Hung Parliament: How likely is it?

This page first posted 27 November 2009, revised 7 February 2010

There is a chance that the next election will result in a hung parliament. But how likely is this to happen? And if it does happen, what is the chance that the Liberal Democrats will be able to choose the next government. This analysis attempts to answer these questions by simulating possible election results.

1. Summary of Results (updated 7 Feb 2010)

We make a simulation of the possible general election results. These random simulations are centered around the level of party support shown by current opinion polls (as at 7 February 2010). There is also a "volatility" or uncertainty factor introduced to allow for change in national opinion support from now until the election.

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.

2. Model description

All the data in Sections 2 to 4 are based on opinion polls as at 27 November 2009.

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:

It gives the support figures for the three major parties, and the total for the other parties. Call them: Con, Lab, Lib, Oth.
The support for each party group is between 0% and 100%.
The total support over all four parties must equal 100%.
The average support over all the scenarios for any one party should equal its current opinion poll support.
There should be random variation in a party's support.
The size of the random variation should be controllable.

The model we have chosen for this is called the Multivariate Beta Distribution. Its mathematical details are described at the end of this document for anyone who is interested in the technicalities. It has the properties we need.

Party	Support
Con	39.8%
Lab	28.1%
Lib	17.8%
Oth	14.3%

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.

3. Scenario generation and results

Using the model and parameters above we can run some simulations of general election support for the parties. We made 1,000 simulations, and passed the results into the Electoral Calculus prediction model. The model then tells us how many seats are won by each party. We then look at these seats to see what the result would mean in terms of the largest party and parliamentary arithmetic.

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
37.5	23.7	21.5	17.3	346	208	64	32	Con	Con majority
38.6	30.6	14.7	16.1	322	267	30	31	Con	Con - Lib coalition
35.8	41.8	13.9	8.6	213	382	26	29	Lab	Lab majority
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
36.4	32.2	10.2	21.2	294	303	20	28	Lab	Fragmented parliament

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 Scenarios	Chance %
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%
Fragmented parliament	8	0.8%

Note that these figures are as at 27 November 2009. More updated figures are shown in section 1.

4. Volatility Estimation

We have used a volatility of 5% when calculating the results. We can re-run the simulation under different volatility assumptions and also compare the results with the implied market prices from the Betfair politics market.

Outcome	3% Vol	4% Vol	5% Vol	6% Vol	7% Vol	Betfair
Con majority	68%	64%	62%	60%	57%	70%
Lab majority	2%	4%	10%	12%	18%	7%
Hung parliament	30%	32%	29%	28%	25%	24%

Betfair prices are implied percentages from the average of bid and ask prices as at 27 Nov 2009.

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
Con majority	77%	72%	70%	63%	62%	70%
Lab majority	1%	3%	7%	12%	14%	7%
Hung parliament	22%	25%	24%	26%	25%	24%

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%.

Appendix: Multivariate Beta Distribution

The distribution is a simple generalisation of the one-dimensional Beta distribution. The n-dimensional Beta can be thought of as having its support on the (n-1)-simplex Δ(n-1), where

Δ(n-1) = { x = (x_i)_i=1..n : 0 <= x_i <= 1, Σ_ix_i = 1 }.

The distribution is parameterised by n parameters β₁,...,β_n, and the distribution has density

f(x) = Γ(B) Π_i Γ(β_i)^-1 x_i^β(i)-1, where B = Σ_iβ_i.

The ith co-ordinate X_i has mean and variance

E(X_i) = β_i / B, and Var(X_i) = β_i ( B - β_i ) / B²(B+1).

For modelling elections, we have the opinion poll support levels p_i 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

B = 1 / (4 σ²) - 1, with β_i = p_i B.

Then the mean and variance of X_i are

E(X_i) = p_i, and Var(X_i) = 4 σ² p_i ( 1 - p_i ).

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

X = (X₁,..,X_n) = (Γ₁,..,Γ_n) / Σ_iΓ_i, where Γ_i is a Gamma random variable Γ(β_i).

Useful links

Wikipedia article on the Beta distribution.
Wikipedia article on the Gamma distribution.
Betfair (Sports) section on politics