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Norman, John M. & Clarke, Stephen R. (2006). Dynamic programming in : optimizing order for a sticky . Journal of the Operational Research Society. 458(2006) : 1678-1682.

Copyright © 2006 Palgrave Macmillan Ltd.

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Dynamic programming in cricket: optimizing batting order for a sticky wicket

JM Norman11 and SR Clarke2 1 University of Sheffield, Sheffield, UK; and 2Swinburne University, Hawthorn, Victoria, Australia

In cricket, a rain-affected pitch can make batting more difficult than normal. Several other conditions such as poor light or an initially lively pitch, may also result in difficulties for the batsmen. In this note, we refer to all of them as 'sticky '. On sticky wickets, lower order batsmen are often sent into 'hold the fort' until conditions improve. In this paper, a stochastic dynamic programming model is used to examine the appropriateness of this policy. The model suggests that the tactic is often optimal when the sticky wicket persists until the end of the day's play, but not often when the sticky wicket is transitory. In some circumstances, it is worthwhile, on a normal wicket near the end of the day, to send in a lower order batsman to hold the fort (a night watchman): when the wicket is sticky, this tactic is even more worthwhile.

Keywords: sports; cricket; dynamic programming; strategies

Introduction In this paper, we examine the correctness of the policy of sending in a lower order batsman on a sticky wicket. In cricket, the difficulty of batting depends to a large extent on the state of the pitch. This can change during the course of the match, particularly if it is affected by rain. A rain-affected or 'sticky' The model wicket may be very difficult to bat on until it has dried out. It may We adopt the model formulated in Clarke and Norman (2003), take several overs to dry out and return to normal batting which is described again here, in slightly adapted form. For condition. When his side was faced with a sticky wicket, Sir Don simplicity, we assume that decisions are made at the start of each Bradman (Bradman, 1997) would adopt the policy of 'sending in over. Again, for simplicity, we consider two types of batsman. In tail-end batsmen to hold the fort until the wicket improved'. While practice, teams usually contain six or seven recognized batsmen, this tactic reduces the exposure of the good batsmen to the poor selected primarily for their batting skills, and four or five lower conditions, it increases the chance they will be left at the end of order batsman (or duffers) generally selected for their the without a batting partner. and with less reliable batting ability. In the model, recognized While covered wickets now reduce the frequency of rain- batsmen gradually improve their expected performance as they affected pitches, there are still many other situations in today's gain experience of the conditions (play themselves in), so their cricket that produce variable batting conditions. The first session rate of increases and their chance of dismissal decreases of a test match often produces a lively green wicket that is in the early part of their innings. In the model, duffers do not expected to become less dangerous after the first hour or so. At improve, and their expected performance depends only on the start of any day, overcast weather might provide ideal whether or not the pitch is sticky. Each batsman has an assumed conditions for . At any time cloud cover might rating x which is an indicator of expected performance and may result in poor light. The opposing team's main strike fast change only at the end of an over, depending on the pitch and the bowler(s) may become less dangerous as tiredness sets in, and number of overs batted. During the over, a batsman with a rating ultimately must be replaced by possibly more benign bowlers. In of x scores r (x) runs but may be dismissed (on the last ball of such cases, the use of lower order batsman to protect the the over) with probability p(x). At the start of the next over, his recognized batsmen from the more dangerous conditions may be rating is x '. Note that r (x) must be even to avoid changes of end advantageous to the team. In the remainder of this paper, the term during an over. r (.) could also be thought of as the expected 'sticky wicket' is used to refer to any situation where the playing number of runs, for a distribution of runs that contained only even conditions for the batting side are temporarily more difficult than values. usual.

