Robust optimization, game theory, and variational inequalities

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.Includes bibliographical references (p. 193-109).We propose a robust optimization approach to analyzing three distinct classes of problems related to the notion of equilibrium: the nominal variational inequality (VI) problem over a polyhedron, the finite game under payoff uncertainty, and the network design problem under demand uncertainty. In the first part of the thesis, we demonstrate that the nominal VI problem is in fact a special instance of a robust constraint. Using this insight and duality-based proof techniques from robust optimization, we reformulate the VI problem over a polyhedron as a single- level (and many-times continuously differentiable) optimization problem. This reformulation applies even if the associated cost function has an asymmetric Jacobian matrix. We give sufficient conditions for the convexity of this reformulation and thereby identify a class of VIs, of which monotone affine (and possibly asymmetric) VIs are a special case, which may be solved using widely-available and commercial-grade convex optimization software. In the second part of the thesis, we propose a distribution-free model of incomplete- information games, in which the players use a robust optimization approach to contend with payoff uncertainty.(cont.) Our "robust game" model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative, distribution-free equilibrium concept, for which, in contrast to ex post equilibria, existence is guaranteed. We show that computation of "robust-optimization equilibria" is analogous to that of Nash equilibria of complete- information games. Our results cover incomplete-information games either involving or not involving private information. In the third part of the thesis, we consider uncertainty on the part of a mechanism designer. Specifically, we present a novel, robust optimization model of the network design problem (NDP) under demand uncertainty and congestion effects, and under either system- optimal or user-optimal routing. We propose a corresponding branch and bound algorithm which comprises the first constructive use of the price of anarchy concept. In addition, we characterize conditions under which the robust NDP reduces to a less computationally demanding problem, either a nominal counterpart or a single-level quadratic optimization problem. Finally, we present a novel traffic "paradox," illustrating counterintuitive behavior of changes in cost relative to changes in demand.by Michele Leslie Aghassi.Ph.D

Democracy after Deliberation: Bridging the Constitutional Economics/Deliberative Democracy Divide

This dissertation addresses a debate about the proper relationship between democratic theory and institutions. The debate has been waged between two rival approaches: on the one side is an aggregative and economic theory of democracy, known as constitutional economics, and on the other side is deliberative democracy. The two sides endorse starkly different positions on the issue of what makes a democracy legitimate and stable within an institutional setting. Constitutional economists model political agents in the same way that neoclassical economists model economic agents, that is, as self-regarding, rational maximizers; so that evaluations of democratic legitimacy and stability depend on the extent to which the design of institutional rules and practices maximize individual utility by promoting efficient schemes of collective choice. Deliberative democrats, on the other hand, understand political agents as communicative reason-giving subjects who justify their preferences and positions on issues that jointly affect them in a process of consensus-directed discourse, or deliberation; so that evaluations of democratic legitimacy and order depend on the degree to which institutional norms and practices promote deliberation and draw upon deliberated public judgment. I argue that despite the numerous incompatibilities between constitutional economics and deliberative democracy—which amount to a 'deep divide'—an opportunity to produce a genuine synthesis of the two approaches arises inasmuch as it is possible to overcome several points of opposition in their separate research programmes. The central thesis of the dissertation is that it is possible to construct a bridge spanning the divide between constitutional economists and deliberative democrats, and that Dewey and Bentley's transactional view can facilitate this bridge-building project. Pursuant to this end, the points of opposition between the v research programmes are mediated by way of five concepts which, on balance, favor deliberative democracy and its feasible institutionalization

Description of Courses, 1979-80

Official publication of Cornell University V.71 1979/8