Dynamic programming in cricket: choosing a night watchman

In cricket, when a batsman is dismissed towards the end of a dayʹs play, he is often replaced by a lower-order batsman (a ʹnight watchmanʹ), in the hope that the remaining recognised batsmen can start their innings on the following day. A dynamic programming analysis suggests that the common practice of using a lower-order batsman is often sub-optimal. Towards the end of a dayʹs play, when the conventional wisdom seems to be to use a night watchman, it may be best to send in the next recognised batsman in the batting order. Sending in a night watchman may be good judgement when there are several recognised batsman and several lower order batsmen still to play (say four of each). However, with smaller numbers (two of each, for example), then, with very few overs left to play, it may be better to send in a recognised batsman.