Nash equilibrium reconsidered and an option for maximin

In this paper we will discuss some peculiarities of Nash equilibrium which are at odds with its standard applications: (a) the underlying dynamic interpretation, (b) the incentive independency if equilibria are mixed, and (c) the unprofitability if equilibrium and maximin are mixed and the game is 2-by-2. Maximin is proposed as an alternative solution concept in relevant situations.Institute of Economics, University of Hamburg, Von-Melle-Park 5, D-2000 Hamburg 13, Germany. This is a substantially revised version of the paper Anti-Nash and an Option for Maximin (Institute of Economics, University of Aarhus, Memo 1989-16), which the author presented at the Fourth Annual Congress of the European Economic Association, Augsburg, September 2–4, 1989.     

On statistical approaches to the study of the “Law of Initial Values”

This paper is concerned with the well-known Law of Initial Values (Wilder, 1931, 1967). First, possible causes of this phenomenon of initial value dependence are discussed. The limitations of a structural relationships based method for testing its effect in empirical data sets in social and behavioral science research are next discussed. Within the popular structural equation modeling methodology, a multiple-indicator model is then focused on, which allows studying the Law of Initial Values. The described approach is illustrated using data from a cognitive intervention study (Baltes, Dittmann-Kohli & Kliegl, 1986).     

On the conceptualization of theoretical constructs

An attempt is made to bridge the gap between the theorists and modellists approaches to the conceptualization of theoretical constructs. The former approach assumes that every theoretical construct has a surplus meaning over and above its operationalization and measurement as a concept. The latter approach denies this: it only accepts theoretical constructs as functions of relations between observations, i.e. measured concepts.This gap can be bridged by explicating the definition of a theoretical construct before making attempts to operationalize and measure it. An explicated definition should cover all relevant aspects (or: facets) of a construct, systematically brought together in an ideal type model. On this basis, operationalization of the construct can be directly linked to model elements, so that any possibly remaining surplus value is made explicit. The distance between theoretical construct and the measured concept can then be made as small as seems feasible; in principle, the distance can be made zero. The manner in which conservative ideology is conceptualized in a longitudinal research project in the Dutch electorate is presented as an example of this strategy.     

Logic from space

This is one of the essential results of an inquiry into spacetime-sociology which was outlined and begun with in a research memorandum of the Institute for Advanced Studies, Vienna (IHS) 1992. It contains a proof that classical logic, or Boole's laws of thought, can be derived from an original concept of orientation that has emerged in human cognition during paleolithic worship activity. The original concept of orientation represents a genetic structure of orientation in both physical and social space. This is of a considerable importance. Because the original concept of orientation in its exact mathematical form is a symmetry in space, strictly a subgroup of the rotation-groupD ₃. Thus it belongs to the fields of mathematics and physics entirely and has nothing to do with logistics, linguistics or sociology. On the other hand binary logic is an important aspect of language and thought and seems to have little in common with the perception of objects in space. So there is a connection between logic and orientation in space. Mathematically this is represented by an isomorphism between the spatial symmetryD _2d of the Mandala of the quartered circle and the symmetry of classical logic.     

On functionals of order statistics

Summary Let g_n be real functions,U _ni, 1in, the ordered sample ofn independentU(0,1) distributed random variables, andc _ni(), 1in, 01 be (known) real numbers,n=1, 2, ... The random quantity , 01, is studied. Based on a method proposed byShorack [1972] the main result is the weak convergence of to Gaussian processes, where , 01. The convergence is with respect to theSkorokhod [1956]-topologiesM ₂,M ₁ onD (I) and the -topology onC(I), depending on the conditions imposed on thec _ni().     

On the flexible functional forms

This note proposes a general structure of the so-called flexible functional forms able to describe direct utility functions. It is obtained by solving the functional equation:

Eine Verallgemeinerung des „Gesetzes seltener Ereignisse”

Zusammenfassung Beim bekannten Gesetz seltener Ereignisse betrachtet mane große Zahl unabhängiger Ereignisse, die jeweils mit der gleichen kleinen Wahrscheinlichkeit eintreten können. Unter geeigneten Annahmen ist die Zahl der eingetretenen Ereignisse näherungsweise poissonverteilt. [Vgl. z.B. Morgenstern, p. 39, 1968;Waerden, p. 47, 1957].Ein oft angegebenes Beispiel hierfür ist die Häufigkeitsverteilung von Unfällen, die jeweils nur eine Person oder ein Fahrzeug betreffen. In dieser Arbeit gehen wir von Unfällen mit zwei Beteiligten aus. Auf diese Weise erhält man eine Verallgemeinerung des Gesetzes seltener Ereignisse auf gewisse abhängige Ereignisse.Außerdem wird ein neuer Beweis für die allgemeine Formel der Momente der Poissonverteilung gegeben.Damit ist ein Grenzwertsatz für eine spezielle Folge schwach abhängiger Zufallsgrößen bewiesen.

