Combinatorial Integer Labeling Thorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations

Tucker's well-known combinatorial lemma states that for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set f§1;§2; ¢ ¢ ¢ ;§ng with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set f§1;§2; ¢ ¢ ¢ ;§ng. Using a constructive approach we prove two combinatorial theorems of Tucker type, stating that under some mild conditions there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set f0; 1gn+q for some integral vector q. These theorems will be used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions.Sperner lemma;Tucker lemma;integer labeling;simplicial algorithm;discrete nonlinear equations

Freedom, Anarchy and Conformism in Academic Research

In this paper I attempt to make a case for promoting the courage of rebels within the citadels of orthodoxy in academic research environments. Wicksell in Macroeconomics, Brouwer in the Foundations of Mathematics, Turing in Computability Theory, Sraffa in the Theories of Value and Distribution are, in my own fields of research, paradigmatic examples of rebels, adventurers and non-conformists of the highest caliber in scientific research within University environments. In what sense, and how, can such rebels, adventurers and non-conformists be fostered in the current University research environment dominated by the cult of 'picking winners'? This is the motivational question lying behind the historical outlines of the work of Brouwer, Hilbert, Bishop, Veronese, Gödel, Turing and Sraffa that I describe in this paper. The debate between freedom in research and teaching, and the naked imposition of 'correct' thinking, on potential dissenters of the mind, is of serious concern in this age of austerity of material facilities. It is a debate that has occupied some of the finest minds working at the deepest levels of foundational issues in mathematics, metamathematics and economic theory. By making some of the issues explicit, I hope it is possible to encourage dissenters to remain courageous in the face of current dogmasNon-conformist research, economic theory, mathematical economics, 'Hilbert's Dogma', Hilbert's Program, computability theory

Applications of a generalization of a set intersection theorem of Von Neumann

Includes bibliographical references.by Robert M. Freund

Equilibria on a congested transportation network

Bibliography: leaves 26-29.Supported in part by the Transportation Advanced Research Program of the U.S. Department of Transportation under contract. DOT-TSC-1058 Supported in part by the National Science Foundation under grant. 79-26225-ECSby Hedayat Z. Aashtiani and Thomas L. Magnanti

A geometric-combinatorial approach to index and stability in bimatrix games

This thesis provides a new geometric-combinatorial construction to characterise the Nash equilibria of a non-degenerate bimatrix game and their indices. Considering a non-degenerate m x n bimatrix game, the construction yields an (m — 1)-simplex X^ that is simplicially divided into (m — l)-simplices, reflecting the best reply structure of player II. Each (m — 1)-simplex in the triangulation is divided into best reply regions of player I. This yields a division of XA into regions with labels 1,..., m. In this representation, the Nash equilibria are represented by completely labelled points, and the index is the local orientation of the m regions around completely labelled points. For a missing label of player I, the Lemke-Howson algorithm follows paths in XA that are defined by m — 1 labels of player I. This representation of bimatrix games is shown to be related to Sperner's Lemma in dimension m — 1. In particular, the existence of Nash equilibria in non-degenerate bimatrix games is equivalent to Brouwer's fixed point theorem. The construction yields a new strategic characterisation of the index, conjectured by Hofbauer (2000). It is shown that a Nash equilibrium in a non-degenerate bimatrix game has index +1 if and only if one can add strategies to the game such that the equilibrium is the unique equilibrium of the extended game. The construction can be extended to outside option equilibrium components in bimatrix games. The characterisation for such components is shown to be similar to the well-known Index Lemma. As a consequence, index zero boundary labellings allow triangulations that do not contain a completely labelled simplex. The game theoretic counterpart applies to outside option equilibrium components. It is shown that an outside option equilibrium component is hyperessential if and only if it has non-zero index. This question had been open for some time. It is also shown how equilibrium components of arbitrary index can be constructed by means of outside options in bimatrix games

A Discrete Multivariate Mean Value Theorem with Applications

AMS classifications: 47H10; 54H25; 55M20; 90C33; 91B50Discrete set;mean value theorem;fixed point;algorithm;equilibrium;complementarity