Generic Finiteness Research Articles

Stationary Equilibria in Stochastic Games: Structure, Selection and Computation

This paper is the first to introduce an algorithm to compute stationary equilibria in stochastic games, and shows convergence of the algorithm for almost all such games. Moreover, since in general the number of stationary equilibria is overwhelming, we pay attention to the issue of equilibrium selection. We do this by extending the linear tracing procedure to the class of stochastic games, called the stochastic tracing procedure. From a computational point of view, the class of stochastic games possesses substantial difficulties compared to normal form games. Apart from technical difficulties, there are also conceptual difficulties, for instance the question how to extend the linear tracing procedure to the environment of stochastic games.We prove that there is a generic subclass of the class of stochastic games for which the stochastic tracing procedure is a compact one-dimensionalpiecewise differentiable manifold with boundary. Furthermore, we prove that the stochastic tracing procedure generates a unique path leading from any exogenously specified prior belief, to a stationary equilibrium. A well-chosen transformation of variables is used to formulate an everywhere differentiable homotopy function, whose zeros describe the (unique) path generated by the stochastic tracing procedure. Because of differentiability we are able to follow this path using standard path-following techniques. This yields a globally convergent algorithm that is easily and robustly implemented on a computer using existing software routines. As a by-product of our results, we extend a recent result on the generic finiteness of stationary equilibria in stochastic games to oddness of equilibria.

A Generic Finite State Compiler for Tagging Rules

We describe a novel method of compiling ranked tagging rules into a "bimachine", i.e. a deterministic finite state device composed of two finite automata: a left-to-right one and a right-to-left one. The actual compilation is based on algorithms for finite state acceptors rather than transducers, which guarantees determinizability and the efficiency of compilation. The compiler has been used in a number of applications within a speech synthesis system.

On pluricanonical maps for threefolds of general type, II

This note mainly studies the generic finiteness of \phi_m of a complex projective 3-fold of general type. A new result on the classification to bicanonical pencil for Gorenstein 3-folds is attached in the last section.

Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes

Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes

On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms

Consider nonempty finite pure strategy sets S1,…,Sn, let S=S1×⋅⋅⋅×Sn, let Ω be a finite space of “outcomes,” let Δ(Ω) be the set of probability distributions on Ω, and let θ: S→Δ(Ω) be a function. We study the conjecture that for any utility in a generic set of n-tuples of utilities on Ω there are finitely many distributions on Ω induced by the Nash equilibria of the game given by the induced utilities on S. We give a counterexample refuting the conjecture for n≥3. Several special cases of the conjecture follow from well known theorems, and we provide some generalizations of these results.

On the Generic Finiteness of Equilibrium Outcomes in Plurality Games

In this paper we prove that for generic (noncooperative) voting games under plurality rule the set of equilibria that induce a mixed distribution over the outcomes (i.e., with two or more candidates elected with positive probability) is finite and, furthermore, each of these equilibria is regular. From that we deduce the finiteness of the set of equilibrium distributions over outcomes. Furthermore we offer an example (S. Govindan and A. McLennan, 1997, “On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms,” mimeo) that shows the impossibility of extending such results to a general framework, even just to voting games. Journal of Economic Literature Classification Numbers: C72, D72.

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.

Sraffian Indeterminacy in General Equilibrium

The indeterminacy claim for competitive price systems made by Sraffa (1960) is examined by placing Sraffa's work in an intertemporal general equilibrium model. We show that indeterminacy occurs at a natural type of equilibrium. Moreover, the presence of linear activities instead of a differentiable technology is crucial and the indeterminacy is constructed, as in Sraffa, by fixing some or all of the economy's aggregate quantities. On the other hand, an extra condition, that some factors have inelastic excess demand is necessary, and, unlike Sraffa's model, relative prices must be allowed to vary through time. Sraffian indeterminacy and the generic finiteness of the number of equilibria are reconciled by showing that indeterminacy occurs at a measure-zero set of endowments. We use an overlapping-generations model to show that these endowments nevertheless arise systematically and that indeterminacy does not occur when relative prices are constant through time. Near the beginning of his Production of Commodities by Means of Commodities (1960), Sraffa specifies a system of equations that prices must obey if the capital invested in each sector of an economy is to earn the same rate of profit. The system contains one more unknown than equation, leading Sraffa to conclude that prices, and wage and interest rates in particular, are indeterminate. Sraffa's book has since been read as arguing that some outside, non-economic consideration must therefore set prices and the distribution of income. Part of the controversy the book has provoked is due to Sraffa's apparent agreement with the neoclassical emphasis on optimization and competitive markets. The standard Sraffian system of equations is nothing other than the first-order conditions that hold for price-taking, profit-maximizing firms using linear activities. But if Sraffa's assumptions are in accord with neoclassical theory, how can his assertions of indeterminacy be correct? I examine Sraffa's indeterminacy argument and his critique of neoclassical economics by embedding his equations in a general equilibrium model. As we will see, the indeterminacy claim, properly interpreted, correctly describes the equilibria with strictly positive prices. Moreover, the indeterminacy demonstrated here closely duplicates the indeterminacy Sraffa claimed. First, it is prices alone that are indeterminate. I follow Sraffa's striking procedure of fixing the quantities of goods produced: an infinity of (normalized) price vectors support a single vector of aggregate quantities. Second, Sraffa seemed to claim, and his advocates (e.g. Robinson (1961)) have certainly claimed that his book is a critique of marginal productivity theory. Correspondingly, indeterminacy in the present model hinges on marginal products not being everywhere well-defined when there are linear activities; with smooth production sets, our indeterminacy arguments do not work. Third, the dimension of the indeterminacy I demonstrate matches Sraffa's accounting of the dimension of indeterminacy.

Generic Finiteness of Equilibrium Outcome Distributions for Sender–Receiver Cheap-Talk Games

This paper shows generic determinacy of equilibria for sender-receiver cheap-talk games: for a generic set of utilities on outcomes, there are a finite number ofreduced formequilibria. A corollary is that generically the set of probability distributions over outcomes generated by equilibria is finite. Because of the identification of terminal nodes for utility purposes, Sard's theorem is not applicable in the way used in Kreps and Wilson [11], and a new proof strategy is developed. It is also shown that the set of sender types is the efficient message space supporting all reduced form equilibria.Journal of Economic LiteratureClassification Number: C72.

Generic finiteness in solving optimal control problems

Generic finiteness in solving optimal control problems