The existence of TU α-core in normal form games

This paper provides a TU α-core existence result in a large class of normal form games. In the oligopoly markets of a homogeneous good, the TU α-core is non-empty if all profit functions are continuous and concave. In a general game, the existence of TU α-core follows from the weak separability, the compactness and convexity of choice sets, and the concavity and continuity of payoff functions. Received: November 1997/final version: August 1998     

Refinements of the β-core and the strong equilibrium and the Aumann proposition

The Aumann proposition establishes that an outcome in the -core of the one-shot gameG is the same as the strong equilibrium utility allocations of the associated supergame. We refine the -core by requiring that the threat strategies used to deter deviations be credible and examine the consequences for the Aumann proposition in the case of the refinement. We show that a refinement of the strong-equilibrium notion is always in the refined -core. The converse is not necessarily true unless the utility allocations are linear.I am grateful to T. Ichiishi for several discussions that helped in the development of this paper. This is part of the author's Ph.D. dissertation submitted at the university of Iowa in December 1985.     

An axiomatization of the τ-value

The τ-value is characterized by three axioms. It is shown that the τ-value is the unique solution concept which is efficient and has the minimal right property and the restricted proportionality property. The minimal right property is weaker than the additivity property, which plays a role in the axiomatic characterization of the Shapley value: together with individual rationality and efficiency additivity implies the minimal right property. The restricted proportionality property says that for games with minimal right vector zero, the dividend given to the players is proportional to the marginal contribution of the players to the grand coalition.     

The periodic points and the invariant set of an ϵ-contractive map

It is shown that the invariant set of an ϵ-contractive map f on a compact metric space X is the same as the set of periodic points of f. Furthermore, the set of periodic points of f is finite and, only assuming that X is locally compact, there is at most one periodic point in each component X. The theorems are applied to prove a known fixed-point theorem, a result concerning inverse limits, a result about periodic points of compositions, and a result showing that ϵ-contractive maps on continua are really contraction maps with a change in metric. It is shown that all our results hold for locally contractive maps on compact metric spaces.     

On the consistency of the MLE for Ornstein–Uhlenbeck and other selfdecomposable processes

In this paper we give easy to verify conditions for the strong consistency of the maximum likelihood estimator (MLE) in the case when data is sampled from a parametric family of selfdecomposable distributions. The difficulty arises from the fact that standard conditions for the consistency of the MLE are based on the pdf, which, for most selfdecomposable distributions, is not available in a closed form. Instead, our conditions are based on properties of the Lévy triplet (i.e. the Lévy measure, the Gaussian part, and the shift) of the distribution. Further, we extend out results to certain selfdecomposable stochastic processes, and, in particular, we give conditions in the case when the data is sampled from a Lévy or an Ornstein–Uhlenbeck process.     

The exit time of planar Brownian motion and the Phragmén–Lindelöf principle

In this paper a version of the Phragmén–Lindelöf principle is proved using probabilistic techniques. In particular, we will show that if the p  th moment of the exit time of Brownian motion from a planar domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to ∞ along some sequence more rapidly than e^|z|2p$e^{{|z|}^{2p}}$. We also provide a method of constructing domains whose exit time has finite pth moment. This allows us to give a general Phragmén–Lindelöf principle for spiral-like and star-like domains, as well as a new proof of a theorem of Hansen. A number of auxiliary results are presented as well.     

The rational approximation of convex functions of the class Lip α

It is proved that if a function f(x) is convex on [a, b] and f ∈ Lip_K(f)α, 0<α<1, then the least uniform deviation of this function from rational functions of degree not higher than n does not exceed (v is a natural number; C(α, v) depends only onα andv; K(f) is a Lipschitz constant; and      

The projection onto the center of an ℓ-algebra

In a uniformly complete -algebra with unite>0 the principal band {e} ^dd generated bye is a projection band. We prove that the formula of A. R. Schep for the projection onto this band which holds in the -algebra of order bounded operators on a Dedekind complete vector lattice, in general defines a projection onto the commutant of {e} ^dd . We present some examples to show that the commutant may be strictly larger than the band {e} ^dd , and also conditions which guarantee equality.This paper was written during the fall semester of 1993 while the first author visited the Department of Mathematics of the University of Leiden as a guest of Dr. C. B. Huijsmans, to whom the express his gratitude for the hospitality offered to him.     

On some properties of the ε-core of games with coalition structure

Chang [1991] stated some inclusion relation about kernel, reasonable set, and ε-core of games with coalition structure. We give counterexamples and modify erroneous results. Received: May 1997/Revised version: January 1999     

Extremal Lyapunov exponents: an invariance principle and applications

We propose a new approach to analyzing dynamical systems that combine hyperbolic and non-hyperbolic (“center”) behavior, e.g. partially hyperbolic diffeomorphisms. A number of applications illustrate its power.