1Correspondence: JM Norman, The Management School, University of Sheffield, 9 Mappin Street, Sheffield S1 4DT, UK. E-mail: [email protected]

Table 1 State transitions, scoring rates and dismissal probabilities as a function of batsmen's ratings Type of Batsman Rating at start of Rating at start of Runs scored Probability of dismissal p ( x ) over next over x x > r ( x ) Normal wicket Sticky wicket Duffer (scenario 1) 0 0 2 0.2 0.2 Duffer (scenario 2) 0 0 2 0.2 0.3 Duffer (scenario 3) 0 0 2 0.2 0.4 Recognized batsman, who 1 2 0 0.2 0.3 plays himself in when not on 2 3 0 0.167 0.25 strike as well as when he is 3 4 2 0.125 0.1875 on strike. It takes him four 4 5 4 0.100 0.15 overs to play himself in 5 5 4 0.083 0.125

The numerical data assumed are given in Table 1. A duffer Then b, x, y) always has a rating of zero, while a recognized batsman has a f n (a, rating of 1 initially, rising by one each over (whether he is on = max[A: send out a recognized batsman strike or not) up to a maximum of 5 as he plays himself in. On a if a dismissal occurs (provided a ^ 1) B: send sticky wicket, all dismissal probabilities for a recognized batsman out a duffer (tail-ender) are 50% greater than those for a normal wicket. For a duffer, we consider three scenarios: in the first, the dismissal probability on a if a dismissal occurs (provided b ^ 1)] sticky wicket is the same as for a normal wicket; in the second, it where is 50% greater than on a normal wicket; and in the third, it is A = r(x) + [1 - p(x)] fn-1( a , b , y', x') + p(x) fn- double that for a normal wicket. The r(.) and p(.) values for a normal wicket imply an 1 (a - 1, b, y', 1) expected score for a duffer of 10 and for a recognized batsman of B = r(x) + [1 - p(x)]fn-1 (a, b, y', x') + p(x) nearly 40. A team of six recognized batsmen and five duffers f n-1(a, b - 1, y', 0) would generate an average total score during an innings of about

270 runs, a reasonable figure for test and county matches. Note f n (0, 0, x, y) = r (x) + [(1 - p(x)] fn-1(0, 0, y', x') that scores in the second innings are usually lower than those in the first innings, and are also the more likely to be rain affected. When n = 0, the wicket returns to normal, and we have a sup- Consider the of the side batting on a sticky wicket, at plementary dynamic programming problem concerned with the the start of an over, with n overs left until it is expected that the optimal strategy and maximum expected score for a normal pitch. wicket will return to normal. The batsman on strike has a rating of This dynamic programme is similar to the above formulation, but x, and the batsman at the other end has a rating of y. If there is no with no limit on the number of overs left, since we are concerned dismissal, at the start of the next over their respective ratings will with test cricket. While the generally optimal strategy to this be x' and y', and the strike will rotate to the other batsman. If a problem is well known (always send in the recognized batsmen dismissal occurs, the captain must take a decision regarding the first), here we need the limiting values of the objective function to incoming batsman, who will not be on strike next over, whether use as the starting values f 0(a, b, x, y) for the sticky wicket he should be a recognized batsman or a duffer. This decision problem. These can be found by using backward recursion for a depends on a, the number of recognized batsmen not dismissed, large number of overs, often called value iteration (see, eg, and b, the number of duffers not dismissed, in both cases Hastings (1973) or Smith (1991)). So, for example, f0(4, 5, 1, 1) is excluding those at the wicket. While various objective functions the expected score when opening with two recognized batsmen on are possible in cricket (see, eg, Clarke (1988) or Preston and a normal pitch with four recognized batsmen and five duffers still Thomas (2002)), here we maximize the expected number of runs to come in. The above procedure resulted in a value of 267, in scored. agreement with the 270 calculated earlier by a less exact method. The expected scores are thus in good agreement with typical Let fn (a, b, x, y) be the expected number of runs scored in the remaining part of the innings with n overs remaining until the scores in three- and five-day cricket. In other papers and pitch returns to normal, using an optimal policy, that is, one that presentations, this and related models have been accepted as maximizes the expected runs scored. realistic representations of the game from which worthwhile conclusions could be drawn (see, eg, Clarke and Norman (2003)). However, there are two simplifications in the model which need explanation. Only even numbers of runs are scored in each over and wickets can fall only at the end of an over. In the model, the times at which decisions are made are at the

