Cricket is one of the most popular team sports in the world. Players have multiple roles in a game of cricket, predominantly as batsmen and bowlers. Over the generations, statistics such as batting and bowling averages, and strike and economy rates have been used to judge the performance of individual players. These measures, however, do not take into consideration the context of the game in which a player performed. Furthermore, these types of statistics are incapable of comparing the performance of players across di erent roles. In this paper, we present an approach to quantitatively assess the performances of individual players in single match of One Day International (ODI) cricket. For this, we have developed a new measure, called the Work Index, which represents the amount of work that is yet to be done by a team to achieve its target. Our approach incorporates game situations and the team strengths to measure the player contributions. This not only helps us in evaluating the individual performances, but also enables us to compare players within and across various roles on a common scale. Using the player performances in a match, we predict the player of the match award for the ODI matches played between 2006 and 2016. We have achieved an accuracy of 86.80% for the top-3 positions, which is superior to baseline models and previous works, to the best of our knowledge.
Cricket is majorly played in three formats { Test, ODI and Twenty20 (T20), with ODI being one of the most followed formats. An ODI is a form of limited overs cricket, played between two teams where each team has a combination of batsmen and bowlers making up to 11 players in total. Each team bats for a maximum of 50 overs where an over is de ned as a set of six deliveries bowled by the bowlers of the opponent team. An ODI cricket match starts with a coin toss and the Captain of the side winning the toss chooses to either bat or bowl rst. The team batting rst sets the target score in a single innings, where the ? This work was done when the author was a student at IIIT-Hyderabad. innings lasts until the batting side loses all the 10 wickets, where a wicket refers to a player getting out, or the batting side's quota of 50 overs is completed. The team batting second tries to score more runs than the target score in order to win the match. Similarly, the side bowling second tries to take all the 10 wickets of the opponent team or make them exhaust their overs before they reach the target score in order to win.
In a game of cricket, the batsmen of one team play against the bowlers of the other team, and vice-versa, in order to win the match. Therefore, evaluating the performances of individual players in a game of cricket becomes very critical. It helps in segregating the players who are contributing to the team from the ones who are failing to deliver on the ground. However, evaluating the performances of players is not a straight-forward task. Traditionally, statistics such as batting and bowling averages, and strike and economy rates have been used to assess the performance of individual players. However, these statistics fail to incorporate several important aspects of the game. Runs scored or wickets taken under pressure at crucial stages are of more value as compared to scoring more number of runs or taking more number of wickets. Furthermore, assessing the overall performance of an individual cricketer requires a comprehensive evaluation of his contributions to the team, both in terms of his batting and bowling contributions. However, combining and comparing the batting and bowling performances of a player, on a common scale, is a challenging task and often becomes a subjective decision.
Therefore, in this paper, we propose a methodology to quantitatively assess the performances of individual players in a single game of ODI cricket match. We introduce a new measure, called the Work Index, which represents the amount of work yet to be done by a team to reach their expected target score. Work Index incorporates several important aspects of a game, including the current stage of the match, the progress so far as compared to the initial estimations, the two competing teams' strengths, etc. We measure the Work Index for both the batting as well as bowling teams, namely, the Batting Work Index and the Bowling Work Index. The former denotes the amount of work to be done by the batting team to reach the target, while the latter represents the amount of work to be done by the bowling team to restrict the batting team from reaching the target. Using these two work indices, we calculate a contribution-score for each player which incorporates his batting and bowling contributions towards achieving the team's overall goal.
Furthermore, akin to many other sports, the player of the match title is awarded to the player who played the most signi cant role in a match of cricket. Today, it is chosen by the match committee and the commentators which makes it a subjective decision. Therefore, we propose a methodology to determine the player of the match using the player contribution-scores calculated by our approach. We compare our model with previous works and other baseline models, and the superiority of our model over others further proves the validity of our approach.
In literature, Duckworth and Lewis proposed a real-time measure to estimate
the amount of resources remaining with a team, called as D/L resources, as a
function of number of balls and wickets remaining. They further used the D/L
resources to reset targets in rain interrupted matches [
Pertaining to assessing player performance, Johnston et al. [
Recently, Bhattacharjee et al. [
Therefore, in this paper, we propose a new dynamic measure, called the Work Index. Apart from the runs scored and balls bowled, the work index also incorporates the potentials of the batsmen and bowlers who are remaining to perform. In addition to this, with the use of D/L resources, we propose a method which enables us to calculate the work index in the rst innings also, where a team does not have a xed target to chase. We further use the work index to quantitatively assess the player performances in a match and predict the player of the match. 3
Our methodology to assess the player performances for a given ODI cricket match involves estimating a new measure, called the Work Index. Work Index incorporates several crucial qualitative and quantitative aspects of the game. The three parameters considered in calculating the work index are as follows: { The progress, in terms of runs scored, towards chasing the set target. { The current stand, in terms of scoring rate, of the batting team relative to the initial estimations. { The remaining batting and bowling potentials of the batting and bowling teams, respectively.
