Market frictions create portfolios of complicated nature, constrained zero-income portfolios. They generate no income in the future contingencies but matter to risk sharing whether they are mispriced or not. The no arbitrage conditions of the literature may not be qualified for equilibrium conditions in the presence of constrained zero-income portfolios. Thus 'pricing by arbitrage' is far from completely characterizing asset pricing relations. The purpose of this paper is to provide a unified treatment of both asset pricing and equilibrium when asset markets are subject to portfolio constraints. In particular, 'projective arbitrage' is proposed to examine both asset pricing and the existence of equilibrium under portfolio constraints. To formulate the effect of constrained zero-income portfolios on asset pricing and equilibrium, we introduce the local and global laws of one price as an extension of the law of one price to the case with portfolio constraints. The global law of one price is identified as 'pricing by projective arbitrage.' Moreover, the global law of one price provides an important criterion for judging the capability of various notions of arbitrage to characterize equilibrium asset prices under portfolio constraints. The traditional approach to general equilibrium with incomplete markets breaks down in the face of market frictions simply because the law of one price fails.

Apart from the underlying theme that all the contributions to this volume pertain to models set in an infinite dimensional space, they differ on many counts. Some were written in the early seventies while others are reports of ongoing research done especially with this volume in mind. Some are surveys of material that can, at least at this point in time, be deemed to have attained a satisfactory solution of the problem, while oth ers represent initial forays into an original and novel formulation. Some furnish alternative proofs of known, and by now, classical results, while others can be seen as groping towards and exploring formulations that have not yet reached a definitive form. The subject matter also has a wide leeway, ranging from solution concepts for economies to those for games and also including representation of preferences and discussion of purely mathematical problems, all within the rubric of choice variables belonging to an infinite dimensional space, interpreted as a commodity space or as a strategy space. Thus, this is a collective enterprise in a fairly wide sense of the term and one with the diversity of which we have interfered as little as possible. Our motivation for bringing all of this work under one set of covers was severalfold.

A single condition, limited arbitrage, is shown to be necessary and sufficient for the existence of a competitive equilibrium and the core in economies with any number of markets, finite or infinite, with or without short sales. This extends earlier results of Chichilnisky for finite economies. This unification of finite and infinite economies is achieved by proving that in Hilbert spaces limited arbitrage is necessary and sufficient for the compactness of the Pareto frontier . Limited arbitrage has also been shown to be necessary and sufficient for a resolution of the social choice paradox.