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Summary The well-known law of rare events deals with a large number of independent events, each of which can occur with the same probability. Under suitable assumptions the number of events which occur has approximately a Poisson distribution. [CompareMorgenstern, p. 39, 1968;Waerden, p. 47, 1957].An example which is often used for illustration is the frequency distribution of accidents, in which only one car or one person is involved. In this paper we refer to accidents in which two parties are involved. This leads to a generalization of the law of rare events where cases of dependent events are concerned.In addition a new proof of the general formula as to the moments of Poisson distribution is given.This proves a limit theorem of weakly dependent random variables of a certain sequence.

     

Eine Invarianzeigenschaft für Startverteilungen einer Klasse stochastischer Prozesse

Summary This paper discusses some concepts of mixing for stochastic processes with discrete time. The idea of mixing, which is defined with respect to a starting distribution , means that the tranjectories of the process get out of any set with -measure zero with probability one. Such -mixing processes satisfy an invariance property; an asymptotic event has probability zero under any starting distribution, provided that it has probability zero under the starting distribution . Concerring stationary Markov chains these results imply a weak zero-one-law the relation of which with well-known stronger versions especially for aperiodic Harris chains and with the notions of weak ergodicity and a.s. triviality is studied.     

RAM: A Range Adjusted Measure of Inefficiency for Use with Additive Models,and Relations to Other Models and Measures in DEA

Generalized Efficiency Measures (GEMS) for use in DEA are developed and analyzed in a context of differing models where they might be employed. The additive model of DEA is accorded a central role and developed in association with a new measure of efficiency referred to as RAM (Range Adjusted Measure). The need for separately treating input oriented and output oriented approaches to efficient measurement is eliminated because additive models effect their evaluations by maximizing distance from the efficient frontier (in ₁, or weighted ₁, measure) and thereby simultaneously maximize outputs and minimize inputs. Contacts with other models and approaches are maintained with theorems and accompanying proofs to ensure the validity of the thus identified relations. New criteria are supplied, both managerial and mathematical, for evaluating proposed measures. The concept of approximating models is used to further extend these possibilities. The focus of the paper is on the physical aspects of performance involved in technical and mix inefficiencies. However, an Appendix shows how overall, allocative and technical inefficiencies may be incorporated in additive models.     

Coloured semantic networks for content analysis

This paper adapts a widespread formalism of Knowledge Representation known in the AI literature as J. Sowa'sConceptual Graphs to the purposes of Content Analysis. It is proposed that instead of nested contexts, negation and modalities could be represented by colouring the links and the nodes of the graphs. This colour-based representation of logic enables the replacement of Sowa's game-theoretical semantics by a matching algorithm. It is also claimed that the resulting local application of negation and modalities is close to the behaviour of negation and adverbs in natural language. Above all, the use of colours restores a unique level in semantic networks. These flat semantic networks are appealing to social scientists because they avoid information redundancy and improve connectivity in the network. Two methods of textual analysis that can take direct benefit from semantic networks (semantic paths distributions and texts clustering) are presented. However, flat semantic networks that remain semantically correct are perhaps even more interesting because they may bridge standard networks with connectionnist networks. In such a perspective, Coloured Conceptual Graphs represent an intermediate but promising step.     

Maximal asymptotic power and efficiency of two-sample tests based on generalized U-Statistics

In this article a systematic study is given of the asymptotic behavior of two-sample tests based on U-Statistics with arbitrary antisymmetric kernels . Besides the investigation under the hypothesis and under fixed alternatives we determine the local power as a function of as well as its maximizing value _opt. Moreover formulas for the asymptotic relative efficiency ARE(₂,₁) of the ₂-test with respect to the ₁-test are derived. It turns out that _opt also yields the most efficient test in the sense that ARE(_opt,)1 for all (admissible) kernels .     

Cascaded channels and the equivocation inequality

The variation of the equivocation and the average mutual information over cascaded channels is studied, using the generalized information measures (entropies of degree ) and Renyi information measures (entropies of order ). The equivocation inequality, which indicates that the equivocation can never decrease as we go further from the input on a sequence of cascaded channels, is shown to be satisfied by Renyi entropies (for 0<<1) for all channels and all probability distributions. The generalized entropies are shown to satisfy this inequality for binary symmetric channels for certain probability distributions The relations among the mutual information measures between the different terminals of the sequence of cascaded channels are studied, considering both the generalized and Renyi entropies. A necessary and sufficient condition for transmitting information without any loss across cascaded channels is obtained, when Renyi entropies are applied. The same condition is proved to be sufficient for achieving the equivocation equality (which indicates the case of no information loss across cascaded channels) over the class of binary symmetric channels, when the generalized entropies are applied.     

