Wednesday 7 March 2012

(Ir)Rational Expectations in Cricket

This post is motivated by the New Zealand cricket team’s recent decision to bat first upon winning the toss in the first of the ODI games against South Africa, and Australia’s decision to bat first against Sri Lanka in the game yesterday.

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.


  1. I think the analysis needs to consider the speciality of the captain, and perhaps even captaincy skill. Also, New Zealands cricketing history has recognisable eras, each with differing levels of success. Though I suppose the eras may coincide with changes in captains too.

  2. 1. Bounded rationality -- captains are likely to have captained < 100 games and their own experience is unlikely to give statistically significant evidence that sending the other lot in is beneficial.
    If so, we would expect to see more sending in for more experienced captains.
    You could test this with a variable for number of games played/captained by the winning captain.

    2. Is sending in correlated with day/night games, away games, etc...

    3. How is this for tests?

  3. In tests between all teams since 1970, when winning the toss and batting, it's 293 wins, 295 losses, 1 tie and 311 draws. When losing the toss and being sent in, it's won 142, lost 167, 156 draws. So being sent in is bad for your chances while choosing to bat gives you an even chance. It could be that teams are following WG Grace's advice much more than they should and batting whenever they are in the slightest doubt.

    Looking at NZ only, when choosing to bat it's 14 wins, 32 losses and 22 draws. When sent in it's 14 wins, 19 losses and 26 draws.

  4. Those test stats are particularly surprising when you consider that in tests played in India, the advantage of batting first is considerable. Take matches played in India (and probably Pakistan as well), and the record for teams choosing to bat first must start to look pretty dismal.

  5. It is enough to make you wonder if there is something in the NZ psyche which makes chasing a target rather than setting a target the preferred action. Perhaps we are too mild mannered & meek to embrace batting first. I wouldn't have thought any international sportsperson would be especially timid, competitiveness at an international level should select against such folk, but there is something odd going on with these stats. Perhaps our guys simply lack the confidence and fortitude to successfully set goals for opposing teams.

  6. @Lats.

    Absent a predicted change in pitch between the first and second innings, every team should choose to bat second. In which case, being mild-mannered and meek might be an explanation for why NZ has traditionally punched above its weigth in ODIs (relative to their test-cricket standing). But it can't explain why our record is so abysmal when we *choose* to bat, but actually pretty good when we asked to bat.

    1. Maybe our senior cricketers aren't very good at reading a good batting pitch, and as such make poor decisions when choosing to bat. But that doesn't explain our relatively better performance when this decision is made for us by the opposition. You'd expect opposing captains to make us bat on wickets which are perceived as bowler friendly, so theoretically we ought to perform worse when we are put in to bat. I still think it comes down to the top couple of inches, but am unsure what psychological dynamics are in play.

    2. Lats: In a sensible world, it shouldn't be the case that opposing batters make us bat on wickets that are perceived as bowler friendly; rather it should be that we are made to bat on wickets that are perceived as not likely to become *more* bowler friendly (or, equivalently, less batter-friendly) between the first and second innings. Absent any expected change, the toss winner should always bat second.

      I suspect, or more accurately, Scott from the comment above suspects, that the explanation is that NZ is more guilty than others of falling into the sunk-cost fallacy when the decision to bat first is our own: That is, when we bat first, we feel a need to aim for a very high score when we choose to bat first, and don't adjust our target if we get into early trouble.

      An alternative explanation is that all teams make the error of thinking you should bat first when playing on a high-scoring pitch, but that we play more such games because of our ridiculously small grounds, and batsmen-favouring groundsmen.