UKIP showed a strong performance at the local county council elections on 2 May 2013. According to the BBC's figures, they enjoyed the equivalent of 23% national support. Local election results are notoriously a poor guide to the subsequent general election result, so we will focus on the national opinion poll results. The UKIP poll support level at the time of the local elections was 12%, but it has since increased to around 18%, as at 2 June. This level puts them well ahead of the Liberal Democrats, and is a historical high point for UKIP nationally.

The obvious question is whether these levels of support can translate into UKIP seats at Westminster. Now it is famously difficult for minor parties to win Westminster seats. At the last general election, the three largest minor parties (UKIP, BNP, Green) achieved over 6% of the popular vote, but only the Green party won a single seat (Brighton Pavilion). Under a fully proportional system, UKIP (3.17%) would have won 20 seats, with 12 seats for the British National Party (1.94%), and six seats for the Greens (0.97%). But since the UK does not have a proportional system, the question remains of how many seats might UKIP win next time.

The short answer is that it is still difficult for UKIP to win many Westminster seats. Using the Electoral Calculus Strong Transition Model, we can produce estimates of how many seats they would win under various scenarios. The results are striking: even at the national support level of 18%, UKIP will win hardly any Westminster seats at all. UKIP will only start to gain more than a handful of MPs when their national support goes above 20%.

The methodology for this calculation has been revised since its initial publication. It now adds an additional factor to allow for the fact that seats do not all behave identically at a general election, but instead there is natural variation between them. These projections now allow for that natural variation, which increases the chances of UKIP winning some seats, even at low support levels. The Methodology Appendix below has more details on the new calculation scheme.

The table below shows some scenarios of increasing UKIP strength. The baseline scenario (on line 3) is from the national opinion poll average as at 2 June 2013, which has a Labour lead over the Conservatives of 8%. This table keeps that Lab/Con lead constant, but decreases both major parties by a further 1% each step, and increases UKIP by 2%. For now, these scenarios assume that UKIP gains strength equally at the expense of both major parties.

Con % | Lab % | Lib % | UKIP % | CON seats | LAB seats | LIB seats | UKIP seats |
---|---|---|---|---|---|---|---|

30 | 38 | 9 | 14 | 230 | 366 | 25 | 0 |

29 | 37 | 9 | 16 | 228 | 366 | 27 | 1 |

28 | 36 | 9 | 18 | 225 | 365 | 29 | 2 |

27 | 35 | 9 | 20 | 220 | 364 | 30 | 7 |

26 | 34 | 9 | 22 | 212 | 361 | 30 | 18 |

25 | 33 | 9 | 24 | 198 | 355 | 30 | 38 |

24 | 32 | 9 | 26 | 178 | 344 | 28 | 71 |

23 | 31 | 9 | 28 | 151 | 327 | 25 | 117 |

22 | 30 | 9 | 30 | 121 | 304 | 22 | 174 |

21 | 29 | 9 | 32 | 92 | 274 | 19 | 237 |

20 | 28 | 9 | 34 | 66 | 238 | 17 | 301 |

19 | 27 | 9 | 36 | 43 | 203 | 14 | 361 |

18 | 26 | 9 | 38 | 23 | 174 | 13 | 412 |

We see that UKIP only gets more than a handlful of MPs when its support reaches 20%. Even when it is the most popular single party, it can still win fewer seats than Labour. The figures suggest UKIP would get a majority in the House of Commons when its share of the vote is around 35%. At lower UKIP support levels, the Conservatives are slightly more damaged by UKIP than Labour is.

But this is based on the assumption that UKIP gains equally at the expense of both Labour and the Conservatives, which is a questionable axiom. An alternative set of scenarious can be defined on the basis that UKIP gains solely at the expense of the Conservatives. The table below shows the results of increasing UKIP by 2% and decreasing the Conservatives by 2% each step. Labour and the Liberal Democrats are held constant.

Con % | Lab % | Lib % | UKIP % | CON seats | LAB seats | LIB seats | UKIP seats |
---|---|---|---|---|---|---|---|

32 | 36 | 9 | 14 | 255 | 344 | 23 | 0 |

30 | 36 | 9 | 16 | 240 | 355 | 26 | 1 |

28 | 36 | 9 | 18 | 225 | 365 | 29 | 2 |

26 | 36 | 9 | 20 | 208 | 374 | 31 | 8 |

24 | 36 | 9 | 22 | 186 | 382 | 32 | 21 |

22 | 36 | 9 | 24 | 158 | 386 | 32 | 46 |

20 | 36 | 9 | 26 | 123 | 385 | 30 | 84 |

18 | 36 | 9 | 28 | 76 | 379 | 26 | 140 |

16 | 36 | 9 | 30 | 30 | 367 | 23 | 201 |

14 | 36 | 9 | 32 | 9 | 352 | 19 | 241 |

12 | 36 | 9 | 34 | 5 | 335 | 17 | 265 |

10 | 36 | 9 | 36 | 5 | 319 | 14 | 284 |

8 | 36 | 9 | 38 | 5 | 303 | 13 | 301 |

Again UKIP needs support of about 20% to win more than a handful of Westminster seats. Beyond that, it gains seats more quickly because its votes are concentrated in Conservative areas, rather than being equally diffused. In these scenarios, as one would expect, the Conservatives do very badly and Labour does quite well.

