Homological Methods for the Economic Equilibrium Existence Problem: Coincidence Theorem and an Analogue of Sperner’s Lemma in Nikaido

In this paper, I introduce the theorems in Professor Hukukane Nikaido’s work, “Coincidence and some systems of inequalities,” published in the Journal of Mathematical Society of Japan, 1959, and note the significance of his mathematical methods on the history and the future of mathematical economics. Nikaido (1959) may be considered a compilation of his works of the 1950’s on economic equilibrium existence problems. It also provides, however, his further developments and attempts for mathematical methods in the theory of mathematical economics and an algebraic (algebraic topological) methods based on results of the Vietoris homology theory (the earliest kind of Čech-type homology theories). From Nikaido’s main mathematical results, an analogue of Sperner’s lemma and a coincidence theorem, we may obtain a simple proof for Eilenberg-Montgomery’s theorem for finite dimensional cases. We may also utilize such homological methods for many generalizations of fixed point arguments on multivalued mappings in relation to Lefschetz’s fixed point theorem.