Thinking cricket, fast and slow (and sadistically)

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Would you prefer a batsman who scores 50 every innings, or one who scores 100 half the time and 0 the rest? The ‘expected value’ (batting average) each player offers their team is the same. The decision appears to rest on whether consistency or the potential for a match-defining innings contributes more to your team.

Jarrod Kimber and Andy Zaltzman discussed this conundrum on a recent episode of the ‘Cricket Sadist Hour,’ acknowledging @analytics_jonas as the source of the riddle. The discussion suggests that there is an answer to the question, if sufficient mathematical heft is applied to it. Indeed there may be, but how often in the real world of cricket, or any other occupation, is a difficult decision subject to rational analysis and resolution?

Suprisingly rarely, according to Daniel Kahneman, Nobel prize winner in Economics and author of ‘Thinking, Fast and Slow’. Kahneman challenged the orthodoxy in the social sciences that, “people are generally rational and their thinking is normally sound.” Kahneman and others put judgement to the test by running experiments. What they consistently found were “heuristics and biases.. the simplifying shortcuts of intuitive thinking.”

Following Kahneman’s lead, I won’t try the tricky statistical enterprise of devising an answer to the question. Instead, I will draw on his book to understand why and in which conditions team-mates, fans or even selectors may choose one of these theoretical batsmen over the other.

Question substitution

The numbers (50, 50, 50, 50… and 0, 100, 0, 0, 100, 100…etc) will play a part in these answers, but to begin with, let’s acknowledge that for most of us, much of the time, decisions are not based on numerical analysis, which Kahneman has shown is too effortful and our brains tend to shun.

This is the essence of intuitive heuristics: when faced with a difficult question, we often answer an easier one instead, usually without noticing the substitution.” Kahneman, p12

Instead of attempting the difficult question (which player, based on their pattern of scoring, and the complex structure of a cricket match, will over time contribute more to the team?), we substitute an easier one. In this case, it might be: “Who scored more runs when I last saw them on TV? or, which player has the purer cover-drive? or, which player is being lauded by the ex-player whose opinion I respect?” Kahneman gives examples of how choices are made over political candidates, investment opportunities and charitable donations, but could easily have added cricket team selection to his list.

I will now engage more directly with @analytics_jonas’s original question but introduce some small aspects of context in order to understand how each of the batsmen might, in different circumstances be our preference.

Experience and memory

Our experience of the two batsmen, aggregated over time, will be very similar (if not identical) in terms of the number of runs we see them score. If decision-making were based on lived experience, we may struggle choosing between the consistent batsman and the success/fail player.

However, studies carried out by Kahneman and other psychologists have shown that decisions are not driven by the lived experience, but by the retained memory of the experience.

The memory that the remembering self keeps.. is a representative moment strongly influenced by the peak and the end. (Kahneman, p383)

Taking the ‘peak’ effect first, the success/fail player will clearly have provided greater peaks to plant in the memory than the steady accumulator of half-centuries. The second aspect, the recency effect works equally well for each as around half the time each of the batsmen will have recorded the higher score in their last innings. Combining the ‘peak effect’ and the ‘end effect’, the success/fail player will have created the more favourable impression.

Prospect theory

Put now in the position of selector opting for one of our two players for the next match, rationally there is nothing to choose between them, as over two innings they are, on average, expected to contribute the same number of runs.

Daniel Bernoulli, Swiss scientist of the eighteenth century, refined this understanding by observing that most people dislike risk. So while the expected value of the two batsmen is the same, there is a premium on the player guaranteed to score 50 each innings (or a discount on his alter ego). For the selector this may make sense. There may be only one innings in the match, so better bank 50 runs than risk 0. Even with two innings, we cannot be certain that our success/fail player won’t make two ducks, before bouncing back to form in a later game.

Kahneman added further nuance to Bernoulli’s theory from the empirically observed standpoint that people’s decisions are context dependent. The pattern found, and stated in Prospect theory, is that the decision-maker whose prospects are poor is more likely to gamble. On the other hand, when faced with a situation that offers only an upside, decision-makers tend to be averse to risk and seize the sure thing.