The model says that UKIP is predicted to win hardly any seats if its support is below 20%. This seems quite a high threshold. The history of elections since 1900 shows the Liberal party never won fewer than six seats even when its national support slumped to 2.5% in 1951. More recently, the Liberal Democrats managed to win 52 seats in 2001, with only 18% national support. Why did UKIP win no seats in 2010 with a vote share of 3.17%, and why does it need around 20% support to win a few seats.

The answer is a combination of three separate factors:

**Established local strength**The Liberal party had been one of the two major parties until 1922. It had built up strong local roots in many areas, and was able to maintain some of these roots even as its national support withered. That left some concentrated pockets of Liberal support, which were able to elect Liberal MPs. For instance, four out of the six Liberal seats in 1951 were in the "celtic fringes" of Scotland and Wales.**Strong candidates**Having strong well-known candidates can draw in support from other parties and beat the national trend. This "incumbency factor" was visible at the 2010 election in seats like Eastleigh (Christopher Huhne) and Brent Central (Sarah Teather) where a well-known Liberal Democrat incumbent held the seat against the odds.**Tactical voting**The Liberal Democrats have benefitted from tactical voting in recent years. Where they are in second place to one of the major parties, they can benefit from tactical votes from natural supporters of the third-placed party. Typically they gain tactical Labour votes in the South to "keep the Tories out", and vice versa some tactical Conservative votes in the North.

If UKIP is unlikely to win many seats, then perhaps its existence is an irrelevance which does not affect the calculations of the major parties. This may be true at the level of parliamentary arithmetic in the House of Commons, but the general election result itself is affected by UKIP. We saw above that even if the Lab/Con difference is kept fixed, then the growth of UKIP is negative for the Conservatives and good for Labour. And as a right-of-centre party, it should be expected that UKIP draws relatively more support from the Conservatives than from Labour. This would strengthen the trend for UKIP growth to be bad news for the Conservatives.

Only if UKIP took more votes from Labour than from the Conservatives, would its growth be positive for the Conservatives.

However, it is worth remembering that the main driver of the election result will be the difference in vote shares between the two major parties. Although this difference may well reflect the popularity of UKIP and other minor parties, it is the difference itself which is the key indicator. So the prediction of the general election result is still mostly based on the strengths of the three largest parties.

This analysis has been conducted with the best available models and using national opinion poll support levels. It is the current best estimate of possible general election outcomes, but the growth of UKIP is a new phenomenon and there are a number of assumptions and approximations involved in the calculations. These sources of possible error include:

**Assumption of uniform support**The calculations assume that UKIP support is fairly evenly distributed, or at least it is evenly created from defecting Conservative and Liberal Democrat voters. The calculations have**not**used local election results to estimate possible locations of heavy UKIP support. If UKIP support is concentrated, rather than being evenly spread, then they will get more MPs despite lower levels of popular support.**Assumption of homogeneous defectors**The transition model assumes that defecting voters behave identically, no matter which party they previously supported. Using the May 2013 polls, around 20% of Conservative voters and about half of Liberal Democrats are expected to defect. These defectors are assumed to break evenly between Labour and UKIP. In reality we might expect that the ex-Conservatives would be more attracted to UKIP, and the ex-LibDems to Labour. This might affect the accuracy of the calculations, especially in strong Conservative areas.**Absence of strong candidates**The numerical models have no data on any strong UKIP candidate who might stand at the next general election. If a strong candidate with good local backing emerges, then UKIP could win that seat against the numerical odds. The election of Caroline Lucas (Brighton Pavilion, Green) in 2010 is a useful example of this possibility.

Users of Electoral Calculus can now make their own predictions for UKIP. Simply go to the main user-defined prediction page and select the check-box "Predict UKIP". The display will then allow you to enter the national support figure for UKIP. For convenience, a default value (from the 2010 election) is provided.

Change the UKIP support value, and then press the "Predict Election" button as usual. Please note that UKIP prediction only works for the current 2010 boundary set and not for any other choice of seat boundaries.

The methodology for this calculation has been revised since the initial publication on 5 May. The previous calculation was made using a direct application of the Electoral Calculus Strong Transition Model. This model is designed to predict the most likely winner in each seat, and thus the likely winner of the general election. However it is not particularly suited to predicting the small number of seats won by UKIP, when UKIP's support is relatively low (that is, below 30%). If UKIP were to get exactly, say, 20% in every seat, it would not win any seats even though it would come second or third in almost every seat.

In reality, there is a natural variation between seats at a general election. Local factors and specific candidates mean that parties do better than average in some seats, and worse than average in other seats. (Obviously they can't do better than average in all seats, because the average is defined to be in the middle of their performances.) So even if a new party gets just 20% nationally, it will probably get more than that in some seats, and less than that in other seats. So potentially it might even get as high as 40% in a few seats, but also get close to 0% in a few other seats, preserving the average support at 20%. This means that the party might even win some seats, due to this natural variation, which it was not predicted to be win on the basis of the average poll ratings.

