# Using Bradley–Terry models to analyse test match cricket

Using Bradley–Terry models to analyse test match cricket Abstract In this paper we investigate the use of Bradley–Terry models to analyse test match cricket. Specifically, we develop a new and alternative team ranking and compare our rankings with those produced by the International Cricket Council, we forecast the outcomes of a selected number of test cricket matches and show that our predictions perform well compared to bookmaker predictions. We offer ratings of individual players and use these ratings to predict the results of some recent matches. The general purpose of the paper is to illustrate the potential of Bradley–Terry models, which are effectively models of P(i is preferred to j), and thus can be applied in a number of settings where there are paired comparisons. Popular applications include analysing taste test experiments and modelling sports competitions. More creative examples of applications include statistical modelling of citation exchange among statistics journals, predicting the fighting ability of lizards and estimating driver crash risks. 1. Introduction Suppose that there are a set of entities that we wish to consider according to some common attribute. Pairs of these entities can be compared with respect to some quantifiable aspect of this attribute. Such comparisons are generally referred to as ‘paired comparisons’. An example of such a comparison is a taste test, where a judge tastes two specimens before declaring a preference for a particular specimen. The Bradley–Terry model is a useful model that was developed with a view to analysing the results from a set of paired comparisons. Such models can produce scores for the entities based on a set of results, and can also be used to make predictions for the results of future comparisons. Furthermore, if we instead consider a comparison as a ‘contest’ between two competitors, a variety of possible applications becomes apparent, most notably in sport. In this paper we investigate the use of Bradley–Terry models to analyse test match cricket. There are a number of other papers which have analysed at least one version of cricket. Some recent papers include Scarf & Shi (2005), Scarf et al. (2011), Scarf & Akhtar (2011), Akhtar & Scarf (2012), Perera & Swartz (2012), Davis et al. (2015) and Akhtar et al. (2015). Each of these papers also has numerous references of interest to those in analysing cricket. Specifically, we do the following: We develop a new and alternative team ranking and compare our rankings with those produced by the International Cricket Council (ICC). We forecast the outcomes of a selected number of matches and compare our predictions with bookmaker odds. We offer a method to obtain ratings of individual players. The purpose of the paper is to illustrate the potential of Bradley–Terry models, which are effectively models of P(i is preferred to j). This is an interesting construct in itself, with many potential applications. The Bradley–Terry model (Bradley & Terry, 1952) is named after Ralph A. Bradley and Milton E. Terry, who devised the method in 1952. However, as noted in Simons & Yao(1999), the method was discovered independently in Zermelo (1929), Ford (1957) and Jech (1983). Agresti (2014) discusses the Bradley–Terry model and presents a simple example of how it can be used to rank baseball teams. Applications of the Bradley–Terry model are both plentiful and diverse. Popular applications include analysing taste test experiments (Hopkins, 1954; Bliss et al., 1956) and modelling sports competitions, with racquetball (Strauss & Arnold, 1987), soccer (Hallinan, 2005) and basketball (Cattelan et al., 2013) all receiving attention in this regard. More creative examples of applications include statistical modelling of citation exchange among statistics journals (Varin et al., 2013), predicting the fighting ability of lizards (Whiting et al., 2006) and estimating driver crash risks (Li & Kim, 2000). The structure of the paper is as follows. In Section 2 we offer a brief description of Bradley–Terry models and some of the areas in which they have been successfully applied thus far. In Section 3 we describe the three sources of data we have used to model the aspects of test match cricket, as described in the previous paragraph. Our analyses are described in Section 4 and we conclude our findings in Section 5. We deliberately leave technical content to a minimum in an aim to improve the readability of the paper. Cricket is a notoriously difficult game to model, as it contains many subtle nuances and variables. Accordingly, we believe it is appropriate as a test-bed for the potential of Bradley–Terry models. More generally, we show the capability of these models for predicting contests and ranking individuals. These are useful applications with wide-bearing implications. 2. Bradley–Terry models 2.1. Description For any pair of entities, say i and j, the Bradley–Terry model takes an input as some measure of quality for the respective entities and computes the probability that i is preferred to j. For a system of n entities, the model introduces parameters π1, π2, …, πn, which can be interpreted as some measure of quality of the respective entries, such that \begin{align*} p_{ij}=P(i \;\textrm{is preferred to } j)=\frac{\pi_{i}}{\pi_{i}+\pi_{j}}, \end{align*} where $$\sum _{i=1}^{n}\pi _{i}=1.$$ It follows that \begin{align*} \log \frac{p_{ij}}{p_{ji}}=\lambda_{i}-\lambda_{j}, \end{align*} where $$\lambda _{i}=\log \pi _{i}.$$ For n parameters λ1, λ2, …, λn then \begin{align*} p_{ij}=\frac{\exp(\lambda_{i}-\lambda_{j})}{1+\exp(\lambda_{i}-\lambda_{j})}\, . \end{align*} In many paired comparison experiments there is often a factor that, independent of the attributes of the respective entities, influences the outcome of the experiment. In a taste test, this could refer to some advantage gained by the first sample tasted, or could refer to a perceived home advantage in a sports contest. It is possible to include such factors into a Bradley–Terry model. Let δ ⩾ 0 be the advantage gained by entity i from the external effect in question. Then we may write \begin{align} \log \frac{p_{ij}}{p_{ji}}=\lambda_{i}-\lambda_{j}+\delta\! . \end{align} (2.1) In order to fit our Bradley–Terry models we use the BradleyTerry2 package in R, see Turner & Firth (2012). 3. Data In this paper we consider three different sets of data, each of which are described below. The data to be considered are (i) results of test matches since 2004, (ii) bookmakers odds for selected test matches and (iii) ball-by-ball data for selected test matches. The data are for a relatively long period of time. In such a period the relative abilities of teams may change. In this paper we do not pursue the fitting of a ‘dynamic’ Bradley–Terry model but note that there has been recent work in this area, see Cattelan et al. (2013). We also note that the ICC rankings have remained fairly consistent throughout the time period considered, perhaps making this less of an issue. Table 1. Extract of data of test match results taken from ESPN Cricinfo Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 Table 1. Extract of data of test match results taken from ESPN Cricinfo Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 3.1. Test match results We consider data collected from ESPN Cricinfo which comprises of the results of 442 test matches between 8 March 2004 and 30 June 2014. For each match, the data recorded contains the home team, the away team, the outcome (indicated as 1 for a home win, 0 for an away win and 0.5 for a draw), which ground the match was played at, which team batted first (1 if the home team batted first, 0 for the away team) and the start date of the match. A one-off test match between an ICC World XI and Australia in October 2005 has been excluded from the data since this is the only test that an ICC World XI has played and it is inappropriate to compare them with regular test playing nations. Table 1 contains an example extract of the data taken from ESPN Cricinfo. Due to security concerns, Pakistan has not hosted a test match since February 2009, with their ‘home’ matches instead being played at neutral venues. The majority of these matches have taken place in the United Arab Emirates and, given the regularity with which they have played there and the relative similarity to Pakistani conditions, we have decided to treat Pakistan as the home side in such matches. There was, however, a test series played between Australia and Pakistan in England in 2009, but Pakistan was officially stated to be the ‘home’ team. Bangladesh and Zimbabwe are often perceived to be far weaker than the eight other test-playing nations. As shown in Table 2, they have played fewer games than the other sides, and have worse records than the other teams, between them winning just 7 of their 75 test matches included in the data, four of which have come in matches between the two sides. As a result of this gulf in quality, the stronger teams often field weakened line-ups when playing against either of these sides and so results in these matches are often not a reflection of the true ability of the teams. Table 2. Summary of team performances over 442 test matches between 2004 and 2014 Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% Table 2. Summary of team performances over 442 test matches between 2004 and 2014 Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% The most notable example of this is regarding Bangladesh’s tour of West Indies in 2009. A contract dispute led to several leading West Indies players boycotting the match and so their side featured numerous uncapped players. Bangladesh won both test matches in the series and these remain their only victories over a major international team. After consideration, matches involving Bangladesh and Zimbabwe have been excluded from the analysis, leaving a dataset of 372 test matches played between eight major international teams. Initial analysis showed that parameter estimates concerning these teams suffered from huge standard errors, rendering them unusable. Table 3 gives a summary of team performances excluding matches involving Bangladesh and Zimbabwe. Table 3. Summary of team performances over 372 test matches excluding matches involving Bangladesh and Zimbabwe Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% Table 3. Summary of team performances over 372 test matches excluding matches involving Bangladesh and Zimbabwe Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% 3.2. Bookmakers’ odds Data regarding the average pre-match odds of all test matches between March 2012 and June 2014 were taken from Oddsportal. The odds that were taken were those of a home win, away win and a draw. Average odds were gathered for a total of 64 matches. 3.3. Ball-by-ball data Further data regarding test matches was collected from Cricsheet. The data consists of detailed information on all test matches played since 2009. Basic information about each match, such as the competing teams, the location, the dates and the result were provided along with information about each delivery (a delivery or ball in cricket is a single action of bowling a cricket ball toward the batsman) in the match, including the bowler, the batsman, how many runs were scored and whether or not a wicket was taken. The information extracted from the data comprised of ball-by-ball details of every match played between Australia, England, India and South Africa between December 2009 and August 2013. An extract of the data is given in Table 4. Using this data, it was possible to compute traditionally referenced statistics for cricket players such as batting and bowling averages for each player. Tables 5 and 6 give these statistics for the top 15 run scorers and wicket takers, respectively, from the matches considered. Table 4. Extract of ball-by-ball data Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Table 4. Extract of ball-by-ball data Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Table 5. Batting summary statistics for the top 15 run-scorers for data considered Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Table 5. Batting summary statistics for the top 15 run-scorers for data considered Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Table 6. Bowling summary statistics for the top 15 run-scorers for data considered Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 Table 6. Bowling summary statistics for the top 15 run-scorers for data considered Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 4. Analysis 4.1. An alternative team ranking system The ICC, the governing body of international cricket, currently uses a method devised by David Kendix to rank its teams in each form of the game. Although the precise methodology that is used to compute these rankings is not freely available, several aspects of the method are known. The rankings are based on a rating points value assigned to each team depending on their results. These take account of the results of matches in the past 3–4 years, with greater weighting given to more recent matches. Points are assigned after each match, with the amount of points available dependent on the respective ratings of the two teams. Furthermore, a team earns bonus points for winning a series of games (such as the Ashes). A simple Bradley–Terry model, of the form (2.1) was fitted to estimate the ‘ability’ of each team based on the data described in Section 3.1. We set West Indies as our reference category and thus this team is nominally given an ability rating of 0. Note that any team may be set to be the reference category, the results will remain the same. Matches that resulted in draws were considered as half a win for each side. Table 7 contains the ability ratings for the teams considered based on a simple Bradley–Terry model (2.1) where parameters for batting first, and playing at home have been included. More details as to the fitted model are given in the next section. Table 8 contains the specific Bradley–Terry fit, with standard errors. Also reported is the ICC ratings and rankings. We see that the systems produce similar results. South Africa enjoy a sizeable rating advantage at the top of each ranking system, with Australia, England and India occupying the next three spots. Note that the ICC rankings consider data from 36–48 months prior to the date given. Table 7. Comparison of rankings from ICC and the Bradley–Terry model. Model fitted using data from 8 March 2004 and 30 June 2014. ICC rankings as at 30 June 2014 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Table 7. Comparison of rankings from ICC and the Bradley–Terry model. Model fitted using data from 8 March 2004 and 30 June 2014. ICC rankings as at 30 June 2014 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Table 8. Initial Bradley–Terry model fitted to data described in Section 3.1. Model fitted using data from 8 March 2004 and 30 June 2014. West Indies set as reference category. Estimates given as log-probability ratios as given in (2.1) Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 Table 8. Initial Bradley–Terry model fitted to data described in Section 3.1. Model fitted using data from 8 March 2004 and 30 June 2014. West Indies set as reference category. Estimates given as log-probability ratios as given in (2.1) Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 4.2. Forecasting match outcomes In this section we consider how Bradley–Terry models can be used to forecast the outcomes of test cricket matches. We again use the data given in Section 3.1 to form our models. 