One-nil down in the series with one match to play, the opportunity of having the century-maker is more appealing. One-up, at the same stage of the series, and the cautious choice is preferred. As explored in a previous post on declaration decisions, when a skipper is on top in a Test match, losing feels worse than winning feels good.

It is important to emphasise that Kahneman’s work is descriptive of the decisions people (Test selectors?) make and not prescriptive of the decisions they should make. Indeed, Kahneman emphasises that these systematic biases, based in this case on decision makers’ reference points (i.e. their context), can create unfavourable outcomes, with good opportunities passed over and poor choices pursued.

Thinking slow

While we wait for advanced cricket analytics to tell us whether absolute consistency trumps its opposite, selectors will have to pick players, pundits will continue to push the interests of their proteges and we followers of the game will invest great hopes in our favourites. By making some effort to examine our preferences we might reveal biases and so move towards more informed selections of, lobbying for and championing of players.

Batting statistics adjusted for fielding mistakes

As Karun Nair surged to a triple-hundred in his third Test innings, commentators dared the Indian selectors to drop the young batsmen for the next Test, when the three more senior players, whose injuries had opened the way for his debut, will have returned to fitness. That’s a recurring selection dilemma – form versus seniority; promise against proven ability. Nair’s situation raises another dilemma, one that I find even more interesting, but suspect selectors do not.

Measurement of the impact of fielders has become topical. It has found official recognition in Cricket Australia’s publication of a fielding index. To appreciate the broader scope of the subject and its potential there’s no better source than Jarrod Kimber’s ESPNCricinfo post, Why doesn’t cricket have proper metrics for fielding?

Now that fielding performance is being subjected to more intense analytical scrutiny, it follows that its impact on batting and bowling performance also needs to be understood. This post presents some options for adjusting batting statistics to take account of certain aspects of fielding performance, drawing on data collected from the India v England series and reported in the Declaration Game post, A series of missed opportunities.

The end of series batting tables showed the dominance of Virat Kohli, the impact of Nair’s mammoth innings and the continuing prolific run-scoring of Root, Pujara, Bairstow and Vijay. But in a series of 49 dropped catches and missed stumpings, how dependent were the batsmen on the competence of the fielders? I have assessed the impact of missed chances on the output of the 18 players who scored more than 100 runs across the series.

42 missed chances were distributed across these 18 batsmen. No distinction is made between chances of different levels of difficulty. All chances that went to hand (or body), or flew between fielders stationed close together are counted, but not those that looped just out of reach or through areas where one might have expected captains to have placed fielders. Also excluded are missed run outs and missed opportunities relating to umpiring decisions or to the operation of the Decision Review System. 

The bar to the far right of the chart represents Alastair Cook who benefited from the highest number of misses: six; Ben Stokes is one place to his left. 

More interesting than a count of drops is how the missed chances impacted on batting performance. A measure of this is the number of runs scored, had each of the innings ended when the batsman gave their first chance. The full height of the column in the graph below shows the total number of runs scored by the player in the series. The filled blue part of the column shows the number of ‘chanceless’ runs accumulated by each batsman; runs scored after a missed chance are depicted by an unfilled (white) area.

On this measure, Root supplants Kohli as the most prolific batsman in the series, with the Indian captain falling to third place behind Pujara. Nair and Jennings have the highest proportion of their runs bitten off by this metric. At the other extreme, Rahul and Patel were unaffected having not benefited from any missed chances.

With four comfortable victories, India’s batsmen had fewer innings than England’s. Standardisation can be achieved by converting the measure into a batting average – the ‘chanceless’ average – by dividing by the number of dismissals.

Patel, Kohli and Rahul are the three players who maintain ‘chanceless’ averages (orange columns) above 50. Kohli’s average when only chanceless runs are counted falls 43%. But it is Nair with the steepest drop from a conventional average of 160 to just 17.

