Within this model we estimate the chance of a hung parliament is about 38%, with odds of 3-2 against.
If there is a hung parliament, there are a number of possible outcomes. No party on its own would have a majority in the House of Commons. In that case the Liberal Democrats might join forces with one of the major parties to build a two-party coalition which has a majority in parliament. Depending on the exact outcome the Lib Dems might be able to do this with only one of the major parties, with either, or with neither.
We can look at all the possibilities for either majority or coalition governments. The approximate chances of the possibilities are:
|Possible Outcome||Odds||Chance %||Comment|
|Con majority government||11-10 against||48%||Conservatives govern on their own|
|Lab majority government||6-1 against||14%||Labour govern on their own|
|Con - Lib coalition||17-3 against||15%||Lib Dems only have enough seats to support the Conservatives|
|Lab - Lib coalition||6-1 against||14%||Lib Dems only have enough seats to support Labour|
|Lib Dem choice of government||12-1 against||8%||Lib Dems have enough seats to choose which party governs|
|Fragmented parliament||125-1 against||0.8%||Lib Dems do not have enough seats to form a two-party coalition|
These probabilities depend on both the current level of opinion poll support for all the parties and the estimated level of uncertainty. They are subject to change.
In many of the hung parliament scenarios, the possibilities open to Nick Clegg and the Liberal Democrats are more limited than one might suspect. In about 4 cases out of every 5, the Lib Dems do not have enough seats to choose which of the major parties forms the government. They only have the choice of entering a coalition with the largest party or standing back and letting that party form a minority government. If they stay out of government, they not only lose a share of power but also run the risk of being "punished" by voters if there is a subsequent general election.
The chances that Labour continues in government are higher than the raw opinion poll figures suggest. There is approximately a chance of 29% (5-2 against) that Labour is the largest party and can govern either with an outright majority or in coalition with the Lib Dems. In this case, the Lib Dems do not have a choice which party to support because they do not have enough seats to make a coalition with the Conservatives.
Reassuringly, there is only a very small chance that the outcome is so finely balanced that no two-party coalition with the Lib Dems is possible. This means that even if there is a hung parliament, it is quite likely that there would be a workable coalition if the Lib Dems chose to have one.
To make these calculations we need to simulate the (random) possible outcomes of the general election. Although these are random, that does not mean they are completely uncertain and arbitrary. We know the current level of opinion poll support for the major parties, so we will assume that the actual election outcome is likely to be near those levels, but with a degree of random variation. In some cases, it will be close to the opinion poll figures, and in other cases it will be further away.
To describe this quantitatively, we need a model for the general election support levels. This model must have the following properties:
|The parameters we use are based on the average of opinion poll results in November 2009. These are shown in the table
on the left
We choose a volatility level (one standard deviation of uncertainty) of 5%. This is a rather arbitrary choice, but we will see some evidence for it below.
Some example scenarios are shown here. There are 650 seats, so a party or coalition needs 326 to have a majority in the House of Commons.
|Party Support %||Party Seats won||Largest|
|Con %||Lab %||Lib %||Oth %||Con||Lab||Lib||Oth||Party||Outcome|
|38.6||30.6||14.7||16.1||322||267||30||31||Con||Con - Lib coalition|
|35.6||34.5||17.8||12.2||251||324||43||32||Lab||Lab - Lib coalition|
|37.0||30.5||19.9||12.6||292||270||56||32||Con||Lib Dem choice of government|
These are just example scenarios, and they are not equally likely. In fact calculating their relative frequencies is the main purpose of the exercise. The result over 1,000 scenarios is:
|Possible Outcome||Number of|
|Con majority government||617||62%|
|Lab majority government||97||10%|
|Con - Lib coalition||140||14%|
|Lab - Lib coalition||85||8%|
|Lib Dem choice of government||53||5%|
Note that these figures are as at 27 November 2009. More updated figures are shown in section 1.
|Outcome||3% Vol||4% Vol||5% Vol||6% Vol||7% Vol||Betfair|
There is no obvious correspondence between the model's prices and the Betfair prices. However, suppose we increase the Conservative vote share by 1% to 40.8% and decrease the Lib Dem vote share by 1% to 16.8%. If we re-run the simulation using this revised central case, then the table becomes:
|Outcome||3% Vol||4% Vol||5% Vol||6% Vol||7% Vol||Betfair|
Then there is very good agreement between the 5% volatility case and Betfair. This suggests that Betfair market participants have a slightly higher estimate of Conservative support at the expense of the Lib Dems, and that they are pricing the outcomes under an implied volatility of 5%.
The distribution is parameterised by n parameters β1,...,βn, and the distribution has density
The ith co-ordinate Xi has mean and variance
For modelling elections, we have the opinion poll support levels pi for the ith party. We also define the volatility σ to be the standard deviation of uncertainty of a hypothetical party which has 50% support. Then we define
Then the mean and variance of Xi are
So the distribution has expectation equal to the current support level (as desired). Its volatility is defined as its standard deviation, which is the square root of its variance. The volatility is proportional to σ, which is the maximum possible volatility (achieved for a party with 50% support). As a party's support decreases, so does its volatility.
For simulation of the Multivariate Beta distribution, we can use the result that it can be represented in terms of gamma random variables. That is