Rating players in test match cricket

In general, the evaluation of player performance in test cricket is based on measures such as batting and bowling averages. These measures have a number of limitations, among which is that they fail to take into account the context in which runs are made or conceded and wickets are taken or given away. Furthermore, batting and bowling averages do not allow the comparison of performances in these two disciplines; this is because batting and bowling performances are measured using different metrics. With these issues in mind, we develop a new player rating system for test cricket. We use multinomial logistic regression to model match outcome probabilities session by session. We then use these probabilities to measure the overall contribution of players to the match outcome based on their individual batting, bowling and fielding contributions during each session. Our measure of contribution has the potential for rating players over time and for determining the ‘best’ player in a match, a series or a calendar year. We use results from 104 matches (2010–2012) to illustrate the method.     

Integer optimisation for the selection of a Twenty20 cricket team

The attraction of Twenty20 cricket for spectators has sparked huge commercial interest in the marketing of the concept. The International Cricket Council has sanctioned world tournaments, the inaugural tournament in South Africa and the recent 2009 event, while the 2008 and 2009 Indian Premier League tournaments have generated profitable returns for both players and franchise owners. This paper describes methods for quantifying a cricket player's performance based on his ability to score runs and take wickets. These performance measures are then used to determine the optimal team using an integer programme. An illustration of the method is provided using data from the inaugural Twenty20 World Cup held in South Africa in 2007.     

A fair method for resetting the target in interrupted one-day cricket matches

A method is described for setting revised target scores for the team batting second when a limited-overs cricket match has been forcibly shortened after it has commenced. It is designed so that neither team benefits or suffers from the shortening of the game and so is totally fair to both. It is easy to apply, requring nothing more than a single table of numbers and a pocket calculator, and is capable of dealing with any number of interruptions at any stage of either or both innings.The method is based on a simple model involving a two-factor relationship giving the number of runs which can be scored on average in the remainder of an innings as a function of the number of overs remaining and the number of wickets fallen. It is shown how the relationship enables the target score in an interrupted match to be recalculated to reflect the relative run scoring resources available to the two teams, that is overs and wickets in combination. The method was used in several international and domestic one-day competitions and tournaments in 1997.     

Dynamic Programming in One-Day Cricket-Optimal Scoring Rates

Using a dynamic programming formulation, an analysis is presented of the innings of the team which bats first (here referred to as the first innings) and the innings of the team which bats second (here referred to as the second innings). This allows a calculation, at any stage of the innings, of the optimal scoring rate, along with an estimate of the total number of runs to be scored (in the first innings) or the chance of winning (in the second innings). The analysis is used to shed some light on possible batting tactics (in terms of the best run rate at any stage of the innings), to quantify the effects of selecting extra batsmen in a side, and to suggest a method for the development of alternative measures of player performance. Results suggest that scoring rates should be more uniform than at present, and that the team batting second has an advantage. Possible extensions to the model are discussed.     

Fair measures of performance: the World Cup of cricket

Luck can play a big part in tournament success, and progress is not necessarily the best measure of performance. A linear model is used to fit least squares ratings to margins of victory in the cricket World Cup. The Duckworth/Lewis rain interruption rules are used to project a winning second innings score and create a margin of victory in runs, equivalent to that used when the team batting first wins. Results show that, while the better teams progressed through the first round of the competition, some injustices occurred in the Super-Six round. This appears to be due to the double counting of selected matches. Ordering teams by average margin of victory gives similar results to the more complicated linear model, and its use as a tie breaker is suggested. Publication of the margin of victory as estimated by the Duckworth/Lewis method for second innings victories in all one-day matches would provide a common margin of victory suitable for analysis.     

An adjusted Duckworth–Lewis target in shortened limited overs cricket matches

This paper presents an investigation of the Duckworth–Lewis (D/L) method for setting fair targets in limited overs cricket matches which have been shortened in duration due to interruptions. Specifically, the D/L method's fundamental structural assumption that the resource utilization table used to calculate adjusted targets is the same for interruptions of either the first or second innings is examined. The analysis is based on the results from over 1100 international matches, 989 of which were completely uninterrupted (excluding those matches where penalties in the form of shortened second innings were imposed) and 61 of which were decided by using the D/L method. Overall, the investigation shows that the D/L method is admirable in its fair modelling of appropriate targets. However, detailed analysis of the uninterrupted matches, employing tobit regression and random walk boundary crossing techniques, indicates that the assumption that resource utilization follows the same pattern in both innings is not sustainable. It appears that actual resource usage during the second innings is more heavily weighted towards the very early and very late overs. To account for this, an adjustment to the D/L method is proposed, whereby separate resource usage tables are employed for each innings. For the first innings, the current D/L method is retained, as it has been constructed based solely on first innings information; however, for the second innings, the resources available with any given number of overs remaining and wickets lost is determined by a simple transformation of the associated resources remaining from the current D/L method. The specific form of the transformation function is based on the cumulative distribution function of a beta distribution. This choice of transformation was based on a combination of structural appropriateness, flexibility and ease of implementation considerations. The adjusted method is evaluated on the 61 matches determined by the D/L method and is seen to predict a reversal of the result in five matches.     

