This paper proposes an alternative method to calculate the revised target in interrupted 50 overs One Day international (ODI) cricket matches. Existing Duckworth Lewis (D/L) method and its modified versions only take available batting resources of the batting team into account and ignore the individual player’s excellence to calculate the revised target. Here, it is worth mentioning that individual player’s excellence varies in reality, and therefore quality of the available resources may affect the revised target significantly. Further in D/L method, revised target calculation depends only on the available batting resources of the batting team and does not consider the available bowling resources of the fielding team. Proposed method overcomes these two shortcomings by taking individual player’s excellence and available bowling resources of the fielding team into account. Individual player’s excellence has been determined by Data Envelopment Analysis (DEA), a well-known non parametric mathematical programming technique. A new idea of “Net Resource Factor” has been introduced to capture both batting and bowling resources to calculate the revised target. To the best of our knowledge, this is the first attempt to incorporate the ability of individual players and bowling resources of the fielding team for calculating the revised target. A comparative analysis between the proposed method and D/L method has been carried out using the data of real ODI matches held in the past. To facilitate ground application of the proposed method, a flow chart and a “Net Resource Factor Table” have been designed.
Cricket, one of the most popular team games in the world, is played in the three formats based on duration difference, i.e. a 5-day test match and two forms of limited overs cricket, viz., 50 over One Day Internationals (ODIs) and Twenty-Twenty (T-20) internationals. First version of limited over cricket format, i.e. ODI was introduced at international level in 1971. In the initial years, many ODIs were played with 40/45/55/60 overs in an innings. Finally, International Cricket Council (ICC) standardized 50 overs as a length of an innings. The first batting team sets a target for the second team with the help of 10 wickets or in maximum 50 overs whichever ends first. The second batting team has to chase the target with 10 wickets in maximum 50 overs. Sometimes, unexpected circumstances like rain, bad light, floodlight failure, etc. enforce a truncated game. As a result, the length of an innings gets shortened. In case of these shortened matches, as two teams have unequal batting and bowling resources, so a revised target is set for the second batting team. In the last two decades, International Cricket Council (ICC) adopted different methods like Average Run Rate (ARR), Most Productive Over (MPO), Discounted Most Productive Over (DMPO), PARAB, World Cup 96 (WC 96), etc.1 for calculating the revised target in the interrupted matches. Apart from these, existing literature indicates the presence of several scholarly works like VJD method [
Though ARR is simplistic in nature and easy to compute, it gives stress only on the run rate and ignores the match situation. In MPO and DMPO methods, the revised target is set by excluding the first batting team’s most economic overs. As a result, they often become more biased to the first batting team because of ignoring the bowling excellence of the first fielding team. PARAB and WC96 are designed on the basis of norms’ table prepared for the overs lost. But in these two methods, numbers of wickets fallen are not considered, thus bringing down their practical significance. As per CLARK method, developed on the basis of dynamic programming model, every innings has 3 stoppages and at every stoppage, the team has different resources. The revised target is calculated on the basis of resources available. But the ambiguity among revised scores at the meeting point of two adjacent stoppages weakens this approach. Thomas [
However, all the resource based approaches including D/L method only focus on the batting resources of the second batting team, numbers of wickets fallen and numbers of overs remaining. Available bowling resources of the fielding team are always ignored. Here, we introduce a new idea of “Net Resource Factor” that takes both batting resources of the batting team and bowling resources of the fielding team into account. While determining available batting resources, none of these methods takes the individual player’s efficiency (quality or efficiency) into account, which differs in different players. Here, it is worthwhile to note that batting team’s capability to chase a particular target depends not only on the remaining overs and numbers of wickets lost, but also on the quality of the batsmen who are at crease and the batsmen who are yet to bat. In a similar fashion, ability of the bowling team to restrict the opponent below the target relies not only on the remaining overs, but also on the quality of the bowlers who are eligible to bowl the remaining overs. Recognizing the importance of this, the proposed method incorporates the individual excellence and remaining bowling resources of the fielding team to calculate the revised target. Here, efficiency of a player has been determined with the help of Data Envelopment Analysis (DEA), a well-known non parametric mathematical programming technique. In addition, a comparative analysis between the proposed method and D/L method has been performed using the data of real ODI matches held in the past. To the best of our knowledge, this is the first attempt to incorporate the ability of individual players and bowling resources of the fielding team for calculating the revised target.