The rst and second parameters capture the quantitative aspects of the game situation in terms of the runs scored and the required run rate as compared to the initial required run rate for the batting team. On the other hand, the third measure captures the quality of the batsmen and bowlers remaining for the batting and bowling teams, respectively. Therefore, work index, a blend of these features, captures the context at a given stage of the match. 3.1
Calculating the rst and second parameters requires us to know the target score the batting team is trying to achieve. In an ODI cricket match, the team batting second has a prede ned target, set by the opponent team, to chase in order to win the match. On the other hand, the team batting rst does not have a xed target to score. Ideally, the team would try to score as many runs as possible, but there are limited resources in terms of the number of overs and wickets. Therefore, at the start of the rst innings, we estimate the target score to be the average number of runs scored in the rst innings of all the matches played in the same country in the past, where the team batting rst was able to successfully defend their score. This estimated target score is further improved after every ball is bowled as per Equation 1, depending upon the actual situation of the match.
newT arget
runsScored + DLrem
initT arget
(1)
where runsScored is the number of runs scored by the batting team at the
current stage of the game, DLrem is the ratio of the D/L resources remaining
with the team [
With a de ned target score to achieve for both the innings, Equations 2 and 3 represent the mathematical formulation of the rst and second parameters of Work Index (as mentioned at the start of Section 3), respectively. k r
runsRemaining=T arget reqRunRate=initRunRate (2) (3) where runsRemaining, reqRunRate and initRunRate represent the number of runs yet to be scored, the average number of runs required per over from the current stage and the average number of runs require per over at the start of the match, respectively, by the batting team to achieve its target score. The higher values of k and r intuitively tell us that the batting team has a lot of work to do to reach the target score, and vice-versa.
Calculating the third parameter requires us to model the player and team
potentials for a given match. Therefore, as proposed in [
Further, we de ne the total batting score of a team to be the summation of the batting scores of all of it's players. And similarly for the total bowling score of a team. However, at a given state of the match, some of the players from the batting team would have got out and some of the players from the bowling team would have bowled a part of their quota of maximum 10 overs (60 balls). Therefore, we de ne the remaining batting score of the batting team as the sum of the batting scores of only those players who haven't got out yet. Similarly, the remaining bowling score of the bowling team is calculated as the sum of all the individual bowlers' bowling scores, weighed by the ratio of number of balls he has remaining to bowl in this match to the maximum number of balls he can bowl in an ODI cricket match, i.e., 60.
Hence, with a de ned method to estimate the total and remaining batting and bowling potentials of the batting and bowling teams, respectively, Equations 4 and 5 represent the mathematical formulation of the third parameter. w b remBatScore=totalBatScore remBowlScore=totalBowlScore (4) (5)
Where totalBatScore and remBatScore represent the total and remaining batting potentials of the batting team. And similarly for the totalBowlScore and remBowlScore. The higher values of b and w tell us that the batting and bowling teams have got good players to perform who can win the match for them, respectively. 3.3
Having formalized all the three parameters of work index, we will now explain the Work Index in detail. Work Index is a dynamic measure which is updated as the innings progresses and takes into account the current state of the match to estimate the work yet to be done. With the help of variables de ned in Equations 2, 3, 4 and 5, the batting and bowling work indices are calculated as per the Equations 6 and 7, respectively.
battingWorkIndex 100 k (r + (1 b) + bowlingWorkIndex 100 k (1=r + b + (1 w) w)) (6) (7) 600 500 ts400 e k c i w fo300 r e b m u200 N 100 0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738394041424344454647484950
Over
Variable k (de ned in Equation 2), de ned as the ratio of the runs remaining to score with respect to the target, is used as a bias in calculating the work index. The lower values of k, generally found at the end of the innings, directly reduces the impact of the other factors in determining the work index. As it can be seen from Figure 1, as a game reaches its nal stages, the batsmen adopt a riskier strategy to score runs at a higher rate, with less concern about losing wickets. The value of a wicket reduces as scoring runs becomes the sole purpose. Similarly, higher values of k, found at the start of the innings, boosts the impact of other factors. This is because, at the initial stages of the innings, the wickets of the batsmen carry a lot more importance. Losing early wickets at the start of an innings puts the batting team into tremendous pressure, as they lose their key batsmen and face the threat of getting all-out, before even completing the quota of 50 overs. Variable r (Equation 3) captures how well is the batting team scoring as compared to the initial estimations. It is directly proportional to the batting work index, as increased required run rate, increases the amount of work to be done, and similarly it is inversely propotional to the bowling work index.
