However, the behaviour is not satisfactory. The historical record shows that the original Liberal party had its low point in 1951 with 2.5% of the national vote, but still kept six seats (though five of those had no Conservative candidate). Interestingly, the 1931 election reduced the Liberals' support from 23% to 7%, but their number of seats only decreased from 59 to 37.

We present an updated variant of the *Electoral Calculus* transition model which has better
behaviour in the event of a steep decline in a party's support, by acknowledging the advantage
of incumbency. The new model, called the *Strong Transition Model* (STM) continues to benefit
from the existing advantages of the original Transition Model, namely that
predicted votes are always positive.

As a simple model, we assume that the *weak* supporters of a party in any given seat are
all those who voted for that party up to a threshold of 20% of the turnout. The *strong* supporters
are those who voted for the party beyond that threshold. There will be no strong supporters in seats
where the party received less than 20% of the votes cast.

In a particular seat where the Lib Dem election vote was 45%, that divides into 20% weak votes and 25% strong votes. As 41.3% of the weak votes stay loyal, they are predicted to now be only 0.413 x 20% = 8.3%. But all the strong votes are assumed to stay loyal, so the total Lib Dem vote is predicted to be 8.3% + 25% = 33.3%.

This compares with the standard transition model which assumes that the loyalty factor of all Lib Dem voters is 12.4/22.7 = 54.6%, so the seat with 45% support is now predicted to have 0.546 x 45% = 24.6% support.

The three models are:

**Additive model**The predicted support in each seat is just the election support less the swing of 10.3%. Note that the predicted support goes negative if the election support level was less than 10%.**Transition model**The predicted support in each seat is just a fraction (0.546) of the election support. The predicted support is always positive, but very strong Lib Dem seats decline substantially.**Strong Transition model (STM)**The predicted support is a fraction (0.413) of the weak election voters plus all of the strong election voters. The line is piecewise linear with a change in slope at 20% (the strong voter threshold). The predicted support is always positive, but strong Lib Dem seats decline less dramatically than under the Transition Model.

*V(j)* = Σ_{k} *TO*(*k*)*V*(*k*,*j*) / *TO*,

and its strong vote shares are

*V _{S}*(

*P _{W}*(

*P _{S}*(

- the previous national results,
*V*(_{S}*j*) and*V*(_{W}*j*) =*V*(*j*) −*V*(_{S}*j*). - the predicted national results,
*P*(_{S}*j*) and*P*(_{W}*j*) from above. - the previous seat results,
*V*(_{S}*i*,*j*) = max(*V*(*i*,*j*) −*α*, 0 ), and*V*(_{W}*i*,*j*) =*V*(*i*,*j*) −*V*(_{S}*i*,*j*).

The Transition Model applied to this extended party list then gives the predicted weak and strong votes for each party in each seat. We then just add up the weak and strong votes to get the total predicted vote for each party in that seat.

We see that the three models are qutie similar for support levels above the general election result of 23%. This is expected because all the models behave additively for gaining parties. For declining vote shares, the additive model is most favourable to the Lib Dems. This includes the prediction that the Lib Dems will win 2 seats even if they get zero votes nationally, which is over-optimistic. On the other hand, the transition model predicts zero Lib Dem seats even if they get 10% of the votes.

The strong transition model lies between these extremes. It does predict zero seats for zero votes, but it only predicts zero seats for support levels of 3% or less. For current support levels of 12.4%, it predicts 15 Lib Dem seats, which is only a few less than the (over-optimistic) additive model.

- Always gives positive vote predictions, which avoids distortions when votes go negative.
- Predicts zero seats for zero votes
- Reasonable seat predictions even for significant drops in support
- Intuitive model of incumbency and local strength
- Behaves proportionally for weak seats, and additively for strong seats.

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