Quick single: Maths and the opening partner problem
England have tried: the leading scorer in county cricket, the fresh graduate of every ECB age group team, an Australian, a man who survived a medical emergency, England’s most prolific number 3 batsman in decades and a bald Yorkshireman. With so many options tried, but no solution found, they could do worse than look now to maths. There’s a formula that could help find Alastair Cook an opening partner.
It’s the solution to the Optimal Stopping Problem. Its role is to assist in situations that feature the following characteristics:
- An agent has many options to choose from but can test only one option at a time.
- Once an option has been tested and discarded it’s very difficult to go back to it later.
- If an early choice is selected for ‘keeps’ then the agent would remain ignorant of what all the other options could have offered and whether they would have been superior.
- But if the agent keeps testing more and more options in search of a better one, the best option may get discarded.
The method is also known as the ‘secretary problem’, recalling a time in the last century when recruiting the right personal assistant was the sort of issue that bedevilled business men. For not unconnected reasons, it is now talked about as an aid to singletons trying to find a life partner.
Alastair has Alice as his life partner, but Straussy has long gone (from whites and track suits, anyway) and the search is on to find an opening partner. The Optimal Stopping Problem solution says that the agent (A Cook) should estimate the total number of partners he would be likely to try out in his (post-Straussy) Test career. In 29 Tests since the former captain’s retirement, Cook had, on average a new partner every six Tests. If his career continues for another five years, he could appear in 60 more Tests. That would equate to 10 more opening partners at the current attrition rate – and 15 in total.
The next step of the solution is to identify the number of partners that should be tested in order to get a feel for the quality of the field. Research has shown that the square root of the total number of potential partners (3.87) gives a strong probability of getting that feel but, to be certain, the agent should divide the total number of potential partners by 1/e (ie 35% of 15 = 5.25). So, Cook should test five partners and, following the theorem, identify the best of those and then keep changing partners until another one matches that standard.
When Cook opened with Trott, he completed the testing phase. His task now is to identify the best of the five and find another partner to match that standard. Who of Compton, Root, Carberry, Robson and Trott was the best? None, of course, made a compelling case, but with two Test centuries and one 50 in nine matches and an average of 32, I think Nick Compton has the edge. The mathematical solution for Cook is that when he comes across another opening partner who can emulate Compton’s record, he should toss aside his promiscuity and settle into a long term opening partnership.
In (at least) one respect, selecting an opening partner differs from the classic Optimum Stopping Problems: it is of course possible to reselect a previously discarded partner. That provides a very neat solution to Cook and England’s dilemma: call up Nick Compton.
Addendum: I am grateful to Seamus Hogan for this contribution:
@seamus_hogan: @chrisps01 Drawing on a paper by Weitzman (1979), you could add that ECB should try high-variance openers first!
I interpret this to mean that Alex Hales should be given a run in the Test team.
For more on the Optimum Stopping Problem, listen to the interview with Matt Parker in this episode of the BBC’s ‘More or Less‘.