Sulla dipendenza continua della probabilità di rovina dal caricamento dei premi

Si studia un modo di approssimare la probabilità di rovina relativa a un caricamento ₀ con le probabilità di rovina relative a una successione di caricamenti ( _k ) _k , che approssimano ₀ quandok tende all'infinito.

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Summary In this paper we study a way of approximating the probability of ruin related to a loading ₀, by the probabilities of ruin related of a sequence of loadings ( _k ) _k which «approximate» ₀ ask converges to infinity.

     

A numerical representation of semiorders on a countable set

Let be a semiorder on a countable setX and letx⁰ y if and only if either there existsx withxxy or there existsx withxxy. Then ⁰ is a preference relation with transitive indifference, which can be represented by a utility functionf of the usual sort. It is well known that is represented by a pair of real-valued functionsu, v, in the sense thatxy if and only ifu(x)>v(y). We prove that there exists a pair of functionsu, v, representing , such thatu+v is the utility function which represents in the usual sense. Moreover it is easily seen that, for such a pair of functionsu, v, we havex⁰ y if and only if eitheru(x)>u(y) or (u(x)=u(y) andv(x)>v(y)).

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Sommario Consideriamo unsemiordine su un insiemeX numerabile e poniamox⁰ y se e solo se esistex tale chexxy, oppure esistex tale chexxy. In questo caso ⁰ è unordine debole, che può essere rappresentato da una funzione di utilitàf nel senso usuale. D'altra parte è rappresentato da una coppia di funzioniu, v, nel senso chexy se e solo seu(x)>v(y). In questo lavoro si prova che ammette una rappresentazioneu, v tale chex⁰ y se e solo seu(x)+v(x)>u(y)+v(y). Si dimostra altresì che, con riguardo ad una siffatta rappresentazioneu, v di , riescex⁰ y se e solo seu(x)>u(y) oppure (u(x)=u(y) ed anchev(x)>v(y)).

     

Some brief notes on the “unfinished business” of qualitative inquiry

The paper argues that there are a variety of implicit issues in qualitative inquiry that need to be addressed if the area is to develop in some normal science sense. This unfinished business is concerned with a deeper investigation of basic terms that are now simply taken for granted, such as theme and pattern. It also includes the need to develop rules which will assist in making and justifying how qualitative interpretations are made from the implicit processes of inference. Specific suggestions are made for accomplishing these issues.     

Estimation problems for rectangular distributions (or the taxi problem revisited)

The problem of estimating the unknown upper bound on the basis of a sample of sizen from a uniform or rectangular distribution on [0, ] has considerable interest. This or the analogous discrete version is variously known as the Taxi-problem or the German bomb (or Tank) problem and has a long history. The emphasis here is on estimation of through the lengths of the observed gaps or spacings which seem natural for this problem.     

Who consults investment analysts?

This paper shows that the questions: Who consults investment analysts? and Who goes to the El Farol bar in Santa Fe? are similar. Thus, investors use a mixed optimal strategy. The demand for consulting services is also characterized.Received: 25 September 2002, Accepted: 26 May 2003, JEL Classification: D81, C70Jacob Paroush: The author is grateful to Yaw Nyarko, Yigal Milchtaich and an anonymous referee for helpful comments and suggestions.     

Estimation of a regression coefficient after two preliminary tests of significance

Summary The estimation of the regression coefficient of a population, defined byE (y)= +x, incorporating two preliminary tests of significance has been discussed. The experimenter has two random samples of different sizes from two such populations, as defined above, with regression coefficients ₁ and ₂ respectively, where ₂ may possibly be equal to ₁. Besides this, it is also conjectured that the common conditional variance ² of the two populations has a specified value ₀ ² . The two preliminary tests are used to resolve these two uncertainties.The author has rejoined Lucknow University, Lucknow India on Oct. 4, 1976 after availing leave for two years.     

An Efficiency Comparison of European Banking Systems Operating under Different Environmental Conditions

The paper investigates the operating efficiency differences of a sample of commercial banks across 10 European countries. First, the paper analyzes the technical efficiency of each country sample following the basic Data Envelopment Analysis (DEA) model incorporating only banking variables. Then, a complete DEA model is introduced, incorporating environmental factors together with the banking variables of the basic model. The comparison between the two models shows that country-specific environmental conditions exercise a strong influence over the behavior of each country's banking industry.     

Eine simultane Charakterisierung der geometrischen Verteilung und der logistischen Verteilung

Summary LetE denote the generating function of a non-degenerate probability distribution on the positive integers, letF be a non-degenerate distribution function, and let be a real valued function on the interval (0, 1]. In the present paper the solutions (E, F, ) of the functional equationE(F(x)/E()=F(x+()), –<x<+, (0, 1] are given. It is shown that ifF is symmetric about zero,E andF belong to a solution of the functional equation if and only ifF is a logistic distribution andE is the generating function of a geometric distribution.