Duckworth–Lewis and Twenty20 cricket

Originally designed for 1-day cricket, this paper considers the use of the Duckworth–Lewis method as an approach to resetting targets in interrupted Twenty20 cricket matches. The Duckworth–Lewis table is reviewed and an alternative resource table is presented. The alternative table is constructed using observed scoring rates from international Twenty20 matches. A desideratum of a resource table is monotonicity in both the rows and columns corresponding to wickets and overs respectively. Consequently, a Gibbs sampling scheme related to isotonic regression is applied to the observed scoring rates to provide a non-parametric resource table. Taking into account the more aggressive batting style of Twenty20 compared to 1-day cricket, the resultant resource table is seen to possess sensible features. A discussion is provided concerning the use of the Duckworth–Lewis method applied to Twenty20.     

Extending the range of player-performance measures in one-day cricket

The author's previous paper on this topic showed that the Duckworth/Lewis methodology has considerable potential for providing relevant and objective measures of performance in one-day cricket. The present paper evaluates how these measures can apply in the longer term. This assessment is based upon several series of matches involving the main players from most of the major cricketing nations. Although data used are not exhaustive, they are of sufficient quantity to provide strong indications of the long-term viability of the measures, thus leading to the realistic expectation of inclusion of the methodology into the existing player-rankings or the development of an independent set of performance indicators that reflect more validly the value of players' inputs to matches. Such measures are expected to be of valuable use to team management, media commentators and to the public.     

A criterion for comparing and selecting batsmen in limited overs cricket

The batting average statistic has been used almost exclusively to assess the worth of a batsman. It reveals a great deal about the potential performance of batsmen in cricket played at the first class level. However, in the one-day game, strict limits on the number of balls bowled have introduced a very important additional dimension to performance. In the one-day game, it is clearly not good enough for a batsman to achieve a high batting average with a low strike rate. Runs scored slowly, even without the loss of wickets, will generally result in defeat rather than victory in the one-day game. Assessing batting performance in the one-day game, therefore, requires the application of at least a two-dimensional measurement approach because of the time dimension imposed on limited overs cricket. In this paper, we use a new graphical representation with Strike rate on one axis and the Probability of getting out on the other, akin to the risk–return framework used in portfolio analysis, to obtain useful, direct and comparative insights into batting performance, particularly in the context of the one-day game. Within this two-dimensional framework we develop a selection criterion for batsmen, which combines the average and the strike rate. As an example of the application, we apply this criterion to the batting performances of the 2003 World Cup. We demonstrate the strong and consistent performances of the Australian and Indian batsmen as well as provide a ranking of batting prowess for the top 20 run scorers in the tournament.     

On Simple MIX game

We consider zero-sum game which is called Simple MIX game. Each of two players (I and II) draws a number (x andy respectively) according to a uniform distribution on [0, 1]. After observing his number each player can then choose to offer or not offer to exchange his number for the other player's number. Conditions for an exchange are the following: 1) both players must offer for a trade to occur with certainty; 2) if only one player offers, a trade occurs with probabilityp. A player's payoff is equal to 1, 0 or — 1 if the value of the number which he finally gets is greater, equal or less than the number of his opponent. In the present paper we shall investigate Simple MIX game in which both of the players can obtain additional information about the opponent's number. Besides, we consider two-stage variant of this game.     

An analysis of strategy in the first three innings in test cricket: declaration and the follow-on

This paper analyses declaration and the follow-on decisions in test cricket. We model the match outcome given the end of first, second and third innings positions; data on 391 test matches, from the period 1997 to 2007, are used to fit the models. We then investigate how declaration strategy should vary from innings to innings, and how the nature and strength of the covariate effects vary. As the match progresses, the explanatory power of the covariates increases (from 44% at the end of the first innings to 80% at the end of the third). Home advantage and the effects of team strengths decrease. Overs-remaining, or equivalently overs used, and the number of runs by which the reference team lead their opponents remain important throughout. The follow-on decision problem is also briefly considered, and surprisingly, we find that the decision to enforce the follow-on or otherwise has no effect on match outcome.     

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

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 wickets’. 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.     

To bat or not to bat: An examination of match outcomes in day-night limited overs cricket

The tradition of tossing a coin to decide who bats first in a cricket match introduces a randomly assigned advantage to one team that is unique in sporting contests. The potential importance of the toss rule in determining cricket match results has been the subject of some investigation, which is further advanced in this paper that utilizes a data set relating to the increasingly popular, but contentious, day-night form of limited overs cricket as played at international level. We employ logit regression models to examine the effects of winning the toss and choice of batting order on the likelihood of a match victory, while controlling for home advantage and (relative) team quality. Our findings suggest that winning the toss and batting first increases the probability of winning whereas winning the toss and bowling first does not.     

