Player-aware resource compensation in interrupted cricket matches

Disadvantages of generalized resource allocation

After introducing the D/L method, criticism started pouring, and a revised method was introduced in 2004 (Clarke & Allsopp, 2001). Even after the perceived improvements, the new method is well criticized by critiques. A survey of the D/L method shows that this method is biased. Studies show that the D/L method favors the team batting first, and teams also use that for gaining an unfair sporting advantage. That study also shed light on how D/L is influenced by wickets rather than runs and run rate (Shah et al., 2015). Another criticism of the D/L method is that the flow of resources is not the same in both innings, and the team batting 2nd can also benefit from that by quickly achieving the D/L par score. In 2009, SE Stern generated separate resource tables differently (Stern, 2009).

Researchers behind the D/L method again defended their mechanism by calling other methods unfair and giving examples of the behavior of the D/L method in some matches with comparison to other proposed methods (Duckworth & Lewis, 2015). Stern also criticized the D/L approach with examples that this method is unfair and the reactive approach of calculating the target is unsuitable. He compared some other proposed methods with D/L and concluded that although D/L is simpler, it still competes with others (Stern, 2016). In the same year, a researcher also proposed a methodology that evaluates the form of teams and home advantage. This method has over by over prediction accuracy of 82% in the 2nd inning (Asif & McHale, 2016). The core issue with D/L is that it starts forecasting after 20 overs. Recently a poster paper claimed that neural networks, Naive Bayes, and Random Forest have better accuracy than the D/L method. Besides optimizing the resource table, they claimed that their model has the same accuracy at ten overs as D/L has at 20 (Abbas & Haider, 2017).

Influence of T20 on scoring patterns

When T20 cricket arrived, the primary issue with the D/L method was that it was not designed according to the game’s shortest format. Scoring patterns are different in both formats, and batters are less cautious about losing their wicket. A simulator is proposed by researchers for twenty-twenty international (T20I) cricket matches. The good thing about it is that it considers players but does not account for its current form. But for the second innings, these simulation results are dynamic but biased (Davis, Perera & Swartz, 2015). Studies show that the starting of a match has a different approach than the later part. Usually, powerplay overs add more runs to the game than regular overs because of the aggressive nature of opening batters and the applied fielding restrictions (Silva, Manage & Swartz, 2015). McHale and Asif (2013) declared D/L better than all previously proposed methods, but they suggested that the best method to predict the outcome of a cricket match should be the one that works in T20I and ODIs. They proposed a method in 2013 and came up with a better approach in 2016 (McHale & Asif, 2013). Historical data of players and venues has its worth, and by using that, a match can be predicted (Cricviz, 2019).

Importance of condition and players

Usually, conditions have a significant impact on players’ performances in cricket. That’s why a region-based ranking method is proposed (Daud et al., 2018). Some researchers developed a simulator that takes some historical and instantaneous data points to simulate the game. They also take individual characteristics of batters but leave the bowlers for future work (Sankaranarayanan, Sattar & Lakshmanan, 2014). Statistical algorithms such as Naïve Bayes are also used to predict the match’s outcome with the help of historical data (Singh, Singla & Bhatia, 2015). A team’s performance depends on how the team is formulated, and it also indicates their preferences to win matches. A method has also been proposed to calculate the number of deliveries expected to be bowled in a competition based on the players involved and their strength (Tyagi et al., 2020). In 2017, a method was proposed to calculate the precedence. This can show the strengths and weaknesses of a team and could also be helpful to understand their strategy (Ahmad et al., 2017). Some people have treated player prediction as a classification problem, while others have done the same using regression analysis (Kalpdrum & Niravkumar, 2018; Anik et al., 2018).

Over aggregation

Individually for all teams, each over of the innings was assigned a value or worth by the following formula: (1)${\displaystyle {\mathrm{O}}_{\mathrm{W}}=\frac{\text{Runs Scored in that over}}{\text{Total Runs Scored}:}\times \mathrm{C}}$

In Eq. (1) C is a constant, and its value is 50. This value indicates that deliveries and wickets have an equal share in total resources at the beginning of the inning. The assumption that led to Eq. (1) was that each team could have a different resource usage strategy, giving them the freedom to change or continue their playing style.

Figure 2 validates the assumption mentioned above as it can be seen that Bangladesh has a very different trajectory than Australia and Pakistan. Also, it can be seen that initial overs contribute less to Pakistan’s inning, and Pakistan’s over weightage is almost similar to Australia. Still, in the last part of the inning, Pakistan’s trajectory is more inclined toward Bangladesh’s scoring pattern.

