In their coverage of the Wellington-Auckland game in the HRV cup last Friday, Sky Sport introduced WASP—the “winning and score predictor” for use in limited-overs games, either 50-over or 20-20 format. In the first innings, the WASP gives a predicted score. In the second innings, it gives a probability of the batting team winning the match.

I am very happy about this as it is based on research by my former doctoral student, Scott Brooker, and me. Not surprisingly, the commentators didn’t go into any details about the way the predictions are calculated, so I thought I would explain the inner workings in a wonkish blog post.

The first thing to note is that the predictions are not forecasts that could be used to set TAB betting odds. Rather they are estimates about how well the average batting team would do against the average bowling team in the conditions under which the game is being played given the current state of the game. That is, the "predictions" are more a measure of how well the teams have done to that point, rather than forecasts of how well they will do from that point on. As an example, imagine that Zimbabwe were playing Australia and halfway through the second innings had done well enough to have their noses in front. WASP might give a winning probability for Zimbabwe of 55%, but, based on past performance, one would still favour Australia to win the game. That prediction, however, would be using prior information about the ability of the teams, and so is not interesting as a statement about how a specific match is unfolding. Also, the winning probabilities are rounded off to the nearest integer, so WASP will likely show a probability of winning of either 0% or 100% before the game actually finishes, even though the result is not literally certain at that point.

The models are based on a database of all non-shortened ODI and 20-20 games played between top-eight countries since late 2006 (slightly further back for 20-20 games). The first-innings model estimates the additional runs likely to be scored as a function of the number of balls and wickets remaining. The second innings model estimates the probability of winning as a function of balls and wickets remaining, runs scored to date, and the target score.

The estimates are constructed from a dynamic programme rather than just fitting curves through the data. To illustrate, in the first innings model to calculate the expected additional runs when a given number of balls and wickets remain, we could just average the additional runs scored in all matches when that situation arose. This would work fine for situations that have arisen a lot such as 1 wicket down after 10 overs, or 5 wickets down after 40 overs, etc.), but for rare situations like 5 wickets down after 10 overs or 1 wicket down after 40 it would be problematic, partly because of a lack of precision when sample sizes are small but more importantly because those rare situations will be overpopulated with games where there was a mismatch in skills between the two teams. Instead, what we do is estimate the expected runs and the probability of a wicket falling on the next ball only. Let V(b,w) be the expected additional runs for the rest of the innings when b (legitimate) balls have been bowled and w wickets have been lost, and let r(b,w) and p(b,w) be, respectively, the estimated expected runs and the probability of a wicket on the next ball in that situation. We can then write
V(b,w) =r(b,w) +p(b,w) V(b+1,w+1) +(1-p(b,w)))V(b+1,w)
Since V(b*,w)=0 where b* equals the maximum number of legitimate deliveries allowed in the innings (300 in a 50 over game), we can solve the model backwards. This means that the estimates for V(b,w) in rare situations depends only slightly on the estimated runs and probability of a wicket on that ball, and mostly on the values of V(b+1,w) and V(b+1,w+1), which will be mostly determined by thick data points. The second innings model is a bit more complicated, but uses essentially the same logic.

Now many authors have applied dynamic programming to analyse sporting events including limited overs cricket (see my previous post on this here), although I don’t know of any previous uses of such models in providing real-time information to the viewing public. Scott’s and my main contribution, however, is in including in our models an adjustment for the ease of batting conditions. I have previously blogged about our model for estimating ground conditions, here. Without that adjustment, the models would overstate the advantage or disadvantage a team would have if they made a good or bad start, respectively, since those occurrences in the data would be correlated with ground conditions that apply to both teams. Using a novel technique we have developed, we have been able to estimate ground conditions from historical games and so control for that confounding effect in our estimated models.

In the games on Sky, a judgement is made on what the average first innings score would be for the average batting team playing the average bowling team in those conditions, and the models’ predictions are normalised around this information. At this stage, I believe this judgement is just a recent historical average for that ground, but the method of determining par may evolve.

I gather that the intention is to unveil more graphics around the use of WASP throughout the season, with the system fully up and running by the time of the international matches against England. It’s going to be interesting listening to what the commentators make of the WASP. Last Friday’s game wasn’t the best showcase, since when Auckland came to bat in the second innings, their probability of winning was already at 92% and quickly rose higher. It was fun, though, hearing the commentators ask Wellington captain, Grant Elliot, who was wired for sound while fielding, what he thought their chances were given that WASP had the Auckalnd Aces at 96% at that point. Grant's reply was lovely: "Sometimes even pocket aces lose". This is worth remembering when (as will inevitably happen), a team has a probability of winning in the 90s but still goes on to lose.