The remainder of this paper is organized as follows. In Section 2, theoretical background and mathematical model are demonstrated. The proposed method is illustrated through a real match in Section 3. In Section 4, a comparative study between the method and the D/L method has been carried out and the implications have been demonstrated. The paper is concluded by discussing the summary of contribution and future research avenues in Section 5.
Following assumptions have been used to describe the method:
1) Total batting resources and bowling resources of a team are 100 units each assuming that a team can use its resources up to 100% level in the respective batting and bowling departments.
2) In reality all batsmen and bowlers may have different capabilities. So, individual player may contribute a different percentage of resources to teams’ total resources.
3) Traditionally, there is a four years gap in the occurrence of most of the major sports events like World Cup (played in every form of game e.g. cricket, football, Hockey), Olympics etc. Following this convention of 4 year cycle, a player’s efficiency prior to any match is based on the performance of the last 4 years.
4) Since, players have different ability in terms of batting and bowling skills, all the batsmen, bowlers and all-rounders have been classified in 10 classes, i.e.
5) A player without any prior international match experience is placed in
6) A team loses its batting or bowling resources exponentially with time due to the effect of match condition, pitch, weather, batting, bowling or fielding excellence [
7) For a bowler, the resources are consumed uniformly in maximum overs a bowler can bowl in a match i.e. 10 overs in a 50 over match. If the game is shortened due to rain, then bowling resource consumption will be uniformly truncated considering the new maximum number of overs a bowler can bowl in that match.
8) If the match is stopped permanently, a bowler’s maximum over limit for computing his remaining bowling resources will be determined as per the length of the first innings. But it will maintain the aforementioned assumption of uniform consumption of bowling resources per over.
9) As it is not known when the batsman will be out, batting resource consumption on per over basis is not considered.
10) A team’s net resource factor in any match situation depends on its remaining batting resources, remaining bowling resources of the opponent team, overs to be bowled and length of the first innings.
Here, we have summarized the procedure to calculate the revised target. First step is to determine the efficiency of each player of two teams using DEA; categorize them as per “Players’ Category Table” and allocate resource according to the category. Second step is the computation of individual player’s contribution in team’s total batting or bowling resources. Subsequently the following information will be collected:
1) At the point of interruption, list of batsmen who got out, batsmen at the crease, batsmen yet to bat for the second batting team, and bowling summary (the overs already bowled, wickets taken and total runs conceded) of every bowler for the second fielding team.
2) In case of temporary interruption (the interruption in the second innings after which second innings resumes but the length is reduced), remaining overs left as per the revised length of the second innings.
3) In case of permanent interruption (the interruption after which match is stopped permanently), remaining overs that would have been bowled in the absence of interruption.
On the basis of this information we obtain the remaining batting resource of the second batting team and available bowling resource on the basis of individual player’s possession in team’s total batting or bowling resource and calculate the net resource factor. Finally, revised target will be calculated with the help of Net resource factor.
In this paper, individual player’s excellence has been evaluated based on the efficiency score computed using DEA. It is a nonparametric mathematical programming technique to calculate the relative efficiency of Decision Making Units (DMUs) that transform multiple inputs into multiple outputs. DEA helps to compare the efficiency of one DMU to the efficiency of other DMUs in the same peer group based on the level of the inputs used and outputs produced. These DMUs can be business units, hospitals, educational institutions, individuals etc. On the basis of Farrell’s [
In this work, DEA has been used to classify cricket players in different categories according to their level of efficiency. Players are considered as DMUs producing multiple outputs by taking multiple inputs. In case of batsmen, Sharp et al. [
Let, there be n DMUs (here, n individual players) each having m inputs to produce s outputs. Let,
notes the value of the
Batsman | Bowler | |
---|---|---|
Input measures | 1) Number of matches a batsman played (Ma) 2) Number of innings in which a batsman bat (Ba) 3) Number of innings in which a batsman got out (Out) | 1) Number of matches a bowler played (BMa) 2) Number of innings in which a bowler bowled (Bow) |
Output measures | 1) Strike rate (SR) 2) Average (Batavg) 3) Highest individual score (HS) 4) Number of 4s (4s) 5) Number of 6s (6s) 6) Number of centuries (Ce) 7) Number of half centuries (HC) 8) Total number of runs scored by the batsman in the last 4 years (RUN) | 1) Total number of wickets taken by the bowler in the last 4 years (Wkt) 2) Bowling Economy rate (Ecr) 3) Strike rate (BSR) 4) Bowling average (Bowavg) |
pearance of zero weights, a positive non-Archimedean infinitesimal constant
Subject to
Based on the efficiency score
After the categorization and resource allocation as per the “Players’ Category Table”, individual player’s possession in corresponding team’s total bowling and batting resources is determined. For the second batting team, remaining batting resources comprise the resources of the batsmen who are at the crease and yet to bat till the point of interruption.
