Predicting the winning percentage of limited-overs cricket using the Pythagorean formula

1Introduction

The Pythagorean Win-Loss formula, which was developed by Bill James in the early 1980s, is a sports analytics formula that can be effectively used to calculate winning percentages for sporting events. This calculation is based on the average number of runs scored (RS) and the average number of runs allowed (RA) by a team. In particular, the Pythagorean Win-Loss formula with Pythagorean exponent γ is given as follows.

One can directly predict the expected number of matches a team will win in a future series based on the observed winning percentage for the period so far. For example, the observed winning percentage of the matches played from the end of the last ODI cricket world cup to the beginning of the next world cup can be used to predict the number of matches expected to be won by a team in the next world cup. One can do a better prediction using the Pythagorean winning percentage, which can be calculated using Equation (1). The added value of the Pythagorean winning percentage over the regular winning percentage is that it takes into account more accurate defensive and offensive strengths of the teams by using the actual runs scored and actual runs allowed. The regular winning percentage does not have a way to accommodate the actual scores and it uses the binary outcome of a game (win or lose). However, with the appropriate Pythagorean exponent (γ), the Pythagorean winning percentage utilizes additional information of the actual scores, which in return provides predictions that are more practical. Initially, this formula was used by baseball statisticians, and the Pythagorean exponent was assumed to be 2. However, a later empirical study by Miller (2006) has shown that a better agreement can be attained when the exponent was about 1.82. Miller (2006) was also the first to show the theoretical justification of the formula by assuming independent Weibull distributions for runs scored and runs allowed. Often this formula is used to predict the end of the season standing of a team, based on its performance halfway through the season (Miller, 2006). Several researchers have used this Pythagorean formula to determine the winning percentages of other sports as well by finding appropriate γ values for the specific sport. According to Schatz (2003), statistician Daryl Morey’s research has proven that the Pythagorean formula can be applied to all the major sports, with different exponents for each; in particular, he has shown that the γ value for National Football League (NFL) was about 2.37. Dayaratna and Miller (2012) have shown that the Pythagorean Win-Loss formula can be used as an evaluative tool in hockey. They have further shown that the maximum likelihood estimate of γ is almost always slightly above 2 for all three National Hockey League (NHL) seasons they considered. Oliver (2004) has shown that the appropriate value of γ for basketball is around 14. Rosenfeld et al. (2010) have used the formula to predict overtime outcomes in the National Football League (NFL), Major League Baseball (MLB), and National Basketball Association (NBA). Heumann (2016) described a new version of the Win-Loss formula, which was called pairwise Pythagorean win. Chen and Li (2018) have introduced a shrinkage factor to increase the accuracy of the Pythagorean Win-Loss formula.

Application of the Pythagorean formula to cricket has been more limited, as the authors are aware of only a single relevant study. In that study, Vine (2016) used the Pythagorean formula to estimate the winning percentage of the Twenty20 Cricket Big Bash League, an Australian domestic competition. Vine (2016) has further shown that the appropriate γ value for the Twenty20 Cricket Big Bash League was 7.41. Vine (2006) also suggested that for the cases where the second batting team wins the match without using all the allocated resources, the dependency issue between runs scored and runs allowed could be eliminated by extrapolating the second team’s run total, by using the Duckworth–Lewis resources table. It is obvious that, as in any other sport, there could be a significant difference in the performance parameters between international level games and regional level games such as Big Bash League. Our effort in this paper is therefore to extend Vine’s results to international limited over cricket. Specifically, the objective of this paper is to calculate the winning percentage of the top 10 teams of the International Cricket Council (ICC); in particular, the goal is to estimate the appropriate Pythagorean exponent for both of the limited-over cricket formats, Twenty20 and One Day International (ODI).

2Description of data

In this study, we have used data from ODI matches played by the top 10 national teams of the International Cricket Council (ICC) during the time period between the 2015 and 2019 ICC World Cups. These teams are Afghanistan, Australia, Bangladesh, England, India, New Zealand, Pakistan, Sri Lanka, South Africa, and West Indies. For Twenty20 cricket, in order to have sufficient data, we used data for all matches played by the above 10 teams between 2012 and 2018. Matches with no results and matches shortened due to weather interruptions were removed from the analysis.

