Genericity with Infinitely Many Parameters

Abstract

Genericity analysis is widely used to show that desirable properties that fail in certain "knife-edge" economic situations nonetheless obtain in "typical" situations. For finite-dimensional spaces of parameters, the usual notion of genericity is full Lebesgue measure. For infinite dimensional spaces of parameters (for instance, the space of preferences on a finite-dimensional commodity space, no analogue of Lebesgue measure is available; the lack of such an analogue has prompted the use of less compelling topological notions of genericity. Christensen (1974) and Hunt, Sauer and Yorke (1992) have proposed a measure-theoretic notion of genericity, which Hunt, Sauer and Yorke call prevalence, which coincides with full Lebesgue measure in Euclidean space and which extends to infinite-dimensional vector spaces. This notion is not directly applicable in most economic settings because the relevant parameter sets are small subsets of vector spaces -- especially cones or order intervals -- not vector spaces themselves. We adapt the notion to economically relevant environments by defining two notions of prevalence relative to a convex set in a topological vector space. The first notion is very easy to understand and apply, and has all of the properties one would desire except that it is not closed under countable unions; the second notion contains the first and has all the good properties of the first notion except simplicity; it is closed under countable unions. We provide four economic applications: 1) generic existence of equilibrium in financial models, 2) generic finiteness of the number of pure strategy Nash equilibria and Pareto inefficiency of "non-vertex" Nash equilibria for games with a continuum of actions and smooth payoffs, 3) generic regularity of exchange economies when some agents are constrained to have 0 endowment of some goods, 4) generic single-valuedness of the core of transferable utility games.

The Limits of ex post Implementation

“…10 A Borel set A ⊂ U is finitely prevalent in U if the relative complement U \ A is finitely shy in U. Hunt, Sauer, and Yorke (1992) and Anderson and Zame (2001) have argued that finite prevalence and prevalence, which is a generalization, provide a sensible measure-theoretic notion of "largeness" for infinite-dimensional spaces of parameters. In particular, if E = R n , then B = U \ A is finitely prevalent in U if and only if the Lebesgue measure of A is 0.…”

Identification of Nonparametric Simultaneous Equations Models With a Residual Index Structure

“…With minor modification, our arguments below demonstrate that even the set F * ⊂ F * S (defined in (S.4)) is residual in F . As discussed by Mas-Colell (1985), these topological definitions of genericity are standard in infinitedimensional spaces but fall short of fully satisfactory notions of "typical" (see also Hunt, Sauer, and Yorke (1992), Anderson andZame (2001), or Stinchcombe (2002)). Classification of our density conditions according to alternative notions of genericity is a potentially interesting direction for future work.…”

Genericity and Robustness of Full Surplus Extraction

“…Relaxing this assumption, we should study the genericity/non-genericity of FSE in the space of general priors supported on an arbitrary number of types. Following this view, HN prove that FSE priors are "negligible" (non-generic) in a geometric sense (i.e., they are contained in a proper face), and a measure-theoretical sense (i.e., they are contained in a finitely shy set as defined in Anderson and Zame (2001)). 2 In this paper, we also relax CM's common-knowledge assumption of a fixed finite number of types, and yet we prove that FSE is topologically generic.…”

“…8 Anderson and Zame (2001) point out some weakness of the residual (resp. meager) set as the notion of genericity (resp.…”

“…nongenericity). 9 This follows because an open set cannot be finitely shy (see Anderson and Zame (2001)). Thus, this observation holds in any topology which is finer than the weak topology.…”