By including only the runs scored before giving the first chance of an innings, this measure has the drawback of giving no credit for runs that played a part in the match result. I have calculated a second alternative measure of batting performance: the batting average per chance (orange columns). Total runs scored are divided by dismissals plus missed chances.

Kohli, the dominant figure of the series with the bat, returns to the top of the list, followed by Patel. Nair, showing how he made England pay for their errors in the 5th Test, rises to third-place. Cook is near the bottom of the list, having managed just 23 runs per chance.

Adjusting measures of batting performance in this way offers some insights: it shows how certain players’ success relied upon the opposition making fielding errors, while others enjoyed no good fortune of that kind at all, and some failed to capitalise on the good luck that came their way. In this series, there is also a pronounced levelling out of individual batting performance when chances given are taken into account. The range for batting average of the players in this sample fell from a factor of 11 separating top from bottom with conventional batting average, to eight using the chanceless batting average. This type of analysis may, with a far larger sample, start to factor out elements of luck in batting performance measures. 

A single Test match series does provide far too small a sample for drawing statistically robust conclusions. Yet, it is exactly the sample most pressing on the attentions of international team selectors, particularly when assessing the contribution of players new to the Test arena. My contention is that selectors and other observers are better served by a batting measure that attempts to control for the varied dose of luck experienced by players than the conventional and crude batting average. 

 

 

Future stars of the Under 19 World Cup

Sarfaraz Khan, Gidron Pope, Alzarri Joseph, Avesh Khan, Jack Burnham. Names that have earned recognition for performances at the Under 19 World Cup this month. But will they, and their peers at this tournament, be the successors to Brendon McCullum, Mitch Johnson, MS Dhoni and Kumar Sangakkara in the wider consciousness of world cricket?

An analysis of previous Under 19 World Cup participants will not tell us specifically whether, say, Keemo Paul will become better known for his exploits as a senior than junior international cricketer. It will, though, cast some light on the development of international cricketers.

For every member of a full nation squad at the Under 19 World Cups of 2000, 2002, 2004 and 2006, I have recorded the highest level of senior cricket attained in their career. The ten year elapse since the most recent tournament reviewed makes it unlikely that any of the 555 players will reach a new peak. Unlikely, but not impossible: Stephen Cook, graduate of the 2002 tournament, made his Test debut this year.

Four levels of senior cricket have been identified, in ascending order: i) professional limited overs (List A or national T20 tournament), ii) first class, iii) international limited overs (ODI or T20) and iv) Test. With very few exceptions, this grading represents progress in a player’s career – ie he will have played the form of cricket considered lower than the level I have taken to be the highest level he attained.

Within each level, there is a broad range of attainment, measured by appearances. For example, from the 2000 tournament, grouped together at the first class level are Mark Wallace (England) with 249 appearances and Gareth Irwin (New Zealand) who played a single first class match in 2002/03. (Irwin is one of the exceptions to my hierarchy, as he did not appear in professional limited overs fixtures.) It might be fairer, therefore, to think of each group as containing players who have passed a common threshold, rather than attaining the same level.

The summary analysis of the 555 players shows that 45% have gone on to play international cricket (not all with the nation they represented at the Under 19 age group). 5% have not played any professional senior cricket.

I would have hypothesised that the conversion rate of under 19 internationals to senior internationals would have increased over this period; this being a reflection of the more structured approach taken towards the development of youth cricketers. The results don’t support that hypothesis: the proportion of under 19 players going on to play international cricket has varied: 2000 – 48%; 2002 – 40%; 2004 – 46%; 2006 – 42%.

There are some stark country-by-country differences. The youngsters of Bangladesh and Zimbabwe have had a higher likelihood of becoming full internationals, two-thirds in the latter case – perhaps reflecting that selection in those countries is from a smaller pool of players. On the other hand, barely one-quarter of those who have appeared at under 19 World Cups for England have played for the senior team. Unsurprisingly, England, with its 18 first class counties has had no players fail to reach the senior professional game – nor did Pakistan, South Africa and India.