beginning of each over, when the captain of the batting side must scenario 1, (p(0) normal, others raised decide whether to send in a recognized batsman or a duffer if a 50%) dismissal occurs. A dismissal, if one occurs, happens at the end of n = 1,2 n = n = 20 n = 30 n = 40 3n = 10 an over so that it is known who will be at the bowler's end at the RRRRR RRSDD RRSDD RRRSD RRRRS RRRRR start of the next over (the incoming batsman). For the same reason, RRRRR RRRDD RRDDD RRADD RRRRD RRRRR only an even number of runs in an over may be scored. We claim RRRRR RRRDD RRDDD RRDDD RRRDD RRRRD that these two simplifications result in a tractable model while RRRRR RRRDD RRDDD RDDDD RRDDD RRRSD retaining the essential features of the game. RRRR RRRD RSDD RDDD RADD RRJD RRR RRR RDD RDD RDD RRD Results Table 3 Optimal strategy with n sticky wicket overs left for scenario 2, (every p(.) raised by 50%) Since the optimal strategy at each stage (number of overs) depends on three variables (a, b, and the rating of the other batsman), a n = 1,2 n = 3 n = 1 0 n = 20 three-way table is necessary to show the optimal strategy. RRRRR RRRSD RRRRA RRRRR RRRRR RRRRD RRRRS RRRRR However, it turns out that the dependence on the rating of the other RRRRR RRRRD RRRSD RRRRR batsman falls into one of only seven categories, so that the third RRRRR RRRRD RRRDD RRRRS dimension can be conveyed symbolically, using seven letters. We RRRR RRRR RRRD RRRA use a matrix format to show the policy as a function of a and b for RRR RRR RRR RRR each stage. Table 4 Optimal strategy with n sticky wicket overs left for

scenario 3, (p(0) doubled, others raised Number of recognized 1 xxxxx 50%) Number of duffers n=1 n = 2 n=10 n = 20 remaining b 1 2 3 4 RRRRR RRRRR RRRRR RRRRR 5 RRRRR RRRRR RRRRR RRRRR batsmen remaining a 2 xxxxx RRRRR RRRRR RRRRR RRRRR 3 xxxxx RRRRR RRRRR RRRR RRRRR RRRR RRRR RRRR RRRR 4 xxxxx RRR RRR RRR RRR 5 xxxx 6 xxx from scratch when normal playing conditions return. A sticky The entries in the matrices have the following meanings: wicket does not suddenly become a normal wicket between one over and another and a recognized batsman may play himself in • R send in a recognized batsman; just as well on a sticky wicket as on a normal wicket. Thus, we • D send in a duffer; make the limiting values of /0(a, b, x, y) take account of the • S send in a duffer unless the other batsman is a duffer, in which actual ratings of the two batsmen at the end of the sticky wicket case send in a recognized batsman; period. • J send in a recognized batsman unless the other batsman has a In the second scenario, duffers are sent in very rarely and in the rating of 3 or more; third scenario, duffers are not sent in at all. Only in the first • A send in a duffer unless the other batsman has a rating of scenario is there much chance of duffers being used, but even then, 0or1; they are not used often, even though in this scenario they perform • B send in a recognized batsman unless the other batsman has a just as well on a sticky wicket as they do on a normal wicket, rating of 4 or 5; while recognized batsmen do worse. • C send in a recognized batsman unless the other batsman has a rating of 1, 2 or 3; and Case 2 • n is the number of overs left until the wicket gets back to normal. We now assume that when play resumes on a normal wicket, recognized batsmen have to play themselves in, as would certainly Case 1 be the case if a sticky wicket persisted for the rest of the day's play We suppose, first, that a sticky wicket does not last until the end of but improved overnight. In this case, the values of a and b in /0(a, the day's play. It might then be reasonable to suppose that a b, x, y) would be either 0 or 1, according to whether the batsmen recognized batsman would not need to play himself in on strike and at the bowler's end were duffers or recognized

Table 2 Optimal strategy with n sticky wicket overs left for batsmen.

