Let’s label our groups by \(g\), with turnout \(T_g\), population \(N_g\) and Dem preference \(P_g\). If the votes cast for dems are \(V_D\) out of the total votes \(V\), then post-stratified preference is
\(\begin{equation} P = \frac{V_D}{V}=\frac{\sum_g T_g N_g P_g}{\sum_g T_g N_g} \end{equation}\)
splitting into dem leaners (denoted by \(g\in D\)) and others (\(g \in R\)):
\(\begin{equation} P = \frac{\sum_{g\in D} T_g N_g P_g + \sum_{g \in R} T_g N_g P_g}{\sum_{g \in D} T_g N_g + \sum_{g \in R} T_g N_g} \end{equation}\)
Suppose we boost turnout in Dem leaning groups by x (so for a 1% boost, x would be 0.01):
\(\begin{equation} P(x) = \frac{\sum_{g\in D} (T_g + x) N_g P_g + \sum_{g \in R} T_g N_g P_g}{\sum_{g \in D} (T_g + x) N_g + \sum_{g \in R} T_g N_g} =\frac{x\sum_{g \in D}N_g P_g + V_D}{x\sum_{g \in D} N_g + V} =\frac{V_D}{V}\frac{1 + x\sum_{g \in D}N_g P_g/V_D}{1 + x\sum_{g \in D} N_g/V} = P \frac{1 + x\sum_{g \in D}N_g P_g/V_D}{1 + x\sum_{g \in D} N_g/V} \end{equation}\)
Usually, we’re curious about what \(x\) we need for a certain \(P(x)\). For example, \(P(x)=0.5\) is the level required to “flip” a state. So let’s call the \(P(x)\) we’re hoping for \(P_h\) and write \(P_h =P\times(1 + \delta)\) or \(\delta = \frac{P_h - P}{P}\). Also, just to simplify things, we’ll define \(a = \sum_{g \in D}N_g P_g/V_D\) and \(b = \sum_{g \in D} N_g/V\). So we have
\(\begin{equation} P\times(1 + \delta) = P \frac {1 + ax}{1 + bx} \end{equation}\)
which we can solve for \(x\):
\(\begin{equation} x = \frac{\delta}{a - (1+\delta)b} \end{equation}\)
We can understand this formula a bit. We need a boost that is proportional to \(\delta\), the gap we need to make up. Making up ground is easier when Dem preference is high in the groups we are boosting (\(a - b\), more or less).
Let’s look at a simple example. Imagine 800 voters in group A with 50% voter turnout and leaning 75/25 toward Dems and 1000 voters in group B with 60% voter turnout and leaning 65/35 toward Republicans. A Democrat would get 280 votes from the Dem leaners and 210 votes from the R leaners, for a total of 490. The Republican would get 120 votes from the Dem leaners and 390 from the R leaners, for a total of 510. So \(P_0=\frac{V_D}{V} = \frac{490}{1000} = 0.49\)
How much do we need to boost turnout in group A to flip the state, that is to get P_h = 0.5? Plugging the numbers above into the equation for \(P(x)\) (\(\delta = 0.02\); \(a = 560/510 = 1.1\); \(b = 800/1000 = 0.8\)), we get \(x = 0.11 = 11\%\). That is, we’d need to boost turnout in group A by 11%, from 50% to 61%.