I also looked at whether performance at the under 19 World Cup was a good predictor of future prospects by narrowing the analysis to the top run scorer and wicket taker for each of the ten nations at the four tournaments. 50 of the 81 players in this sample (64%) have played senior international cricket, compared to 55% of the total population, which is less of an increase than I would have expected. The outlier is New Zealand’s Jonathan McNamee, who was their top scorer at the 2000 tournament, but has no senior professional record.

At the team level, success in the under 19 tournament has not been associated with having teams choc full of future international cricketers. Looking at the eight finalists in these four tournaments, 43% (Test: 31%; Limited over: 12%) of their squad members went on to play senior international cricket, compared with 45% (35%; 10%) of the total.

I was also interested in understanding the proportion of players who reach Test level who have been participants at the junior World Cup. My method provides an estimate, not a precise figure. I extracted the number of Test debutants for each nation in the period 2002-2012. The chart below shows the number of players in the four under 19 World Cups who went on to play Test cricket and the proportion they are of the total debutants in the eleven year period. It provides a rough, rather than definitive, picture as some participants in those four tournaments had debuts before and after the eleven year period; and some players from the 2008 and 2010 tournaments probably had debuts during the period.

Approximate though this analysis is, it does show that England and Zimbabwe are outliers. Around half the Test debutants from the other eight nations had played in the four under 19 World Cups. For England, that figure was below one-quarter. At the other extreme, those players accounted for over 90% of Zimbabwe’s Test debutants.

There are positive and negative connotations to these two extremes. England’s position could be evidence that it performs poorly at identifying future talent, or that its junior cricketers mature at a later age. It could be a strength that international selection remains open to players emerging from outside of the elite juniors. England may have the resources to invest in a broader base of juniors, making precise selection at 19 difficult. Experience of international cricket as a teenager may be a poor one, having a negative impact on English juniors, or their development is interrupted by injury. The opposite to each of these arguments can be made for Zimbabwe. The data cannot help us with this key point. I would be interested in the views of readers.

In conclusion, the data analysis shows:

• Unless from England or India, an Under 19 World Cup participant has close to, or better than, an evens chance of senior international cricket.
• The first class game should definitely be within reach – if not already attained.
• Having a strong tournament (relative to your teammates), desirable in its own right, boosts by a modest amount a player’s likelihood of moving onto senior international cricket.
• At Test level, there is a heavy dependence on Under 19 World Cup graduates, with around one-half of the debutants in the years following tournaments having participated in the junior World Cup.
• England and Zimbabwe are, respectively, less and more likely to choose Test debutants from Under 19 World Cup players.

The curse of Cook

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Six opening partners tried and rejected in three years – a turnover of one opening batsman per Test match season. The inability to find a player to rise to the challenge of opening the innings alongside Alastair Cook is the most prominent of the selection problems besetting England. This post does not venture a solution (although I have provided a mathematical response), it looks at the impact on those six selected, then rejected batsmen.

The impact of playing with Cook and then being dropped is assessed in a narrow, statistical fashion. The first graph shows the ten innings batting average in first class matches before and after each player’s brief career as a Test opener. (NB Adam Lyth’s post-Cook average is based on the six first-class innings he has played to date)

Across the group, there is a reduction in batting average of 40 runs per completed innings (56%). Joe Root has the sharpest reduction. He and Trott are the only members of the group who played Test cricket before opening with Cook; and Root is the only member of the group who played Test cricket after opening with Cook.

Three of the players (Compton, Root, Robson) may have harboured hopes that their Test opening careers would continue when they returned to first class cricket. Compton, for example, played four innings (including a century and a fifty) before his supplanting by Joe Root was made clear by the selection of an England side for a warm-up match. Selecting ten innings from his return to the Somerset side or from his official relegation from the England side makes little difference to this ten innings average (48.2 v 47.6).

