main
HeNine 2 years ago
parent acdeb9c490
commit b93c1cc72e

@ -136,25 +136,25 @@
\section{Combat Pool Steady State}
\begin{multicols}{2}
The combat pool starts with the proficiency stat, i.e., \stat{STR} or \stat{DEX}. However, after rolling, the change in pool size is determined by \stat{CON}. After a number of turns, the expected size of the pool should converge on a number $E[d]$.
The combat pool starts with the proficiency stat, i.e., \stat{STR} or \stat{DEX}. However, after rolling, the change in pool size is determined by \stat{CON}. After a number of turns, the expected size of the pool should converge on a number $\text E[d]$.
We start by assuming that $E[d]$ is a number such that
We start by assuming that $\text E[d]$ is a number such that
\begin{equation}
E[d_{n+1}] = E[d_{n}] \label{equ:dice_ss_a1}
\text E[d_{n+1}] = \text E[d_{n}] \label{equ:dice_ss_a1}
\end{equation}
where $n$ is the turn after which the steady state is achieved.
To compute $E[d_n]$ for any $n$ we can use
To compute $\text E[d_n]$ for any $n$ we can use
\begin{equation}
E[d_n] = E[d_{n-1}] - p_f \cdot E[d_{n-1}] + \stat{con} \label{equ:dice_ss_a2}
\text E[d_n] = \text E[d_{n-1}] - p_f \cdot \text E[d_{n-1}] + \stat{con} \label{equ:dice_ss_a2}
\end{equation}
where $p_f = \stat{dt}/12$ is the probability of failure.
Using assumptions in \cref{equ:dice_ss_a1,equ:dice_ss_a2} we write the following equations:
\begin{align}
E[d] & = E[d] - p_f \cdot E[d] + \stat{con} \\
p_f\cdot E[d] & = \stat{con} \\
E[d] & = \frac{\stat{con}}{p_f}
\text E[d] & = \text E[d] - p_f \cdot \text E[d] + \stat{con} \\
p_f\cdot \text E[d] & = \stat{con} \\
\text E[d] & = \frac{\stat{con}}{p_f}
\end{align}
As $p_f$ approaches 1, the steady state becomes just \stat{CON}, meaning the dice pool is entirely discarded and refreshed by \stat{con} on every turn. On the other hand, as $p_f$ approaches 0, the expected dice pool size increases. Specifically at 0, it shoots off toward infinity, as the dice pool loses no dice and \stat{con} dice get added toward it on every turn.

Loading…
Cancel
Save