Descriptive statistics of total runs scored for ODI matches is shown in Table 1. It can be seen that the lowest total was 74 (Pakistan), and the highest was 512 (India). The lowest median score was 212 (West Indies), and the highest median score was 323 (England). Note that some of these scores were extrapolated scores by using the Duckworth-Lewis method for matches in which the second batting team won with overs and wickets are still remaining.

Descriptive statistics of total runs allowed in ODI matches is shown in Table 2. The highest total was 481 (Australia), and the lowest was 92 (New Zealand). The highest median was 316(Sri Lanka), and the lowest median was 229 (Afghanistan).

Descriptive statistics of total runs scored in Twenty20 matches is shown in Table 3. These totals ranged from 60 (New Zealand and West Indies) to 333(New Zealand). Median scores ranged from 132 (Afghanistan) to 166(India). Again, some scores were extrapolated by using the Duckworth-Lewis method, for matches in which the second batting team won without using all the resources allocated to it.

Table 4 shows the descriptive statistics for runs allowed in Twenty20 matches. The number of runs allowed is ranged from 67 (New Zealand) to 263 (Sri Lanka). Median scores ranged from 148 (Pakistan) to 171 (Bangladesh).

3Methodology

We applied the Pythagorean Win-Loss formula to cricket, following Miller (2006) assumption that runs scored and runs allowed follow independently distributed Weibull distributions with a common shape parameter. The probability density function of the Weibull distribution is given as follows.

where γ is the shape parameter, α is the scale parameter, and β is the location parameter.

As explained by Miller (2006), number of runs scored and the number of runs allowed in a given match follow independent Weibull distributions with parameters (α_RS,β, γ)  and  (α_RA, β, γ) respectively. By denoting the mean of runs scored as RS and mean of runs allowed as RA for a given team, we can show that

where RS = α_RSΓ (1 + γ^-1) + β and RA = α_RAΓ (1 + γ^-1) + β.

Therefore, the winning percentage of the team is given by,

We used both the maximum likelihood method and the least squares method for parameter estimation. The next subsection describes the parameter estimation and the other results using the Weibull models for runs scored and runs allowed.

3.2Goodness of fit of the weibull distribution

The chi-squared goodness of fit test was used to evaluate the appropriateness of the Weibull distribution for modeling the runs scored and runs allowed for each team separately. The estimates of α_RS, α_RA, and γ using both the maximum likelihood method and the least squares method were used as parameter estimates to perform the chi-squared goodness of fit test.

Figure 1. through Fig. 4. show some representative plots to describe the appropriateness of the Weibull fit for the observed data. The complete list of plots can be found in the appendix.

Fig. 1

Weibull Distribution Fit for Runs Scored and Runs Allowed for Australia using Least Squares Method (ODI).

Fig. 2

Weibull Distribution Fit for Runs Scored and Runs Allowed for New Zealand using Maximum Likelihood Method (ODI).

Fig. 3

Weibull Distribution Fit for Runs Scored and Runs Allowed for England using Least Squares Method (Twenty20).

Fig. 4

Weibull Distribution Fit for Runs Scored and Runs Allowed for South Africa using Maximum Likelihood Method (Twenty20).

As can be seen from Table 10, all the p-values are above 0.05, which indicates that the use of Weibull distribution for runs scored and runs allowed is appropriate for ODI data. Table 11 shows the same for Twenty20 data, and the results are consistent, meaning that the use of Weibull distribution for runs scored and runs allowed is appropriate for Twenty20 data as well.