Remaining bowling resources of the second fielding team depend on the overs left in each bowler’s quota and its bowling resource consumption per over. Now, determination of maximum over limit for a bowler and its bowling resource consumption per over depends on the type of interruption occurred in the match.
According to assumption (8), a bowler’s resource is uniformly consumed in the maximum overs a bowler can bowl in a match. In case of a rain interrupted match, the interruption can be either permanent or temporary. In
Categories | Efficiency (q) | Resource of the category |
---|---|---|
P1 | 0.9 ≤ q ≤ 1.0 | 100 |
P2 | 0.8 ≤ q < 0.9 | 90 |
P3 | 0.7 ≤ q < 0.8 | 80 |
P4 | 0.6 ≤ q < 0.7 | 70 |
P5 | 0.5 ≤ q < 0.6 | 60 |
P6 | 0.4 ≤ q < 0.5 | 50 |
P7 | 0.3 ≤ q < 0.4 | 40 |
P8 | 0.2 ≤ q < 0.3 | 30 |
P9 | 0.1 ≤ q < 0.2 | 20 |
P10 | 0 ≤ q < 0.1 | 10 |
case of temporary interruption, length of the second innings is shortened leading to the new maximum over limit of a bowler’s bowling quota. As a bowler can bowl less number of overs compared to an uninterrupted match, bowling resource consumption per over will be higher compared to that in case of an uninterrupted match. If a bowler does not finish the quota, some of its resource will remain unused. For example, a P1 category bowler can bowl maximum 10 overs in a 50 over match and possesses 15 units of its team’s total bowling resources. So the bowler’s bowling resource consumption per over in a full length match is 1.5 units per over or 10% of its total bowling resource per over. For the temporary interruption, if the match is shortened to 30 overs, a bowler can bowl maximum 6 overs in that match. Then the bowler’s bowling resource consumption per over is 15/6 = 2.5 units per over or 100/6 = 16.66% of his total bowling resource per over. Now, if a bowler bowls 4 overs, then 2.5 × 2 = 5 units of his bowling resource or 16.67 × 2 = 33.34% of his bowling resource will remain unused.
In case of permanent interruption, when game is not continued after the point of interruption, the maximum over limit a bowler can bowl will be decided based on the length of the first innings. Here, length of the first innings signifies the maximum over limit of the first innings for the first batting team as per the rule of the International Cricket Council (ICC). In the case of interruption in the first innings, it may be shortened from 50 overs to any specific over limit. Suppose, in a match due to the temporary interruption in first innings, length of the first innings is reduced to 45 overs. The match is stopped permanently after 30 overs of the second innings. Let, a P1 category bowler possess 18 units of his team’s total bowling resource and has already bowled 6 overs in the second innings till the point of interruption. To determine the remaining bowling resource, we consider bowling quota for a bowler same as the maximum over limit of an individual bowler in the first innings, i.e. 45/5 = 9 overs. Here, bowler’s bowling resource consumption per over will be 18/9 = 2 units per over or 100/9 = 11.11% of his total bowling resource per over. For this bowler, 2 × 3 = 6 units of his bowling resource or 11.11 × 3 = 33.33% of bowling resource will be treated as the remaining resource.
