I recently posted a working paper showing how to decompose overall volatility in election results into party switching, turnout switching and population replacement. I showed that across 104 election pairs, party switching was the most important factor for electoral change in 97% of cases. This flies in the face of a lot of conventional wisdom that there are no swing voters and that elections are won by energizing the base.

But there was one remaining possibility that I wanted to address. OK, perhaps turnout isnâ€™t the main driver of electoral change in general, but what if itâ€™s really important for some parties? Perhaps the fate of parties that rely on lower turnout demographic groups is driven more by turnout switching?

So this raised the question, how can I take a change in one partyâ€™s vote share between two elections and split it into the parts attributable to party switching, turnout switching, and population replacement?

The obvious answer would simply be to look at net flows. If a party gains 10,000 previous non-voters and loses 5,000 of its previous voters to non-voting, they have a net gain of 5,000 newly mobilized voters. But I think this gives a misleading picture of non-votingâ€™s importance. If your party gains a net 5,000 newly mobilized voters but all the other partiesâ€™ gain 20,000, itâ€™s possible that your partyâ€™s share of the vote has actually fallen due to turnout switching because the denominator of all voters has increased faster than your number of votes. By contrast, recruiting a net 5,000 voters from another party directly increases your partyâ€™s vote share without affecting the denominator.

I couldnâ€™t find an existing decomposition of changing vote shares that really addressed this, so Iâ€™ve created a new one. Iâ€™d be interested if anyone knows if this decomposition of a changing proportion exists anywhere else already. I use it for vote share changes here but it looks like it would be very widely applicable.

Suppose we observe a mean \(P\) at two discrete times. A mean can be expressed as a numerator \(n\) divided by a denominator \(d\). We can define the change in the mean as follows:

\[ P_{\Delta} = P_2 - P_1 = \frac{n_2}{d_2} - \frac{n_1}{d_1} \\ \]

For the rest of this paper, we talk about the example of a population proportion, but this decomposition would work for any ratio statistic.

Now letâ€™s suppose all changes in the numerator and denominator between the two timepoints are categorized exclusively into either type A or type B so that:

\[ P_{ \Delta } = \frac{ n_1 + n_{ \Delta A} + n_{ \Delta B } }{ d_1 + d_{ \Delta A } + d_{ \Delta B } } - \frac{ n_1 }{ d_1 } = \frac{n_1 + n_{\Delta A} + n_{\Delta B} }{ d_2} - \frac{n_1}{d_1} \]

Perhaps A is the entries into a population and B is the exits (and the existing population members do not change). Or A might just be the net effect of entries and exits from the population and B might be change within existing population members (in which case B would not affect the denominator).

We want to be able to partition the overall change \(P_{\Delta}\) into the change attributable to A and B.

\[ P_{\Delta} = P_{\Delta A} + P_{\Delta B} \]

We start by defining \(n_{\Delta A 0}\) as the A numerator required to not affect the numerator-denominator ratio from \(P_1\) given the observed A denominator changes \(d_{\Delta A}\):

\[ \frac{n_{\Delta A 0}}{d_{\Delta A}} = \frac{n_1}{d_1} \]

Rearranging gives us:

\[ n_{\Delta A 0} = n_1 \cdot \frac{ d_{\Delta A}}{d_1} \]

We then define \(n_{\Delta A \epsilon}\) as the difference between the numerator change to keep the proportion constant(\(n_{\Delta A 0}\)) and the numerator change observed (\(n_{\Delta A }\)):

\[ n_{\Delta A \epsilon} = n_{\Delta A } - n_{\Delta A 0} \]

