In ODI cricket, ease-of-batting conditions vary not only from game to game, based on the type of soil, the size of the ground and recent weather, but also within a game. In day-night games in particular, the quality of the lights used in the second half of the game, and the amount of dew that settles after the sun goes down can affect whether it gets easier or more difficult to bat between the first and second innings. The players involved in a match have some information about the likely change in playing conditions over the course of a match, in part based on previous matches at the same ground. Since the decision of which team bats first is not entirely random, but is a choice made by the team winning the (random) toss, and because the toss is equally likely to be won by the stronger team as the weaker team, we would expect to see the team winning the toss winning more than 50% of the time, and for this to be true whether the toss-winning team chooses to bat first or second.
Now consider the data. In the 2,240 completed ODI games played between two top-8 teams since ODI cricket began in the 1970s, the team winning the toss has won 1127 times, a winning percentage of only 50.3%, which seems to indicate that there is essentially no difference between setting a target or in chasing one, either in terms of the ease of batting conditions of any advantage from knowing the target score.
It is a different story, however, when you look at the breakdown in these games between when the toss-winning team chooses to bat first and when it chooses to bat second. The team winning the toss has chosen to bat 2nd 1167 times and has had a winning percentage of 51.9. In contrast, the team winning the toss has chosen to bat first the remaining 1083 times, and had a winning percentage of only 48.8%, which is not suggestive of rational expectations.
Now consider New Zealand’s record. When we lose the toss and are asked to bat first, our record is 64 wins from 135 games, a winning percentage of 47.4, a result that seems about right given that New Zealand’s overall record in ODI matches (about average against top-8 teams), and the disadvantage of losing the toss. But when we choose to bat first, our record is 36 wins from 127 games, a winning percentage of only 28.3.
If I were ever entrusted with teaching an econometrics course (a purely hypothetical thought experiment as my econometric-specialising colleagues would never allow that to occur), here is a question I would love to ask:
- Come up with some alternative theories to explain the above numbers, including at least one theory that is consistent with rational expectations on the part of captains;
- suggest ways in which these different hypotheses could be tested with data.
I can come up with a model under which the New Zealand data would be consistent with rational decision making, but I don’t believe it. For the data overall, I think that captains make a classic error in statistical inference: on average, the higher the first-innings score, the more likely it is that the batting team has put in a better-than-average performance relative to the bowling, and the more likely it is that the conditions are relatively easy for batting. The first of these imply that the higher the first-innings score the greater is the probability that the team batting first will win, but I hypothesise that captains misunderstand the source of that relationship and mistakingly believe that you are more likely to win if you bat first on an easy pitch.
For ODIs, we need to reverse WG Grace’s famous dictum to: “When you win the toss – bowl. If you are in doubt, think about it, then bowl. If you have very big doubts, consult a colleague – then bowl.”
But that still doesn’t explain New Zealand’s miserable record when we choose to bat first. I welcome all hypotheses in the comments: Testing these will be the topic for a future Honours project.