A situation may arise where the reduced length of second innings leads to the determination of unequal maximum over limit for the bowlers of second fielding team. To develop the procedure of determining the maximum over limit for individual bowler in a quantified manner, we have assumed that the captain of the second fielding team would decide the best bowlers based on the performance measures, i.e. wickets taken, the economy rate and the strike rate in the ongoing match till the point of interruption. The captain would rationally allocate the largest shares of unequal maximum bowling quotas to the best bowlers. Though in case of real match scenarios captain can call any bowler to bowl, we have assumed that the captain uses a ranking method to determine the rank of bowlers by considering number of wickets taken, the economy rate and the strike rate in the current match into account. This assumption will help to design a simplified model along with capturing the reality match. As less economy rate and strike rate is always preferred [
In reality, if a bowler bowled only in a few occasions in past or didn’t bowl at all, there is a very low chance that such bowler would be called to bowl. In order to eliminate the error in determining the remaining resources of these bowlers who rarely bowl, we assume that the remaining overs of these bowlers will be zero. In our method the bowlers who have not taken any wicket for the specified time span, i.e. 4 years are recognized as the bowlers of this category. The general formula to compute the bowling resource consumption per over for a bowler can be expressed as follows:
As per assumption (10), team’s net resource factor (NRF) depends on four factors:
Team’s remaining batting resources;
Opponent team’s remaining bowling resources;
Overs to be bowled;
Length of the first innings.
Here, fundamental idea is that the batting resources of the batting team decrease exponentially with the overs. But this reduction effect is partly compensated by the exponentially decreasing bowling resources of the opponent team. So in any match situation, NRF of the batting team stands different to the conventional remaining batting resource. This NRF will be applied to Duckworth Lewis target for modifying and setting a new revised target. Here two scenarios which are illustrated below:
Scenario 1
Let, first batting team plays a certain number of overs. After that temporary interruption occurs. For this reason, its innings does not continue after that point. Now, new over limit of the innings determined for the second batting team is the length of the first innings. As the second batting team is aware of the shortened innings, it can focus on increasing the run rate from the beginning of the innings. On the other hand, first batting team may lose its opportunity to make a higher score due to sudden interruption. To get rid of the unfair advantage given to the second batting team, the NRF of the first batting team can be designated as
Scenario 2
If the permanent interruption takes place in the second innings, the NRF of the second batting team is calculated as follows:
where,
α: Remaining batting resource of the batting team;
β: Remaining bowling resource of the bowling team.
As per D/L method, if a match is shortened or stopped permanently due to the interruption, then D/L method is used to calculate the revised target using the following equation:
where, w: wickets already lost;
Revised target of the second batting team for scenario 1 and 2 can be expressed by (3) and (4), respectively.
As per our method, if a team has less remaining batting resource compared to the remaining bowling resource of the bowling team, then it can be deduced that it has less net resource and it has to chase a higher target than the revised target set by D/L method. On the other hand, if a team has more remaining batting resource compared to the remaining bowling resource of the bowling team, then it has more net resource factor and hence revised target will be less. Thus the efforts of the batting team and bowling team up to the point of interruption of the match are recognized.
Here, we illustrate our method with the help of a real match held between South Africa and West indies in Champions trophy 2013 played at Sophia Gardens, Cardiff, England on 14th June 2013. It was a very crucial match as the winning team would eventually qualify for the semifinal. From the beginning of the match the progress of the match had been often interrupted by rain and as a consequence, length of each innings was reduced to 31 overs. South Africa, batting first, scored 230 runs. In the second innings, the game was permanently stopped due to rain after 26・17 overs, i.e. 26 overs and 1ball. Till then West Indies scored 190 runs by losing 6 wickets. But the revised target by D/L method was 191 runs. So the match ended in tie. As South African team had the better run rate compared to West Indies in the tournament, it qualified for the next round. But, according to proposed method, the revised target would have been 189 and as a result West Indies would have emerged as winner by 1 run2. The mechanism of our proposed method is described below in details. All the data related to the input and output measures of players are collected for the 4 year span staring from 13th June 2009 to 13th June 20133.
