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Application: Calculating Partisan Bias

One definition of partisan bias is the extra seat share received by one party when the average vote is split equally between both. We estimate the distribution of partisan bias conditional on $\beta$ and $\gamma$ , then use their distributions to determine a probability interval.

To get one draw from the distribution of partisan bias:

• Draw $\beta$ and $\gamma$ from their respective distributions. These represent conditions in the electoral system up until the election.⁵

• Calculate the mean and variance of the vote share in each district conditional on the draws of $\beta$ and $\gamma$ .

• Calculate the grand mean vote (with whatever weights are being used) and subtract it from 0.5; this is the value of $\delta$ we use to adjust the mean vote to 0.5.

• Determine the expected seat share $P(v_i>0.5\vert\beta,\gamma,\delta)$ for each district and take the weighted mean. Twice the (weighted mean minus 0.5) is the partisan bias conditional on $\beta$ and $\gamma$ .

Repeating this procedure gives us the distribution of the partisan bias.