A Bayesian perspective of middle-batting position in ODI cricket

1Introduction

Game of cricket is played in three distinct formats: Test Cricket, One Day International (ODI) and Twenty20 (T20). One-day cricket was introduced in the 1960s as an alternative to the Test Cricket characterized by more aggressive batting, colorful uniforms and fewer matches ending in draws. ODI cricket is limited to fifty overs. The biggest event in ODI cricket takes place after every four years when the World Cup of Cricket (WCC) is organized by the International Cricket Council (ICC) which is the global governing body for cricket games. Later in 2003, T20 form of the cricket game was introduced with focus on gaining wider audience and with + emphasis on power hitting. Cricket in T20 format is limited to twenty overs. The present research is related to the game of ODI. Swartz et al., 2009 describe the play structure and related terminologies in detail.

Batting is the heart of cricket and bowling is its backbone. Cricket knowledge tells us that batting is more difficult early in a player ‘s innings but becomes easier as players familiarize themselves with the pitch conditions. A player’s batting ability is primarily assessed based on their batting average. However, in a cricket match a player never begins an innings while batting to the best of his ability as it takes time to adjust both physically and mentally to the specific match conditions. This process is nicknamed as getting eye-in which is shaped by factors such as the pitch conditions and opposition strategies. According to cricket experts a batsman is supposed to get his eye-in after he looks very comfortable in hitting boundaries (sixes and fours) and this confidence comes to a batsman after spending some time on the crease which enables him to read the pitch conditions and the bowler’s strategy against him. In other words, a batsman is said to be ‘set’ when he gets his eye-in. Batsmen are regularly seen to be dismissed early in their innings while still familiarizing themselves with the field conditions, which suggests that the probability of a batsman being dismissed on their current score (called the hazard) remains high regardless of their current score. Quantifying how well a batsman is expected to perform at a given stage of his innings identifies his batting potential could assist in the team selection strategy.

India lost the semi-final game of ICC ODI world cup 2019 (WC19) against New Zealand. Team India (TI) was super favourite and was expected to be champion for the third time in ODI world cup cricket history. Despite good play, excellent form of each player and presence of a dynamic opener (Rohit Sharma), an aggressive captain (Virat Kohli), the calm strategist (MS Dhoni), along with the other exceptionally talented players TI could not enter the final game.

There may have been a variety of reasons for India’s defeat but according to the great cricket experts like Sunil Gavaskar, Sachin Tendulkar and other seasoned cricketers it was the unsuitable player-selection at the crucial ‘number four batting position’ (N4), since almost two years prior to the WC19. According to the experts, consistent chopping and changing of TI players in the playing eleven and failure to fix mid-batting order led to loss in self-confidence of the batsman and hence his ability to play with a clear mind set, which is known to hamper the best playing ability (BPA) of a cricketer. N4 is a huge responsibility for any batsman in ODI cricket because he is expected to instantaneously pick up the pace of the innings with which the top order batsmen are batting or have batted. In other words, batsman at N4 needs to assess the batting conditions very quickly and take the ‘least time’ to get his eye-in which represents the transitional time interval between the initial playing ability and BPA. The uncertainty and instability for N4 started after the great number four Yuvraj Singh left the Indian ODI squad twice, first due to his health condition after ODI ICC world cup 2011 and then again in June 2017.

Though WC19 preparation started a couple of years ago yet TI remained indecisive regarding player at the crucial N4. From August 2017 to July 2019, India tried and tested seven batsmen, namely Ajinkya Rahane, Ambati Raydu, KL Rahul, Dinesh Kartik, Rishabh Pant, Vijay Shankar and Kedar Jadhav, who had played a minimum of two innings at N4. Rishabh Pant and Vijay Shankar were finally positioned for mid-batting order in the WC19. MS Dhoni, Manish Pandey and Hardik Pandya have also played at N4 on some occasions but they were never considered as front runners for N4 by the TI’s management committee. Post WC19, Shreyas Iyer has consistently played at N4.

