# League probability projections – why so volatile?

This season’s Premier League title race is captivating. Four teams, led by unfancied Leicester City, all have a reasonable chance of winning. How do we know this? Firstly, betting markets give us a good guide to the likely champions. But there are also now many models that calculate the probability of season-end outcomes by simulating thousands of different potential future scenarios – with their results published and discussed on twitter.

One particular challenge raised against these models is that the calculated probabilities can vary significantly from week to week . This weekend’s results are a great example. In my model (explained here), prior to this round of matches I’d estimated Leicester City’s probability of winning the title at 16% and Man City’s at 55%. At the end of the weekend these had swung to Leicester (31%) and Man City (27%). Why the big shift?

To explain, I’ll go through the factors that affect a team’s week to week probability changes.

1. The number of games left to play. As the number of matches played increases there are fewer potential final league outcomes. This means that individual results have a bigger impact on a team’s probability as the season progresses.
2. Current points and relative position in league table. Obviously for a team’s title chances to change at all they need to have a chance in the first place. Aston Villa beating Norwich didn’t do a whole lot to change title probabilities.
3. Actual points won or lost in the particular match compared to expected points.
4. Assessment of team’s strength in future matches. Performance in the last match may cause a re-evaluation of that team’s rating.
5. Other teams’ results.

To illustrate these factors, I’ll go through how they impacted Leicester and Man City’s probabilities this weekend.

If Leicester had beaten Man City away early in the season, say in the 3rd round of matches, their title chances (according to my model) would have increased from 0.02% to 0.10% (no one gave them a chance back then!), whereas favourites Man C would have reduced from 56.5% to 48.7% – much smaller changes because there is so much more of the season to go.

But Leicester’s position before the weekend was top, with 50 points compared to Man City’s 47. However they weren’t favourites because Man City were still expected to get more points in remaining matches – as they’re rated a better team.

Man City were expected to win 2.00 points per match, whereas Leicester were expected to win 1.48 per match. This is more than enough for Man City to overtake Leicester – hence title probabilities of 55% and 16%.

The model expected Man City to win 2.22 points in the match against Leicester, compared with Leicester’s 0.59. So Leicester’s victory delivered a 2.41 increase in points above expected and Man C correspondingly 2.22 below. But because they’re both title contenders it had the double whammy effect of a 4.63 expected points swing between these two rivals.

This left us, after the match, with Leicester on 53 points but Man C still on 47. Because there are only 13 matches left, this change makes a big difference. Man City are still rated as the better team, with expected points slightly reduced to 1.95 per match and Leicester increased to 1.56 per match – but because the points gap has increased and the number of matches has reduced their relative chances have changed significantly.

The final factor is other teams’ performance. Arsenal and Spurs are genuine title contenders too, so their victories further impacted the probabilities of their rivals.

The chart below illustrates the effect of these factors on Man City’s title probability fall over the weekend.

This example uses my model, other models rate teams differently but probability changes can be just as volatile – and the same also applies to the betting markets. The chart below shows volatility in the implied probability for Betfair’s Man City title odds.

So volatility in title probability from match to match is a mathematical property of this type of modeling uncertainty, rather than a flaw in the method. Whether the underlying assessment of team strength is always robust (as Leicester have confounded all season) is another matter.