Models of inter-election change in partisan vote share

For a two-party electoral competition in a districted legislature, the change in mean vote share for party A from one election to the next is commonly referred to as swing. A key question, highly relevant to election forecasting and the measurement of partisan gerrymandering, is: “How do we expect the swing to be distributed across the districts as a function of previous vote share?”. The literature gives two main answers: uniform swing and proportional swing. Which is better has been unresolved for decades. Here we (a) provide an axiomatic foundation for desirable properties of a model of swing; (b) show axiomatically that using uniform swing or proportional swing is a bad idea, (c) provide a simple swing model that does satisfy the axioms, and (d) show how to integrate a reversion to the mean effect into models swing. We show that all the above models can be expected to work well when (a) elections are close, or (b) when we restrict to data where swing is low, or (c) when we eliminate the cases where the model is most likely to go wrong. We show on a large US Congressional dataset that in addition to its superior axiomatic properties, our new model provides an overall equal or better fit on five indicators: mistakes about directionality of change, mistakes in winner, estimates that are outside the [0..1] bounds, mean-square error, and correlation between actual and predicted values. We recommend replacing the uniform and proportional swing models with the new model.