Last week we unveiled our new-and-improved General Election 2017 model. Our first look at its output suggests that the Conservatives are heading towards a huge majority, which is consistent with many predictions.

It is apparent to most observers that the best bet the opposition parties have of preventing this is to coordinate together in what has been termed a 'Progressive Alliance'. In this article, we use our model to look at the arithmetic of such a victory. We will follow this up with a region-by-region analysis of how this can be achieved.

**What is needed for a majority?**

It should be obvious how many seats are needed for a majority. There are 650 seats, however the Speaker takes a neutral position, so the number needed for a majority should be 325 (leaving 324 opposition, plus the Speaker). However, it is more complicated than that.

We also need to appreciate that the main aim of the Progressive Alliance is to prevent the 'hard Brexit' Theresa May has pursued. Therefore we need to look at which parties are likely to support a hard Brexit, as they will vote with May (we are assuming there are no Conservative rebels).

Of the parties with a realistic chance of having Westminster representation after the election, the most Eurosceptic of these are the Democratic Unionists, Ulster Unionists (both Northern Irish parties) and of course UKIP. Our model currently has their expected seats as 7.99, 0.87 and 0.48 respectively. If we treat this as 9 seats then the Conservatives realistically only need 313 seats (with 9 in support, 321 opposition and 7 neutrals).

The remaining parties are all likely to oppose a hard Brexit, and therefore can be considered to make up the Progressive Alliance. This means the parties involved are Labour, the SNP, the Liberal Democrats, Plaid Cymru, the SDLP, the Green party and Alliance, plus the anti-hard Brexit independent Lady Hermon. Of course, there are key differences between (and within!) these parties, but for the sake of this article we'll assume they put this behind them to prevent a Conservative majority. We now need to see how this alliance can make up its own majority.

**Safe seats**

Of course, the Progressive Alliance could simply just try to win every seat possible. However, budgets are limited and key campaigners can only be in one place at a time. So some seats need to be focused on more than others. Some seats will be so safe for one party that their opposition will waste very little resources in contesting it (in fact, there are five seats in our model which have a 100% chance for the favourite).

We are going to measure the grip a particular party has on a seat by its "Probability lead", that is, the difference in probability between the favourite and second-favourite. So for example, if a seat has the following probabilities:

- Party A: 60%
- Party B: 30%
- Party C: 10%

We then need to identify the probability difference level above which the Progressive Alliance can consider a seat to be safe. If it is too low then it leaves too few seats to fight for, if it is too high then resources will be too-thinly spread.

Below is a graph showing how many Conservative seats would be safe for each probability difference. The total Conservative seats includes DUP, UUP and UKIP seats. The red line indicates the 322 seats we estimate as needed for a majority. You can use the arrow keys to zoom in on the area near the intersection.

We are going to say that the probability boundary for a safe seat is 97.5%. This leaves the following number of safe seats:

- Conservatives - 270
- Labour - 59
- SNP - 26
- DUP - 5
- Sinn Féin - 4
- Independent (Lady Hermon) - 1
- Speaker - 1

**Conclusion**

This is, of course, a monumental uphill struggle - as we pointed out last time, it is in fact more likely the Conservatives set the record for the most seats held by one party than it is they fail to make a majority. However, the election still hasn't happened yet, and so a route to victory, however slim, needs to be plotted.

We will be following this article with a region-by-region breakdown to see where the key battles for the Progressive Alliance will be, so stay tuned!