Dynamic programming in cricket: choosing a night watchman

In cricket, when a batsman is dismissed towards the end of a day's play, he is often replaced by a lower-order batsman (a ‘night watchman’), in the hope that the remaining recognised batsmen can start their innings on the following day. A dynamic programming analysis suggests that the common practice of using a lower-order batsman is often sub-optimal. Towards the end of a day's play, when the conventional wisdom seems to be to use a night watchman, it may be best to send in the next recognised batsman in the batting order. Sending in a night watchman may be good judgement when there are several recognised batsman and several lower order batsmen still to play (say four of each). However, with smaller numbers (two of each, for example), then, with very few overs left to play, it may be better to send in a recognised batsman.     

To run or not?: Some dynamic programming models in cricket

In cricket, particularly near the end of an innings, batsmen of different abilities need to manage the rate at which they score runs. Either batsman can choose to bat aggressively or defensively, which alters their chances of scoring runs or being dismissed. Since they change ends when they score a run and at the end of an over, by scoring an odd or even number of runs the two batsmen also determine which of them will face the next ball. It may be worthwhile to refuse a run to keep the slower or lower scoring batsman from the strike. Some dynamic programming models are developed which could be used to maximise the total number of runs scored.     

Optimal challenges in tennis

The use of technology in sport to assist umpires has been gradually introduced into several sports. This has now been extended to allow players to call upon technology to arbitrate when they disagree with the umpire's decision. Both tennis and cricket now allow the players to challenge a doubtful decision, which is reversed if the evidence shows it to be incorrect. However, the number of challenges is limited, and players must balance any possible immediate gain with the loss of a future right to challenge. With similar challenge rules expected to be introduced in other sports, this situation has been a motivation to consider challenges more widely. We use Dynamic Programming to investigate the optimal challenge strategy and obtain some general rules. In a traditional set of tennis, players should be more aggressive in challenging in the latter stages of the games and sets, and when their opponent is ahead. Optimal challenge strategy can increase a player's chance of winning an otherwise even five-set match to 59%.     

A modified Duckworth–Lewis method for adjusting targets in interrupted limited overs cricket

In this paper we present a modified Duckworth/Lewis method. The key modification is an improved functional form for the model describing the runs to be scored in an innings. In the course of our work we compare several alternative methods for resetting targets in limited overs cricket that have been proposed in the literature and conclude that the Duckworth/Lewis method is the most viable. Our analysis also suggests that it is reasonable to use a single method for both the 50-over and 20-over formats of the game.     

A Robust Heuristic for Batting Order Optimization Under Uncertainty

Baseball teams are faced with a difficult scheduling problem every day: given a set of nine players, find the optimal sequence in which they should bat. Effective optimization can increase a team's win total by up to 3 wins per season, and 10% of all Major League teams missed the playoffs by 3 or less wins in 1998. Considering the recent $252 million contract given to one player, it is obvious that baseball is a serious business in which making the playoffs has large financial benefits. Using the insights gleaned from a Markov chain model of baseball, we propose a batting order optimization heuristic that performs 1,000 times faster than the previous best heuristic for this problem. Our algorithm generates batting orders that (i) are optimal or near-optimal, and (ii) remain robust under uncertainty in skill measurement.     

A successful operational research intervention in one-day cricket

After its introduction to one-day cricket in 1997, the Duckworth/Lewis method has spread to become the standard rule throughout senior levels of the game for resetting the target in matches shortened after their start due to rain or other causes. This paper reviews how the method is working following its application to more than 400 known cases. A comparison of the Duckworth/Lewis model with several years of data from international matches is made that confirms the suitability of the underpinning model, but the data indicate that there is a need to update the model's parameters. An upgrading of the model is suggested to cope with high-scoring matches when the basic model's assumptions begin to break down. Although the price of the enhanced model is loss of simplicity and complete reliance on software, cricketing authorities have begun to believe the price to be worth paying. Data from actual matches are used throughout to illustrate the implementation of this successful use of Operational Research principles to the solution of a practical problem in one-day cricket.     

Modelling match outcomes and decision support for setting a final innings target in test cricket

^** E-mail: p.a.scarf{at}salford.ac.uk This paper considers the problem of timing the declaration ofthe third innings in test cricket. Data on the outcomes of recenttest matches are analysed in order to develop simple decisionsupport tools. The first of these tools presents match outcomeprobabilities given the position of the match at a potentialdeclaration point. These probabilities are determined usinga multinomial logistic regression model that is fitted to thetest match data. This idea is then extended to consider progresstowards a declaration—match outcome probabilities areconsidered as a function of target aimed for and run-rate. Thedecision tools have been implemented on a spreadsheet and examplecalculations are presented. The modelling described has thepotential for practical use in test matches.