Rating Cricketers – Further Thoughts

In a previous post, I said that a team’s chance of winning a match can be modelled as a function of the following factors:

- how many runs the team scores
- how many runs the other team scores
- how many wickets the team loses
- how many wickets the other team loses
- how quickly the team scores runs
- how quickly the other team scores runs

Actually, you can reduce a team’s chance of winning a match to:

- how quickly the team scores runs
- how quickly the other team scores runs
- how long the team stays in
- how long the other team stays in

For example, assume in one-day internationals that an average team will score 200 runs and stay in for 250 balls. For a team to bat better than average it has to either score faster than the average team or stay in longer than the average team. For a team to bowl better than average it has to either make the other team score slower than the average team or bowl the other team out faster than the average team. If a team is below average in all of these facets it would be expected to lose more than it wins.

The value of a batsman then is:

Value of how quickly batsman scores relative to average batsman + Value of how long batsman stays in relative to average batsman

And the value of a bowler is:

Value of how slowly bowler concedes run relative to average bowler + Value of how quickly bowler gets batsmen out relative to average bowler

The first part of these formulas appears relatively easy to calculate. In the long-term, for a batsman, the value of how quickly they score is:

Average runs scored by batsman – Average strike rate of all batsmen * Average balls faced by batsman

Let’s say the average strike rate of all batsmen is 0.75. Then a batsman who averages 40 off 60 balls is, on average, creating a value of -5 runs relative to the average team from how quickly they score. A batsman who averages 30 off 35 balls is, on average, creating a value of 3.75 runs relative to the average team from how quickly they score.

But hang on, one batsman is averaging 40 and the other batsman is averaging 30; doesn’t that automatically make the first batsman more valuable? Well no, if the batsman averaging 40 is scoring slower than the average batsman then they are taking away value the longer they are batting in terms of keeping pace with the average team. But the batsman averaging 40 is offsetting this negative effect by virtue of being able to stay in longer than the average batsman, and therefore helping their team stay in longer than the average team.

What is the value of staying in? This appears more difficult to calculate, as it is likely to depend on how many overs are left and how many wickets have been lost. Essentially you would need to work out when a batsman typically comes into the innings and calculate how many more or less runs the team is expected to score as a result of the batsman staying in longer or shorter than the average batsman. Let’s assume the average batsman stays in for 25 balls. For the batsman who averages 40 runs off 60 balls, their value from staying in longer than the average batsman is equal to the extra runs the team is expected to score by not being one wicket further down for those 35 extra balls. If, as a result of this, the team can be expected to score more than an extra 5 runs (recall the batsman was costing the team 5 runs relative to the average team from their slow batting), then that batsman is still making a positive contribution.

For a bowler, the value of how slowly they concede runs is:

Average economy rate of all bowlers * Average balls bowled by bowler - Average runs conceded by bowler

And the value of getting batsmen out would be the difference in runs the other team is expected to score as a result of the bowler getting batsmen out faster or slower than the average bowler.

Alas, I don’t think I have the patience to work out the figures. But the main message I wanted to get across here is that batting and bowling averages in themselves are not useful except as an indication of how long a batsmen can stay in or how quickly a bowler can get batsmen out, and that you can not evaluate a batsman or bowler without taking into account their strike rate or economy rate (a batsman with an average of 40 but a strike rate of 50 in one-day internationals is not a useful batsman). It’s working how to best weight the fall of wickets relative to the pace of scoring which is the tricky part.

P.S. Yes I know that strength of opposition and pitch conditions matter to performance. I'm assuming that, in the long run though, the effects of these on a cricketer's performance should roughly even out.