An Interior-Point Path-Following Method to Compute Stationary Equilibria in Stochastic Games

Subgame perfect equilibrium in stationary strategies (SSPE) is the most important solution concept used in applications of stochastic games, which makes it imperative to develop efficient numerical methods to compute an SSPE. For this purpose, this paper develops an interior-point path-following method (IPM), which remedies a number of issues with the existing method called stochastic linear tracing procedure (SLTP). The homotopy system of IPM is derived from the optimality conditions of an artificial barrier game, whose objective function is a combination of the original payoff function and a logarithmic term. Unlike SLTP, the starting stationary strategy profile can be arbitrarily chosen and IPM does not need switching between different systems of equations. The use of a perturbation term makes IPM applicable to all stochastic games, whereas SLTP only works for a generic stochastic game. A transformation of variables reduces the number of equations and variables of by roughly one half. Numerical results show that our method is more than three times as efficient as SLTP

Constant inapproximability for Fisher markets

We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD = P. In fact, we prove that computing any approximation better than 1/11 is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility function

Essays on optimization and incentive contracts

Includes bibliographical references (p. 167-176).Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2007.(cont.) In the second part of the thesis, we focus on the design and analysis of simple, possibly non-coordinating contracts in a single-supplier, multi-retailer supply chain where retailers make both pricing and inventory decisions. Specifically, we introduce a buy-back menu contract to improve supply chain efficiency, and compare two systems, one in which the retailers compete against each other, and another in which the retailers coordinate their decisions to maximize total expected retailer profit. In a linear additive demand setting, we show that for either retailer configuration, the proposed buy-back menu guarantees the supplier, and hence the supply chain, at least 50% of the optimal global supply chain profit. In particular, in a coordinated retailers system, the contract guarantees the supply chain at least 75% of the optimal global supply chain profit. We also analyze the impact of retail price caps on supply chain performance in this setting.In this thesis, we study important facets of two problems in methodological and applied operations research. In the first part of the thesis, motivated by optimization problems that arise in the context of Internet advertising, we explore the performance of the greedy algorithm in solving submodular set function maximization problems over various constraint structures. Most classic results about the greedy algorithm assume the existence of an optimal polynomial-time incremental oracle that identifies in any iteration, an element of maximum incremental value to the solution at hand. In the presence of only an approximate incremental oracle, we generalize the performance bounds of the greedy algorithm in maximizing nondecreasing submodular functions over special classes of matroids and independence systems. Subsequently, we unify and improve on various results in the literature for problems that are specific instances of maximizing nondecreasing submodular functions in the presence of an approximate incremental oracle. We also propose a randomized algorithm that improves upon the previous best-known 2-approximation result for the problem of maximizing a submodular function over a partition matroid.by Pranava Raja Goundan.Ph.D

On the Complexity of Market Equilibria and Revenue Maximization

This thesis consists of two parts. In the first part, we concentrate on the computation of Market Equilibria and settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utilities. We prove that the problem is PPAD-complete when the Constant Elasticity of Substitution parameter $\rho$ is any constant less than -1. Building on this result, we introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory, and prove that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. In the second part, we study Revenue Maximization. We begin by resolving the complexity of the revenue-optimal Bayesian Unit-demand Item Pricing problem when the buyer's values for the items are independent. We show that the problem can be solved in polynomial time for distributions of support size 2; but its decision version is NP-complete for distributions of support size 3. Next, we study the optimal mechanism design problem for a single unit-demand buyer with item values drawn from independent distributions. We show that, for distributions of support-size 2 and the same high value, Item Pricing can achieve the same revenue as any menu of lotteries. On the other hand, we provide simple examples where randomization improves revenue. Finally, we show that unless the polynomial-time hierarchy collapses, namely P^{NP}=P^{#P}, there is no universal efficient randomized algorithm that implements an optimal mechanism even when distributions have support size 3

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order