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Abstract

Optimization is the foundation for mathematical models of decision making. Many of these models involve tradeoffs between exploration, exploitation and consideration of competitors' reactions. Their analysis also relies on representing conditions for optimality. This dissertation provides new theory of how agents may behave in competitive situations including conditions for learning and strategic interactions that may inhibit learning. In the context of a single decision maker, the dissertation also provides conditions for the existence of optimal dual solutions that are easily characterized and have a meaningful economic interpretation. The first part of the dissertation considers the impact of competition on the design and implementation of dynamic learning strategies. In monopoly pricing situations, firms should optimally vary prices to learn demand. The variation must be sufficiently high to ensure complete learning. In competitive situations, however, varying prices provides information to competitors and may reduce the value of learning. Such situations may arise in the pricing of new products such as pharmaceuticals. Chapter 1 shows that firms in competitive markets can learn efficiently by setting prices which involve adding noise to myopic estimation and best-response strategies. Chapter 2 then discusses how complete learning may not be the desired outcome when actions reveal information quickly to competitors. The chapter provides a setting where this effect can be strong enough to stop learning. Surprisingly, firms may optimally reduce any variation in prices and choose not to learn demand. The result can be that the selling firms achieve a collaborative outcome instead of a competitive equilibrium. The result has implications for policies that restrict price changes or require disclosures. The second part of the dissertation investigates the interplay between the existence of interior points and singular dual functionals, which occur in abstract optimization. While there is growing interest in solving infinite-dimensional optimization problems, many of the intuitions and interpretations common to finite dimensions do not extend to infinite dimensions. For instance, a dual solution in finite dimensional models is represented by a vector of ``dual prices'' that index the primal constraints and have a natural economic interpretation. In infinite dimensions, we show that this simple dual structure, and its associated economic interpretation, may fail to hold for a broad class of problems with constraint vector spaces that are Riesz spaces (ordered vector spaces with a lattice structure) that are either sigma-order complete or satisfy the projection property. In these spaces we show that the existence of interior points required by common constraint qualifications for zero duality gap (such as Slater's condition) imply the existence of singular dual solutions that are difficult to find and interpret. We call this phenomenon the Slater conundrum: interior points ensure zero duality gap (a desirable property), but interior points also imply the existence of singular dual solutions (an undesirable property). Riesz spaces are the most parsimonious vector-space structure sufficient to characterize the Slater conundrum. Finally, we provide sufficient conditions that "resolve" the Slater conundrum; that is, guarantee that in every solvable dual there exists an optimal dual solution that is not singular.

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