We should not be surprised that players’ first class averages drop after a tough period as rookie Test match openers. They had been picked as form players – all six had short-term averages exceeding their career average when brought into the team – and their strong form had been interrupted by the stiffer challenge of Test cricket. In Trott’s case, his return to first-class cricket involved more than just re-finding form with the bat, but psychological health, too. The fall away in their performance, however, is noteworthy for its abruptness and consistency across the group.

To test whether it is a short-term effect, I have also compared their batting average for the last full season of first-class cricket before their selection as Cook’s partner and the first full season of first-class cricket following their demotion from the captain’s sidekick. In all cases except Root’s, the seasons assessed were England county seasons.

In this analysis the average fall in batting performance is less severe and is less consistent across the group. Root, the only player to remain in the Test team, maintained his pre-selection season average and Carberry’s varied downwards by fewer than five runs per completed innings.

All of the six players struggled for most of the innings they opened alongside their captain in Tests matches. Once out of the team (or in Root’s case, batting lower in the order), they were unable to regain their earlier productivity.

Alex Hales is strongly favoured to be Cook’s next opening partner. His current 10 innings first-class batting average is 36.0 – lower than all of his predecessors (although Hales may have further innings in the County Championship and in the UAE to improve on this before the Tests against Pakistan). Hales will, of course, be aiming to repel the curse of Cook that leaves batsmen under-performing when dropping back into county cricket. The surest way of doing this is by scoring so many runs for England that he stays in the team, opening alongside the captain.

Quick single: Alternating keepers

Jonny Bairstow has been scoring runs at a gallop for Yorkshire. Meanwhile, the England top order has regularly given Test opponents a three wicket headstart. Bairstow’s call-up for the third Test (along with some shuffling of the order to accommodate him) aims to channel his strong form into a stiffening of the England line-up for the rest of the Ashes series.

Bairstow is also Yorkshire’s first choice wicket-keeper. He joins Jos Buttler in the team. The England keeper has made just 58 runs in four innings so far in the series. Neither player demands inclusion on the basis of keeping ability, and so needs regular runs to justify selection. It might seem that Bairstow’s return puts pressure on Buttler. It could be seen quite differently, though. A trip back to international football in the 1970s and 1980s will be used to illustrate.

Ron Greenwood became the England football manager in 1977. Two competitive matches later and England had failed to qualify for its second consecutive World Cup Finals. In Brooking and Keegan, the manager had two high class players. The team’s other stars were goalkeepers: Peter Shilton and Ray Clemence. Greenwood adopted a policy, from the qualifying campaign for the 1980 European Championships, until the final warm-ups ahead of the 1982 World Cup, of alternating Shilton and Clemence – as well as giving Joe Corrigan an occasional cap. Greenwood’s rationale was to ensure both players maintained international experience, which he could achieve without weakening the team. There was some recent historical justification for this unusual selection policy. At the 1970 World Cup Finals, England’s first choice ‘keeper, Gordon Banks fell ill before the quarter-final with West Germany. His replacement, Peter Bonetti, had just a handful of caps. His inexperience was exposed by the West Germans in their late three goal rally that eliminated England.

The England cricket team, under Peter Moores and now Trevor Bayliss are already coming close to emulating one feature of Greenwood’s approach to player selection as England manager. For a few matches in 1977, Greenwood, seeking a cohesive team, picked six players from the League Champions Liverpool as well as Kevin Keegan, who had recently moved from Liverpool to play in Germany. Six of the England squad that toured West Indies earlier this year were from the county champions, Yorkshire, which is the source of four players in the current Test squad. 

But it’s Greenwood’s more idiosyncratic selection policy of alternating keepers that could provide an inspiration for the England cricket team. The aim would not be to have two players with deep international experience capable of keeping wicket for England in the next World Cup. The objective would be to have at least one wicket-keeper, fresh and injury-free for that tournament. Buttler is the first choice keeper for Tests, ODIs and T20s. By the end of the next English season, he could be called on to play 17 Tests, over 20 ODIs, sundry T20 internationals and a World T20 tournament. For one player to fulfil the lynchpin role of wicket-keeper for the entirety of the itinerary, particularly a player whose batting is key to the team in limited overs matches, presents a real risk of burn-out or injury through physical stress. 