4.2.1. Initial model Table 8 contains an initial Bradley–Terry model fitted to the test match data where we have included home advantage and batting first as order effects, described algebraically by a simple extension of (2.1). The West Indies have been used as our reference category. Surprisingly, and perhaps against common perception, the model has provided little evidence to suggest that batting first has an effect on the outcome of test cricket matches. Excluding the order effect of batting first (consequently this will also be removed from subsequent models considered in this paper), we now consider home advantage in more detail. Table 9 shows the win probabilities for matches by fitting a Bradley–Terry model excluding home advantage. It can be seen that Australia are favoured against all opposition, since they have been computed as the strongest team by the Bradley–Terry model. Similarly, West Indies, are considered unfavourable in matches against any opponent by the model. Table 10 gives the updated win probabilities after fitting a Bradley–Terry model including home advantage. Here we can see that Australia are no longer favourites in all possible matches. When enjoying home advantage, England, India and South Africa are all slightly favoured in matches against Australia. Similarly, West Indies are no longer considered outsiders for all matches, being favoured when playing at home to New Zealand. Omitted from study is the effect of winning the coin-toss. This is naturally another factor which could be given due consideration. Table 9. Probabilities of victory for the teams in the left hand column against teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 Table 9. Probabilities of victory for the teams in the left hand column against teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 Table 10. Probabilities of victory for the home teams in the left hand column against away teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 Table 10. Probabilities of victory for the home teams in the left hand column against away teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 4.2.2. Predicting draws So far, the probabilities computed assume that there is a winner, i.e. that there is no chance of a draw. In order to derive a method for computing the probability of a draw in any given match, the unique nature of draws in test cricket need to be considered. In many sports, such as soccer, rugby and hockey, a draw is often synonymous with a tie, an outcome whereby both teams end up with the same score. As a result, the probability of a draw may be considered as a function of the abilities of the respective teams, with similarly able teams more likely to draw with each other than teams with greatly differing abilities. In cricket, however, a draw refers to an outcome whereby the allocated time for the match has elapsed before either team could win. Therefore, it may be argued that probability of a draw does not depend as heavily on the teams’ respective abilities. Work in Allsopp & Clarke (2002) is alignment with this claim. For a team to win a match, they necessarily must take twenty wickets over the course of the match, and so any aspects of a match which influence the chance of taking wickets are likely to contribute to the probability of a draw. One such aspect is the weather, where lengthy rain interruptions are clearly conducive to the probability of a draw. Furthermore, it is often considered that cloudier conditions can increase the chances of wickets being taken, and thus reducing the chances of a draw. Weather is not considered in this paper, but would be an interesting topic for future research. Another such factor is the state of the pitch, where flatter pitches are considered to be more difficult to take wickets on. In a similar manner day–night games might also affect the number of wickets taken. Also, if the quality of bowling is weak compared to that of the batting, wickets are less likely to be taken, and thus the probability of a draw will increase. Table 11. Fit of logistic regression model to predict draw Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Table 11. Fit of logistic regression model to predict draw Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Extensions to the Bradley–Terry model have been offered to accommodate the possibility of ties. These extensions consider the probability of a tie as a function of the difference in ability between the two teams. Since in cricket there may be other confounding variables which more heavily dominate the probability of a draw, such extensions are inappropriate for computing the probabilities for draws in test match cricket, and a different method needs to be devised. We fit a logistic regression model with the home team and away team as explanatory variables, with the probability of a draw occurring as the dependent variable. The parameters estimated from this model can be interpreted as ‘draw abilities’, with each team having two separate draw abilities for whether they are playing at home or away. The probability of a draw in a match with team i at home to team j, θij, given the home draw ability of team i, $$\lambda _{i}^{(h)}$$, and the away draw ability of team j, $$\lambda _{j}^{(a)}$$ is computed as follows: \begin{align*} \theta_{ij}=\frac{\exp\left(\lambda_{i}^{(h)}+\lambda_{j}^{(a)}\right)}{1+\exp\left(\lambda_{i}^{(h)}+\lambda_{j}^{(a)}\right)}\, .\nonumber \end{align*} Table 11 contains the the parameters of the fitted logistic regression model. A Bradley–Terry model is then fitted, using all matches that did not end in a draw. The abilities from this model are then used to compute win probabilities as in previous sections. These probabilities are then scaled using the previous calculated draw probabilities so that for any combination of home team and away team, we now have probabilities for any possible outcome. Thus, we have, if the Bradley–Terry model gives abilities λ1, λ2, …, λ8 for the eight respective teams considered and δ is the order effect of playing at home, the probability of team i winning at home to team j as \begin{align*} p_{ij}^{(h)}=(1-\theta_{ij})\left[\frac{\exp(\lambda_{i}-\lambda_{j}+\delta)}{1+\exp(\lambda_{i}-\lambda_{j}+\delta)}\right] . \nonumber \end{align*} Combining the methodology of this section with Section 4.2.1, Table 12 contains the match outcome probabilities as computed on 30 June 2014 for each possible match for the data as described at the outset of the paper. These probabilities reflect historical form, and on current form the home win probabilities for India appear to be quite low. Table 12. Match outcome probabilities evaluated on 30 June 2014 for each possible match Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Table 12. Match outcome probabilities evaluated on 30 June 2014 for each possible match Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Table 12. Continued. Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 Table 12. Continued. Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 The gambling industry makes heavy use of statistical modelling to compute their odds, although the precise methodology is confidential. Figure 1 displays a comparison of the outcome probabilities computed by the model with the implied probabilities based on average bookmakers odds, taken from Oddsportal, for a selection of matches that have taken place over the past few years (the data is described in Section 3.2). This selection covers matches from 2012 to 2014 where pre-match odds are available. We can see that whilst the probabilities for home wins and away wins are strongly correlated between bookmakers' odds and the Bradley–Terry model, there is a much weaker correlation in the predicted probabilities for draws. This could be because people are less inclined to bet on draws. Further work could be conducted to investigate the potential on betting on draws as a winning strategy. Fig. 1. View largeDownload slide Plot of average bookmaker odds against Bradley–Terry-predicted odds. Fig. 1. View largeDownload slide Plot of average bookmaker odds against Bradley–Terry-predicted odds. 4.3. Rating individual players The following section presents a method by which bowlers and batsmen can be rated based on their performance. The data used in this section is the ball-by-ball test match data described in Section 3.3. 4.3.1. Wickets The quality of any given delivery in cricket will initially be judged on whether or not a wicket is taken. Thus, an appropriate method of determining the ability of a bowler is by considering the likelihood that they will take a wicket on any given delivery. Similarly, a batsman can be judged on the likelihood that they will not be dismissed on a delivery. A logistic regression model was fitted for the dataset described in Section 3.3 to compute parameters for each player on the likelihood of a wicket. The model takes the bowler and the batsman as independent variables, with the probability of a wicket occurring as the dependent variable. Thus, the estimated parameters from the fitted model are used to inform ‘wicket-taking ability’, for bowlers, and ‘wicket-preservation ability’, for batsman. These abilities are computed by the inverse logit function. Of course it can be argued that there is more to batting than this, but it is difficult to account for and model all nuances of cricket. Thus, the ‘wicket-taking ability’ for bowler i is given by \begin{align*} \phi_{i}^{(w)}=\frac{\exp{(\omega_{i})}}{1+\exp{(\omega_{i})}} \nonumber \end{align*} where ω1, ω2, …, ωn are obtained from a logistic regression. Similarly the ‘wicket-preservation ability’ for batsman i is given by \begin{align*} \psi_{i}^{(w)}=\frac{\exp{(\mu_{i})}}{1+\exp{(\mu_{i})}}, \nonumber \end{align*} where μ1, μ2, …, μm are obtained from a logistic regression. 4.3.2. Runs If a wicket is not taken, then a delivery is judged by whether the batsman scored any runs, and if so, how many. Therefore, another way of assessing a player’s quality is by analysing the amount of runs scored, if the player is a batsman, or the amount of runs conceded, if the player is a bowler. Table 13 contains the distribution of the number of runs scored per ball as observed in the dataset described in Section 3.1. The mean number of runs per ball is 0.5101, and the variance is 1.2213. Table 13. Distribution of number of runs scored per ball Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Table 13. Distribution of number of runs scored per ball Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Table 14. Rankings of batsmen for different values of α. Also given, in order of left to right, runs, number of balls faced, number of times out, batting average and strike rate Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 Table 14. Rankings of batsmen for different values of α. Also given, in order of left to right, runs, number of balls faced, number of times out, batting average and strike rate Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 We may model the number of runs scored per delivery using a negative binomial regression, with independent variables corresponding to the bowler and the batsman. The model produces parameters representing each player’s contribution to the amount of runs scored on any given delivery. These parameters can be used to inform a ‘run-scoring ability’ for batsmen and a ‘run-prevention ability’ for bowlers. For example suppose the negative binomial regression gives parameters γ1, γ2, …, γn for each of our n bowlers. The ‘run-prevention ability’ for bowler i is given by $$\phi _{i}^{(r)}=\exp {(\gamma _{i})}$$. Similarly, if we have parameters μ1, μ2, …, μm for the batsmen then the ‘run-scoring ability’ for batsman i is given by $$\psi _{i}^{(r)}=\exp {(\mu _{i})}$$. 4.3.3. Computing rating values Combining the results of Sections 4.3.1 and 4.3.2, a rating can now be assigned to each player as a function of these two abilities. We can compute a rating for a bowler’s overall ability as \begin{align} \phi_{i}=\phi_{i}^{(r)}\left(1-\phi_{i}^{(w)}\right) . \end{align} (4.1) Similarly, we can compute a rating for batsmen as \begin{align} \psi_{i}=\psi_{i}^{(r)}\left(1-\psi_{i}^{(w)}\right) . \end{align} (4.2) Thus, a low rating value indicates quality for a bowler, whereas a higher rating value indicates a better batsman. Table 15. Rankings of bowlers for different values of β. Also given, from left to right, number of runs conceded, number of balls bowled, number of wickets taken, bowling average, economy and strike rate Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 Table 15. Rankings of bowlers for different values of β. Also given, from left to right, number of runs conceded, number of balls bowled, number of wickets taken, bowling average, economy and strike rate Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 In test match cricket, a batsman’s ability to preserve his wicket is usually valued greater than his ability to score quickly, although there are certain situations in which aggressive batting is preferred. Similarly, a bowler’s ability to take wickets is treated with greater significance compared to how economical they are. There are, however, very few situations in test cricket in which more economical bowling is valued greater than wicket taking ability. Proposed here is the modification of the ratings formulae (4.1) and (4.2) to include a parameter that allows for one to place more emphasis on one of the two ability parameters used to compute the ratings. This is analogous to the proposal of Barr & Kantor (2004) who devised a criterion by which to measure the performance of batsmen in one-day cricket, allowing one to alter the emphasis placed on aggressive batting. First we consider the ratings of batsmen. By introducing a parameter 0 < α < 1, that represents the emphasis placed between the run-scoring and the wicket-preservation abilities, we gain flexibility in our rating system that allows us to consider the quality of a batsman in different scenarios. The updated final rating formula for batsman is thus given by \begin{align*} \psi_{i,\alpha}=\left(\frac{1}{2}\psi_{i}^{(r)}\right)^{\alpha}\left(1-\psi_{i}^{(w)}\right)^{1-\alpha}\, \!