John G. Kemeny

Finite Markov chains

Markov chains are one of the most useful classes of stochastic processes, being • simple, flexible and supported by many elegant theoretical results • valuable for building intuition about random dynamic models • central to quantitative modeling in their own right You will find them in many of the workhorse models of economics and finance. In this lecture we review some of the theory of Markov chains. We will also introduce some of the high quality routines for working with Markov chains available in QuantEcon.jl. Prerequisite knowledge is basic probability and linear algebra.

Mathematical models in the social sciences

As the need for more substantial mathematical training has increased among social science students, the lack of any adequate textbook between the very elementary and the very advanced levels has become crutial. The authors, long-time experts in this field, have answered the need with this volume, and the MIT Press has repsonded by bringing it into renewed circulation.Mathematical Models in the Social Sciences investigates and teaches the formation and analysis of mathematical models with detailed interpretations of the results. These models are self-contained, with the necessary mathematics included in each chapter. A vast range of topics in the social sciences and a wide variety of mathematical techniques are covered by the models. Ample opportunity is also provided for the students to form their own models. Republication of this book provides social science and mathematics students with a text that is the analogue of mathematical methods textbooks used in the study of the physical sciences and engineering. Prerequisites are kept to a minimum; a course in finite mathematics and a semester of calculus are all that is necessary.The chapters cover these main topics (and employ the mathematical approach parenthetically indicated): methodology; preference rankings (an axiomatic approach); ecology (two dynamic models); market stability (a dynamic model); a Markov chain model in sociology; stabilization of money flow (an application of discrete potential theory); branching processes; organization theory (applications of graph theory); and optimal scheduling (a problem in dynamic programming).

Fair Bets and Inductive Probabilities

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness. On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnaps presentation has been criticized for just this reason. The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinettis weaker criterion but Shimonys criterion of fairness is also satisfied.

Potentials for denumerable Markov chains

A unified treatment of potentials for general Markov chains is presented. Basic properties of potentials associated with transient chains are reviewed; these behave like Newtonian potentials. Potentials for recurrent chains are developed; they behave like logarithmic potentials. The discrete analog of the Newtonian potential classical theory is obtained by applying these results to random walls in 3 dimensions. Since the results are applicable to Markov chains in general, discrete versions of the more general theory are obtained. (L.N.N.)

Generalization of a fundamental matrix

Abstract It is shown that, for a finite ergodic Markov chain, basic descriptive quantities, such as the stationary vector and mean first-passage matrix, may be calculated using any one of a class of fundamental matrices. New applications of the use of these operators are discussed.

A new approach to semantics – Part I

This is the first in a series of articles outlining a new approach to Semantics. The novelty in the approach is that the concept of an interpretation of a logical system is taken as the central concept of Semantics. I hope to show that by means of this approach a satisfactory definition can be given for such controversial concepts as analyticity, and at the same time the approach leads to a unified foundation for formalized Semantics. As the possibility of such definitions has been questioned in recent years, it is an important task to try to give precise definitions. Clearly, this is the task of those philosophers who believe that concepts like analyticity should play a fundamental role in Semantics. On the other hand, the philosophers who have criticized these concepts will now be able to tell just exactly why these definitions are unacceptable to them — instead of being forced to talk in generalities. It seems to me that no further progress is possible until we have precise definitions available for discussion. The fundamental semantic concepts fall into two classes: those, like truth, for which Tarski has offered definitions; and those additional ones, like analytic truth, for which we have Carnaps proposed definitions.

Finite Continuous Time Markov Chains

In a recent book by the authors (see footnote on page 101) a systematic method was developed for computing the basic descriptive quantities of finite Markov chains. Matrix expressions were obtained for these quantities in terms of certain basic matrices, easily obtainable from the transition matrix.In this note, corresponding expressions are given for the basic quantities for finite continuous time chains.

The Exponential Function

The first two results are simple exercises in mathematical induction. The third result can be established by bounding the expression by 1 +x/n from below and by 1 or 1/(1 -x/n) from above. Finally, AIV is a standard theorem in the calculus. As motivation for the discussion I suggest a consideration of compound interest and radioactive decay. At simple interest, of rate x, a dollar yields (1 +x); if compounded twice, it yields (1 +x/2)2; and if we keep halving the time interval, we arrive at the formula (1 +x/2n)2 . If x is negative, then the same formula applies to decay. The function E(x) defined below may be thought of as interest compounded continuously or as continuous decay.

Game-Theoretic Solution of Baccarat

The famous gambling game of baccarat has been the subject of many mathematical studies. There is, however, considerable variation in the solutions offered and it is now clear that these variations have come from trying to solve a game of strategy without a precise meaning for the solution of such a game. The theory of games has made this precise; it is now interesting to carry out the solution of baccarat in terms of the modern concepts of game theory and to compare the solution obtained with the earlier attempts based upon the more vague ideas previously available. Baccarat is usually played by three men, one banker playing against a pair of players. But there is a popular variant of the game, known as chemin de fer, in which the banker plays against a single player. This is the version commonly discussed in the literature, and we too will take this two-person version of baccarat as our subject.

Markov processes in learning theory

Consideration is given mathematical problems arising in two learning theories—one developed by Bush and Mosteller, the other developed by Estes. The theory of Bush and Mosteller leads to a class of Markov processes which have been studied in considerable detail (see [1] and [7]). The Estes model can be treated as a Markov chain, i.e., a Markov process with a finite number of states. For an important class of special cases, it is shown that the Bush-Mosteller model is, in a sense, a limiting form of the Estes model. The limiting probability distributions are derived for the cases treated in both models.