Jerzy Łoś

Semantic Representation of the Probability of Formulas in Formalized Theories

In probability theory, or rather in its foundations, there has long been a trend in favour of identifying events, i.e., objects to which probability is ascribed, with formulas of certain theories. Without adducing arguments in favour of that idea I shall confine myself to mentioning its principal representatives, namely J. M. Keynes, J. Nicod, H. Jeffreys, H. Reichenbach, R. Carnap, and in Poland J. Lukasiewicz and K. Ajdu-kiewicz.

Reswitching of techniques and equilibria of extended von Neumann models

Introduction. The purpose of this paper is to convince the reader that the right way of studying the paradoxes of capital theory is through the equilibria of extended von Neumann models.

The Approximative Horizon in Von Neumann Models of Optimal Growth

For families of dynamic programming problems, the concept of e-horizon (approximative horizon) is introduced. Intuitively h is an e-horizon, e >0, iff the knowledge of conditions for h steps ahead, allow us to make decisions with the final efect differing from the optimal by not more than e.

Von Neumann models of open economies

An economic equilibrium is a relationship of quantities and prices. Depending on the considered economy equilibria have different properties. We shall distinguish here those which are based on positive homogenity, saying that an equilibrium is homogeneous in prices iff quantities remain in equilibrium when prices are multiplies by any positive factor. Obviously in this case the equilibrium does not depend on the exact values of prices, but rather on relative proportions of prices of different commodities. An analogous definition applies to the homogenity in quantities.

Extremal Properties of Equilibria in von Neumann Models

For any weak equilibrium of a von Neumann model there are defined linear programming problems with, the property that the equilibrium intensity vectors and/or price vectors are their optimal solutions. Some applications to special type of models are discussed.

Warsaw fall seminars in mathematical economics, 1975

An open von Neumann model with, consumption.- Contribution to the theory of existence of von Neumann equilibria (II).- Cournot-Bertrand mixed oligopolies.- La grange multipliers for the problem of stochastic programming.- Von Neumann models defined by transformations and by production cones.- Von Neumann models of open economies.- Reswitching of techniques and equilibria of extended von Neumann models.- Quasi-Leontief models.- Coalition games without players. An application to Walras equilibria.- The outline of a general disequilibrium dynamic model with personal income and wealth distributions.

Common Extension in Equational Classes

Publisher Summary The purpose of this chapter is to give a detailed solution of the problem related to the equational classes. The problem of common extension for a class of algebra ଅ 0 is in finding the necessary and sufficient conditions under which for two given algebras A and B in ଅ 0 , there exists an algebra C that is also in ଅ 0 and contains isomorphic images of A and B. The chapter discusses two necessary conditions for equational classes—algebra of constants and algebra of psuedoconstants—and shows by means of suitable examples that they are not sufficient. The chapter then provides a necessary and sufficient condition. The chapter also denotes a conjunction of equations that is called “a system of equations” and denoted by U (x l , …, x n ) . The chapter assumes that no other variables occur in U except those in the brackets. It assumes the same when writing α (x 1 , .. , x n ), β (y 1 , …, y m ), and so on.

Limit Solutions of Sequences of Statistical Games

The present paper deals with the so-called “empirical Bayes approach” proposed and studied by Herbert Robbins. His papers, especially paper [3], have inspired us to show the notion of “empirical Bayes approach” fits into the framework of game theory by means of simple operations on games. We show it by examples of testing two simple statistical hypotheses (Example 5; the original version of this example was presented by H. Robbins in [3]). At the end we prove some easy lemmas concerning the relations between Bayes, minimax and limit solutions.