In a system in which noncooperative agents share a common resource, we propose the ratio between the worst possible Nash equilibrium and the social optimum as a measure of the effectiveness of the system. Deriving upper and lower bounds for this ratio in a model in which several agents share a very simple network leads to some interesting mathematics, results, and open problems.

Abstract. This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

Although the study of market equilibria has occupied center stage within Mathematical Economics for over a century, polynomial time algorithms for such questions have so far evaded researchers. We provide the first such algorithm for the linear version of a problem defined by Irving Fisher in 1891. Our algorithm is modeled after Kuhn’s primaldual algorithm for bipartite matching. 1

We consider the problem of online keyword advertising auctions among multiple bidders with limited budgets, and study a natural bidding heuristic in which advertisers attempt to optimize their utility by equalizing their return-on-investment across all keywords. We show that existing auction mechanisms combined with this heuristic can experience cycling (as has been observed in many current systems), and therefore propose a modified class of mechanisms with small random perturbations. This perturbation is reminiscent of the small time-dependent perturbations employed in the dynamical systems literature to convert many types of chaos into attracting motions. We show that the perturbed mechanism provably converges in the case of first-price auctions and experimentally converges in the case of second-price auctions. Moreover, the point of convergence has a natural economic interpretation as the unique market equilibrium in the case of first-price mechanisms. In the case of second-price auctions, we conjecture that it converges to the “supply-aware” market equilibrium. Thus, our results can be alternatively described as a tâtonnement process for convergence to market equilibrium in which prices are adjusted on the side of the buyers rather than the sellers. We also observe that perturbation in mechanism design is useful in a broader context: In general, it can allow bidders to “share ” a particular item, leading to stable allocations and pricing for the bidders, and improved revenue for the auctioneer.

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the complexity of four-player Nash equilibria [21], settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems: • Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. • The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time. Our results also have a complexity implication in mathematical economics: • Arrow-Debreu market equilibria are PPAD-hard to compute.

We examine the marriage of recent probabilistic generative models for social networks with classical frameworks from mathematical economics. We are particularly interested in how the statistical structure of such networks influences global economic quantities such as price variation. Our findings are a mixture of formal analysis, simulation, and experiments on an international trade data set from the United Nations. 1

The language of game theory—coalitions, payo¤s, markets, votes— suggests that it is not a branch of abstract mathematics; that it is motivated by and related to the world around us; and that it should be able to

ABSTRACT __________________________________________________________________________ This paper evaluates the performances of three of the most prominent multisectoral static applied general equilibrium models used to predict the impact of the North American Free Trade Agreement. These models drastically underestimated the impact of NAFTA on North American trade. Furthermore, the models failed to capture much of the relative impacts on different sectors. Ex-post performance evaluations of applied GE models are essential if policymakers are to have confidence in the results produced by these models. Such evaluations also help make applied GE analysis a scientific discipline in which there are well-defined puzzles with clear successes and failures for competing theories. Analyzing sectoral trade data indicates the need for a new theoretical mechanism that generates large increases in trade in product categories with little or no previous trade. To capture changes in macroeconomic aggregates, the models need to be able to capture changes in productivity.

In this paper we consider the classic problem of finding the market equilibrium prices under linear utility functions. A notion of approximate market equilibrium was proposed by Deng, Papadimitriou and Safra [5]. Using this notion, we present the first fully polynomial-time approximation scheme for finding a market equilibrium price vector. The main tool in our algorithm is the polynomial-time algorithm of Devanur et al. [6] for a variant of the problem in which there is a clear demarcation between buyers and sellers. Their algorithm is used as a subroutine in our algorithm.

A substantial amount of economic activity involves problem solving, yet economics has few, if any, formal models to address how agents of limited abilities find good solutions to difficult problems. In this paper, we construct a model of heterogeneous agents of bounded abilities confronting difficult problems and analyze their individual and collective performance. By heterogeneity, we mean differences in how individuals represent problems internally, their perspectives, and in the algorithms they use to generate solutions, their heuristics. With this model, we find that a collection of bounded but diverse agents can locate optimal solutions to very difficult problems. We can also calculate the marginal benefits to adding additional problem solvers. We find that problem solving firms can exhibit arbitrary returns to scale, that the order that problem solvers are applied to a problem can matter, and that the standard story of decreasing returns to scale is unlikely.