\!\!. \nonumber \end{align*} The run-scoring ability has been multiplied by $$\frac {1}{2}$$ to ensure that the reference point for the player rating for all values of α remains at $$\frac {1}{2}$$. A similar parameter 0 < β < 1 is introduced for bowlers, to allow for emphasis to be placed on either wicket-taking ability or run-prevention ability yielding the rating formula: \begin{align*} \phi_{i,\beta}=\left(\frac{1}{2}\phi_{i}^{(r)}\right)^{\beta}\left(1-\phi_{i}^{(w)}\right)^{1-\beta}\, \!\!\!. \nonumber \end{align*} We have imposed a qualifying criterion for a player to receive a rating, with batsman needing to have played a minimum of 10 innings to qualify for a rating, and bowlers requiring to have delivered 720 balls to qualify. Table 14 contains rankings of a selection of batsmen for different values of α and Table 15 contains rankings of bowlers for different values of β evaluated from the data described in Section 3.3. If we accept α = 0.3 and β = 0.3 to be suitable parameter values then we may argue that it appears that RJ Harris is the best bowler and CA Pujara is the best batsman. In Tables 16 and 17 we compare the player rankings we have computed with the batting and bowling averages obtained from the matches considered. Of the 15 batsman with the highest batting averages, all of them are within the top 15 for the player ratings, with the exception of AB de Villiers, who is in 16th position. Table 16. Comparison of the batting average and Bradley–Terry player ranking for players in the ‘Top 15’ for batting average Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Table 16. Comparison of the batting average and Bradley–Terry player ranking for players in the ‘Top 15’ for batting average Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Table 17. Comparison of the bowling average and Bradley–Terry player ranking for players in the ‘Top 15’ for bowling average Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Table 17. Comparison of the bowling average and Bradley–Terry player ranking for players in the ‘Top 15’ for bowling average Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Table 18. Comparison of Bradley–Terry model predictions with actual results. Bookkeeper odds also provided. HW = Home Win, AW = Away Win, D = Draw Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Table 18. Comparison of Bradley–Terry model predictions with actual results. Bookkeeper odds also provided. HW = Home Win, AW = Away Win, D = Draw Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Similarly, we can see that TT Bresnan is the only bowler who ranks in the top 15 for bowling average but not for player rating, where he ranks 17th. Interestingly, the player who is ranked 3rd, MA Starc, does not rank within the top 15 for bowling average. We can use these rankings to select a hypothetical ‘World XI’, comprising of the best players from each of the teams considered. Typically, a team will select seven batsmen and four bowlers. Furthermore, one of the batsmen must also be able to play as a wicketkeeper. Since no attempt has been made in this paper to assess the wicket-keeping ability of players, we will select the wicketkeeper with the strongest batting rating, which is AB De Villiers. Note that this is not wholly inappropriate since it is a common strategy for teams to select their wicketkeepers with great consideration for their batting ability. Adding to our selection the top 6 ranked batsmen and the top 4 ranked bowlers, we have a World XI as CA Pujara, HM Amla, AN Cook, IR Bell, M Vijay, MJ Clarke, AB De Villiers, VD Philander, MA Starc, RA Jadeja and RJ Harris. 4.4. Using player ratings to predict outcomes We have seen how player ratings can be used either to rank players for general interest, or to inform selection decisions for teams. It will now be shown how these ratings could be used to predict the outcome of a match, given the players which have been selected. 4.4.1. Deriving team ratings from player ratings First, we will look at how the player ratings given in Section 4.3.3 can be used to form overall ratings for each team. We propose that each team have a separate rating for bowling and batting. From this point onwards, any reference to a batsman’s rating refers to the rating produced from taking α = 0.3 in equation (4.1), and, similarly all reference to bowling ratings refer to the ratings produced where β = 0.3 in equation (4.2). These are taken as example values. One could examine approaches to estimate these parameters, or to update them during over time. This is a possible area for future investigation. Table 18. Continued. Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Table 18. Continued. Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Given that any player in a team may be expected to bat, we have considered a team’s batting rating, BATteam to simply be the sum of each player’s batting rating. If a player has not reached the qualification criterion of having played 10 innings, they are not assigned a batting rating and therefore do not contribute to the team rating. It is, however, inappropriate to compute the bowling rating of a team in such a manner. Typically four players in a team will be expected to bowl the majority of overs in a match, with other ‘part-time’ bowlers occasionally alleviating the workload on the main bowlers. Therefore, we postulate that a team’s bowling rating should be based only on the top four individual bowler ratings within their team. We define a team’s bowling rating as follows: \begin{align*} BOWL_{team}=\sum_{i=1}^{4}\frac{1}{4\phi_{i,0.3}}, \nonumber \end{align*} where i = 1, 2, 3, 4 are indices representing the four players with the best bowling rating in a selected team. Since a low rating indicates greater quality for bowlers, it is necessary to compute the team bowling rating as a sum of reciprocal values of the individual bowler ratings, so that a higher rating indicates a better bowling attack. Multiplying by 1/4 simply ensures that the reference value for the ratings of individual bowlers remains at 1/2. 4.4.2. Predicting outcomes based on player selection We have fitted a Bradley–Terry model to the results of the matches in the dataset described in Section 3.1, with home advantage, batting quality and bowling quality each being considered as ‘order effects’, with a parameter for random effects also included. The model is described below. For teams A and B with respective team batting abilities BATA and BATB, and respective team bowling abilities BOWLA and BOWLB, the probability of team A winning at home to team B is given by \begin{align} p_{AB}^{(h)}=\frac{\exp{(\delta_{bat}(BAT_{A}-BAT_{B})+\delta_{bowl}(BOWL_{A}-BOWL_{B})+\delta_{home})}}{1+\exp{(\delta_{bat}(BAT_{A}-BAT_{B})+\delta_{bowl}(BOWL_{A}-BOWL_{B})+\delta_{home})}}, \end{align} (4.3) where δhome is the order effect representing the advantage from playing at home, and δbat and δbowl are order effects representing the influence that team batting and bowling abilities have on the expected outcome of a match, respectively. Table 18 contains the Bradley–Terry predicted match outcomes and bookmakers’ predictions for matches between 2013 and 2014 where pre-match bookmakers’ odds were available. We make the following remarks. The Bradley–Terry model predicts the correct outcome more often than the outcome suggested from the bookmakers’ average odds, although it must be recognised that the bookmakers also perform very well. There are examples when both the bookmakers and the model get it wrong: see games on 1 August 2013, 21 August 2013 and 14 October 2013 for example. An interesting difference can be observed for the game held on 21 November 2013. The bookmakers have a more even spread across all possible outcomes, whilst the Bradley–Terry model gives a more pronounced (correct) prediction for a home win. A big discrepancy occurs for the game held on 3 December 2013, where the Bradley–Terry model correctly predicts a draw whilst the bookmakers placed a lot of confidence in a win for the home side. Indeed on inspection, for the games where the bookmakers have struggled to offer a definite outcome (by putting equal probabilities on a win, lose and a draw), the Bradley–Terry model seems more able to pick out an outcome, and this is often the correct outcome. 5. Conclusion The objective of this paper was to use Bradley–Terry models to analyse various aspects of test cricket. The main areas of investigation were ranking teams, predicting match outcomes and rating individual players. The Bradley–Terry model was used to devise an alternative ranking system for test cricket. Aside from the obvious computational differences, the consideration of home advantage distinguished this system from that employed by the ICC. The strong correlation between the results of this system and the official ICC rankings indicates that it provides an accurate reflection of the abilities of the respective teams at any given point. The Bradley–Terry model was also used to produce a system that predicts the outcome of test matches based on previous results. The predicted outcomes were compared to bookmakers’ odds, showing a strong correlation between the predicted probabilities of home wins and away wins, but only a moderate correlation with the predicted probabilities of draws. We also produced an individual player rating system for batsmen and bowlers. The player ratings derived were then used to inform batting and bowling ratings for a team, given the players that were selected. A Bradley–Terry model was then used to investigate whether these team ratings could be used to predict matches, successfully predicting the result of 8 out of 10 test matches. More generally, this paper has shown the potential for Bradley–Terry models in wider settings. Problems of rating, ranking and evaluating can be tackled by fitting such models and these applications have wide-bearing use. References Agresti , A. 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Google Scholar CrossRef Search ADS Bradley , R. A. & Terry , M. E. ( 1952 ) Rank analysis of incomplete block designs: I. The method of paired comparisons , Biometrika, pp. 324 -- 345 . Cattelan , M. , Varin , C. & Firth , D. ( 2013 ) Dynamic Bradley–Terry modelling of sports tournaments . J. R. Stat. Soc.: Series C (Appl. Stat.), 62 , 135 -- 150 . Google Scholar CrossRef Search ADS Davis , J. , Perera , H. & Swartz , T. B. ( 2015 ) A simulator for twenty20 cricket . Aust. & N. Z. J. Stat., 57 , 55 -- 71 . Google Scholar CrossRef Search ADS Ford , L. R. ( 1957 ) Solution of a ranking problem from binary comparisons . Am. Math. Mon., 28 -- 33 . Hallinan , S. E. ( 2005 ) Paired comparison models for rankinational soccer teams . PhD Thesis , Worcester Polytechnic Institute . Hopkins , J. ( 1954 ) Incomplete block rank analysis: some taste test results . Biometrics, 10 , 391 -- 399 . Google Scholar CrossRef Search ADS Jech , T. ( 1983 ) The ranking of incomplete tournaments: a mathematician’s guide to popular sports . Am. Math. Mon., 246 -- 266 . Li , L. & Kim , K. ( 2000 ) Estimating driver crash risks based on the extended Bradley–Terry model: an induced exposure method . J. R. Stat. Soc.: Series A (Stat. Soc.), 163 , 227 -- 240 . Google Scholar CrossRef Search ADS Perera , H. P. & Swartz , T. B. ( 2012 ) Resource estimation in t20 cricket . IMA J. Manag. Mathematics, 24 , 337 -- 347 . Google Scholar CrossRef Search ADS Scarf , P. & Akhtar , S. ( 2011 ) An analysis of strategy in the first three innings in test cricket: declaration and the follow-on . J. Operational Res. Soc., 62 , 1931 -- 1940 . Google Scholar CrossRef Search ADS Scarf , P. & Shi , X. ( 2005 ) Modelling match outcomes and decision support for setting a final innings target in test cricket . IMA J. Manag. Mathematics, 16 , 161 -- 178 . Google Scholar CrossRef Search ADS Scarf , P. , Shi , X. & Akhtar , S. ( 2011 ) On the distribution of runs scored and batting strategy in test cricket . J. R. Stat. Soc.: Series A (Stat. Soc.), 174 , 471 -- 497 . Google Scholar CrossRef Search ADS Simons , G. & Yao , Y.-C. ( 1999 ) Asymptotics when the number of parameters tends to infinity in the Bradley–Terry model for paired comparisons . Annals. Stat., 27 , 1041 -- 1060 . Google Scholar CrossRef Search ADS Strauss , D. & Arnold , B. C. ( 1987 ) The rating of players in racquetball tournaments . Appl. Stat., 163 -- 173 . Turner , H. & Firth , D. ( 2012 ) Bradley–Terry models in R: the BradleyTerry2 package . J. Stat. Software, 48 , 1 -- 21 . Google Scholar CrossRef Search ADS Varin , C. , Cattelan , M. & Firth , D. ( 2013 ) Statistical modelling of citation exchange among statistics journals . arXiv: 1312.1794 . Whiting , M. J. , Stuart-Fox , D. M. , O’Connor , D. , Firth , D. , Bennett , N. C. & Blomberg , S. P. ( 2006 ) Ultraviolet signals ultra-aggression in a lizard . Anim. Behav. 72 , 353 -- 363 . Google Scholar CrossRef Search ADS Zermelo , E. ( 1929 ) Über den begriff der definitheit in der axiomatik . Fundamenta Mathematicae, 14 , 339 -- 344 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Management Mathematics Oxford University Press

# Using Bradley–Terry models to analyse test match cricket

, Volume Advance Article – Jan 8, 2018
21 pages

Publisher
Oxford University Press
ISSN
1471-678X
eISSN
1471-6798
D.O.I.