The present paper focuses on the analysis and comparison of the batting performances of all the eight TI players who have played at number four position in ODIs after July 2017 till December 2020, initially through survival probabilities which are estimated, for the first time in this paper, through Kaplan Meir method and subsequently through Bayesian formulation adapted from Stevenson and Brewer (2017). The present paper is motivated by the idea of ‘initial’ and ‘eye in’ states to decide about the choice of the middle order batsmen. Concept of ‘positional cohort’ with respect to N4 batsmen is explored through Bayesian discrete event modelling in the present paper. Bayes methodology takes care of the so called ‘curse of dimensionality’ and use of statistical models easily handles the challenges which are posed by the unpredictable and inconsistent nature of the game of cricket. Analytical study of N4 assessment is important for any cricket team to select a robust line-up of batsmen.

Objective of the present work is to predict the batting abilities for the mid-order batsmen. The paper is organized in six sections. Literature review related to the statistical studies on cricket performances is presented in Section 2. Section 3 describes the N4 backdrop for the data selection. Analysis strategy and the proposed model are outlined under Section 4. Section 5 undertakes empirical survival analysis of the batsmen at N4. Section 6 is devoted to the posterior analysis of the data. Section 7 concludes the study.

2Existing parametric and non-parametric methods

Data rich nature of cricket has been utilized by numerous studies to optimize player and team performance. See for instance Clarke, 1988; Clarke & Norman, 1999; Preston & Thomas, 2000. Clarke & Norman, 2003; Swartz et al., 2009; Norman & Clarke, 2010 focus on the decision making in cricket. Davis et al. (2015) explore a novel metric for the notion of ‘expected run differential’. Jayalath, 2018 focuses on the identification of significant predictors of ODI games, which have potential to influence the outcome of a game. Goel et al., 2021 study match prediction under dynamic updation as the game of cricket progresses.

Elderton & Wood, 1945 support the claim that a batsman’s score could be modeled using a geometric progression. Kimber & Hansford, 1993 state that geometric assumption does not necessarily hold for all players, due to its difficulty in fitting the inflated number of scores of 0 appearing in many players’ career records. Bracewell & Ruggiero, 2009 proposed to model the players’ batting scores based on the ‘Ducks ‘n’ runs’ distribution, using a beta distribution to model scores of zero, and a geometric distribution to describe the distribution of non-zero scores.

An advantage of estimating hazard function is that it allows us to observe how a player’s dismissal probability (and hence their batting ability) varies over the course of their innings. While Kimber & Hansford, 1993 found batsman were more likely to get out early in their innings, due to the sparsity of data at higher scores these estimates quickly become unreliable and the corresponding estimated hazard function is observed to jump erratically between the scores. Cai et al., 2002 address this issue by using parametric smoother on the hazard function. However, given the underlying function is still a non-parametric estimator the problem of data sparsity remains an issue and continues to distort the hazard function at higher scores. Damodaran, 2006 provided a method which allows for within-innings comparisons, but lacks a natural cricketing interpretation. On the other hand, Koulis et al., 2014 proposed a Bayesian stochastic model for evaluating performance based on the player’s form, however his focus remained only on inter-innings comparison in terms of batting ability, rather than intra-innings comparison.

Swartz et al., 2009 proposed simulator for simulation of one-day cricket matches, where the outcome probabilities are estimated from historical data which produce realistic results, with a caution for continuously updating the database. Brewer, 2008 proposed an alternative Bayesian parametric model to estimate a player’s current batting ability (via the hazard function), given his score, using single change-point model. This allows for smooth transition in the hazard between the batsman’s ‘initial’ and ‘eye in’ states, rather than the sudden jumps of Kimber & Hansford, 1993 and of Cai et al., 2002. Thus, one assumes that batsmen are more susceptible early in their innings and tend to perform better as they score more runs.

3Data

Survival times of players who get out during the match are completely known. However, survival times of the batsmen who remain at crease by the end of match are said to be censored. One can refer to Lee and Wang, 2003 for statistical definitions and concepts related to survival times and survival distribution. ODI innings of all the players who were declared as front runners for N4 either by the TI management or by the selectors at different points of time are accessed (Table 1). Ajinkya Rahane had emerged as the best choice for N4 during TI’s tour to South Africa in January 2018, owing to the seaming conditions at the pitch. He then played five innings in the six match ODI series. In July 2018, KL Rahul had played two innings at N4 against England. In the same series, Dinesh Kartik was brought up at N4 to play a single inning against England and thereafter he continued to play at N4 in Asia Cup, Dubai during September 2018. After Asia Cup, Ambati Raydu assumed N4 until the start of WC19. An interestingly surprise call was made by TI’s chief selector for N4 to be assigned to Vijay Shankar. However, due to an injury incurred by him, just after playing two games in WC19, Rishabh Pant was declared N4 batsman for WC19.

