In a recent post on Political Betting, antifrank listed the best odds available from the various bookies for bands of seats which the main parties might win in the General Election.
Using that data as a starting point, we can estimate the implied probability for any particular result. For example, let's take the Conservative seat total. The best odds quoted for the low end were:
Under 100 200/1
100 - 124 200/1
125 - 149 150/1
Converting these odds to probabilities, and adding them up in turn, this means that the bookies think there is a 0.49% chance of the Tories winning less than 100 seats, a 0.995% chance of them winning less than 125, and a 1.65% chance of them winning less than 150 seats. Or, to look at it the other way round, the bookies are saying the Tories have a probability of 0.9951 of winning at least 100 seats, 0.9905 of winning at least 125, and 0.9835 of winning at least 150.
Continuing the process, we can find the cumulative probability for the upper figure in each of the bands quoted, right up to 450 or more. (We have to make a correction, to allow for the bookies' profit margin of around 15%, to ensure that our probabilities add up to 1.00). And it is then a simple matter to fit a curve to the resulting data points, to give us a smooth interpolation of the probabilities for any number of seats in the range. (Note for pointy-heads: I used a degree-6 polynomial fit). The results are as follows (Click on the graph for a higher resolution version)
It is now a simple matter to read off the probability for the Conservatives winning at least N seats, for any value of N. For example, the markets are saying that there is around a 50% chance of Cameron winning 350 seats or more.
I have also marked a few points which show the implied probabilities for other markets. (The ones I've shown are all bets on the Conservatives winning either a majority of a certain size, or winning individual seats - see below).
By seeing where they lie on the chart, you can see if they are good value or not, compared with what the seat bands are telling us. If you are betting for the Conservatives, you want the point to be as far below the line as possible (i.e. lower implied probability, which means better odds). Conversely, if you are betting against the Conservatives, you want the point to be as far above the line as possible.
Take the Betfair Party Seats line, for which the current mid-point and last matched price is 360 for the Tories, i.e. this is the figure which punters on that market think corresponds to a probability of 0.5. If you look at the graph, you'll see that this is about 10 seats higher than the seat bands market implies for a probability of 0.5; this indicates that buying the Tories on this market is not good value compared with the seats band. The same is true for the Paddy Power and Ladbrokes markets for majority of over 50 and over 100 (however, in these cases you need to allow for the bookies' profit margin). On the other hand, betting on the other side (for a majority of less than 50, on Paddy Power at odds of 2.5, implying a probability of 0.4 that the Tories will NOT get a majority of at least 50), looks good value compared with the seats band (To avoid clutter, I haven't shown this on the chart).
We can also do something very interesting for assessing spread bets. Since we now have the cumulative probability, we can trivially deduce the implied probability for exactly N seats, for any N (it's just the difference between the probability of at least N and at least N-1). And if we sum the product of those probabilities times the number of seats, we get the price which the seat band markets imply should be the mid-point for the spreads (i.e. the expected value). This is currently very substantially less than the actual mid-point for the Tories: SPIN's mid-point is 359.5, the weighted value of the seat-bands market is about 340. This is not a surprise when you look at the shape of the chart, which has a long flat tail at the low end of the distribution. But it does mean that if you want to bet on a Tory seats value of greater than 360, you may be better off betting on the seats bands than on SPIN.
I must emphasise that none of this means that the other markets are necessarily wrong ; what I have done here is simply reflect what one particular betting market (the seat bands) implies is the probability distribution. We'll never know whether it is correct; this is not a physics experiment, you can't simply repeat this general election hundreds of times under identical conditions to find out how many times each outcome occurs. This is where one has to apply one's own judgement; personally, I think the distribution is too skewed towards the low end, and that the actual probability distribution should be more weighted to a Tory majority in the 25 to 100 region. Others may disagree: that is the essence of political betting.
Finally, a note on individual constituencies. This is much more speculative, because of individual variations (although not every seat can buck the trend!) I've marked a few examples, where I've taken the implied probability from the best available odds in the market, and related these to the corresponding Tory majority at which the seat would fall, everything else being equal. (I used Anthony Wells' excellent UK Polling Report website for the data. )
Take, for example, Broxtowe, the constituency of PB regular Dr. Nick Palmer MP. According to UK Polling Report, the Conservatives would have 214 seats if the 2005 election vote shares were repeated with the new boundaries,and Broxtowe is Tory target 42. Therefore, if everything else were equal, the probability of Broxtowe going Blue would be the same as the probability of the Tories winning 214+42=256 seats in all. Looking at the chart, the probability of this is around 0.9, so we would expect the odds to be no better than decimal 1.11 (1/9). In fact you can get 1.20 (1/5) from Ladbrokes, implying a probability of just 0.83 (and that's without making allowance for Ladbrokes' profit margin). This means that either a bet on the Conservatives winning Broxtowe is very good value compared with the seat bands, or there is some specific reason why Broxtowe is less likely to fall to the Tories than other targets which would fall to similar swings.
Conversely, applying the same method to a seat like Birmingham Selly Oak, you get a data point which falls far above the line on the chart; this suggests (in the absence of any other information) that it is bad value to bet on the Tories winning here; it could, therefore, be very good value to bet on Labour retaining it.