Using the equivalent identities for B, we can rewrite \(P_{\Delta}\) as:

\[ P_{\Delta} = \frac{n_1 }{ d_2} + \frac{n_{\Delta A 0} + n_{\Delta A \epsilon} }{ d_2} + \frac{n_{\Delta B 0} + n_{\Delta B \epsilon} }{ d_2} - \frac{n_1}{d_1} \\ \]

and then:

\[ P_{\Delta} = \frac{n_1 + n_{\Delta A 0} + n_{\Delta B 0}}{ d_1+ d_{\Delta A} + d_{\Delta B}} + \frac{n_{\Delta A \epsilon} }{ d_2} + \frac{n_{\Delta B \epsilon} }{ d_2} - \frac{n_1}{d_1} \\ \]

Since we defined \(n_{\Delta A 0}\) to maintain \(\frac{n_{\Delta A 0}}{d_{\Delta A}}=\frac{n_1}{d_1}\), it follows that:

\[ \frac{n_1 + n_{\Delta A 0} + n_{\Delta B 0}}{ d_1+ d_{\Delta A} + d_{\Delta B}} = \frac{n_1}{d_1} \]

which allows the following simplification:

\[ P_{\Delta} = \frac{n_{\Delta A \epsilon} }{ d_2} + \frac{n_{\Delta B \epsilon} }{ d_2} \\ \]

where we have now partitioned \(P_{ \Delta}\) into parts attributable to A and B. Substituting in observed variables gives us:

\[\begin{equation} \label{finaldecomp} P_{\Delta} = \frac{n_{\Delta A } - \big ( n_1 \cdot \frac{ d_{\Delta A}}{d_1} \big ) }{ d_2} + \frac{n_{\Delta B } - \big ( n_1 \cdot \frac{ d_{\Delta B}}{d_1} \big )}{ d_2} \\ \end{equation}\]

So we can now define the two additive components of change:

\[ P_{\Delta A} = \frac{n_{\Delta A } - (n_1 \cdot \frac{ d_{\Delta A}}{d_1}) }{ d_2} \\ \]

\[ P_{\Delta B} = \frac{n_{\Delta B } - (n_1 \cdot \frac{ d_{\Delta B}}{d_1}) }{ d_2} \\ \]

This decomposition can be applied again to the change components themselves so that we might for instance decompose the A changes further to:

\[P_{\Delta A} = P_{\Delta C} + P_{\Delta D}=\frac{n_{\Delta C } - (n_1 \cdot \frac{ d_{\Delta C}}{d_1}) }{ d_2} + \frac{n_{\Delta D } - (n_1 \cdot \frac{ d_{\Delta D}}{d_1}) }{ d_2}\]

To show a quick example, letâ€™s imagine we have a group of 80 people, 40 of whom are women. By the next time point, 10 people leave, 6 of whom are women. In addition, 110 new people join the group but only 11 of these are women. We go from a 50% female group to 25% by the second timepoint a change in the proportion of -0.25.

When we substitute in these numbers into the equation we get:

\[ P_{\Delta} = \frac{-6 - \big ( 40 \cdot \frac{-10}{80} \big ) }{ 180} + \frac{11 - \big ( 40 \cdot \frac{110}{80} \big )}{ 180} \\ \]

\[ P_{\Delta} = \frac{-1}{ 180} + \frac{-44}{ 180} = -0.0055 + -0.2444 \\ \] In other words, the overall -0.25 in the proportion of women is able to be decomposed into a -0.0055 effect from differential exits and a much larger -0.244 from differential entries.

So what happens when I apply this to the 664 party-election observations in the data? Party switching is a larger contributor to changes in party vote shares 78% of the time, meaning that party switching is more important for most parties as well as most elections.

But maybe certain types of parties rely more on turnout? The following graph shows thatâ€™s not the case. Higher numbers on this graph indicate that party switching contributed more to change than turnout switching. On average, parties in **every family** (e.g.Â conservative, social democrat, christian democrat, radical right etc) in **all 18 countries** is more affected by party switching than turnout switching.

So my results continue to show that elections are overwhelmingly won and lost by party switching rather than turnout switching. This is true for parties everywhere on the political spectrum and all around the world.