An Ordinal Minimax Theorem

In the early 1950s Lloyd Shapley proposed an ordinal and set-valued solution concept for zero-sum games called \emph{weak saddle}. We show that all weak saddles of a given zero-sum game are interchangeable and equivalent. As a consequence, every such game possesses a unique set-based value.Comment: 10 pages, 2 figure

Computing dominance-based solution concepts

Two common criticisms of Nash equilibrium are its dependence on very demanding epistemic assumptions and its computational intractability. We study the computational properties of less demanding set-valued solution concepts that are based on varying notions of dominance. These concepts are intuitively appealing, always exist, 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 set. Based on a unifying framework proposed by Duggan and Le Breton, we formulate two generic algorithms for computing these concepts and investigate for which classes of games and which properties of the underlying dominance notion the algorithms are sound and efficient. We identify two sets of conditions that are sufficient for polynomial-time computability and show that the conditions are satisfied, for instance, by saddles and primitive formations in normal-form games, minimal CURB sets in two-player games, and the minimal covering set in symmetric matrix games. Our positive algorithmic results explain regularities observed in the literature, but also apply to several solution concepts whose computational complexity was previously unknown

Computing Dominance-Based Solution Concepts

Two common criticisms of Nash equilibrium are its dependence on very demanding epistemic assumptions and its computational intractability. We study the computational properties of less demanding set-valued solution concepts that are based on varying notions of dominance. These concepts are intuitively appealing, they always exist, 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. We propose two generic algorithms for computing these concepts and investigate for which classes of games and which properties of the underlying dominance notion the algorithms are sound and efficient