10.1093/imaman/dpx013
Publisher site
See Article on Publisher Site

### Abstract

Abstract In this paper we investigate the use of Bradley–Terry models to analyse test match cricket. Specifically, we develop a new and alternative team ranking and compare our rankings with those produced by the International Cricket Council, we forecast the outcomes of a selected number of test cricket matches and show that our predictions perform well compared to bookmaker predictions. We offer ratings of individual players and use these ratings to predict the results of some recent matches. The general purpose of the paper is to illustrate the potential of Bradley–Terry models, which are effectively models of P(i is preferred to j), and thus can be applied in a number of settings where there are paired comparisons. Popular applications include analysing taste test experiments and modelling sports competitions. More creative examples of applications include statistical modelling of citation exchange among statistics journals, predicting the fighting ability of lizards and estimating driver crash risks. 1. Introduction Suppose that there are a set of entities that we wish to consider according to some common attribute. Pairs of these entities can be compared with respect to some quantifiable aspect of this attribute. Such comparisons are generally referred to as ‘paired comparisons’. An example of such a comparison is a taste test, where a judge tastes two specimens before declaring a preference for a particular specimen. The Bradley–Terry model is a useful model that was developed with a view to analysing the results from a set of paired comparisons. Such models can produce scores for the entities based on a set of results, and can also be used to make predictions for the results of future comparisons. Furthermore, if we instead consider a comparison as a ‘contest’ between two competitors, a variety of possible applications becomes apparent, most notably in sport. In this paper we investigate the use of Bradley–Terry models to analyse test match cricket. There are a number of other papers which have analysed at least one version of cricket. Some recent papers include Scarf & Shi (2005), Scarf et al. (2011), Scarf & Akhtar (2011), Akhtar & Scarf (2012), Perera & Swartz (2012), Davis et al. (2015) and Akhtar et al. (2015). Each of these papers also has numerous references of interest to those in analysing cricket. Specifically, we do the following: We develop a new and alternative team ranking and compare our rankings with those produced by the International Cricket Council (ICC). We forecast the outcomes of a selected number of matches and compare our predictions with bookmaker odds. We offer a method to obtain ratings of individual players. The purpose of the paper is to illustrate the potential of Bradley–Terry models, which are effectively models of P(i is preferred to j). This is an interesting construct in itself, with many potential applications. The Bradley–Terry model (Bradley & Terry, 1952) is named after Ralph A. Bradley and Milton E. Terry, who devised the method in 1952. However, as noted in Simons & Yao(1999), the method was discovered independently in Zermelo (1929), Ford (1957) and Jech (1983). Agresti (2014) discusses the Bradley–Terry model and presents a simple example of how it can be used to rank baseball teams. Applications of the Bradley–Terry model are both plentiful and diverse. Popular applications include analysing taste test experiments (Hopkins, 1954; Bliss et al., 1956) and modelling sports competitions, with racquetball (Strauss & Arnold, 1987), soccer (Hallinan, 2005) and basketball (Cattelan et al., 2013) all receiving attention in this regard. More creative examples of applications include statistical modelling of citation exchange among statistics journals (Varin et al., 2013), predicting the fighting ability of lizards (Whiting et al., 2006) and estimating driver crash risks (Li & Kim, 2000). The structure of the paper is as follows. In Section 2 we offer a brief description of Bradley–Terry models and some of the areas in which they have been successfully applied thus far. In Section 3 we describe the three sources of data we have used to model the aspects of test match cricket, as described in the previous paragraph. Our analyses are described in Section 4 and we conclude our findings in Section 5. We deliberately leave technical content to a minimum in an aim to improve the readability of the paper. Cricket is a notoriously difficult game to model, as it contains many subtle nuances and variables. Accordingly, we believe it is appropriate as a test-bed for the potential of Bradley–Terry models. More generally, we show the capability of these models for predicting contests and ranking individuals. These are useful applications with wide-bearing implications. 2. Bradley–Terry models 2.1. Description For any pair of entities, say i and j, the Bradley–Terry model takes an input as some measure of quality for the respective entities and computes the probability that i is preferred to j. For a system of n entities, the model introduces parameters π1, π2, …, πn, which can be interpreted as some measure of quality of the respective entries, such that \begin{align*} p_{ij}=P(i \;\textrm{is preferred to } j)=\frac{\pi_{i}}{\pi_{i}+\pi_{j}}, \end{align*} where $$\sum _{i=1}^{n}\pi _{i}=1.$$ It follows that \begin{align*} \log \frac{p_{ij}}{p_{ji}}=\lambda_{i}-\lambda_{j}, \end{align*} where $$\lambda _{i}=\log \pi _{i}.$$ For n parameters λ1, λ2, …, λn then \begin{align*} p_{ij}=\frac{\exp(\lambda_{i}-\lambda_{j})}{1+\exp(\lambda_{i}-\lambda_{j})}\, . \end{align*} In many paired comparison experiments there is often a factor that, independent of the attributes of the respective entities, influences the outcome of the experiment. In a taste test, this could refer to some advantage gained by the first sample tasted, or could refer to a perceived home advantage in a sports contest. It is possible to include such factors into a Bradley–Terry model. Let δ ⩾ 0 be the advantage gained by entity i from the external effect in question. Then we may write \begin{align} \log \frac{p_{ij}}{p_{ji}}=\lambda_{i}-\lambda_{j}+\delta\! . \end{align} (2.1) In order to fit our Bradley–Terry models we use the BradleyTerry2 package in R, see Turner & Firth (2012). 3. Data In this paper we consider three different sets of data, each of which are described below. The data to be considered are (i) results of test matches since 2004, (ii) bookmakers odds for selected test matches and (iii) ball-by-ball data for selected test matches. The data are for a relatively long period of time. In such a period the relative abilities of teams may change. In this paper we do not pursue the fitting of a ‘dynamic’ Bradley–Terry model but note that there has been recent work in this area, see Cattelan et al. (2013). We also note that the ICC rankings have remained fairly consistent throughout the time period considered, perhaps making this less of an issue. Table 1. Extract of data of test match results taken from ESPN Cricinfo Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 Table 1. Extract of data of test match results taken from ESPN Cricinfo Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 Home Away Result Ground Bat First Start Date Sri Lanka Australia 0 Galle 0 8 March 2004 New Zealand South Africa 0.5 Hamilton 0 10 March 2004 West Indies England 0 Kingston 1 11 March 2004 Sri Lanka Australia 0 Kandy 0 16 March 2004 New Zealand South Africa 1 Auckland 0 18 March 2004 West Indies England 0 Port of Spain 1 19 March 2004 Sri Lanka Australia 0 Colombo (SSC) 0 24 March 2004 New Zealand South Africa 0 Wellington 1 26 March 2004 Pakistan India 0 Multan 0 28 March 2004 West Indies England 0 Bridgetown 1 1 April 2004 Pakistan India 1 Lahore 0 5 April 2004 West Indies England 0.5 St John’s 1 10 April 2004 Pakistan India 0 Rawalpindi 1 13 April 2004 3.1. Test match results We consider data collected from ESPN Cricinfo which comprises of the results of 442 test matches between 8 March 2004 and 30 June 2014. For each match, the data recorded contains the home team, the away team, the outcome (indicated as 1 for a home win, 0 for an away win and 0.5 for a draw), which ground the match was played at, which team batted first (1 if the home team batted first, 0 for the away team) and the start date of the match. A one-off test match between an ICC World XI and Australia in October 2005 has been excluded from the data since this is the only test that an ICC World XI has played and it is inappropriate to compare them with regular test playing nations. Table 1 contains an example extract of the data taken from ESPN Cricinfo. Due to security concerns, Pakistan has not hosted a test match since February 2009, with their ‘home’ matches instead being played at neutral venues. The majority of these matches have taken place in the United Arab Emirates and, given the regularity with which they have played there and the relative similarity to Pakistani conditions, we have decided to treat Pakistan as the home side in such matches. There was, however, a test series played between Australia and Pakistan in England in 2009, but Pakistan was officially stated to be the ‘home’ team. Bangladesh and Zimbabwe are often perceived to be far weaker than the eight other test-playing nations. As shown in Table 2, they have played fewer games than the other sides, and have worse records than the other teams, between them winning just 7 of their 75 test matches included in the data, four of which have come in matches between the two sides. As a result of this gulf in quality, the stronger teams often field weakened line-ups when playing against either of these sides and so results in these matches are often not a reflection of the true ability of the teams. Table 2. Summary of team performances over 442 test matches between 2004 and 2014 Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% Table 2. Summary of team performances over 442 test matches between 2004 and 2014 Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% Team Matches Wins Losses Draws Win % Loss % Draw % Australia 118 69 28 21 58.47% 23.73% 17.80% Bangladesh 55 4 42 9 7.27% 76.36% 16.36% England 131 59 35 37 45.04% 26.72% 28.24% India 107 46 28 33 42.99% 26.17% 30.84% New Zealand 87 24 39 24 27.59% 44.83% 27.59% Pakistan 81 26 33 22 32.10% 40.74% 27.16% South Africa 99 47 28 24 47.47% 28.28% 24.24% Sri Lanka 93 36 29 28 38.71% 31.18% 30.11% West Indies 93 14 50 29 15.05% 53.76% 31.18% Zimbabwe 20 3 16 1 15.00% 80.00% 5.00% The most notable example of this is regarding Bangladesh’s tour of West Indies in 2009. A contract dispute led to several leading West Indies players boycotting the match and so their side featured numerous uncapped players. Bangladesh won both test matches in the series and these remain their only victories over a major international team. After consideration, matches involving Bangladesh and Zimbabwe have been excluded from the analysis, leaving a dataset of 372 test matches played between eight major international teams. Initial analysis showed that parameter estimates concerning these teams suffered from huge standard errors, rendering them unusable. Table 3 gives a summary of team performances excluding matches involving Bangladesh and Zimbabwe. Table 3. Summary of team performances over 372 test matches excluding matches involving Bangladesh and Zimbabwe Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% Table 3. Summary of team performances over 372 test matches excluding matches involving Bangladesh and Zimbabwe Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% Team Matches Wins Losses Draws Win% Loss% Draw% Australia 116 67 28 21 57.76% 24.14% 18.10% England 125 53 35 37 42.40% 28.00% 29.60% India 99 39 28 32 39.39% 28.28% 32.32% New Zealand 74 14 39 21 18.92% 52.70% 28.38% Pakistan 76 22 32 22 28.95% 42.11% 28.95% South Africa 93 41 28 24 44.09% 30.11% 25.81% Sri Lanka 78 23 29 26 29.49% 37.18% 33.33% West Indies 83 8 48 27 9.64% 57.83% 32.53% 3.2. Bookmakers’ odds Data regarding the average pre-match odds of all test matches between March 2012 and June 2014 were taken from Oddsportal. The odds that were taken were those of a home win, away win and a draw. Average odds were gathered for a total of 64 matches. 