Parliamentary arithmetic

The pie chart shows probabilities that particular groups of parties would both have a parliamentary majority and effectively control the government under assumptions listed below. The categories are effectively (but not perfectly) ordered by the number of Labour seats (increasing clockwise from the top) and Conservative seats (increasing anti-clockwise from top).

Outcomes with Labour getting most seats are presented on the left of the pie chart and defined as follows (starting from the top and working anti-clockwise):

  1. Lab maj: Labour overall majority (323 or more seats)
  2. Lab+DUP: hung parliament and Labour and the DUP combined have a majority
  3. Lab+LD: Labour and the Liberal Democrats have a majority, but Labour and DUP do not
  4. Lab+LD+DUP: Labour would not have a majority with either LD or DUP alone, but all three together would
  5. Lab+LD+DUP+SDLP+PC+Green+Hermon: Labour would not have a majority even with both LD and DUP, but would with one or more of SDLP, Plaid Cymru, Greens, and Sylvia Hermon (independent MP for North Down)
  6. Lab+SNP: Labour would have a majority with the SNP alone, but Lab+LD+DUP+SDLP+PC+Green+Hermon would not have a majority

Outcomes with the Conservatives getting most seats are presented on the right of the pie chart and defined as follows (starting from the top and working clockwise):

  1. Con maj: Conservative overall majority (323 or more seats)
  2. Con+DUP: hung parliament and the Conservatives and DUP combined have a majority
  3. Con+LD: Con and LD have a majority but Con and DUP do not
  4. Con+LD+DUP: The Conservatives would not have a majority with either LD or DUP alone, but all three together would
  5. SNP kingmakers or wreckers: despite the Conservatives being the largest party, Con+LD+DUP would not have a majority, Lab+LD+SNP would, and in the vast majority of these cases a Conservative-led government supported by LD and DUP would win parliamentary votes if SNP abstain, even if MPs from all other parties vote against

Assumptions driving the typology

Note that the following are working assumptions and many are disputed. They are not confident predictions, but necessary simplifications. We welcome comments and the assumptions are subject to revision.

  1. For any given possible election outcome assumptions are made as to which particular grouping of parties would actually control government when more than one grouping has a majority.
  2. The probabilities are over groupings that command a majority, but for them to be probabilities over possible governing groups we need to assume that there will be a government following the election and not just an immediate second election.
  3. Parties are cohesive in the sense that their MPs will all vote together and there will be no party splits, and any rebellions would not change the outcome on key votes (on budgets, Queen’s speeches or confidence motions).
  4. For the purposes of calculating whether a particular party or group of parties has a majority we assume Sinn Fein will win 5 seats but not take them, and so 323, not 326, seats are required for a majority.
  5. We ignore the roles of the Speaker and Deputy Speakers, presuming that they would not alter the government formation process, in part because they could be changed post election. We are currently counting the Speaker as a Conservative.
  6. We are not trying to work out whether or not two or more parties would form a coalition versus reaching some sort of confidence and supply agreement. We are assuming that the difference between coalitions and pacts within a grouping would be less important than the issue of which grouping of parties sustains government. Hence we refer to groupings rather than coalitions.
  7. In assessing which grouping of parties would sustain the government under which seat outcomes we are concerned with whether they could command a parliamentary majority, and so, presumably, a confidence vote, which is important for the operation of the Fixed-term Parliaments Act. The key exception is the “SNP kingmakers or wreckers” scenario where SNP abstention would allow Con+LD+DUP to win a confidence vote.
  8. The calculations depend on seat totals only, not on which particular MPs and candidates within a party get elected.
  9. Groupings will build up either around Labour or around the Conservatives, and considerations of ideological proximity and compactness apply. The idea that two-party groupings are more likely than three parties governing together, and that three more likely than four, etc, also applies.
  10. If the Conservatives or Labour are just short of a majority they will seek DUP support if that is sufficient, if not then LD and if that is not sufficient then both.
  11. If Labour wins the most seats and Lab+LD+DUP would not command a majority then these parties might also look for support from Greens, SDLP, Plaid Cymru and Sylvia Hermon (independent MP for North Down).
  12. If Labour wins the most seats, only if Lab+LD+DUP+SDLP+PC+Green+Hermon as a grouping is short of a majority would Labour look to SNP to sustain them in government.
  13. If the Conservatives win the most seats and Con+LD+DUP would not command a majority, and if UKIP win more seats than LD, then the government would depend on Con+DUP+UKIP. But this option will not appear in the pie chart if UKIP are extremely unlikely to get more seats than LD.
  14. UKIP would only be party of a government grouping with Conservatives and DUP but no other party.
  15. The DUP will hold their current seats and win back Belfast East.
  16. We take at face value the SNP claim that they would not do a deal to keep the Conservatives in power. But they have not ruled out abstaining on a vote of confidence in a Tory-led administration.
  17. A grand coalition between the Conservatives and Labour is always theoretically possible but it is not possible to say that in any particular parliamentary seat outcome it would be preferred to any other viable grouping.

We think this is a sufficient set of assumptions to carve out the set of possible seat-total outcomes into a reasonably defensible set of situations with one grouping having a governing majority in all but one of the situations. Moreover, we feel that each situation is defined using reasonably politically informed assumptions, so that there is good reason to believe that in each situation the assigned grouping will in effect govern. Further detail on probabilities is presented in the list of statistics at the bottom of the Latest Forecast page.

Acknowledgements: We particularly would like to thank Richard Coggins, Philip Cowley, Iain McLean, Petra Schleiter and Jon Tonge for their comments and advice on these assumptions and issues. We were able to incorporate some but not all of their suggestions. Defects are our responsibility.

21 thoughts on “Parliamentary arithmetic”

  1. The DUP have said they are quite happy to do business with Labour, but the Lib Dems in my view sound distinctly cool on any sort of deal involving the SNP…

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