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4Methodology

Table 1 presents the scores at which the batsmen got out and their not-out scores in ODIs played by them during July 2017-December 2020. Latter are marked with * in the table. In this paper not-out scores are considered as the right censored survival data and the remaining scores as complete survival data.

4.1Kaplan meir estimates

We adopt a non-parametric approach for generating survival function for individual players. Survival probabilities are generated using product-limit method developed by Kaplan & Meier, 1958. The survival probability estimates are graphed as survival distribution known as Kaplan-Meir (KM) curves. KM curves are used for comparing batsmen’s performance. KM curves (Figs. 1–5) represent survival probability over the study period for all players under study. In the present paper, survival events represent total runs scored by an individual player before getting out in an innings. The career period, considered for the present analysis, is regarded as the survival time. The ODI in which a player does not get out is regarded as a right censored survival time. For making a fair comparison we have considered pairs of players with equivalent number of innings. For example, Shreyas Iyer having played 11 innings is compared with Ambati Raydu’s last 11 innings out of his total 14 innings played at N4 before WC19. Similarly, survival probabilities through KM estimates are compared for Dinesh Kartik, Rishabh Pant and KL Rahul taking a pair of two with their latest survival times or batting scores at N4. Thus, five pairs of players are assessed through KM analysis.

Fig. 1

KM curves for batting scores of Shreyas Iyer and Ambati Raydu.

Fig. 2

KM curves for batting scores of Dinesh Kartik and KL Rahu.

Fig. 3

KM curves for batting scores of KL Rahul and Rishabh Pant.

Fig. 4

KM curves for batting scores of Dinesh Kartik and Rishabh Pant.

Fig. 5

KM curves for batting scores of Vijay Shankar and Kedar Jadhav.

5Results

5.2Posterior analysis

Bayesian analysis of the model described in section 4 is undertaken in OpenBUGS software. Bayes estimates are obtained through posterior distributions which are in form of complex integrals and difficult to compute without computing. Smith & Gelfand,1993 showed that simulation can be used to draw samples from such complex integrals. Metropolis et al., 1953 and Hastings, 1990 developed Markov Chain Monte Carlo (MCMC) algorithms which iteratively produced sequential samples of dependent draws. The dependency appears as autocorrelation since the long stream of numbers are generated from a Markov Chain (Spiegelhalter et al., 1996). MCMC simulated chains converge to the desired posterior distribution. In the present work MCMC algorithm is utilised to simulate 20,000 data points out of which the initial 2000 generated data points are discarded as burn-in to make dependency in the posterior samples redundant. Such trimmed samples are then aggregated to obtain posterior summaries. Box-plots based on posterior summaries for effective batting average, m_i for each player is plotted in Figs. 6–11. Box plot represent the dispersion in the value of effective batting average obtained through equation (1) and (2) for each inning of batsman estimated under Bayesian setup. Effective batting average for each inning can be given as posterior mean like we have presented for components (μ_1, μ₂and L) of effective batting average in Table 2 but in order to represent the dispersion in MCMC generated posterior sample values for effective batting average and to reflect on batsman’s consistency through number of innings in which effective batting average remains above combined effective batting average of all the innings represented by central line authors presented box-plots instead of mean of posterior sample. Posterior summary for all components of effective batting average (m_i) i.e., μ_1, μ₂and L are obtained and presented in Table 2. Performance analysis of each batsman is demarcated by posterior mean and posterior probabilities. It is evident from Table 2 that Shreyas Iyer leading with highest estimated values for μ₁ and μ₂ Posterior mean and probability for L<20 is higher for Rishabh Pant and Dinesh Kartik indicating that these batsmen get less number runs to get their ‘eye-in’ and play their shots with highest ability. Posterior density and trace plots are shown in Figs. 12 and  13, respectively. Density plots and trace plots are diagnostic tools for detecting convergence in generated MCMC sample for the parameter under interest. If the shape of the density plot is bell curved and distribution is unimodal then the parameter under interest is said to achieve convergence. Convergence diagnostics for MCMC generated samples for the parameters under interests adopted are well utilised from OpenBUGS software (https://www.mrc-bsu.cam.ac.uk/wp-content/uploads/2021/06/OpenBUGS_Manual.pdf).