3.3. Ball-by-ball data Further data regarding test matches was collected from Cricsheet. The data consists of detailed information on all test matches played since 2009. Basic information about each match, such as the competing teams, the location, the dates and the result were provided along with information about each delivery (a delivery or ball in cricket is a single action of bowling a cricket ball toward the batsman) in the match, including the bowler, the batsman, how many runs were scored and whether or not a wicket was taken. The information extracted from the data comprised of ball-by-ball details of every match played between Australia, England, India and South Africa between December 2009 and August 2013. An extract of the data is given in Table 4. Using this data, it was possible to compute traditionally referenced statistics for cricket players such as batting and bowling averages for each player. Tables 5 and 6 give these statistics for the top 15 run scorers and wicket takers, respectively, from the matches considered. Table 4. Extract of ball-by-ball data Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Table 4. Extract of ball-by-ball data Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Match Innings Batting Team Bowler Batsman Runs Wicket SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa JM Anderson GC Smith 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 1 0 SA Eng 2009-12-16 1 South Africa SCJ Broad GC Smith 0 1 SA Eng 2009-12-16 1 South Africa SCJ Broad HM Amla 3 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 SA Eng 2009-12-16 1 South Africa SCJ Broad AG Prince 0 0 Table 5. Batting summary statistics for the top 15 run-scorers for data considered Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Table 5. Batting summary statistics for the top 15 run-scorers for data considered Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Batsman Runs Balls Faced Innings Times Out Batting Av. Strike Rate AN Cook 2435 5333 44 41 59.39 45.66 MJ Clarke 2263 3668 44 41 55.20 61.70 IR Bell 2024 4200 41 36 56.22 48.19 KP Pietersen 2015 3354 40 40 50.38 60.08 HM Amla 1899 3634 25 21 90.43 52.26 IJL Trott 1537 3391 41 40 38.43 45.33 SR Tendulkar 1480 2713 36 33 44.85 54.55 SR Watson 1353 2553 36 37 36.57 53.00 MJ Prior 1347 2216 38 33 40.82 60.79 MEK Hussey 1317 2418 28 27 48.78 54.47 JH Kallis 1225 2303 25 23 53.26 53.19 GC Smith 1169 2175 25 24 48.71 53.75 MS Dhoni 1048 1902 32 27 38.81 55.10 RT Ponting 983 1789 27 24 40.96 54.95 CA Pujara 933 1701 16 11 84.82 54.85 Table 6. Bowling summary statistics for the top 15 run-scorers for data considered Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 Table 6. Bowling summary statistics for the top 15 run-scorers for data considered Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 Bowler Runs Conc. Balls Bowled Wickets Bowling Av. Economy Strike Rate JM Anderson 3072 6156 109 28.18 2.99 56.48 GP Swann 3338 6779 103 32.41 2.95 65.82 PM Siddle 2255 4522 79 28.54 2.99 57.24 SCJ Broad 2105 4401 74 28.45 2.87 59.47 DW Steyn 1618 2951 67 24.15 3.29 44.04 M Morkel 1545 3110 56 27.59 2.98 55.54 R Ashwin 1884 3878 55 34.25 2.91 70.51 NM Lyon 1760 3293 48 36.67 3.21 68.6 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 RJ Harris 1019 2102 45 22.64 2.91 46.71 Z Khan 1164 2322 41 28.39 3.01 56.63 TT Bresnan 1171 2315 39 30.03 3.03 59.36 PP Ojha 1198 2863 36 33.28 2.51 79.53 I Sharma 1806 3414 34 53.12 3.17 100.41 MG Johnson 1265 2049 33 38.33 3.70 62.09 4. Analysis 4.1. An alternative team ranking system The ICC, the governing body of international cricket, currently uses a method devised by David Kendix to rank its teams in each form of the game. Although the precise methodology that is used to compute these rankings is not freely available, several aspects of the method are known. The rankings are based on a rating points value assigned to each team depending on their results. These take account of the results of matches in the past 3–4 years, with greater weighting given to more recent matches. Points are assigned after each match, with the amount of points available dependent on the respective ratings of the two teams. Furthermore, a team earns bonus points for winning a series of games (such as the Ashes). A simple Bradley–Terry model, of the form (2.1) was fitted to estimate the ‘ability’ of each team based on the data described in Section 3.1. We set West Indies as our reference category and thus this team is nominally given an ability rating of 0. Note that any team may be set to be the reference category, the results will remain the same. Matches that resulted in draws were considered as half a win for each side. Table 7 contains the ability ratings for the teams considered based on a simple Bradley–Terry model (2.1) where parameters for batting first, and playing at home have been included. More details as to the fitted model are given in the next section. Table 8 contains the specific Bradley–Terry fit, with standard errors. Also reported is the ICC ratings and rankings. We see that the systems produce similar results. South Africa enjoy a sizeable rating advantage at the top of each ranking system, with Australia, England and India occupying the next three spots. Note that the ICC rankings consider data from 36–48 months prior to the date given. Table 7. Comparison of rankings from ICC and the Bradley–Terry model. Model fitted using data from 8 March 2004 and 30 June 2014. ICC rankings as at 30 June 2014 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Table 7. Comparison of rankings from ICC and the Bradley–Terry model. Model fitted using data from 8 March 2004 and 30 June 2014. ICC rankings as at 30 June 2014 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Team ICC Rating ICC Ranking Bradley–Terry Ranking South Africa 127 1 1 Australia 115 2 2 India 112 3 4 England 107 4 3 Pakistan 100 5 5 Sri Lanka 89 6 7 New Zealand 87 7 6 West Indies 87 8 8 Table 8. Initial Bradley–Terry model fitted to data described in Section 3.1. Model fitted using data from 8 March 2004 and 30 June 2014. West Indies set as reference category. Estimates given as log-probability ratios as given in (2.1) Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 Table 8. Initial Bradley–Terry model fitted to data described in Section 3.1. Model fitted using data from 8 March 2004 and 30 June 2014. West Indies set as reference category. Estimates given as log-probability ratios as given in (2.1) Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 Variable Estimate Std. Error p Australia 1.76447 0.32832 < 0.001 England 1.28945 0.30601 < 0.001 India 1.27814 0.32324 < 0.001 New Zealand 0.22980 0.34062 0.4999 Pakistan 0.87838 0.34854 < 0.05 South Africa 1.33587 0.32945 < 0.001 Sri Lanka 0.89390 0.34861 < 0.05 Home 0.55484 0.11635 < 0.001 Batting first 0.04885 0.11655 0.6751 4.2. Forecasting match outcomes In this section we consider how Bradley–Terry models can be used to forecast the outcomes of test cricket matches. We again use the data given in Section 3.1 to form our models. 4.2.1. Initial model Table 8 contains an initial Bradley–Terry model fitted to the test match data where we have included home advantage and batting first as order effects, described algebraically by a simple extension of (2.1). The West Indies have been used as our reference category. Surprisingly, and perhaps against common perception, the model has provided little evidence to suggest that batting first has an effect on the outcome of test cricket matches. Excluding the order effect of batting first (consequently this will also be removed from subsequent models considered in this paper), we now consider home advantage in more detail. Table 9 shows the win probabilities for matches by fitting a Bradley–Terry model excluding home advantage. It can be seen that Australia are favoured against all opposition, since they have been computed as the strongest team by the Bradley–Terry model. Similarly, West Indies, are considered unfavourable in matches against any opponent by the model. Table 10 gives the updated win probabilities after fitting a Bradley–Terry model including home advantage. Here we can see that Australia are no longer favourites in all possible matches. When enjoying home advantage, England, India and South Africa are all slightly favoured in matches against Australia. Similarly, West Indies are no longer considered outsiders for all matches, being favoured when playing at home to New Zealand. Omitted from study is the effect of winning the coin-toss. This is naturally another factor which could be given due consideration. Table 9. Probabilities of victory for the teams in the left hand column against teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 Table 9. Probabilities of victory for the teams in the left hand column against teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 A E I NZ P SA SL WI A 0.608 0.611 0.804 0.719 0.600 0.701 0.845 E 0.3922 0.5032 0.7255 0.6223 0.4916 0.6016 0.7784 I 0.3892 0.4968 0.7230 0.6193 0.4885 0.5986 0.7763 NZ 0.1962 0.2745 0.2770 0.3840 0.2679 0.3636 0.5707 P 0.2815 0.3777 0.3807 0.6160 0.3699 0.4782 0.6808 SA 0.4002 0.5084 