Having Bairstow in the squad offers the option of resting Buttler, if not from alternate matches, then regularly at the tail-end of series. In limited overs matches, Buttler and Bairstow could swap roles, allowing the Lancashire keeper to play as a specialist batsman. If England can farm the use of these two versatile cricketers their careers could be prolonged and their effectiveness when selected enhanced. Buttler, in these circumstances, if well managed, would not see Bairstow’s elevation to the squad as a threat, but an opportunity to become an even stronger all-round wicket-keeper batsman. 

What this scenario does depend upon, of course, is the new man – Bairstow – scoring enough runs, at the right times, to justify his retention. If his old technical flaws return in the face of the Australia attack, the plan should not be implemented. That does not necessarily mean Buttler must tackle England’s demanding fixture list unsupported. There’s another player, already a squad member, capable of dove-tailing with the number one keeper’s need for rest and relief. Sam Billings may turn out to be more than Joe Corrigan was to Shilton and Clemence. 

Quick single: Jonny Bairstow form and flaws

Picking players in form is one of the national selectors’ least important duties. Far more significant to identify the player with the talent and technique to thrive at international level, than to pluck the name from the top of the county (state, province, etc) averages. 

Sometimes, though, with a cricketer who might, just might, have the right stuff for the international game, the timing of their selection can have a long-lasting impact. Jonny Bairstow, in mid-summer 2015, is in the form of his life. He has a three-figure average in the County Championship with hundreds scored in three of his last fiur innings. In the last but one match, he was joined in the middle by Tim Bresnan, with the score 191-6. The pair added 366 together (the third highest partnership for the seventh wicket in first class history), with Bairstow 219* at the declaration. The week before, replacing Jos Butler in the England limited overs squad, his innings of 83* won the deciding match in the series against New Zealand. 

Bairstow isn’t in the England squad for the first Ashes Test and is playing again for Yorkshire this week at Edgbaston. 

The highlight of his 14 Test matches came at Lord’s in 2012 with two belligerent and brave innings against South Africa, which took England close to a victory against the team that replaced them as the number 1 ranked Test team. The stronger impression created by Bairstow’s Test batting career to date is of a player hampered by technical flaws. Initially, during his debut series against the West Indies, it was his ability to play the short-ball that concerned. Dismissals (bowled, lbw and caught off a leading edge) playing across the line of full, straight deliveries became the focus of doubts about Bairstow’s suitability for Test cricket. 28 was Bairstow’s highest score in his last 8 Test innings (preceded by 64), the most recent of which was in the final Test of the 2013/14 Ashes whitewash. 

How should the England selectors weigh up Jonny Bairstow’s current run of good form with the evidence of his early experiences of Test cricket? In one sense, the existence of clear flaws in his batting in 2012-14 clarifies the matter as the selectors should be looking for assurance that those issues have been resolved. That, however, assumes that problems exposed at Test level would be apparent in the county game, where the bowling subjects techniques to less strict examination. 

That Bairstow favoured the legside was well known when he made his England debut and is far from a unique preference – witness Cook, Trott, etc. But was he falling to straight deliveries in Test cricket because, starved of balls directed at his pads, Bairstow was forced to find runs somewhere? Playing for Yorkshire, Bairstow may have defended those balls safely, knowing that juicier morsels would arrive soon. 

It seems probable that, facing Australia’s strong and deep pace bowling attack, England’s middle order will need reinforcement with new players, by the second half of the Ashes series. If Jonny Bairstow remains in the form he has shown for the past six weeks, his case will be persuasive. It will come though with some unease about frailties that division one county championship attacks lack the expertise to probe. The selectors will, I believe, have to accept that we won’t know if Bairstow is ready for the rigours of Test cricket, without trying him out there again. Weighing on their minds may be a similar calculation, albeit featuring different variables, that was made when Jonathan Trott was reintroduced to Test cricket in April. 