West Indies batsmen | Input measures | Output measures | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Ma | Ba | Out | HS | SR | Batavg | 4s | 6s | Run | Ce | HC | ||
C. Gayle | 43 | 43 | 40 | 125 | 95・31 | 30.02 | 130 | 46 | 1201 | 1 | 7 | |
J. Charles | 13 | 13 | 13 | 130 | 85・90 | 34.69 | 48 | 11 | 451 | 2 | 1 | |
D. Smith | 16 | 16 | 16 | 107 | 74・81 | 31.37 | 50 | 4 | 502 | 1 | 3 | |
M. Samuels | 37 | 35 | 31 | 126 | 71・07 | 31.38 | 96 | 17 | 973 | 2 | 4 | |
D. Bravo | 55 | 52 | 44 | 100* | 69・76 | 32.45 | 117 | 28 | 1428 | 1 | 9 | |
K. Pollard | 62 | 58 | 54 | 119 | 94・95 | 30.68 | 110 | 79 | 1657 | 3 | 6 | |
D. Bravo | 44 | 39 | 38 | 77 | 77・87 | 22.78 | 52 | 20 | 866 | 0 | 4 | |
D. Sammy | 70 | 59 | 42 | 84 | 103・06 | 24.00 | 78 | 50 | 1008 | 0 | 4 | |
S. Narine | 30 | 21 | 18 | 36 | 83・19 | 11.27 | 20 | 5 | 203 | 0 | 0 | |
T. Best | 9 | 6 | 4 | 8 | 48・00 | 6.00 | 2 | 0 | 24 | 0 | 0 | |
R. Rampaul | 42 | 22 | 16 | 86* | 80・41 | 14.37 | 23 | 8 | 230 | 0 | 1 | |
*Denotes not out.
South African bowlers | Input measures | Output measures | ||||
---|---|---|---|---|---|---|
BMa | Bow | Wkt | Ecr | Bowavg | BSR | |
C. Ingram | 26 | 1 | 0 | 17.00 | - | - |
H. Amla | 55 | 0 | 0 | 0.00 | - | - |
A. Devilliers | 56 | 0 | 0 | 0.00 | - | - |
J. Duminy | 50 | 34 | 17 | 5.00 | 39.29 | 47.10 |
F. Duplessis | 35 | 9 | 2 | 5.68 | 71.00 | 75.00 |
D. Miller | 22 | 0 | 0 | 0.00 | - | - |
R. McLaren | 24 | 23 | 34 | 5.05 | 27.32 | 32.40 |
R. Peterson | 35 | 34 | 50 | 4.89 |
C. Morris | 1 | 1 | 2 | 3.57 | 12.50 | 21.00 |
---|---|---|---|---|---|---|
D. Steyn | 43 | 43 | 57 | 4.70 | 30.47 | 38.80 |
L. Tsotsobe | 45 | 45 | 73 | 4.79 | 24.27 | 30.30 |
Using the input and output data, the efficiency of players has been determined with the help of DEAP V2.1, DEA software by Coelli [
When match stopped permanently because of rain, South African bowlers bowled 26 overs and 1 ball, i.e. 26・17 overs. Only 4・83 overs, i.e. 4 overs and 5 ball of West Indies innings were left. In an innings of 31 overs, a bowler can bowl maximum 7 overs and four bowlers can bowl maximum 6 overs each.
West Indies batsmen | Efficiency | Player’s categorization based on efficiency | Resource allocation as per category | Individual resource allocation in total team’s resource (in units) (approximated to the nearest integer) |
---|---|---|---|---|
C. Gayle | 1.000 | P1 | 100 | 11 |
J. Charles | 1.000 | P1 | 100 | 11 |
D. Smith | 1.000 | P1 | 100 | 11 |
M. Samuels | 0.942 | P1 | 100 | 11 |
D. Bravo | 1.000 | P1 | 100 | 11 |
K. Pollard | 1.000 | P1 | 100 | 11 |
D. Bravo | 0.712 | P3 | 80 | 8 |
D. Sammy | 0.895 | P2 | 90 | 10 |
S. Narine | 0.544 | P5 | 60 | 7 |
T. Best | 0.099 | P10 | 10 | 1 |
R. Rampaul | 0.032 | P3 | 80 | 8 |
South African bowlers | Efficiency | Player’s categorization based on efficiency | Resource allocation as per category | Individual resource allocation in total team’s resource (in units) (approximated to the nearest integer) |
---|---|---|---|---|
C. Ingram | 0.099 | P10 | 10 | 2 |
H. Amla | 0.099 | P10 | 10 | 2 |
A. Devilliers | 0.099 | P10 | 10 | 2 |
J. Duminy | 0.683 | P4 | 70 | 13 |
F. Duplessis | 0.099 | P10 | 10 | 2 |
D. Miller | 0.099 | P10 | 10 | 2 |
R. McLaren | 1.000 | P1 | 100 | 19 |
R. Peterson | 0.943 | P1 | 100 | 19 |
C. Morris | 0.099 | P10 | 10 | 2 |
D. Steyn | 0.852 | P2 | 90 | 18 |
L. Tsotsobe | 1.000 | P1 | 100 | 19 |
West Indies batsmen | Batting status (out or not out) till the point of interruption | Individual resource allocation in total team’s resource | Total remaining batting resource for west indies team (in units) |
---|---|---|---|
C. Gayle | Out | 11 | 34 units |
J. Charles | Out | 11 | |
D. Smith | Out | 11 | |
M. Samuels | Out | 11 | |
D. Bravo | Out | 11 | |
K. Pollard | Out | 11 | |
D. Bravo | Not out | 8 | |
