InflGame.domains.rd.

InflGame.domains.rd.beta_distribution(bin_points, alpha_value, beta_value)

Compute a beta distribution for resources on the unit interval.

Uses the beta probability density function to model resource distribution on the interval (0, 1). The beta distribution is useful for modeling bounded resources with various shapes controlled by the alpha and beta parameters.

The distribution is defined as:

\[R(b) = \frac{b^{\alpha-1}(1-b)^{\beta-1}}{B(\alpha, \beta)}\]

where \(B(\alpha, \beta)\) is the beta function:

\[B(\alpha, \beta) = \int_0^1 t^{\alpha-1}(1-t)^{\beta-1} dt = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\]

Parameters:

bin_pointsnp.ndarray | torch.Tensor

Points where the distribution is evaluated, typically on the interval (0, 1).

alpha_valuefloat

Alpha parameter \(\alpha\) of the beta distribution. Must be positive. Controls the shape of the distribution at \(b=0\).

beta_valuefloat

Beta parameter \(\beta\) of the beta distribution. Must be positive. Controls the shape of the distribution at \(b=1\).

Returns:

np.ndarray

Computed resource distribution values (probability density) at the specified bin_points.

Notes

• When \(\alpha = \beta = 1\), the distribution is uniform

• When \(\alpha > 1\) and \(\beta > 1\), the distribution is unimodal

• When \(\alpha < 1\) and \(\beta < 1\), the distribution is bimodal (U-shaped)

Examples

>>> bin_points = np.linspace(0.01, 0.99, 100)
>>> resources = beta_distribution(bin_points, alpha_value=2, beta_value=5)