## Sunday, June 19, 2016

### Converting seat probabilities to an election prediction

I have been reflecting for much of the election campaign that the overall winner market may have  been more favourable for the Coalition than the probabilities from the individual seats market. While the tally of seat favourites has consistently pointed to the Coalition with a majority of seats, for much of the election period the sum of individual seats probabilities has suggested a hung parliament (albeit with the Coalition ahead).

In looking at the probabilities, I have had to deal with a well known problem: bookmaker odds typically over-estimate the likelihood of long-shot events. This is known as the long-shot bias. Typically in an election campaign, the highest odds a bookmaker offers for an unlikely winner is around \$151. Ignoring the bookmaker's over-round for a moment, this equates to a 0.00662 probability of this candidate winning a seat. (Note: if I accounted for the bookmaker's over-round, the probability would be higher).

Because we have 150 seats, and if we assume each has a candidate with a similar probability (ie. 0.00662), what is the likelihood that one of these candidates gets up? In probability, the likelihood that one of k independent events occurring, given the probability α for each of those events is:

If the bookmaker odds accurately reflected the probability of an unlikely outcome (rather than being subject to the long-shot bias), we would expect to see a previously unheard of candidate from a fringe party get up in the House of Representatives in two out of every three elections. It just does not happen.

Okay, now that we have had the long digression on the long-shot bias, we can return to the probabilities from the individual seats.

Rather than tallying favourites, a more accurate way of estimating the likely party distribution of seats from the bookmaker's individual seats odds is to sum the probabilities from each seat. To manage the book-maker's long-shot bias, I have excluded each candidate with a probability of less than 10 per cent. I have normalised the remaining candidates such that the probabilities for each seat sums to one. And then for each day under analysis, I have summed the the 150 seats to get the most likely election outcome as assessed by punters on that day.

For much of the election campaign, this approach has predicted a hung parliament (albeit with the Coalition ahead); as can be seen in the next chart.

In tabular form, the data for the above chart follows. The critical figure is 76 seats, which a party needs to govern in its own right.

Any Other Coalition Green Independent Katter Labor NXT
2016-05-12 0.0 74.468745 2.938078 1.897020 0.784416 68.599665 1.312077
2016-05-13 0.0 74.468745 2.938078 1.897020 0.784416 68.599665 1.312077
2016-05-14 0.0 74.468745 2.938078 1.897020 0.784416 68.599665 1.312077
2016-05-15 0.0 74.394052 2.938091 1.897028 0.784419 68.674327 1.312083
2016-05-16 0.0 74.649053 2.938203 1.897101 0.784449 68.551374 1.179819
2016-05-17 0.0 74.919578 2.938197 1.897096 0.784448 68.280865 1.179816
2016-05-18 0.0 74.951657 2.937508 1.991990 0.784264 68.155041 1.179540
2016-05-19 0.0 74.912952 2.945767 1.997590 0.786469 68.174365 1.182856
2016-05-20 0.0 74.779872 2.946194 1.997880 0.786583 68.306444 1.183028
2016-05-21 0.0 74.729046 2.944697 1.958591 0.786183 68.399056 1.182427
2016-05-22 0.0 74.777564 2.943944 1.958090 0.785982 68.352296 1.182124
2016-05-23 0.0 74.806434 2.943884 1.958050 0.785966 68.323565 1.182100
2016-05-24 0.0 74.984144 2.941717 1.956609 0.785387 68.150913 1.181230
2016-05-25 0.0 75.042677 2.941714 1.956607 0.785387 68.092386 1.181229
2016-05-26 0.0 75.045152 3.164480 1.956672 0.785413 67.867015 1.181268
2016-05-27 0.0 75.302236 3.270579 1.991936 0.785128 67.516947 1.133174
2016-05-28 0.0 75.370120 3.267782 1.958420 0.785212 67.485170 1.133296
2016-05-29 0.0 75.529159 3.267473 1.958235 0.785138 67.326807 1.133189
2016-05-30 0.0 75.529159 3.267473 1.958235 0.785138 67.326807 1.133189
2016-05-31 0.0 75.546244 3.262639 1.956012 0.784246 67.206329 1.244530
2016-06-01 0.0 75.723402 3.265761 1.957883 0.784997 66.998857 1.269100
2016-06-02 0.0 75.421845 3.208630 1.956630 0.924229 67.173650 1.315016
2016-06-03 0.0 75.460929 2.944299 1.939280 0.944448 67.396525 1.314520
2016-06-04 0.0 75.738862 2.879525 2.016399 0.931730 67.214227 1.219256
2016-06-05 0.0 75.738862 2.879525 2.016399 0.931730 67.214227 1.219256
2016-06-06 0.0 75.738862 2.879525 2.016399 0.931730 67.214227 1.219256
2016-06-07 0.0 75.729243 2.824903 2.042580 0.931514 67.252787 1.218973
2016-06-08 0.0 75.775177 2.825329 2.042888 0.931654 67.078320 1.346633
2016-06-09 0.0 75.891196 2.822167 2.159221 0.932125 66.959233 1.236057
2016-06-10 0.0 76.383699 2.823712 2.117320 0.932635 66.505900 1.236734
2016-06-11 0.0 76.150329 2.819242 2.269215 0.931159 66.331946 1.498108
2016-06-12 0.0 76.150329 2.819242 2.269215 0.931159 66.331946 1.498108
2016-06-13 0.0 76.135327 2.140131 2.270217 0.931570 66.991438 1.531318
2016-06-14 0.0 76.660399 2.140315 2.270412 0.931650 66.465775 1.531449
2016-06-15 0.0 77.125082 2.054944 2.272520 0.932515 66.134875 1.480063
2016-06-16 0.0 77.118447 2.052858 2.269558 0.931568 66.150864 1.476704
2016-06-17 0.0 77.127473 1.927562 2.201963 0.946004 66.290524 1.506474
2016-06-18 0.0 77.275149 1.926875 2.201177 0.945667 66.076621 1.574512
2016-06-19 0.0 77.942031 1.927295 2.201657 0.945873 65.215495 1.767650

By way of contrast, the method of tallying favourites has had the Coalition in a winning position throughout the campaign.