# Transition Model 2004

The model has been modified on 28 October 2007 to become the Strong Transition Model, which itself has been superseded by modern regression methods. This article is now of historical interest only.

Since 8 July 2004, predictions have been based on a new model of voter movement. This new model is quite similar in effect to the simple additive Uniform National Swing model, but has two major advantages:

• Predicted share of the vote cannot go unreasonably low or negative
• Predicted share of the vote cannot go unreasonably high or above 100%
Under the additive model, both of these features could theoretically happen and the first often happened in practice.

The new model avoids these problems, whilst retaining the good properties of the old model, and making very similar predictions for which party will actually win each seat. Backtesting each model against the last four elections showed quite good performance from both, with the Transition Model being slightly, but not significantly, more accurate.

Below is a technical description of both models, plus the results of Backtesting, for those who may be interested.

## 1. Introduction and Notation

Suppose we have parties $i=1,\ldots,n$, and seats $k=1,\ldots,N$. We measure all party votes as a percentage of turnout, so support values are real numbers between zero and one.

Let us define our observables as

• $E(i)$ national (percentage) support for party $i$ at the last election,
• $C(i,k)$ (percentage) support for party $i$ in seat $k$ at the last election, and
• $P(i)$ current (percentage) support for party $i$ at the moment, as measured by the opinion polls.
Our aim to estimate the unknown variable of current support for each party in each seat:
• $A(i,k)$ current support for party $i$ in seat $k$.
There is no way we can be certain of this figure (without holding an election now), but we can estimate it in a variety of ways.

The simplest model is that of additive Uniform National Swing. It says that if a party is up 3% nationally, that it will be up 3% in each seat. In symbols, $$A(i,k) = C(i,k) + P(i) - E(i).$$

This model, although simple, is quite robust, easy to implement, and seems to work well in practice. One problem with the model is that the prediction $A(i,k)$ can be higher than 100% or (more commonly) negative. Whilst this may not alter who wins the seat, it is an unattractive feature.

For example, suppose Labour's support has dropped by 6% nationally, but in a particular seat their support at the last election was only 4%. Then their predicted vote in that seat will be -2% which is obvious nonsense.

## 3. Transition Model

The Transition Model aims to get around this problem, without moving too far from the overall behaviour of the Additive Model.

The general idea is to categorise parties into those whose support has increased and those whose support has decreased. The two groups of parties are handled separately.

• Parties who decline will be assumed to decline multiplicatively. That is, if their national supports drops by a tenth (say from 40% to 36%), then their vote in each seat will drop by a tenth.
• Parties who increase will gain those voters from the declining parties in proportion to the amount of that party's increase compared with the other increasing parties.
Note that this is not a simple multiplicative model for the increasing parties. They only increase votes in any one seat to the extent that the declining parties have lost votes.

To express the model in symbols we need to define two quantities. They are

1. Share(i) is the relative national share of gains made by party i relative to all the gaining parties: $$\hbox{Share}(i) = {\max\left( P(i)-E(i), 0\right) \over \sum_{j=1}^n \max\left( P(j)-E(j), 0\right)}.$$
2. SwingVoters(k) is the fraction of voters in seat $k$ who have changed parties $$\hbox{SwingVoters}(k) = \sum_{j=1}^n C(j,k) \max\left(1 - {P(j)\over E(j)}, 0\right).$$

The predicted support levels are now $$A(i,k) = {C(i,k) P(i)\over E(i)},\qquad\hbox{if party i declines},$$ $$A(i,k) = C(i,k) + \hbox{Share}(i)\times\hbox{SwingVoters}(k),\qquad\hbox{if party i increases}.$$

It is called a transition model, because we are modelling the transition of each voter from their vote at the election to their vote now. Probabilistically, we are saying that a voter from party $i$

• will definitely stay with party $i$, if party $i$ increases; otherwise
• will randomly stay with party $i$, with chance $P(i)/E(i)$, or else
• will change to party $j$, with chance $\hbox{Share}(j)(1-P(i)/E(i))$.
This is not fully perfect, as defectors from a particular party may be more likely to vote for some parties rather than others, but it does eliminate the problems of the Additive Model.

## 4. Comparison and Backtesting

To test the two models, they were each run over the last four elections. In each case the model was told the full results of the previous election, plus the total votes received for each party at the new election, and asked to predict the detailed results of the new election. The results were:

Seats WrongAve Vote Error