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To fill a vacancy, follow the maths

17 December 2021

David Wilbourne considers how formulae can inform recruitment — and our view of God


REVISITING my university maths during lockdown, I was fascinated by the “Secretary Problem”. Imagine that ten candidates have applied for a PA post and are interviewed in random order. You have the option to appoint the candidate sitting nervously in front of you, or to move on to interview the next one. At what stage does the universal expectation that something better will come along have to give way to the realism that things are unlikely to get better, so best to plump for the reasonably impressive candidate quaking before you? Jump too soon, and you could miss out on a whole realm of talent; leave it too late, and the last might well be far worse than even the first.

Maths geeks puzzled over the problem, the Optimal Stopping Algorithm, for years, until, in 1961, the British statistician Dennis Lindley came up with a solution. To spare you some fast footwork that involves probability and integral calculus, Lindley’s solution concludes that your best chance of appointing the best candidate comes after you have interviewed 1/e (the inverse of Euler’s number) of the field — that is, 36.788 per cent. You simply treat the first 36.788 per cent as a sample, from which the best candidate sets the bar; you then appoint the next candidate who comes along who is better than that candidate, and dismiss the rest of the interviewees.

Lindley proved that his 37-per-cent-method algorithm was the best approach, provided that you will be completely happy with the best person and completely unhappy with anyone else. A simpler solution would be one that will give you someone as high up the ranking of candidates as possible, even if not necessarily the best.

In 2006, the psychologist Neil Bearden calculated that the best strategy for selecting the highest-ranking candidate, compared with the theoretically best candidate possible, was to appoint after interviewing √n candidates, where n is the total number who have applied. With ten candidates, the √n method will, on average, get you someone about 75 per cent perfect; with 100 candidates, about 90 per cent.

I HAVE spent a lot of my life interviewing candidates for school and church posts, and am frustrated by what seems a very imperfect system; so, why not follow the maths for once?

Let us take as an example a vacant parish, which runs an excellent advert in the Church Times, resulting in 16 applications; so, n=16, √n=4. When I was a diocesan director of ordinands, I assumed that all candidates who crossed my horizon were disturbed: 95 per cent disturbed by God, five per cent by God-knows-what. But assume that all candidates who have applied for this post are, in some sense, following a call, which may be shallow or deep. Whatever it is, they have taken the trouble to explore and imagine themselves into this particular context, feel that it might harness them, and have spent a day or three honing their application form.

So, take all the 16 application forms, shuffle them so that they are in random order, and interview the first four. Decide on which of the four was the best, select him/her as the benchmark, and then carry on interviewing and appoint the first person who is better.

The 19th-century clerical wit Sydney Smith, although baffled by how clergy became bishops, nevertheless felt that ordination at least gave him a ticket in the episcopal lottery. With the departure of the Archbishops’ appointments secretary/process champion, Caroline Boddington (News, 5 November, Comment, 19 November), imagine what fun you could have deploying the mathematical rigour of Bearden’s √n solution with a vacant diocese.

Say 100 priests, deciding that self-promotion was no longer counter-gospel, go Geordie and apply to be Bishop of Newcastle; so, n=100, √n=10. Shuffle their applications, interview the first ten in the pile, let the best set the benchmark, and then interview the rest until one greater than s/he comes along. Like Acts 1.26 with a maths degree.

I guess that it is wise to keep your cunning interview process, and particular interview order, secret, otherwise those in the sample batch might feel miffed that they were being used in setting the bar rather than fielding an appointee. It could be, however, that the best candidate was the best of the sample batch, and no better came along; so those who discover that they were among the original √n should not lose heart.

The possibility that the best candidate might turn down the post when offered could be addressed by slightly tweaking the system and making your sample batch a little less than √n, and then simply appointing the best candidate who says yes. Having lost his first wife, the 17th-century mathematician and astronomer Johannes Kepler decided to date 11 women as prospective wives. Four centuries ahead of his time, he proposed to candidate no. 4 (the best after √11), but she declined, and so he proposed to candidate no. 5, who accepted.

Rather than n being the number of candidates/dates considered for a post/matrimony, or number of houses/cars to be considered for purchase, many mathematicians equate n with the actual time that you are on the look-out, with the fair assumption that the number of candidates/dates/houses/cars will average out over that time.

THAT strikes me as a rather neat way of considering God’s search for the Messiah. Being God, almighty, omniscient, etc., he will be well acquainted with the Optimal Stopping Algorithm; and also, being God, the Lord of all time and up to speed with the doctrine of predestination, he will be well acquainted with the span that Homo sapiens has on earth.

So, by the principles of the Optimal Stopping Algorithm, the time between the appearance of the first Homo sapiens on earth and God’s choosing Jesus will be the square root of that span, King David setting the bench mark from an incredibly strong √n field, which included Abraham, Moses, Joshua, Ruth, Elijah, Elisha, Esther, Susannah, and John the Baptist.

Creationists have never really bettered the highly complex calculations of the much maligned 17th-century Archbishop Ussher, who computed that the world began at about nightfall on 23 October 4004 BC. Sir Isaac Newton came in at 4000 BC, and the much married Johannes Kepler at 3992 BC. So, taking 4000 as √n, the span on earth of Homo sapiens extrapolates to an encouraging 16 million years, which will see me out. Even using Lindley’s perfectionist n/e computation (and God, being God, is probably pro-perfectionist), if n/e = 4000, then n = 4000 multiplied by e = 10,873.127 years, with the end date of Homo sapiens at AD 6873.127, or just before noon on 15 February 6873.

It all depends on whether God, being God, has an unequivocal loyalty to the biblical span, or latterly has become more sympathetic to the span proposed by evolutionary biologists: that Homo sapiens, as we know and love him/her, first appeared on earth about 100,000 years ago. In that case, √n or n/e works out at 97,979, with Homo sapiens span either equal to a mind-blowing 959,984,441 years, or a healthy 266,334.53 (recurring).

I am not sure whether Mother Julian of Norwich was acquainted with the Optimal Stopping Algorithm, but her “All shall be well, all shall be well, all manner of things shall be well” strikes me as spot on. In any case, this quasi-mathematical proof that we’ll be around for some time yet should allow humankind to relax a little this Christmas — an antidote to the twin heresy that emerges at every crisis, namely, that (a) we have caused it, and (b) we can fix it.

Of the Father’s love begotten,
Ere the worlds began to be,
He is Alpha and Omega,
He the source, the ending He,
Of the things that are and have been,
And that future years shall see,
Evermore and evermore.

The Rt Revd David Wilbourne is an Assistant Bishop in the diocese of York. His book
Just John: The authorised biography of John Habgood, Archbishop of York, 1983-1995 (Books, 1 May 2020; Podcast, 15 May 2020) is published by SPCK at £19.99 (Church Times Bookshop £18); 978-0-28105-828-0.

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