Quick single: the best is the enemy of the good Bishoo

Devendra Bishoo flights the ball down the line of Steve Smith’s off-stump. Smith advances to drive back past the bowler. The ball pitches feet in front of where Smith was expecting it to, grips and turns past his wafted bat. The ball careers into Ramdin’s gloves and the keeper completes the stumping before Smith can even turn to try to regain ground.

Bishoo floats a ball under Brad Haddin’s nose, as it descends it swerves to his legside. Haddin is waiting to lean on the ball, killing it dead. But the ball zips past his outside edge and clips the top of off stump.

Two of Bishoo’s wickets in the first Test at Roseau – archetypal leg-spin dismissals.

Bishoo is a very typical leg-spinner. He’s short, slight, loose of limb and can look vulnerable alongside his towering teammates and heavily padded, big bat waving opponents. He turns the ball just the one way and even when bowling well, as he has against Australia, on a helpful pitch, he offers long-hops, full tosses and batsmen are not intimated into staying in their crease. He is a risk-reward bowler: 4 for 177 in the first innings of his previous Test against England.

Just over twenty years ago, Shane Warne’s career took off. With it, we hoped would come a revival of leg-spin bowling – certainly in those parts of the cricket globe where it was dormant. In many respects, Warne managed that feat. Leg-spin bowling was attempted by thousands of children who may only have wanted to bowl fast, if play cricket at all. It was a positive, aggressive choice.

Yet all the imitators have failed to mature into emulators of Warne’s achievements. For years after his retirement, Australia struggled to find a spin bowler, let alone a leg-spin bowler. Was Warne a case of the best being the enemy of the good? Warne wasn’t an archetypal leg-spin bowler. He combined the control of an off-spin bowler (Test career economy rate of 2.6 rpo) with the big and unpredictable turn, flight and pace of a leggie. Maybe he set the bar too high.

Bishoo is a bowler that leg-spinners at all levels of the game will identify with: mixing unplayable balls with deliveries that batsmen can stroke at will to the boundary. He seems to have an equanimity about being bashed for four or six that is an essential part of the leggies’ make-up. Not for them the standing with hands on hips, before ordering a close fielder out to sweep the boundary that’s just been breached. Successful leggies simply challenge the batsman again, this time with a little more flight, or a slight change to the angle of rotation.

Legs-spin bowler, Adil Rashid, sits on the verges of the England Test team. Another typical leggie risk-reward bowler – although mitigated a little by his batting. Can the England selectors find space for this attacking option in the team? Coming off a Test series with New Zealand, when runs were conceded at over four per over, I think it unlikely. All the more reason to appreciate Bishoo – certainly on days like today when he foxed the best of Australia’s batting, but also on days when the pitch doesn’t help, the outside edge isn’t threatened and his major mode of dismissal is the other typical of leg-spinners: caught at deep midwicket.

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. 

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For more on the Optimum Stopping Problem, listen to the interview with Matt Parker in this episode of the BBC’s ‘More or Less‘. 

Quick single: reserve forces

One of the conceits of England’s last period of ascendancy in Test cricket (2010-12) was that so rich were its resources of fast bowlers that its reserve corps would be first choices for any other Test team. Finn, Onions, Tremlett, with Dernbach and Meaker still in the phase of getting ODI experience.

This difficult to falsify, but hard to validate argument was briefly put to some sort of test. Stuart Broad and James Anderson were rested/given injury relief for the third Test of a series against West Indies, that England led 2-0, at Edgbaston in June 2012. The match was ruined by rain. Batting first the West Indies scored their highest total of the series, with Dinesh Ramdin scoring a ton at seven and Tino Best a rollicking 95. England’s understudies underwhelmed.

