Essays on Games and Decisions

Abstract

In the first chapter, a solution concept for two-person zero-sum games is proposed with players' preferences only assumed to satisfy Independence. To each player, there is a set of "admissible" strategies assuring him minimum guarantees. Moreover, rationality requires players to reject non-admissible strategies from any further consideration. Additional knowledge assumptions allow iterated elimination of non-admissible strategies. This leads to a pair of strategy sets, one for each player, whose cross product are the "consideration equilibria". Consideration equilibria always exist and include Nash equilibria if any. Further, consideration equilibria and Nash equilibria (or, minimax strategies) coincide if players' preferences additionally satisfy Continuity. Three examples are analysed for illustration. The second chapter investigates the implications of additivity type axioms in economic theory. In several areas of microeconomic theory, axiomatic characterizations have been provided for the respective objects of study to possess lexicographic structures. We introduce the concept called "graded halfspace" which is an abstraction of lexicographic structures. Then, we formulate and establish a geometric result called the "Decomposition Theorem". This result characterizes graded halfspaces as the convex cones which are elements of some partition, of a given Euclidean space, consisting of a pair of mutually reflecting convex cones and a subspace. Thus, the Decomposition Theorem formalizes the following intuitive idea: an object defined over a convex domain is additive, if and only if, it has a lexicographic structure. To illustrate this geometric approach, we present four applications ranging over decision theory, social choice, convex analysis and linear algebra. In the third chapter, we consider pre-norms on the Euclidean space which are functions that satisfy the definition of a norm except that a vector and and its reflection through the origin may have different values. Then, we characterize those binary relations on the Euclidean space which admit a pre-norm as a (utility) representation. The notion of the "dual" of such a binary relation is introduced. For any such binary relation, its "second dual" - the dual of the dual - is identical to itself. Further, such a binary relation is "self dual" if and only if it is spherical - the Euclidean norm is a representation. The duality theory allows us to generalize the Holder's inequality to arbitrary pre-norms. Binary relations which admit a norm as a representation are also characterized. We specialize our theory to characterize binary relations which admit a p-norm as a representation. Thus, the classical inequalities due to Minkowski and Holder follow as corollaries of the general theory.