Fig. 6

Box plot of Ajinkya Rahane’s effective average m_i.

Fig. 7

Box plot of Ambati Raydu’s effective average m_i.

Fig. 8

Box plot of Shreyas Iyer’s effective average m_i.

Fig. 9

Box plot of Rishabh Pant’s effective average m_i.

Fig. 10

Box plot of KL Rahul’s estimated effective average m_i.

Fig. 11

Box plot of Dinesh Kartik’s estimated effective average m_i.

Fig. 12

Density plots.

Fig. 13

Trace plots.

6Discussion and conclusion

Cricket fraternity comprises of different experts each of whom have a distinct analysis strategy for the outcomes of matches based on their individual logic. Cricket is game of uncertainties and dynamics of a particular game changes very rapidly which also depend on the form of players on a particular match day. Selectors pick players in TI considering their performance in domestic cricket and in Indian Premier League (IPL). It is always difficult for captain and team management to drop or pick when there is close competition among two players. Here, we present a statistical approach which gives an unbiased opinion on comparison of performances for closely competing players. This analytic approach might augment the existing selection strategies utilised by the cricket analytic experts to support selection committee of Board of Cricket for Control in India (BCCI).

The present paper proposes a simple survival analysis approach which allows analysis of individual and pairwise player-assessment at a time (Fig. 1–Fig. 5). Such an assessment, undertaken for the first time in sports related research, enables quantification of how players’ abilities differ, taking into account the fact that the information about any player is limited by the finite number of innings that they have played.

Also, Bayesian modelling approach is adopted for analysing initial and final playing abilities of a group of Indian batsmen who have played at the fourth position for team India in ODI cricket. Table 2 represents posterior estimates which reveals Shreyas Iyer to have highest BPA. Criteria for comparison and selection of N4 batsman is based on the final batting average ^′m₂^′ which is the indicator of best playing ability of a batsman. Another important indicator i.e., number of runs required to transit from initial playing ability to final or best playing ability is the e-folding time ‘L’ which is used for performance ranking whenever two batsmen show a tie on the final batting average. The latter one will be more relevant criteria of comparison in case of shorter formats like T20s. The present analysis for ODIs show that Shreyas Iyer has the best final batting average and so is the corresponding posterior probability. As far as future scope for this research is concerned, we can assess performance of different batsman in T20’s by setting up threshold of 30 runs and 10 runs for μ₂ and L, respectively.

The present work proposes modelling for runs scored in a ODI by an Individual batsman who have potential to sustain themselves for longer period on the field, in pursuit of devising strategies for selection of the right players in the middle order batting position. Initial playing ability is the intensity with which batsman plays when he first arrives at the crease and is shy of playing his shots and is sometimes tentative because gets beaten by the swing offered by the fast bowlers and tries to adjust the pace, bounce and turn offered by the pitch. Therefore, we are able to see the low posterior estimates (Table 2) for initial playing ability. Adapting quickly to pitch conditions and understanding the opposition plan of attacking are the key to get your ‘eye-in’ quickly and score runs. Such an adaption of batsman is quantified in the present research which analyses the performance of batsman playing ability fragmented into three parts i.e., initial playing ability, final playing ability and the transition time between the two through adopted Bayesian model. Box-plots for effective batting average, m_i, of individual player’s innings (Fig. 6–Fig. 11) represents variation in batting performance or consistency of the batsman. Also, box plots indicate how many times out of total innings considered for a batsman, he had played above the combined posterior mean of his effective batting average. Shreyas Iyer, Rishabh Pant and Dinesh Kartik are observed to be more consistent and their effective average of individual innings remains above combined effective batting average of all the innings played.