Player weightage allocation

The weightage of a player indicates how much that player will contribute to the team’s total score in the next match in the form of a percentage. The Player Weightage Allocation is a way to identify the importance of a player in his team. Dataset(i) is the source of this experiment, and it helped in calculating the weightage of a player using past performances. One crucial factor involved in this process is calculating the weightage of players based on playing conditions. In order to measure the condition-wise impact, a list of 10 countries was created. That list includes Pakistan, India, Sri Lanka, Australia, England, South Africa, New Zealand, West Indies, UAE, and Bangladesh. Nine of the chosen countries were selected based on the official ODI ranking. Since there were no matches played in Afghanistan, Afghanistan is not part of the list. In the recent past, UAE has hosted a significant number of matches. That’s why the UAE was also included in the list.

Only innings played in that particular country were considered for each country in the list. Since cricket teams also have bowlers and they don’t have batting as their core skill, there was a possibility that there could be bias if all the innings of a batter were in the training dataset while all the data of a bowler was included in the testing dataset. In order to remove this bias, each player’s innings data was separated into two parts. One part consists of 80% of the innings, while the other has 20% of his inning data. These separate datasets of all players were combined to form two complete datasets for training and testing. If a player had played only one inning in any country, in that case, he was not considered a part of any of both datasets.

At the time of model selection, there are several options to consider. One thing that impacted the model selection decision is that the dataset is divided into ten parts. Each part must be evaluated on its accuracy and will have no impact on others. From the accuracy and prediction point of view, it can be a linear or non-linear model, but after some analysis, we found out that linear regression is working well with our data, and having different algorithms for different countries might not be a good approach because of different calculation criteria. Previously, in Net Run Rate and team or player ranking calculation, International Cricket Council didn’t use different criteria based on playing condition or country. D/L also maintains that in their approach and PRCICM is also based on that approach.

In order to check multicollinearity, VIF was used on each country’s dataset separately, and the score for each parameter in their respective dataset is presented in Table 1. According to (O’brien, 2007), this score is satisfactory, and there is no need to reevaluate the independent variables.

The training dataset of each country was then passed to multiple linear regression separately to find out the weightage of each parameter in the dataset. Runs were used as the dependent variable, while career average, average of last five matches, and average in the host country were independent variables. Since each country has separate datasets for training and testing, multiple linear regression generates separate equations for every country. The generated equation for each country is given below in Table 2.

This equation can predict the score of any player in any of the countries mentioned in Table 2 with the help of the player’s stats. If a player is on his international debut, then the player’s record that he is replacing will be used as his stats. Some of the variables negatively relate to the score in some countries. For instance, the average of the last five matches in Pakistan has the strongest negative relationship compared to other negative values, and West Indies has two negatively impacting parameters. There could be a slight chance that the predicted score of a player turns out to be a negative value if any of the players have a bigger value of the negatively impacting parameter. Although none of the predicted values was negative during testing, to remove this exception, the absolute value of the prediction was taken. It will have no impact on normal predictions but assign the negative value as a non-negative to the player.

The predicted score of each player in a team was used to calculate G50. G50 represents the total expected score of a team if they utilize all their resources. A team in cricket has 11 players, so G50 for a team will be the sum of all 11 players playing on any given day. Suppose S is the predicted score of each player in a team, then for each predicted score, the equation below shows the formula to calculate G50. (2)${\displaystyle \mathrm{G}50=\sum _{\mathrm{K}=1}^{11}\left|{\mathrm{S}}_{\mathrm{K}}\right|}$ In order to calculate the weightage of each player, his expected score is divided by G50. Suppose W is the weightage of each player then for each player P in the playing eleven: (3)${\displaystyle {\mathrm{W}}_{\mathrm{P}}={\mathrm{S}}_{\mathrm{P}}\mathrm{\ast \; D}/\mathrm{G}50}$

In Eq. (3) D is a constant, and its value is 50. This weightage was combined with over’s worth, already stored in the dataset (ii), then used in the PRCICM method to determine how many resources a team has before and after an interruption.

Resource compensation

PRCICM method is a resource compensation algorithm that takes values of lost resources and then assigns a score that the team under consideration gets for those resources. PRCICM method changes the underlying working of the DLS method and uses a different method to assign resources to each wicket and over. Instead of using a one-size-fits-all approach, the PRCICM method treats each team and player separately and uses a generalized but more dynamic approach than DLS.