The following year, over a sequence of six Tests against Australia, England brought in Woakes, Tremlett, Stokes and Rankin to support Broad and Anderson. Stokes had a successful game with the ball at Sydney, but none of the others looked deserving of a run in the England side, let alone their major rivals’ teams.

The point of recounting this recent history is to recognise the risk of lauding the strength in depth of a country’s cricket players when, by definition, those reserve forces haven’t had the chance to prove themselves at Test level. With that caution in mind, consider New Zealand.

As with England, their attack is led by a complementary pair: Tim Southee and Trent Boult. Chief support has been provided by Neil Wagner. At Lord’s, Matt Henry made a promising debut, weeks after his surprise World Cup Final appearance. Doug Bracewell, who took nine wickets in New Zealand’s first defeat of Australia in Australia in 26 years, is also in the touring party. Arriving for the short-form cricket should have been the Black Caps’ fastest bowlers: Adam Milne and Mitch McGlenaghan; Milne has had to withdraw due to injury. Fourth seamer duties can fall to all-rounders Corey Anderson, James Neesham and Grant Elliot. In the background, leading wicket taker in the Plunket Shield last season was 20 year old Jacob Duffy.

The absence of Southee or Boult would be a major hindrance, with perhaps only Bracewell offering anything close to their control and new ball penetration. But if their form and fitness holds, as established internationals in their mid-twenties, they have plenty of time to apprentice one or two of the younger bowlers and build a succession.

Three years on from Tino Best’s assault on England’s fast bowling reserve forces, that’s something England may only now just have under way with Ben Stokes and Mark Wood.

Quick single: Head Coach – without preconditions

Those of us wearied and annoyed by the ECB’s management of its limited pool of international standard male cricketers, enjoyed a little spiteful satisfaction last week. It seemed that Andrew Strauss’s decision to exclude Kevin Pietersen from England selection for the foreseeable future – in all likelihood ending his Test career – may deter the better qualified candidates for the Head Coach role. Pre-conditions, constraining who could and couldn’t be selected, made the role unattractive.

Today, however, it seems that Yorkshire Coach, Jason Gillespie, may want the role after all. Does that make the Australian weak-willed or status-hungry?

I am pretty sure neither is the case. It may be that Gillespie prefers the job with the KP question resolved without his involvement and so no comeback on him. More likely, I would argue, coaches with international aspirations are pragmatic beasts.

The teams they coach are only intermittently at full strength. Injuries, squad rotation, the lure of the T20 tournaments that clash with international commitments, or even retirement to earn more playing in another country’s domestic competition all have to be worked around.

Gillespie will also be very aware of the circumstances his peers, should he be appointed, work under. Duncan Fletcher began his stint with India with four batting greats, three of whose careers were in clear decline, installed in the team, holding up the development of the next generation of batsmen. Fletcher managed that succession to their timescale, rather than his.

Fletcher made it into the post-Tendulkar era, surely expecting to hold greater authority, but soon found the voluble Ravi Shastri appointed Team Director for most of his last year in the job. Fletcher would not have lasted long insisting on coaching without preconditions.

In the West Indies, Phil Simmons inherits a ‘West Indies first’ policy. On the one hand, it’s the strong backing that an international coach would want; on the other it may restrict flexibility the coach could take advantage of when the players pursuing T20 contracts make themselves available.

Being told to manage without KP is a far simpler task than that facing the coaches of South Africa, Zimbabwe and Sri Lanka, each of whom will confront preconditions set by politicians, not just cricket administrators. And it’s unlikely those preconditions will be as overt and as easy to abide by as ‘just don’t pick Kevin.’

Even Gillespie’s former teammate, Darren Lehmann, took on the Australia coaching role with limited room for manoeuvre. The Ashes squad had already been selected; the first-choice opener had been sent to the ‘A’ team as a disciplinary measure. Lehmann waited, worked with the players he was given and began to shape the team culture.

Strauss’s decision to exclude Pietersen will not have narrowed the field of potential head coaches. Whoever gets the role will understand that it’s not the sort of job that comes without preconditions.