D. Sammy | Not out | 10 | |
S. Narine | Not out | 7 | |
T. Best | Not out | 1 | |
R. Rampaul | Not out | 8 |
Computation procedure of remaining bowling resource for the South African team involves more complexity.
According to our method West Indies should have won the match by 1 run and qualified for semifinal instead of South Africa.
To test the effectiveness of the method, we have applied the method on four other international ODI matches where D/L method had been applied due to the interruption4.
Here, we have observed that the scenario 1 described in section 2.7 is similar in case of match 1 and 4 whereas scenario 2 is applicable for match number 2 and 3. From
Inclusion of remaining bowling resource of the second fielding team in determining the revised target helps to bring forth more accurate results. In case of the first match, as second fielding team Pakistan had more available bowling resource compared to the remaining batting resource of West Indies, net resource factor of the second batting team was low. As a consequence, the revised target was set higher, compared to the target determined by D/L method. It justifies the effort of bowlers of Pakistan as they were able to keep more bowling resource on hand compared to remaining batting resource of the West Indies team till the point of interruption.
South African bowlers | Wickets taken in this match | Economy Rate | Strike rate | weighted average of number of wickets, inverse of economy rate and inverse of strike rate | Rank in the match as per the weighted average | Maximum overs a bowler can bowl as per ranking |
---|---|---|---|---|---|---|
C. Ingram | 0.000 | 0.000 | 0.000 | 0.000 | 7 | 6 |
H. Amla | 0.000 | 0.000 | 0.000 | 0.000 | 7 | 6 |
A. Devilliers | 0.000 | 0.000 | 0.000 | 0.000 | 7 | 6 |
J. Duminy | 0.000 | 9.660 | 0.000 | 0.030 | 6 | 6 |
F. Duplessis | 0.000 | 0.000 | 0.000 | 0.000 | 7 | 6 |
D. Miller | 0.000 | 0.000 | 0.000 | 0.000 | 7 | 6 |
R. McLaren | 1.000 | 10.730 | 19.020 | 0.380 | 4 | 6 |
R. Peterson | 1.000 | 5.500 | 24.000 | 0.408 | 2 | 6 |
C. Morris | 1.000 | 7.500 | 24.000 | 0.392 | 3 | 6 |
D. Steyn | 2.000 | 5.500 | 18.000 | 0.746 | 1 | 7 |
L. Tsotsobe | 0.000 | 6.160 | 0.000 | 0.054 | 5 | 6 |
South African bowlers | Overs already bowled | Remaining Overs in bowler’s quota | Bowling resource Consumption per over | Remaining bowling resource (in units) (nearest integer) | Total remaining bowling resource for South African team (in units) |
---|---|---|---|---|---|
C. Ingram | 0.00 | 0.00 | 0.33 | 0 | 28 units |
H. Amla | 0.00 | 0.00 | 0.33 | 0 | |
A. Devilliers | 0.00 | 0.00 | 0.33 | 0 | |
J. Duminy | 3.00 | 3.00 | 2.17 | 7 | |
F. Duplessis | 0.00 | 6.00 | 0.33 | 2 | |
D. Miller | 0.00 | 0.00 | 0.33 | 0 | |
R. McLaren | 3.17 | 2.83 | 3.17 | 9 | |
R. Peterson | 4.00 | 2.00 | 3.17 | 6 | |
C. Morris | 4.00 | 2.00 | 0.33 | 1 | |
D. Steyn | 6.00 | 1.00 | 2.57 | 3 | |
L. Tsotsobe | 6.00 | 0.00 | 3.17 | 0 |
Description | Result |
---|---|
Total remaining batting resource for West Indies team | 34 units |
Total remaining bowling resource for South African team | 28 units |
Remaining overs in the second innings | 4.83 overs or 4 overs and 5 balls |
Length of the first innings | 31 overs |
The net resource factor of West Indies team | 1015 |
Revised target as per D/L method | 191 |
Revised target as per proposed method | 1896 |
Score of West Indies when the match stopped permanently | 190 |
Result as per D/L method | Tie |
Result as per proposed method | West Indies win the match by 1 run |
Match No. | Venue | Date | Teams | Target as per D/L method | Target as per proposed method | Result as per D/L method | Result as per proposed method |
---|---|---|---|---|---|---|---|
1 | Barbados, West Indies | 2nd May 2011 | West Indies vs. Pakistan | 154 | 156 | West indies won the match by 1 run | Pakistan should have won the match by 1 run |
