  This paper introduces a novel framework for assessing risk and
decision-making in the presence of uncertainty, the \emph{$\varphi$-Divergence
Quadrangle}. This approach expands upon the traditional Risk Quadrangle, a
model that quantifies uncertainty through four key components: \emph{risk,
deviation, regret}, and \emph{error}. The $\varphi$-Divergence Quadrangle
incorporates the $\varphi$-divergence as a measure of the difference between
probability distributions, thereby providing a more nuanced understanding of
risk. Importantly, the $\varphi$-Divergence Quadrangle is closely connected
with the distributionally robust optimization based on the $\varphi$-divergence
approach through the duality theory of convex functionals. To illustrate its
practicality and versatility, several examples of the $\varphi$-Divergence
Quadrangle are provided, including the Quantile Quadrangle. The final portion
of the paper outlines a case study implementing regression with the Entropic
Value-at-Risk Quadrangle. The proposed $\varphi$-Divergence Quadrangle presents
a refined methodology for understanding and managing risk, contributing to the
ongoing development of risk assessment and management strategies.