Table 10

Goodness of Fit – Weibull Distribution (ODI Matches -Maximum Likelihood Method)

Team	Chi-squared RS	p-value	Chi-squared RA	p-value
AFG	3.58	0.997	0.84	0.999
AUS	4.67	0.989	17.43	0.234
BAN	5.56	0.976	2.68	0.999
ENG	7.96	0.892	3.41	0.998
IND	4.65	0.990	4.50	0.992
NZ	11.67	0.633	7.54	0.912
PAK	7.11	0.930	6.19	0.961
SA	7.67	0.906	2.41	0.999
SL	3.57	0.998	5.48	0.978
WI	4.04	0.995	8.46	0.864

Table 11

Goodness of Fit – Weibull Distribution (Twenty20 Matches -Maximum Likelihood Method)

Team	Chi-squared RS	p-value	Chi-squared RA	p-value
AFG	5.86	0.923	1.66	0.999
AUS	5.40	0.944	3.36	0.992
BAN	4.35	0.976	6.85	0.867
ENG	4.40	0.975	1.35	0.999
IND	3.93	0.985	6.23	0.904
NZ	6.96	0.860	9.50	0.660
PAK	3.69	0.988	1.73	0.999
SA	0.98	0.999	4.87	0.962
SL	4.04	0.983	3.99	0.983
WI	2.03	0.999	8.25	0.766

Note that we have also investigated the goodness of fit of Weibull distribution using least squares estimates. However, the results were closely similar to those obtained using the maximum likelihood method. Therefore, tables showing the least squares results are not included here.

4Discussion and conclusion

In this study, we have shown how Bill James’ Pythagorean formula can be applied to the game of cricket. In particular, we have derived the appropriate exponent γ for the two limited-overs international cricket formats ODI and Twenty20. The maximum likelihood method has resulted in mean γ values of 4.71 for ODI matches and 6.06 for Twenty20 matches. The least squares method resulted in slightly higher values for both formats, with mean γ values 5.01 and 6.56 respectively for ODI and Twenty20. Given the desirable properties such as consistency, invariance, asymptotic normality, and relationship to the sufficient statistics, we suggest that the maximum likelihood estimators are more apposite estimators to be used in practice. In addition to this, the standard deviations of the maximum likelihood estimates of γ were lower than that of the least squares estimates for both ODI and Twenty20 formats. This concludes that the suitable γ values are 4.71 and 6.06 for ODI and Twenty20 respectively. The only comparison that can be found in the literature is the γ value 7.41, which was given by Vine (2016) for the Twenty20 cricket Big Bash League, an Australian domestic competition.

The significance of the findings of this paper is that it was based on international Twenty20 and ODI cricket matches. Note that the independence assumption is valid for the ODI data, but for the Twenty20 data, it was valid only for four teams. Therefore, the application of the method to the Twenty20 data should be done cautiously due to the limitation of the validity of the assumption of independence. It would be an excellent future study to assess the robustness of the independence assumption for the Twenty20 data. Extension of the method to ODI data and incorporation of the Duckworth-Lewis method to extrapolate the runs scored by the second batting team when that team won the match without using all the resources allocated to it are also key contributions of this paper.

As mentioned in the introduction, we have used the data from the matches played during the time period between the 2015 and 2019 Cricket World Cups with the derivation of the γ for ODI matches. Consequently, we used those estimates to predict the number of matches expected to be won by each team in the 2019 World Cup. As a comparison, we have also used the regular observed winning percentages for the given time period to predict the expected win totals (number of matches) for the world cup matches. Based on our analysis, England had the highest Pythagorean winning percentage (0.76) of the 2019 World Cup, and we would have predicted them to win the tournament, which they did. Based on our findings, the Pythagorean formula would have predicted the 2019 World Cup semifinalists to be England, India, South Africa, and New Zealand. Of these, only South Africa failed to make the semifinals, with Australia qualifying instead. We have also quantified the predictive powers of the Pythagorean winning percentage and the regular observed winning percentage by using the actual win totals and implied win totals for 2019 World Cup outcomes. In particular, we have calculated ∑(actual win total-implied win total)2number of mathes where the summation is taken across the top 10 teams of the 2019 World Cup. The resulting outcome for regular observed winning percentage was 5.10, and for the Pythagorean winning percentage, it was 4.71. The lower value for the Pythagorean winning percentage indicates that the prediction based on the Pythagorean expectation is closer to the actual outcome. However, the significance of the improvement should be justified by using a statistical test, and that is left as a future study. This demonstrates the effectiveness of the Pythagorean formula in predicting cricket outcomes. It performs better than the prediction based on the regular observed winning percentage. As cricket continues to grow in popularity, this formula may be used as a prediction tool in the sports betting industry as well.