Results corresponding to those in Tables 2-4 are shown in duffers are still frequently used. This is in line with the common Tables 5-7. practice as used by Bradman, and noted in the introduction. In the It is perhaps not surprising that in the first scenario, it is third scenario, when duffers handle the sticky wicket much worse common for a duffer to be sent in—in this case, duffers bat as than recognized batsmen, it may still be optimal to sacrifice them well on a sticky wicket as they bat on a normal one. However, in when the wicket will improve overnight. the second scenario, when the dismissal probabilities of duffers When n = 1, that is, when there is exactly one over left, then are raised by the same percentage as those of recognized batsmen, invariably a recognized batsman is sent in. He will go to the

bowler's end and will not be dismissed. When n = 2, the batsman compared, using the scoring rates and dismissal probabilities of sent in will be at risk of dismissal in the last over. These features scenario 2. The optimal policies are shown in Tables 8 and 9. In are a consequence of only one dismissal being possible in an over both cases, recognized batsmen have to play themselves in when and only an even number of runs being scored in an over. play resumes at n = 0. Nowadays, it is common practice to send in a night watchman Discussion towards the end of a day, but maybe less common to send in a lower order batsman to hold the fort on a sticky wicket, a policy Tail-end batsmen are often sent in, not only to hold the fort on a which Tables 3, 8 and 9 support. It does seem that much depends sticky wicket, but to hold the fort towards the end of a day's play, on whether a sticky wicket is expected to last until the end of the to save a recognized batsman from having to play himself in day's play. If so, then the use of duffers to hold the fort is very twice, that is, to act as a night watchman. The two policies, for the often optimal, but otherwise not. night watchman situation, and for a sticky wicket that lasts until the end of the day's play but not until the next morning, may be

Table 8 Optimal strategy with n overs left in the day for night watchman policy for scenario 2 Table 5 Optimal strategy with n sticky wicket overs left for n=1 n=2 n=3 n=4 n = 5 scenario 1, (p(0) normal, others raised 50%) RRRRR RRSDD RSDDD SDDDD RRSDD n = 1 n = n = 20 n = 30 n = 40 RRRRR RRRDD RRDDD RDDDD RRSDD 2n = 10 RRRRR RRRDD RRDDD RDDDD RRDDD RRRRR RRSDD RSDDD RRRSD RRRRR RRRRR RRRRR RRRDD RRDDD RDDDD RRDDD RRRRR RRRDD RDDDD RRRDD RRRRS RRRRR RRRR RRRD RRDD RDDD RRDD RRRRR RRRDD DDDDD RRDDD RRRAD RRRRR RRR RRR RRD RDD RRD RRRRR RRRDD DDDDD RRDDD RRRDD RRRRR n = 6 n = 7 n = 8 n = 9 n=10 RRRR RRRD DDDD RDDD RRBD RRRR RSDDD RRRSD RRSDD RRRRS RRRSD RRR RRR DDD RDD RRD RRR RDDDD RRRSD RRDDD RRRRS RRRDD Table 6 Optimal strategy with n sticky wicket overs left for RDDDD RRRDD RRDDD RRRRD RRRDD scenario 2, (every p(.) raised by 50%) RDDDD RRRDD RRDDD RRRRD RRRDD RDDD RRRD RRDD RRRR RRRR n=1 n = 2 n=10 n = 20 RDD RRR RRD RRR RRR RRRRR RRDDD RRRSD RRRRR RRRRR RRRDD RRJDD RRRRR RRRRR RRRDD RRSDD RRRRR RRRRR RRRDD RRDDD RRRRR RRRR RRRD RRDD RRRR RRR RRR RRD RRR Table 9 Optimal strategy with n sticky wicket overs left for Table 7 Optimal strategy with n sticky wicket overs left for scenario 2, when the wicket does not improve until the start of scenario 3, (p(0) doubled, others raised 50%) the next day's play n = 1 n = 2 n = 10 n = 20 n=1 n=2 n=3 n=4 n=5 RRRRR RRDDD RRRRC RRRRR RRRRR RRDDD SDDDD SDDDD RRSDD RRRRR RRRDD RRRRS RRRRR RRRRR RRRDD RDDDD DDDDD RJDDD RRRRR RRRDD RRRRS RRRRR RRRRR RRRDD RDDDD DDDDD RSDDD RRRRR RRRDD RRRRD RRRRR RRRRR RRRDD RDDDD DDDDD RSDDD RRRR RRRD RRRR RRRR RRRR RRRD RDDD DDDD RSDD RRR RRR RRR RRR RRR RRR RDD DDD RSD n=6 n=7 n=8 n=9 n=10 RSDDD RRRSD RRSDD RRRRD RRRSD RDDDD RRADD RRDDD RRRSD RRJDD RDDDD RRSDD RRDDD RRRDD RRSDD RDDDD RRDDD RSDDD RRRDD RRDDD RDDD RRDD RDDD RRRD RRDD RDD RRD RDD RRR RRD

