  The Bell experiment is a random game with two binary outcomes whose
statistical correlation is given by $E_0(\Theta)=-\cos(\Theta)$, where $\Theta
\in [-\pi, \pi)$ is an angular input that parameterizes the game setting. The
correlation function $E_0(\Theta)$ belongs to the affine space ${\cal H} \equiv
\left\{E(\Theta)\right\}$ of all continuous and differentiable periodic
functions $E(\Theta)$ that obey the parity symmetry constraints
$E(-\Theta)=E(\Theta)$ and $E(\pi-\Theta)=-E(\Theta)$ with $E(0)=-1$ and,
furthermore, are strictly monotonically increasing in the interval $[0, \pi)$.
Here we show how to build explicitly local statistical models of hidden
variables for random games with two binary outcomes whose correlation function
$E(\Theta)$ belongs to the affine space ${\cal H}$. This family of games
includes the Bell experiment as a particular case. Within this family of random
games, the Bell inequality can be violated beyond the Tsirelson bound of
$2\sqrt{2}$ up to the maximally allowed algebraic value of 4. In fact, we show
that the amount of violation of the Bell inequality is a purely geometric
feature.
