A couple of years ago, I supervised an Honours research paper by Johnny Sharland (now at the RBNZ) on the efficacy of using a nightwatchman in cricket. For the uninitiated, a nightwatchman is a player (usually a bowler**) who is sent in ahead of a better batsman when a wicket is lost near the end of the day in a multiple-day cricket match. The idea is that a batsman is at most risk of getting out at the start of his innings, so the nightwatchman's job is to protect the better batsman from having to start his innings at the end of the day, and then make a fresh start the next morning.
We wanted to compare the cost to a team of changing the batting order to the benefit of a reduced probability of dismissal for a top-order batsman, to calculate when and if the benefits would outweigh the costs. As it turned out, the benefit-cost calculation turned out to be completely uninteresting for an unexpected reason: We could find no evidence in a database of 200 test matches that having to make a second start in any way increases the probability of a top-order batsman being dismissed early in his innings. That is, the key variable that determines a batsman's probability of being dismissed is how many balls he has faced in that innings, not how many he has faced that morning. The cost-benefit calculation then becomes irrelevant, as there is simply no benefit from using a nightwatchman to balance against the costs.
I thought this made the results uninteresting, and so did not seek to write it up for publication. The latest test between Australia and India, however, makes me think that maybe that decision was wrong. The Australian captain, Michael Clarke, has just earned the unenviable distinction of being the first ever captain to declare his first innings closed and then go on to lose by an innings.* Clarke declared the Australian innings closed when they were 9 wickets down with 5 overs remaining on the first day. It seems that his reasoning was that Australia's last two batsmen were probably not going to be able to score many runs anyway, and by declaring he would force the Indian batsmen to start their innings that night and make a fresh start the next morning. Johnny's results suggest that there was no expected benefit to this declaration, only a cost. Maybe a shortened version of his paper concentrating only on the estimate of the second-start effect would be interesting.
* Explanation for non-cricket fans. Each team bats until 10 of its 11 batsmen have been dismissed. The 10th dismissal signals the end of the "innings" after which the other team gets a chance to bat and score more runs. The two teams get two innings of up to 10 dismissals. A captain, however, can choose not to use it full allotment and declare its innings over before 10 batsmen have been dismissed. If a team scores fewer runs in its two innings combined than the opposition scores in its first innings, the opposition does not need to bat a second time and is said to have "won by an innings".
** For an example of a batsman being used as a nightwatchman, check out this match (which is also notable for featuring Bradman's first test century). Batting on a notorious Melbourne sticky wicket, England sent specialist batsman, Douglas Jardine, in ahead of his normal batting position in order to protect superstar batsman, Wally Hammond.