Set-Valued Solution Concepts in Social Choice and Game Theory: Axiomatic and Computational Aspects

This thesis studies axiomatic and computational aspects of set-valued solution concepts in social choice and game theory. It is divided into two parts. The first part focusses on solution concepts for normal-form games that are based on varying notions of dominance. These concepts are intuitively appealing and admit unique minimal solutions in important subclasses of games. Examples include Shapley’s saddles, Harsanyi and Selten’s primitive formations, Basu and Weibull’s CURB sets, and Dutta and Laslier’s minimal covering sets. Two generic algorithms for computing these concepts are proposed. For each of these algorithms, properties of the underlying dominance notion are identified that ensure the soundness and efficiency of the algorithm. Furthermore, it is shown that several solution concepts based on weak and very weak dominance are computationally intractable, even in two-player games. The second part is concerned with social choice functions (SCFs), an important subclass of which is formed by tournament solutions. The winner determination problem is shown to be computationally intractable for different variants of Dodgson’s rule, Young’s rule, and Tideman’s method of ranked pairs. For a number of tractable SCFs such as maximin and Borda’s rule, the complexity of computing possible and necessary winners for partially specified tournaments is determined. Special emphasis is then put on tournament solutions that are defined via retentiveness. The axiomatic properties and the asymptotic behavior of these solutions is studied in depth, and a new attractive tournament solution is proposed. Finally, necessary and sufficient conditions for the strategyproofness of irresolute SCFs are presented.