At the beginning of a match, each team has 100% resources. Resources were divided into two parts one is for players, and the other is for overs. When either ends, total resources became 0 as the team has either no batter to send in for batting or they have no over available to bat. The worth of each over and every player is calculated using Eqs. (1) and (3), respectively, so at the beginning of any match total worth that any of both teams involved in a match can be calculated using the equation below: (4)${\displaystyle R=\sum O_W+\sum W_P}$ D/L denotes R₁ as the resources of the team batting first, and R₂ represents the resources of the team batting 2nd. If an over has bowled in a match, then the worth of that over will be subtracted from R. Similarly, if any player has lost his wicket, in that case, the worth of that player will also be subtracted from the total resources of the team.

As the match progresses and a team starts consuming overs, their total resources will also reduce, as described above, but what if they don’t lose any wickets even until the 49th over of a match. In order to cope with this anomaly, another formula is proposed that only focuses on the total remaining worth of players and reduces their total worth accordingly. Suppose P_RES is the total worth of all not-out players in a team’s inning at the interruption, and O_B is the total number of overs that the team has faced before the interruption, while C is the total allocated overs to that inning. (5)${\displaystyle P_{LOSS}=P_{RES}\ast O_B/C}$ Equation (5) helps reduce resources so that even if a team has ten wickets in hand at the interruption, the total consumable number of resources represents the extent to which the players in the team can take the target rather than just relying on their possible potential. That is the reason to subtract the value of P_LOSS from R.

PRCICM method allocates total resources to a team using Eq. (4). It then subtracts the value of consumed overs, the worth of each player that has lost their wicket, and P_LOSS from R. This subtraction returns the total resources that the team still has. In order to calculate how many resources a team has been deprived of, the value of R is calculated before and after interruption and then passed on to the proposed equations of D/L. D/L has two equations for two conditions based on resources. If R₂ ≤ R_1, then: (6)${\displaystyle \mathrm{T}=\mathrm{S}{\mathrm{R}}_2/{\mathrm{R}}_1}$ In the case of R₂ ≥ R_1, the proposed equation is: (7)${\displaystyle \mathrm{T}=\mathrm{S}+\mathrm{G}50\left({\mathrm{R}}_2-{\mathrm{R}}_1\right)}$ In Eqs. (6) and (7), T denotes target while R₁ and R₂ are team one and two resources, respectively. D/L uses the average ODI total as G_50, but the PRCICM method has a different way to calculate this value, as described in Eq. (2). With these equations, predefined rules to set the value of T were also used in the PRCICM method. Pseudocode for resource calculation is given below:

Evaluation of allocated resources to players

Multiple linear regression assigned weightage to the parameters under consideration. Since the dataset is divided into the country-wise inning, every country has its weightage for every parameter, and their accuracy was checked separately. Kalpdrum & Niravkumar (2018) predicts scores using classification and sets class boundaries with a difference of 24 runs. That is why the accuracy of the proposed method in this paper is measured using that similar criterion.

A total of 1,896 predictions were made on the testing dataset of each country using the equation of that country mentioned in Table 2. The accuracy of each country’s equation is calculated as shown in Fig. 3.

Every country has accuracy greater than 80% except Pakistan and South Africa. One benefit of the doubt in Pakistan’s case that can be given to the model is that the dataset is small, but that can’t be the case in South Africa’s evaluation. So further investigation indicates that the countries in which batters score 50 or more runs more frequently have less accuracy than others. This is also true for the opposite assumption: countries where batters score 50 or more runs less frequently have more accuracy than others.

Figure 4 shows the percentage of scores greater than 49 in the testing dataset of each country. It is evident that the three countries, Australia, Sri Lanka, and Bangladesh have the most accuracy but have the least percentage of scores greater than 49. In comparison, the three countries with the worst accuracy, Pakistan, India, and South Africa, have the most percentage of 49 plus scores in their respective countries.

This argument gets further validation with another experiment. If the above-written arguments were correct, then there should be an increase in the accuracy of each country’s predictions when there are no scores greater than 49 in the testing and training dataset. This experiment was performed, and Fig. 5 shows the accuracy of the equations. The accuracy of each country’s prediction has increased, but South Africa and India’s accuracy has increased the most. For both these countries, the correct predictions of the models were almost the same in both scenarios. Still, the number of wrong predictions saw a significant drop, resulting in more accuracy.

Predictions are required in a match’s context, and the predicted score is the criteria to assign weightage and calculate G50. That is why the model is also evaluated from a match’s perspective. A maximum of 11 players can be part of a team, so total predictions were divided by 11 to get the total number of matches, and then the correct number of predictions was divided by that number. That can be interpreted as the expected number of correct predictions in a match.