This can be summarised with the apparent paradox that **both** of the following two statements
are true if UKIP has around 20% support:

- There are no seats which UKIP are likely to win.
- UKIP are likely to win some seats.

The explanation of the paradox comes if we consider the likely chance that UKIP has of winning each specific seat. At a support level of 20%, UKIP has only low chances of winning any particular seat. Their most likely seat is Montgomeryshire (CON over LIB) which they have a 25% chance of winning. Though the Conservatives have a 71% chance of holding the seat, so UKIP is not the most likely party to win that seat. Apart from that seat, their chance of winning any seat is less than 20%, with an average chance of just 1.15%.

But the statistical law of averages now comes to help UKIP. As there are 632 GB constituencies, then the expected average (mean) number of seats which are won by UKIP will be 7.2 ( = 632 * 1.15%). This means that on average, UKIP will win around seven seats. It's not possible to say which seats they will be, but it is quite likely that they will win some seats somewhere. (A helpful analogy comes from buying lottery tickets. Any particular ticket is very unlikely to win, but it is also very likely that at least one of the many tickets will win.)

UKIP's support needs to reach 24% before there is any particular seat which they are the most likely party to win. For instance, at 24% they are likely to win Montgomeryshire (65% chance). But even for support levels between 18% and 24%, they are still likely to win some seats somewhere. Even at 16% support level, it is possible but unlikely for them to win a seat, because their expected number of seats won is 0.7.

The model we put forward to represent the natural variation between seats, is a random model for the vote share in each seat. We take the official prediction, from the Strong Transition Model, as the starting point for each seat. But the seat prediction is further modified by a random variation, which is centered on the official prediction. The model used is a Dirichlet distribution (multi-variate beta distribution). This distribution was chosen because it is well-suited to model a vector of non-negative numbers which add up to one. For each seat, we take random samples from this distribution, and record the (random) seat winner. By taking an average over all the possible outcomes, we get the probability for each party of winning the seat. We will not particularly make use of the correlation structure, but for definiteness we assume that each seat behaves independently.

In other words, we drive the seat vote shares directly. These random vote shares are distributed so that their average exactly matches the predicted vote share for that seat, but it is allowed some random variation around that average.

The model requires parameters. The requirement that the random model is centered around
the predicted seat vote shares, specifies most of the model parameters.
There is additionally one free parameter, which controls the amount of
variability around the average. This can be expressed in terms of the standard deviation in support
of a hypothetical party which enjoys 50% average support. Empirical evidence from the 2010 election
suggests that seat variability has a standard deviation of about 5%. This means that a party with
support of *p* will have a standard deviation given by the expression:

where *σ* is the standard deviation at 50% support, for example *σ* = 5%.

In the case of UKIP having a predicted 20% vote share in a particular seat, then its standard deviation from the formula above is 4%. This means that under most random scenarios, its will get between 16% and 24% of the votes in that seat, which is probably not enough to win the seat. But there is a 1% chance that the random variation could lift UKIP's vote share up to 30%. In many closely-fought seats, that could be enough to win the seat. There is even a 0.1% chance of UKIP's vote being 34% in that seat.

For example, suppose UKIP's national support level is 20%. Then they are predicted to win zero seats definitely under the deterministic prediction model. But their closest seat is Montgomeryshire, where they are predicted to be on 24.4%, just 4.4% behind the Conservatives. But their standard deviation is 4.3%, so they only need a swing of less than one standard deviation to win the seat. Unsurprisingly, their chance of winning the seat is 25%. There are about 40 seats where UKIP's chance of winning is more than 5%, if their national support level is 20%. These 40 seats alone contribute 3.6 expected seats (which is half of the total expected seats of 7.2).

The table below shows the difference between the number of seats which UKIP is predicted to win outright (with 0% variability), as well as their expected number of seats won (with 5% variability). For low support levels, the expected number is always higher than the predicted number, because UKIP are not likely to win any seat in particular, but there is a large number of seats which they have a small chance of winning if variability is included in the model.

UKIP national vote share (%) | UKIP Seats predicted to win (0% variability) | UKIP Average Win chance (%) | UKIP Expected number of seats won (5% variability) |
---|---|---|---|

14 | 0 | 0.02 | 0.15 |

16 | 0 | 0.11 | 0.7 |

18 | 0 | 0.39 | 2.5 |

20 | 0 | 1.15 | 7.2 |

22 | 0 | 2.84 | 17.9 |

24 | 5 | 6.03 | 38.1 |

The values in all the tables above are calculated on this basis, using a 5% standard deviation of seat variability. This means that UKIP is predicted to win more seats than the zero-variability model at low support levels. A consequence of this is that these tables will not match the Electoral Calculus headline prediction or user-defined poll predictions, because both of these use the zero-variability model.

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