There is often a case for using a night watchman towards the end Conclusion of a day's play: if the wicket is sticky then the case is stronger In Farewell to Cricket, Bradman (1997) puts the issues clearly. still. In particular, from Tables 8 and 9, we may note that if it is worthwhile to send in a recognized batsman on a sticky wicket 'It is all very well to [be] gallant and heroic, but the captain's job then in the same circumstances it is worthwhile to send in a embodies the welfare of the team, and if his own personal success recognized batsman on a normal wicket. Conversely, if it is is an integral part of victory, he should not act accordingly. On worthwhile to send in a duffer on a normal wicket, then in the several occasions I was compelled to rearrange our batting order same circumstances it is worthwhile to send in a duffer on a as a matter of tactics because of the state of the wicket. It almost sticky wicket. invariably succeeded.

Some were unkind enough to suggest that my purpose was to tool. avoid batting on a wet wicket. Of course it was, but only because such avoidance was in the interests of the team. Cricketers appear reluctant to alter their set batting order despite the fact that batting conditions could be widely different Cricket pitches behave in a variety of ways after rain. The man depending on the earlier as and playing conditions. never lived whose judgement was infallible. Not the least With an early dismissal, a number 3 batsman could be facing a difficulty is to decide how long a wicket will remain bad. Under Australian conditions sufficient rain on a hard wicket, followed fired up fast bowler under cloudy skies conducive to swing by a hot sun, will generally produce a glue pot. Some are worse bowling. Following a good opening partnership, he could be than others. But will it remain sticky for an hour or a day? One striding to the in the middle of a warm afternoon facing a cannot tell. dispirited and tired spin attack. The same batting order may not necessarily be optimal in both cases and cricketers should at least In 1936-1937 against Allen's team, we worked like beavers to try consider a dynamic batting order, which changes subject to and get quick wickets. Then I found out the pitch was drying circumstances. This paper demonstrates that some benefit may be more slowly than anticipated and I had to tell my bowlers not to gained by modifying the batting order near the end of the day get England out' and/or when batting conditions are temporarily different.

We do not expect that modern captains would be brave enough to instruct their team not to try to dismiss the opposition. Acknowledgements—We thank three anonymous referees, whose com- Bradman hints at the criticism captains can invite by following ments helped us to improve this paper. Note: An earlier version of this paper was presented at the seventh Australasian Conference on strategies which could be misconstrued by less informed Mathematics and Computers in Sport, Palmerston North, 2004. observers. The benefits of a strategy that is optimal in the long run may be outweighed by the criticism bound to follow when it does not succeed in the short term. While Bradman was happy to open References the innings with his number 10 and 11 batsman, most observers would probably see this as a sign of weakness by the regular Bradman D (1997). Farewell to Cricket. ETT Imprint: Sydney. Clarke SR (1988). Dynamic programming in one-day cricket—optimal openers. Clearly strong evidence would be needed to convince scoring rates. J Opl Res Soc 39: 331-337. Clarke SR and Norman JM captains this might be a reasonable tactic in some circumstances. (2003). Dynamic programming in cricket: While there are several improvements that could be made to Choosing a night watchman. J Opl Res Soc 54: 838-845. Hastings our model, it is a first step in investigating the merits of al- NAJ (1973). Dynamic Programming with Management ternative batting orders. Allowing dismissals off any ball, and Applications. Butterworth: London. Preston I and Thomas J (2002). Rain rules for incorporating a distribution of runs to allow for a change of ends and probabilities of victory. Statistician 51: 189-202. Smith DK for batsmen during overs, would make the model more realistic, (1991). Dynamic Programming: A Practical Introduction. but probably intractable. While these models may be difficult to Ellis Horwood: Chichester. solve for optimality, a more realistic simulation model could be used to investigate fixed strategies. A simulation model might also carry more weight with cricket players and administrators, Received December 2004; accepted July 2006 and might even be used as a learning after three revisions