Figure 6 shows that every country has nine correct predictions per match except for Pakistan, India, and South Africa, and 2–3 predictions can go wrong in a match. As addressed above, those players who perform better than expectations are usually not so accurately predicted. That means a player with higher ability gets lower weightage than expected. These assigned weights are calculated to use in the PRCICM method, and the PRCICM method deducts the full weightage of a player from the team’s resources when he gets out. Thus, those wrong predictions can’t significantly impact the PRCICM method.

Analysis of PRCICM method in comparison with DLS

ICC uses DLS for resetting targets in interrupted matches. Although the creators of D/L and some others have suggested it for calculating victory points that can be an alternate for NRR, its primary job is to adjust targets based on available and lost resources in a match. PRCICM method is a proposed alternative to the DLS method. With the help of the player’s weightage and over’s worth PRCICM method was applied to all the matches in the cricket world cup 2019 where ICC used the DLS method. Figure 7 shows a comparison of the PRCICM method with the DLS method.

PRCICM method uses players as resources, so every player’s data in playing 11 of both teams must be available. Some players are not available in the dataset, but they were part of actual teams in the aforementioned matches. That’s why their replacement players were used using information from their respective team’s world cup squad. In Afghanistan vs. South Africa, Muhammad Shahzad was considered a part of the Afghanistan team instead of Ikram Alikhil, and Dale Steyn’s stats were used instead of Beuran Hendricks. Suranga Lakmal was the missing player from Sri Lanka vs. Afghanistan match in the dataset, so Jeffrey Vandersay was his replacement for testing purposes. In Pakistan vs. India’s match, Shaheen Afridi and Junaid Khan replaced Wahab Riaz and Muhammad Amir.

In the first match, the match was reduced to 48 overs because of rain interruption. Rain interrupted at 20th over of Afghanistan’s inning, so both methods evaluated the lost resources similarly, and the difference between PRCICM’s target score and DLS’s target is only one run. In this kind of scenario where interruption occurs in the 1st inning, both the DLS and PRCICM method considers the lost resources of the 2nd team because they also have fewer overs to play because of interruption.

The interruption scenario of 2nd match, Sri Lanka vs. Afghanistan, is similar to this. However, the DLS method weighs a bit more in favor of Afghanistan’s loss of 9 overs and reduced Sri Lanka’s target, but the PRCICM method’s resource worth is different than DLS, increasing Sri Lanka’s target. This result is debatable, and based on their interpretation of the game, experts can interpret this differently based on their cricketing instincts. The third match was between Pakistan and India. According to the PRCICM method’s evaluation, Pakistan has lost more resources because of interruption, but DLS assigns a lesser number to lost resources in this case. That is why there is a difference between both models’ allocated scores. That shows how the PRCICM method works typically in these scenarios and does not produce any result that can be deemed an odd result. Although PRCICM uses D/L’s equation, resource allocation of both methods is different, and PRCICM has separate over weightage for each country and uses a very dynamic player weightage allocation method. That gives the PRCICM method the ability to evaluate the team’s resources based on their inning trajectory in the past combined with players’ weightage based on their past performances and the country they are playing in.

In order to get good results from the DLS method, the team has to plan their inning according to DLS’s generalized resource table that is used for every team, but the PRCICM method has the resources generalized for every team separately. That is why who got out impacts the PRCICM method, and it caters to changes in the batting order, but DLS does not bother that. In the scenario of Sri Lanka vs. Zimbabwe match, mentioned by the creators of DLS, when applied to the Indian team that played the group match vs. Pakistan in cricket world cup 2019 and instead of sending Kohli in, Chahal bats at number 3. That impacted the final target by four runs.

As investigated by stern D/L under estimated lower-order partnerships, the match stern mentioned is checked using the same Indian team mentioned above in English conditions. PRCICM method suggests that the Indian lower order did not have what stern expected from the lower order. That is why the PRCICM method produces a result more inclined toward D/L than DLS in that scenario.

PRCICM method failed in one extreme case scenario defined by D/L that if team A and B hit six runs off every ball and then interruption happens at the end of 25th over of 2nd inning, the realistic target is 901 runs. PRCICM methods assigned a target of 904 runs to the Indian team used in previous experiments. From D/L’s perspective, this is a failure because they think if both teams are equal all the time, then, in the end, both should have been equal, but PRCICM is a prediction-based model, and in this scenario, the prediction was India is three runs short.

The famous 2003 world cup final that justifies the D/L method was also passed as a test case to the PRCICM method. India was losing the match by 20 runs according to D/L, but the PRCICM method increased this margin to 66 runs. In this case, the PRCICM method is by far closer to reality.