Mathematical Politics

Abstract

Politics today is largely about the art of messaging to influence the public, but the mathematical theory of messaging -- information and communication theory -- can turn this art into a precise analysis, both qualitative and quantitative, that enables us to gain retrospective understandings of past political events and to make forward-looking future predictions.

Figures (4)

• Figure 1: The probability of winning an election in one year, as a function of today's support rate $p$, for $\sigma=0.2$ (purple) and $\sigma=1.2$ (red). When $p>0.5$, the realised probability of winning a two-candidate election in the future is always higher than $p$, and conversely for $p<0.5$.
• Figure 2: Negative gradient flow of the variance. The three corners of the octant, where the flow attempts to take you to, correspond to states with zero uncertainty.
• Figure 3: If $p_1=0.01$ and $p_2=0.99$, then even if the noisy information is provided by $\xi_t=\sigma x_1 t + \epsilon_t$, indicating that $X=x_1$ is the correct alternative so that $\pi_{1t}$ should approach $1$, a Bayesian thinker will maintain the position $\pi_{1t}\approx0$ for a long time. Four sample paths are shown here for $\sigma=1$.
• Figure 4: There are six alternatives labelled by $X=1,2,3,4,5,6$. If $X=3$ but $p_3=0$, then the mean $X_t$ tends to hop between the closest alternatives $X=2$ and $X=4$, when there is a lot of ($\sigma=5$) reliable information coming out saying that $X=3$ (top panel). In contrast if $p_3=0.01$ but $X=3$, a genuine misunderstanding, then $X=2$ and $X=4$ remain plausible alternatives for a while, but the view will gradually converge to the true alternative $X=3$ (bottom panel). Six sample paths are shown in each case.