0.5115 0.7321 0.6301 0.6096 0.7842 SL 0.2994 0.3984 0.4014 0.6364 0.5218 0.3904 0.000 WI 0.1552 0.2216 0.2237 0.4293 0.3192 0.2158 0.3006 Table 10. Probabilities of victory for the home teams in the left hand column against away teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 Table 10. Probabilities of victory for the home teams in the left hand column against away teams on top row. A: Australia, E: England, I: India, NZ: New Zealand, P: Pakistan, SA: South Africa, SL: Sri Lanka, WI: West Indies A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 A E I NZ P SA SL WI A 0.7376 0.7415 0.8901 0.8110 0.7290 0.8064 0.9115 E 0.5177 0.6393 0.8334 0.7261 0.6244 0.7201 0.8643 I 0.5128 0.6301 0.8306 0.7221 0.6197 0.7161 0.8619 NZ 0.2716 0.3763 0.3810 0.4793 0.3660 0.4719 0.6886 P 0.4130 0.5324 0.5374 0.7663 0.5214 0.6278 0.8067 SA 0.5287 0.6448 0.6494 0.8394 0.7348 0.7289 0.8694 SL 0.4202 0.5398 0.5447 0.7716 0.6415 0.5288 0.8113 WI 0.2265 0.3216 0.3259 0.5772 0.4197 0.3120 0.4125 4.2.2. Predicting draws So far, the probabilities computed assume that there is a winner, i.e. that there is no chance of a draw. In order to derive a method for computing the probability of a draw in any given match, the unique nature of draws in test cricket need to be considered. In many sports, such as soccer, rugby and hockey, a draw is often synonymous with a tie, an outcome whereby both teams end up with the same score. As a result, the probability of a draw may be considered as a function of the abilities of the respective teams, with similarly able teams more likely to draw with each other than teams with greatly differing abilities. In cricket, however, a draw refers to an outcome whereby the allocated time for the match has elapsed before either team could win. Therefore, it may be argued that probability of a draw does not depend as heavily on the teams’ respective abilities. Work in Allsopp & Clarke (2002) is alignment with this claim. For a team to win a match, they necessarily must take twenty wickets over the course of the match, and so any aspects of a match which influence the chance of taking wickets are likely to contribute to the probability of a draw. One such aspect is the weather, where lengthy rain interruptions are clearly conducive to the probability of a draw. Furthermore, it is often considered that cloudier conditions can increase the chances of wickets being taken, and thus reducing the chances of a draw. Weather is not considered in this paper, but would be an interesting topic for future research. Another such factor is the state of the pitch, where flatter pitches are considered to be more difficult to take wickets on. In a similar manner day–night games might also affect the number of wickets taken. Also, if the quality of bowling is weak compared to that of the batting, wickets are less likely to be taken, and thus the probability of a draw will increase. Table 11. Fit of logistic regression model to predict draw Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Table 11. Fit of logistic regression model to predict draw Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Variable Estimate Std. Error p Australia -2.39260 0.53607 < 0.001 England -1.15563 0.39512 < 0.01 India -1.03145 0.40315 < 0.05 New Zealand -0.61504 0.44865 0.17042 Pakistan -0.99483 0.51802 0.05480 South Africa -1.95886 0.50228 < 0.001 Sri Lanka -0.86322 0.45881 0.05991 West Indies -0.81985 0.45231 0.06990 Extensions to the Bradley–Terry model have been offered to accommodate the possibility of ties. These extensions consider the probability of a tie as a function of the difference in ability between the two teams. Since in cricket there may be other confounding variables which more heavily dominate the probability of a draw, such extensions are inappropriate for computing the probabilities for draws in test match cricket, and a different method needs to be devised. We fit a logistic regression model with the home team and away team as explanatory variables, with the probability of a draw occurring as the dependent variable. The parameters estimated from this model can be interpreted as ‘draw abilities’, with each team having two separate draw abilities for whether they are playing at home or away. The probability of a draw in a match with team i at home to team j, θij, given the home draw ability of team i, $$\lambda _{i}^{(h)}$$, and the away draw ability of team j, $$\lambda _{j}^{(a)}$$ is computed as follows: \begin{align*} \theta_{ij}=\frac{\exp\left(\lambda_{i}^{(h)}+\lambda_{j}^{(a)}\right)}{1+\exp\left(\lambda_{i}^{(h)}+\lambda_{j}^{(a)}\right)}\, .\nonumber \end{align*} Table 11 contains the the parameters of the fitted logistic regression model. A Bradley–Terry model is then fitted, using all matches that did not end in a draw. The abilities from this model are then used to compute win probabilities as in previous sections. These probabilities are then scaled using the previous calculated draw probabilities so that for any combination of home team and away team, we now have probabilities for any possible outcome. Thus, we have, if the Bradley–Terry model gives abilities λ1, λ2, …, λ8 for the eight respective teams considered and δ is the order effect of playing at home, the probability of team i winning at home to team j as \begin{align*} p_{ij}^{(h)}=(1-\theta_{ij})\left[\frac{\exp(\lambda_{i}-\lambda_{j}+\delta)}{1+\exp(\lambda_{i}-\lambda_{j}+\delta)}\right] . \nonumber \end{align*} Combining the methodology of this section with Section 4.2.1, Table 12 contains the match outcome probabilities as computed on 30 June 2014 for each possible match for the data as described at the outset of the paper. These probabilities reflect historical form, and on current form the home win probabilities for India appear to be quite low. Table 12. Match outcome probabilities evaluated on 30 June 2014 for each possible match Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Table 12. Match outcome probabilities evaluated on 30 June 2014 for each possible match Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Home Away Home Away Draw A E 0.6155 0.2551 0.1294 A I 0.6974 0.1833 0.1193 A NZ 0.8648 0.1022 0.330 A P 0.7265 0.1993 0.741 A SA 0.6067 0.2802 0.1131 A SL 0.7481 0.1393 0.1126 A WI 0.8414 0.386 0.1200 E A 0.5449 0.1884 0.2667 E I 0.5132 0.1324 0.3544 E NZ 0.7689 0.1096 0.1215 E P 0.6357 0.1193 0.2449 E SA 0.4261 0.2332 0.3407 E SL 0.5534 0.1071 0.3395 E WI 0.6106 0.335 0.3558 I A 0.5074 0.2197 0.2729 I E 0.4103 0.2064 0.3833 I NZ 0.7561 0.1190 0.1249 I P 0.5694 0.1798 0.2508 I SA 0.4061 0.2460 0.3478 I SL 0.5271 0.1263 0.3466 I WI 0.6004 0.365 0.3631 NZ A 0.3093 0.2999 0.3908 NZ E 0.2425 0.2425 0.5151 NZ I 0.2887 0.2194 0.4919 NZ P 0.3896 0.2465 0.3639 NZ SA 0.2335 0.2896 0.4769 NZ SL 0.3390 0.1855 0.4755 NZ WI 0.4474 0.591 0.4935 (continued). Table 12. Continued. Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 Table 12. Continued. Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 Home Away Home Away Draw P A 0.3892 0.2525 0.3582 P E 0.3206 0.1991 0.4803 P I 0.3628 0.1800 0.4572 P NZ 0.6402 0.1847 0.1751 P SA 0.2875 0.2701 0.4424 P SL 0.4049 0.1540 0.4410 P WI 0.4852 0.560 0.4588 SA A 0.6981 0.2047 0.971 SA E 0.6020 0.2468 0.1512 SA I 0.6892 0.1711 0.1397 SA NZ 0.8764 0.843 0.393 SA P 0.6935 0.2189 0.876 SA SL 0.7390 0.1289 0.1320 SA WI 0.8268 0.327 0.1405 SL A 0.4040 0.2440 0.3521 SL E 0.3122 0.2142 0.4736 SL I 0.3737 0.1757 0.4506 SL NZ 0.6825 0.1462 0.1713 SL P 0.4712 0.2023 0.3265 SL SA 0.3125 0.2517 0.4358 SL WI 0.5057 0.421 0.4522 WI A 0.1939 0.4746 0.3315 WI E 0.1697 0.3794 0.4510 WI I 0.1989 0.3729 0.4281 WI NZ 0.4539 0.3874 0.1587 WI P 0.3099 0.3834 0.3067 WI SA 0.1452 0.4413 0.4135 WI SL 0.2489 0.3390 0.4121 The gambling industry makes heavy use of statistical modelling to compute their odds, although the precise methodology is confidential. Figure 1 displays a comparison of the outcome probabilities computed by the model with the implied probabilities based on average bookmakers odds, taken from Oddsportal, for a selection of matches that have taken place over the past few years (the data is described in Section 3.2). This selection covers matches from 2012 to 2014 where pre-match odds are available. We can see that whilst the probabilities for home wins and away wins are strongly correlated between bookmakers' odds and the Bradley–Terry model, there is a much weaker correlation in the predicted probabilities for draws. This could be because people are less inclined to bet on draws. Further work could be conducted to investigate the potential on betting on draws as a winning strategy. Fig. 1. View largeDownload slide Plot of average bookmaker odds against Bradley–Terry-predicted odds. Fig. 1. View largeDownload slide Plot of average bookmaker odds against Bradley–Terry-predicted odds. 4.3. Rating individual players The following section presents a method by which bowlers and batsmen can be rated based on their performance. The data used in this section is the ball-by-ball test match data described in Section 3.3. 4.3.1. Wickets The quality of any given delivery in cricket will initially be judged on whether or not a wicket is taken. Thus, an appropriate method of determining the ability of a bowler is by considering the likelihood that they will take a wicket on any given delivery. Similarly, a batsman can be judged on the likelihood that they will not be dismissed on a delivery. A logistic regression model was fitted for the dataset described in Section 3.3 to compute parameters for each player on the likelihood of a wicket. The model takes the bowler and the batsman as independent variables, with the probability of a wicket occurring as the dependent variable. Thus, the estimated parameters from the fitted model are used to inform ‘wicket-taking ability’, for bowlers, and ‘wicket-preservation ability’, for batsman. These abilities are computed by the inverse logit function. Of course it can be argued that there is more to batting than this, but it is difficult to account for and model all nuances of cricket. Thus, the ‘wicket-taking ability’ for bowler i is given by \begin{align*} \phi_{i}^{(w)}=\frac{\exp{(\omega_{i})}}{1+\exp{(\omega_{i})}} \nonumber \end{align*} where ω1, ω2, …, ωn are obtained from a logistic regression. Similarly the ‘wicket-preservation ability’ for batsman i is given by \begin{align*} \psi_{i}^{(w)}=\frac{\exp{(\mu_{i})}}{1+\exp{(\mu_{i})}}, \nonumber \end{align*} where μ1, μ2, …, μm are obtained from a logistic regression. 