive Axiomatic Models of Strategic Interaction

During its duration the project NAAMSI has followed the expected schedule producing satisfying results and has opened new unexpected avenues of research that have led the project in a different and more promising direction. 1.1. Development of the Research Packages. The research plan included two main research packages: 1.1.1. Decision Theory. Part of NAAMSI dealt with the definition of particular logical systems to represent different kinds of expected utility based on probabilities and possibility measures, and the study of their main semantic and computational properties. The research focused on probabilistic expected utility and possibilistic expected utility. Two classes of logical systems were defined: one for reasoning about and representing probabilistic expectation and the other for reasoning about and representing possibilistic expectation, both in n-person non-cooperative games. These logics make it possible to reason about and represent agents’ expectation in general and in particular in situations of strategic interaction. The probabilistic case is part of a work with Prof. Lluís Godo (Artificial Intelligence Research Institute - CSIC, Spain) whose results were presented in two international conferences (see below). The class of logics defined was characterized in terms of completeness and complexity. In addition, the work offers a functional characterization of the existence of independent expected utilities in strategic interactions. A longer version of the work is being prepared and will soon be submitted for publication to a journal. The logical study of the possibilistic case is part of an ongoing collaboration with Dr. Hykel Hosni (London School of Economics, U.K.) in the framework of possibilistic games (see below). Similar to the probabilistic case, both a completeness and complexity characterization has been studied. The results will soon be submitted for publication. 1.1.2. Game Theory. NAAMSI also dealt with the definition of the notion and investigation of the main mathematical properties of n-person non-cooperative games with complete information based on the possibilistic models of expected utility. The basic notions of a possibilistic game theory have been defined and a characterization theo- rem for the existence of equilibria has been given. Possibilistic expectation has been used to integrate the role of intentions in situations of strategic interactions through possibility distributions. This has made it possible to use possibilistic expected utility to provide an alternative approach to the longstanding problem of selection of multiple equilibria. This study is part of an ongoing collaboration with Dr. Hykel Hosni (London School of Economics, U.K.). The results have been presented at several workshops and conferences (see below). A submission to a journal is in preparation. While the work initially planned was to focus also on expectation and game theory based on belief functions, the project has essentially focused on the possibilistic case. This is due to the fact that while some previous results existed for belief functions, no work had been done for a possibilistic approach to game-theoretic notions and this required developing a new theory from the mathematical point of view, along with a novel interpretation. Focusing on this aspect has made it possible to develop the foundations of a totally novel theoretical approach. In addition, during the development of the project, a new approach to using many-valued logics for representing strategic interactions has emerged. This approach was needed in order to fully understand and develop a theory of games based on many-valued logics and uncertainty measures. Therefore, this required a change in the direction of the project. 1.1.3. New Research Directions. The study of logics and games has been the basis for defining the theory of Lukasiewicz games, a compact representation of strategic interactions based on polynomial functions. This work was carried out in collaboration with Prof. Michael Wooldridge (Department of Computer Science, University of Oxford, U.K.). The theory of Lukasiewicz games makes it possible to offer a symbolic representation of games by means of the class of Lukasiewicz logics. This approach has the property of compactly specifying discrete and infinite cooperative and non-cooperative games. Lukasiewicz games have been extensively studied both from the logical and the computational point of view and the results have been presented in several conferences and workshops (see below). A foundational work on the theory of Lukasiewicz games has been submitted for publication to Transactions in Computational Logic. Based on the work on Lukasiewicz games and on the logics for representing probabilistic expectation, a new kind of games where players mutually try to influence their expectations has been defined. This work, in collaboration, with Lluís Godo (Artificial Intelligence Research Institute - CSIC, Spain) was presented at several conferences and a full article on the subject will soon be submitted. This is the first work that combines the many-valued logic approach to expectation, the many-valued logic approach to game-theory and the new theory of Lukasiewicz games. 1.2. Training Activities. 1.2.1. Funding. With the experience gained during the first year of the project, the researcher has prepared new research projects based on the ideas developed in NAAMSI. One of these projects (in collaboration with Prof. M. Wooldridge, Department of Computer Science, University of Oxford, U.K.) has been recently funded, granting the researcher a post of research assistant at the Department of Computer Science at the University of Oxford. 1.2.2. Networking. During the project, the researcher has both strengthened previous collaborations and started new ones with other European researchers. The works with Prof. Lluís Godo is based on previous joint efforts. The researcher has also established new partnerships with Dr. Hykel Hosni and Prof. Michael Wooldridge.

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