Variables b and w (Equations 4 and 5) represent for the remaining batting and bowling potentials of the corresponding teams. They account for the amount of batting and bowling resources remaining with the batting and bowling teams, respectively. Incorporating individual player's skills into these variables enables us to assess the current game scenario in a detailed way. They enable us to capture those scenarios where a team has lost several wickets yet still has good players remaining in the batting line-up, who can potentially change the game's direction. The parameters, and , represent the relative weightage of the remaining batting potential and the remaining bowling potential, respectively. Also, bowlers have 300 balls to possibly get the batsmen out, whereas the batting team possesses only 10 wickets to score runs. Therefore, losing wickets has signi cant impacts on the batting team. The values of the parameters, and , have been discussed in the experiments Section 4.
In all, work index is a combination of many important aspects of the game that enables it to capture the overall game scenario. 3.4
Cricket is a game of bat and ball. Bowlers from the bowling team take turns in overs to bowl their quota of 50 overs, where, in an over, one of the players from the bowling team bowls 6 deliveries to the batsman at strike. We calculate the Batting Work Index and the Bowling Work Index after each ball is bowled. The di erence between the two consecutive batting work indices is contributed to the batsman who played the corresponding ball. Similarly, the di erence between the two consecutive bowling work indices is contributed to the bowler who bowled the ball. We repeat this process for each ball bowled in the entire match. At the end of the match, the scores attributed to an individual player represents his all-round performance in the entire match. This score not only captures the quantitative amount of runs scored or wickets taken by a player, but also the context in which he made the contributions. Phrases like \Catches win Matches" are very popular in the game of cricket and therefore, the elding e orts of players could also be integrated to capture the players' overall contribution to the match. However, we have tabled it for the future work as of now.
Furthermore, Player of the Match title is awarded to the player who played the most signi cant role in a particular match. At rst, it seems that the player who has the maximum score at the end of the match should be awarded the player of the match. However, player of the match award is almost always (95.83% of the times) given to a player from the winning side. A player from the losing side bags the player of the match award only if his performance is signi cantly better than the others and has contributed to an almost win for the losing side. Therefore, as of now, we choose the players from the winning side only as the potential candidates for the player of match award. We rank them directly based on their scores and the player with the maximum score is awarded the player of the match. Note that some heuristics can be used to capture the very rare player of the match awards from the losing side also. However, for now, we have tabled it for future work. 4
We have studied all the ODI cricket matches played between 1st of January, 2006
and 30th June, 2016. Ball-by-ball data for each match has been taken from the
cricsheet database [
For the potential player of the match candidates, as discussed in Section 3.4, we consider only the players from the winning team. For a given match of cricket, our model outputs a list of player ranked in the descending order of their contribution in the match. Hence, to measure the e ciency of our model, we use an exponential-ranking metric. For a given match, the match score is calculated to be 1=2R 1, where 1 R 11 is the rank at which the player of the match has been predicted by our model. Therefore, the exponential-rank of our model is calculated as the summation of the match scores for all matches. The choice of an exponentially decaying metric over existing metrics such as Mean Reciprocal Rank has been made to increase the penalty for wrong predictions, as there is only one player of the match as compared to top-k relevant outputs.
With the de ned accuracy metric, we use a validation set containing all the matches played between January, 2006 and December, 2012 to nd the most suitable values of the parameters and , de ned in Equations 4 and 5. Table 1 tabulates the exponential-rank of our model on the validation set for multiple values of the parameters. As it can be seen, = 2:0 and = 1:5 yields the best results. Therefore, these values will be considered for the further discussions. innings by estimating the target score using the same approach as discussed in Section 3, and add up the player's batting and bowling contributions for both the innings to calculate a player's overall performance in a match. The players from the winning team are considered to be the potential player of the match in the order of their overall contribution in the match. We implemented their work, to the best of our abilities, to compare their approach against our model. 70 60 50 s40 t i H 30 % 20 10 0
Apart from that, in a game of cricket, the number of runs scored by a player and the number of wickets taken by him are the two major criterion to judge a player's performance. Therefore, we further compared our approach with the two following baseline models which take into account a player's overall contribution in a match{ { Model 1: The overall contribution of a player is the summation of the ratio of the runs he has contributed to the teams total batting score and the ratio of the wickets he has taken to the total of wickets taken by his team. { Model 2: To be able to combine the runs scored and wickets taken by a player, we map one of these into another, i.e., we calculate the weight of a wicket taken by a bowler in terms of the runs scored by a batsman. The weight of a wicket, denoted by !, is calculated as the total number of runs scored in the match divided by the total number of wickets fell down. Therefore, the total contribution of a player in a match is the summation of the number of runs scored by him and ! times the number of wickets taken by him.