4.3.2. Runs If a wicket is not taken, then a delivery is judged by whether the batsman scored any runs, and if so, how many. Therefore, another way of assessing a player’s quality is by analysing the amount of runs scored, if the player is a batsman, or the amount of runs conceded, if the player is a bowler. Table 13 contains the distribution of the number of runs scored per ball as observed in the dataset described in Section 3.1. The mean number of runs per ball is 0.5101, and the variance is 1.2213. Table 13. Distribution of number of runs scored per ball Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Table 13. Distribution of number of runs scored per ball Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Runs No. of balls 0 66196 1 12410 2 3152 3 905 4 5529 5 11 6 254 Table 14. Rankings of batsmen for different values of α. Also given, in order of left to right, runs, number of balls faced, number of times out, batting average and strike rate Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 Table 14. Rankings of batsmen for different values of α. Also given, in order of left to right, runs, number of balls faced, number of times out, batting average and strike rate Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 Batsman Runs Faced Inn. Out Bat. Av. Str. Rate Ranking for value of α 0.1 0.3 0.5 0.7 0.9 AB de Villiers 926 1823 23 22 42.09 50.8 17 16 19 20 24 AJ Strauss 813 1772 26 26 31.27 45.88 27 31 35 36 42 AN Cook 2435 5333 44 41 59.39 45.66 3 3 9 22 34 BJ Haddin 779 1503 30 28 27.82 51.83 36 34 34 24 23 DA Warner 805 1173 25 25 32.2 68.63 38 36 24 9 6 DW Steyn 213 536 20 17 12.53 39.74 49 50 52 53 53 GC Smith 1169 2175 25 24 48.71 53.75 9 7 7 13 19 GP Swann 638 780 31 23 27.74 81.79 43 39 23 3 2 Harbhajan Singh 149 210 13 13 11.46 70.95 57 55 49 31 5 HM Amla 1899 3634 25 21 90.43 52.26 2 2 2 8 18 IJL Trott 1537 3391 41 40 38.43 45.33 13 17 26 28 33 IR Bell 2024 4200 41 36 56.22 48.19 6 4 11 21 31 JE Root 432 1120 12 10 43.2 38.57 4 9 28 42 51 JM Anderson 181 518 32 23 7.87 34.94 56 56 56 56 56 JM Bairstow 361 774 10 10 36.1 46.64 19 21 30 33 39 JP Duminy 264 656 14 12 22 40.24 37 42 46 50 50 KP Pietersen 2015 3354 40 40 50.38 60.08 14 10 4 6 8 M Morkel 197 398 19 15 13.13 49.5 53 52 53 51 38 M Vijay 649 1351 12 12 54.08 48.04 5 5 17 30 44 MA Starc 332 442 13 8 41.5 75.11 35 27 5 2 3 MEK Hussey 1317 2418 28 27 48.78 54.47 15 13 10 14 20 MG Johnson 297 509 19 17 17.47 58.35 50 46 44 32 13 MJ Clarke 2263 3668 44 41 55.2 61.7 12 6 3 4 7 MS Dhoni 1048 1902 32 27 38.81 55.1 23 18 16 15 17 MV Boucher 400 611 11 10 40 65.47 34 25 8 5 4 PD Collingwood 427 962 13 12 35.58 44.39 21 28 33 38 45 PJ Hughes 444 925 22 21 21.14 48 41 43 42 37 29 PM Siddle 497 1125 35 33 15.06 44.18 46 47 47 47 43 R Dravid 767 1742 20 17 45.12 44.03 7 11 20 29 40 RJ Harris 165 272 16 12 13.75 60.66 55 51 48 39 12 RT Ponting 983 1789 27 24 40.96 54.95 24 23 21 16 15 SCJ Broad 541 840 27 26 20.81 64.4 47 45 41 23 10 SK Raina 223 427 11 11 20.27 52.22 44 44 45 41 28 SPD Smith 665 1409 20 18 36.94 47.2 22 22 27 25 26 SR Tendulkar 1480 2713 36 33 44.85 54.55 16 15 14 17 22 SR Watson 1353 2553 36 37 36.57 53 26 26 25 18 21 V Kohli 772 1674 21 19 40.63 46.12 8 12 22 26 36 V Sehwag 922 1118 29 30 30.73 82.47 42 38 12 1 1 VD Philander 198 389 11 9 22 50.9 39 40 39 40 32 We may model the number of runs scored per delivery using a negative binomial regression, with independent variables corresponding to the bowler and the batsman. The model produces parameters representing each player’s contribution to the amount of runs scored on any given delivery. These parameters can be used to inform a ‘run-scoring ability’ for batsmen and a ‘run-prevention ability’ for bowlers. For example suppose the negative binomial regression gives parameters γ1, γ2, …, γn for each of our n bowlers. The ‘run-prevention ability’ for bowler i is given by $$\phi _{i}^{(r)}=\exp {(\gamma _{i})}$$. Similarly, if we have parameters μ1, μ2, …, μm for the batsmen then the ‘run-scoring ability’ for batsman i is given by $$\psi _{i}^{(r)}=\exp {(\mu _{i})}$$. 4.3.3. Computing rating values Combining the results of Sections 4.3.1 and 4.3.2, a rating can now be assigned to each player as a function of these two abilities. We can compute a rating for a bowler’s overall ability as \begin{align} \phi_{i}=\phi_{i}^{(r)}\left(1-\phi_{i}^{(w)}\right) . \end{align} (4.1) Similarly, we can compute a rating for batsmen as \begin{align} \psi_{i}=\psi_{i}^{(r)}\left(1-\psi_{i}^{(w)}\right) . \end{align} (4.2) Thus, a low rating value indicates quality for a bowler, whereas a higher rating value indicates a better batsman. Table 15. Rankings of bowlers for different values of β. Also given, from left to right, number of runs conceded, number of balls bowled, number of wickets taken, bowling average, economy and strike rate Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 Table 15. Rankings of bowlers for different values of β. Also given, from left to right, number of runs conceded, number of balls bowled, number of wickets taken, bowling average, economy and strike rate Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 Bowler Con. Bowled Wkts Bowl Av. Econ. Str. Rate Ranking for value of β 0.1 0.3 0.5 0.7 0.9 A Mishra 570 1202 7 81.43 2.85 171.71 32 32 29 25 20 BW Hilfenhaus 1326 3079 46 28.83 2.58 66.93 16 12 8 6 5 CT Tremlett 493 1029 21 23.48 2.87 49 12 10 9 9 11 DW Steyn 1618 2951 67 24.15 3.29 44.04 3 6 11 20 24 GP Swann 3338 6779 103 32.41 2.95 65.82 22 21 19 16 13 Harbhajan Singh 1229 2586 31 39.65 2.85 83.42 26 25 22 17 9 JL Pattinson 961 1885 32 30.03 3.06 58.91 5 8 10 18 22 JM Anderson 3072 6156 109 28.18 2.99 56.48 15 14 17 14 16 M Morkel 1545 3110 56 27.59 2.98 55.54 11 11 13 12 17 MA Starc 818 1500 26 31.46 3.27 57.69 2 3 7 21 25 MG Johnson 1265 2049 33 38.33 3.7 62.09 18 24 27 29 30 MS Panesar 456 1098 17 26.82 2.49 64.59 20 13 5 3 2 NM Lyon 1760 3293 48 36.67 3.21 68.6 14 20 21 24 26 PL Harris 726 1543 16 45.38 2.82 96.44 30 26 25 22 12 PM Siddle 2255 4522 79 28.54 2.99 57.24 4 5 6 13 21 PP Ojha 1198 2863 36 33.28 2.51 79.53 24 22 15 5 4 RA Jadeja 535 1581 27 19.81 2.03 58.56 13 2 1 1 1 RJ Harris 1019 2102 45 22.64 2.91 46.71 1 1 2 7 19 S Sreesanth 600 896 11 54.55 4.02 81.45 25 29 32 32 32 SCJ Broad 2105 4401 74 28.45 2.87 59.47 17 15 14 10 8 SR Watson 589 1353 15 39.27 2.61 90.2 21 18 16 8 6 ST Finn 1010 1583 31 32.58 3.83 51.06 7 16 23 27 28 TT Bresnan 1171 2315 39 30.03 3.03 59.36 19 17 18 15 14 UT Yadav 625 893 18 34.72 4.2 49.61 9 19 24 28 31 VD Philander 647 1536 30 21.57 2.53 51.2 6 4 3 2 3 In test match cricket, a batsman’s ability to preserve his wicket is usually valued greater than his ability to score quickly, although there are certain situations in which aggressive batting is preferred. Similarly, a bowler’s ability to take wickets is treated with greater significance compared to how economical they are. There are, however, very few situations in test cricket in which more economical bowling is valued greater than wicket taking ability. Proposed here is the modification of the ratings formulae (4.1) and (4.2) to include a parameter that allows for one to place more emphasis on one of the two ability parameters used to compute the ratings. This is analogous to the proposal of Barr & Kantor (2004) who devised a criterion by which to measure the performance of batsmen in one-day cricket, allowing one to alter the emphasis placed on aggressive batting. First we consider the ratings of batsmen. By introducing a parameter 0 < α < 1, that represents the emphasis placed between the run-scoring and the wicket-preservation abilities, we gain flexibility in our rating system that allows us to consider the quality of a batsman in different scenarios. The updated final rating formula for batsman is thus given by \begin{align*} \psi_{i,\alpha}=\left(\frac{1}{2}\psi_{i}^{(r)}\right)^{\alpha}\left(1-\psi_{i}^{(w)}\right)^{1-\alpha}\, \!\!\!. \nonumber \end{align*} The run-scoring ability has been multiplied by $$\frac {1}{2}$$ to ensure that the reference point for the player rating for all values of α remains at $$\frac {1}{2}$$. A similar parameter 0 < β < 1 is introduced for bowlers, to allow for emphasis to be placed on either wicket-taking ability or run-prevention ability yielding the rating formula: \begin{align*} \phi_{i,\beta}=\left(\frac{1}{2}\phi_{i}^{(r)}\right)^{\beta}\left(1-\phi_{i}^{(w)}\right)^{1-\beta}\, \!\!\!. \nonumber \end{align*} We have imposed a qualifying criterion for a player to receive a rating, with batsman needing to have played a minimum of 10 innings to qualify for a rating, and bowlers requiring to have delivered 720 balls to qualify. Table 14 contains rankings of a selection of batsmen for different values of α and Table 15 contains rankings of bowlers for different values of β evaluated from the data described in Section 3.3. If we accept α = 0.3 and β = 0.3 to be suitable parameter values then we may argue that it appears that RJ Harris is the best bowler and CA Pujara is the best batsman. In Tables 16 and 17 we compare the player rankings we have computed with the batting and bowling averages obtained from the matches considered. Of the 15 batsman with the highest batting averages, all of them are within the top 15 for the player ratings, with the exception of AB de Villiers, who is in 16th position. Table 16. Comparison of the batting average and Bradley–Terry player ranking for players in the ‘Top 15’ for batting average Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Table 16. Comparison of the batting average and Bradley–Terry player ranking for players in the ‘Top 15’ for batting average Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Batsman Bat Av. Rank HM Amla 90.43 2 CA Pujara 84.82 1 AN Cook 59.39 3 IR Bell 56.22 4 MJ Clarke 55.20 6 M Vijay 54.08 5 JH Kallis 53.26 8 KP Pietersen 50.38 10 MEK Hussey 48.78 13 GC Smith 48.71 7 R Dravid 45.12 11 SR Tendulkar 44.85 15 AN Petersen 43.46 14 JE Root 43.20 9 AB de Villiers 42.09 16 Table 17. Comparison of the bowling average and Bradley–Terry player ranking for players in the ‘Top 15’ for bowling average Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Table 17. Comparison of the bowling average and Bradley–Terry player ranking for players in the ‘Top 15’ for bowling average Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Bowler Bowl Ave. Rank RA Jadeja 19.81 2 VD Philander 21.57 4 RJ Harris 22.64 1 CT Tremlett 23.48 10 DW Steyn 24.15 6 MS Panesar 26.82 13 P Kumar 27.38 7 M Morkel 27.59 11 JM Anderson 28.18 14 Z Khan 28.39 9 SCJ Broad 28.45 15 PM Siddle 28.54 5 BW Hilfenhaus 28.83 12 JL Pattinson 30.03 8 TT Bresnan 30.03 17 Table 18. Comparison of Bradley–Terry model predictions with actual results. Bookkeeper odds also provided. HW = Home Win, AW = Away Win, D = Draw Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Table 18. Comparison of Bradley–Terry model predictions with actual results. Bookkeeper odds also provided. HW = Home Win, AW = Away Win, D = Draw Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 2 January 2013 SA NZ 0.7648 0.703 0.1740 0.8638 0.474 0.888 HW 3 January 2013 A SL 0.7345 0.841 0.1857 0.6664 0.2069 0.1267 HW 11 January 2013 SA NZ 0.7843 0.706 0.1613 0.8707 0.439 0.854 HW 1 February 2013 SA P 0.6129 0.1572 0.2495 0.7459 0.1564 0.976 HW 14 February 2013 SA P 0.6755 0.1424 0.2010 0.7586 0.1473 0.941 HW 22 February 2013 I A 0.4304 0.2345 0.3577 0.4005 0.2519 0.3476 HW 22 February 2013 SA P 0.6658 0.1680 0.1785 0.7694 0.1397 0.908 HW 2 March 2013 I A 0.4041 0.2212 0.3876 0.4421 0.2271 0.3307 HW 6 March 2013 NZ E 0.1341 0.5921 0.2887 0.816 0.5266 0.3917 D 14 March 2013 I A 0.3293 0.1114 0.5723 0.4891 0.2063 0.3046 HW 14 March 2013 NZ E 0.1269 0.5636 0.3343 0.741 0.4782 0.4477 D 22 March 2013 I A 0.5381 0.1961 0.2820 0.5136 0.2086 0.2778 HW 22 March 2013 NZ E 0.1517 0.5884 0.2812 0.776 0.4608 0.4616 D 16 May 2013 E NZ 0.5319 0.832 0.3907 0.7693 0.333 0.1974 HW 24 May 2013 E NZ 0.4698 0.806 0.4769 0.7796 0.351 0.1853 HW 10 July 2013 E A 0.4113 0.2712 0.3040 0.5359 0.2170 0.2472 HW 18 July 2013 E A 0.4674 0.2549 0.2772 0.5436 0.2205 0.2358 HW 1 August 2013 E A 0.5258 0.2141 0.2741 0.5734 0.1928 0.2338 D 9 August 2013 E A 0.4674 0.2549 0.2772 0.5676 0.1598 0.2727 HW 21 August 2013 E A 0.4113 0.2712 0.3040 0.6068 0.1594 0.2338 D 14 October 2013 P SA 0.1885 0.4541 0.3815 0.1874 0.2642 0.5484 HW 23 October 2013 P SA 0.2796 0.4195 0.3276 0.2302 0.2610 0.5088 AW 6 November 2013 I WI 0.5444 0.1167 0.3222 0.5729 0.351 0.3920 HW 14 November 2013 I WI 0.6382 0.984 0.2385 0.6004 0.350 0.3646 HW 21 November 2013 A E 0.3546 0.3182 0.3426 0.4634 0.3873 0.1493 HW 3 December 2013 NZ WI 0.5141 0.1264 0.2666 0.3109 0.1957 0.4933 D 4 December 2013 A E 0.3644 0.2515 0.4037 0.4642 0.3810 0.1548 HW 11 December 2013 NZ WI 0.5509 0.2191 0.2361 0.3093 0.1651 0.5256 HW 13 December 2013 A E 0.5643 0.2606 0.1792 0.4739 0.3766 0.1494 HW 18 December 2013 SA I 0.5476 0.1292 0.3378 0.6582 0.1927 0.1491 D 19 December 2013 NZ WI 0.5784 0.1671 0.2703 0.3499 0.1583 0.4918 HW 26 December 2013 A E 0.5509 0.2201 0.2457 0.5189 0.3545 0.1266 HW 26 December 2013 SA I 0.4923 0.1759 0.3475 0.6483 0.1611 0.1906 HW 31 December 2013 P SL 0.3972 0.3047 0.3232 0.4051 0.1162 0.4788 D 3 January 2014 A E 0.5542 0.2673 0.1829 0.5288 0.3468 0.1244 HW 7 January 2014 P SL 0.4188 0.2809 0.3222 0.3788 0.1025 0.5187 AW 16 January 2014 P SL 0.3546 0.2852 0.3800 0.4021 0.1337 0.4642 HW 6 February 2014 NZ I 0.2361 0.1882 0.6006 0.1283 0.3669 0.5048 HW 12 February 2014 SA A 0.4698 0.3193 0.2226 0.7244 0.1650 0.1106 AW 14 February 2014 NZ I 0.3793 0.3788 0.2637 0.1702 0.3574 0.4724 D 20 February 2014 SA A 0.3247 0.3365 0.3590 0.7036 0.1907 0.1057 HW (continued). Similarly, we can see that TT Bresnan is the only bowler who ranks in the top 15 for bowling average but not for player rating, where he ranks 17th. Interestingly, the player who is ranked 3rd, MA Starc, does not rank within the top 15 for bowling average. We can use these rankings to select a hypothetical ‘World XI’, comprising of the best players from each of the teams considered. Typically, a team will select seven batsmen and four bowlers. Furthermore, one of the batsmen must also be able to play as a wicketkeeper. Since no attempt has been made in this paper to assess the wicket-keeping ability of players, we will select the wicketkeeper with the strongest batting rating, which is AB De Villiers. Note that this is not wholly inappropriate since it is a common strategy for teams to select their wicketkeepers with great consideration for their batting ability. Adding to our selection the top 6 ranked batsmen and the top 4 ranked bowlers, we have a World XI as CA Pujara, HM Amla, AN Cook, IR Bell, M Vijay, MJ Clarke, AB De Villiers, VD Philander, MA Starc, RA Jadeja and RJ Harris. 4.4. Using player ratings to predict outcomes We have seen how player ratings can be used either to rank players for general interest, or to inform selection decisions for teams. It will now be shown how these ratings could be used to predict the outcome of a match, given the players which have been selected. 4.4.1. Deriving team ratings from player ratings First, we will look at how the player ratings given in Section 4.3.3 can be used to form overall ratings for each team. We propose that each team have a separate rating for bowling and batting. From this point onwards, any reference to a batsman’s rating refers to the rating produced from taking α = 0.3 in equation (4.1), and, similarly all reference to bowling ratings refer to the ratings produced where β = 0.3 in equation (4.2). These are taken as example values. One could examine approaches to estimate these parameters, or to update them during over time. This is a possible area for future investigation. Table 18. Continued. Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Table 18. Continued. Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Bookmakers Bradley–Terry Date Home Away HW AW D HW AW D Result 1 March 2014 SA A 0.4150 0.3215 0.2812 0.7116 0.1840 0.1043 AW 8 June 2014 WI NZ 0.3921 0.2961 0.2870 0.4329 0.3502 0.2169 AW 12 June 2014 E SL 0.4604 0.2146 0.3366 0.5832 0.1032 0.3136 D 16 June 2014 WI NZ 0.3466 0.3504 0.3159 0.4234 0.3915 0.1851 HW 20 June 2014 E SL 0.5258 0.2131 0.2673 0.5443 0.949 0.3608 AW 26 June2014 WI NZ 0.3415 0.2628 0.4212 0.4773 0.3607 0.1620 AW Given that any player in a team may be expected to bat, we have considered a team’s batting rating, BATteam to simply be the sum of each player’s batting rating. If a player has not reached the qualification criterion of having played 10 innings, they are not assigned a batting rating and therefore do not contribute to the team rating. It is, however, inappropriate to compute the bowling rating of a team in such a manner. Typically four players in a team will be expected to bowl the majority of overs in a match, with other ‘part-time’ bowlers occasionally alleviating the workload on the main bowlers. Therefore, we postulate that a team’s bowling rating should be based only on the top four individual bowler ratings within their team. We define a team’s bowling rating as follows: \begin{align*} BOWL_{team}=\sum_{i=1}^{4}\frac{1}{4\phi_{i,0.3}}, \nonumber \end{align*} where i = 1, 2, 3, 4 are indices representing the four players with the best bowling rating in a selected team. Since a low rating indicates greater quality for bowlers, it is necessary to compute the team bowling rating as a sum of reciprocal values of the individual bowler ratings, so that a higher rating indicates a better bowling attack. Multiplying by 1/4 simply ensures that the reference value for the ratings of individual bowlers remains at 1/2. 4.4.2. Predicting outcomes based on player selection We have fitted a Bradley–Terry model to the results of the matches in the dataset described in Section 3.1, with home advantage, batting quality and bowling quality each being considered as ‘order effects’, with a parameter for random effects also included. The model is described below. For teams A and B with respective team batting abilities BATA and BATB, and respective team bowling abilities BOWLA and BOWLB, the probability of team A winning at home to team B is given by \begin{align} p_{AB}^{(h)}=\frac{\exp{(\delta_{bat}(BAT_{A}-BAT_{B})+\delta_{bowl}(BOWL_{A}-BOWL_{B})+\delta_{home})}}{1+\exp{(\delta_{bat}(BAT_{A}-BAT_{B})+\delta_{bowl}(BOWL_{A}-BOWL_{B})+\delta_{home})}}, \end{align} (4.3) where δhome is the order effect representing the advantage from playing at home, and δbat and δbowl are order effects representing the influence that team batting and bowling abilities have on the expected outcome of a match, respectively. Table 18 contains the Bradley–Terry predicted match outcomes and bookmakers’ predictions for matches between 2013 and 2014 where pre-match bookmakers’ odds were available. We make the following remarks. The Bradley–Terry model predicts the correct outcome more often than the outcome suggested from the bookmakers’ average odds, although it must be recognised that the bookmakers also perform very well. There are examples when both the bookmakers and the model get it wrong: see games on 1 August 2013, 21 August 2013 and 14 October 2013 for example. An interesting difference can be observed for the game held on 21 November 2013. The bookmakers have a more even spread across all possible outcomes, whilst the Bradley–Terry model gives a more pronounced (correct) prediction for a home win. A big discrepancy occurs for the game held on 3 December 2013, where the Bradley–Terry model correctly predicts a draw whilst the bookmakers placed a lot of confidence in a win for the home side. Indeed on inspection, for the games where the bookmakers have struggled to offer a definite outcome (by putting equal probabilities on a win, lose and a draw), the Bradley–Terry model seems more able to pick out an outcome, and this is often the correct outcome. 5. Conclusion The objective of this paper was to use Bradley–Terry models to analyse various aspects of test cricket. The main areas of investigation were ranking teams, predicting match outcomes and rating individual players. The Bradley–Terry model was used to devise an alternative ranking system for test cricket. Aside from the obvious computational differences, the consideration of home advantage distinguished this system from that employed by the ICC. The strong correlation between the results of this system and the official ICC rankings indicates that it provides an accurate reflection of the abilities of the respective teams at any given point. The Bradley–Terry model was also used to produce a system that predicts the outcome of test matches based on previous results. The predicted outcomes were compared to bookmakers’ odds, showing a strong correlation between the predicted probabilities of home wins and away wins, but only a moderate correlation with the predicted probabilities of draws. We also produced an individual player rating system for batsmen and bowlers. The player ratings derived were then used to inform batting and bowling ratings for a team, given the players that were selected. A Bradley–Terry model was then used to investigate whether these team ratings could be used to predict matches, successfully predicting the result of 8 out of 10 test matches. More generally, this paper has shown the potential for Bradley–Terry models in wider settings. Problems of rating, ranking and evaluating can be tackled by fitting such models and these applications have wide-bearing use. References Agresti , A. 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### Journal

IMA Journal of Management MathematicsOxford University Press

Published: Jan 8, 2018

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