/
computer-modern.tex
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/
computer-modern.tex
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\documentclass[letterpaper]{article}
\usepackage{amssymb}
\usepackage{url}
\usepackage{hyperref}
% set text colour to #2c2e35
\usepackage{xcolor}
\definecolor{dark}{HTML}{2c2e35}
\color{dark}
% set margins to 1 inch
\usepackage[margin=1in]{geometry}
% set indent to 0
\setlength{\parindent}{0pt}
% set paragraph spacing to 1em
\setlength{\parskip}{1em}
\title{Imitating the Typography From Classic Hindawi Journals
(i.e., from `Fixed Points as Nash Equilibria' (Torres-Mart\'inez, 2006))}
\author{Pach\'a (\url{https://pacha.dev})}
\begin{document}
\maketitle
This uses Computer Modern (TeX default).
Let $Y \subset \mathbb{R}^n$ be a convex set. A function
$v: Y \rightarrow \mathbb{R}$ is \emph{quasiconcave} if, for each
$\lambda \in (0,1)$, we have
$v(\lambda y_1 + (1 - \lambda y_2)) \geq \min\{v(y_1),\: v(y_2) \}$, for all
$(y_1, y_2) \in Y \times Y$.
\emph{[Nash-2.]} Given $\mathcal{G}=\{I,S_i,V^i\}$, suppose each set
$S_i \in \mathcal{H}$ and that objective functions are continuous in its
domain and \emph{quasiconcave} in its own strategy. Then there is a Nash
equilibrium for $\mathcal{G}$.
\emph{[Kakutani.]} Given $X \in \mathcal{H}$, every closed-graph correspondence
$\Phi: X \twoheadrightarrow X$, with $\Phi(x) \in \mathcal{H}$ for all
$x \in X$, has a fixed point, provided that
$\Phi(x) = \prod_{j=1}^{m} \pi_j^m(\Phi(x))$ for each
$x \in X \subset \mathbb{R}^m$.
The article proves that \emph{[Nash-2]} $\rightarrow$ \emph{[Kakutani]}.
Ok, sufficient display of math symbols.
See \url{https://github.com/pachadotdev/varsityblues} for a set of complete
LaTeX templates to be used with R Markdown or Quarto.
\end{document}