InflGame.kernels (InflGame.kernels)#

Module contents#

kernels (InflGame.kernels)#

Pre-made influence kernels from the paper including Gaussian, Multi-variate Gaussian, Dirichlet, and Matt Jones’ kernel.

Dirichlet Influence Kernel#

Mathematical Definitions:#

The Dirichlet influence kernel is defined as:

\[f_i(\alpha, b) = \frac{1}{\beta(\alpha)} \prod_{l=1}^{L} b_{l}^{\alpha_{l} - 1}\]
where:

  • \(\alpha\) is the vector of parameters for the Dirichlet distribution, defined by the param function.

  • \(b\) is the bin point.

  • \(\beta(\alpha)\) is the beta function.

Gaussian Influence Kernel#

Mathematical Definitions:#

The Gaussian influence kernel is defined as:

\[f_i(x_i, b) = \exp\left(-\frac{(b - x_i)^2}{2\sigma_i^2}\right)\]
where:

  • \(x_i\) is the position of agent \(i\)

  • \(b\) is the bin point

  • \(\sigma_i\) is the parameter for agent \(i\)

Jones Influence Kernel#

This influence kernel is from the work of Mathew Jones et al in their paper “Polarization, abstention, and the median voter theorem” (paper).

Mathematical Definitions:#

The Jones influence kernel is defined as:

\[f_i(x_i, b) = \frac{1}{|x_i - b|^{P_i}}\]
where:

  • \(x_i\) is the position of agent \(i\)

  • \(b\) is the bin point

  • \(P_i\) is the parameter for agent \(i\)

Multi Variate Gaussian Influence Kernel#

Mathematical Definitions:#

The multivariate Gaussian influence kernel is defined as:

\[f_i(x_i, b) = \exp\left(-\frac{1}{2} (b - x_i)^T \Sigma_i^{-1} (b - x_i)\right)\]
where:

  • \(x_i\) is the position of agent \(i\)

  • \(b\) is the bin point

  • \(\Sigma_i\) is the covariance matrix for agent \(i\)

Beta Influence Kernel#

Mathematical Definitions:#

The Beta influence kernel is parameterized by mode (m) and concentration (phi):

\[f_i(x_i, b) = \frac{b^{\alpha-1} (1-b)^{\beta-1}}{B(\alpha, \beta)}\]
where:

  • \(x_i\) is the position (mode) of agent \(i\)

  • \(b\) is the bin point

  • \(\alpha = x_i(\phi - 2) + 1\)

  • \(\beta = (1 - x_i)(\phi - 2) + 1\)

  • \(\phi\) is the concentration parameter for agent \(i\) (must be > 2)

  • \(B(\alpha, \beta)\) is the Beta function

Mode-Parameterized Dirichlet Influence Kernel#

Mathematical Definitions:#

The mode-parameterized Dirichlet influence kernel uses the parameterization:

\[\alpha_{(i,l)} = 1 + \sigma \cdot x_{(i,l)}\]
where:

  • \(\sigma > 0\) is a concentration parameter controlling spread

  • \(x_{(i,l)}\) is the position of agent \(i\) in dimension \(l\)

  • The sum \(\alpha_0 = L + \sigma\) where \(L\) is the dimension

The gradient with respect to agent position is:

\[d_{(i,l)} = \sigma \left( \ln(b_l) - \psi_0(1 + \sigma x_{(i,l)}) \right)\]

The Hessian is a diagonal matrix:

\[H_{l,l} = -\sigma^2 \psi_1(1 + \sigma x_{(i,l)})\]