2 | London, England | 9th September 2011 | England vs. India | 218 | 222 | England won by 3 wickets with 7 balls remaining | England Should have scored 4 runs more in remaining 7 balls |
3 | Bloemfontein, South Africa | 17th January 2012 | South Africa vs. Sri Lanka | 176 | 178 | South Africa won the match by 4 runs | South Africa should have won the match by 1 run |
4 | Pallekele, Sri Lanka | 4th November, 2012 | Sri Lanka vs. New Zealand | 105 | 103 | New Zeland won the match by14 runs | New Zeland should have won the match by 16 runs |
resource of the second fielding teams these 4 matches. To see the details, readers may refer to Appendix B.
Consideration of individual excellence is another factor that fine-tunes the revised target. Here, we have compared the available batting resource of the second batting team as per the proposed method to the available resource as per the standard edition of Duckworth-Lewis resource table7 till the point of interruption. We have found that in case of match number 1, 3, and 4 the available batting resource of the second batting team was higher as per our method compared to the D/L method where it went other way round in match number 2.
Thus, our method recognizes the efforts of the team having more resource in hand compared to its opponent team at the point of interruption and brings more fair result. To facilitate cricket practitioners for implementation, flowchart depicted in
This paper presents an alternative method to calculate the revised target in interrupted ODI matches. Though Duckworth-Lewis method provides robust results at any match situation, it doesn’t take into account individual player’s excellence and importance of the second fielding team’s remaining bowling resource. Under the individual resource computation and net resource factor idea, the revised target set for the second batting team will be as fair as the effort of the team preserving more resources compared with the opponent, till the point of interruption is recognized in this method. With the example of a number of real matches where D/L method has been applied in the past, it’s shown that this method gives sensible and practical target considering all the match situations and two teams’ conditions.
As the remaining batting resource plays a crucial role in our method, it may happen that the second batting team can deliberately devise defensive batting strategy for the sake of preserving wickets. However, it is also intrinsic in the existing D/L method where captain of the second batting team can alter their batting strategy accordingly in case of probable interruption. Also, it can compel teams to devise more complex strategies of setting the batting order or bowling orders to optimize their respective resources. From future research point of view, impact of several other factors like fielding excellence and a debutante’s categorization based on his performance in domestic cricket may be taken into account.
The authors wish to express their sincere thanks to the anonymous reviewers for their insightful comments, which have significantly improved the paper.
There are 6 batsmen classified as the batsmen of P1 category, 1 batsman belongs to P2 category, 2 batsmen categorized as batsmen of P3 category and 1 from P5 and P10 each for West Indies cricket team. We have 1 bowler of P2 category, 2 bowler of P3 category, 1 bowler of P4 category, 1 bowler of P8 category and 6 bowlers of P10 category in South African cricket team.
Player description | Individual resource allocation computation in team’s total resources |
---|---|
West Indies batsmen | Individual batting Resource allocation computation in team’s total batting resources |
A batsman of West Indies who belongs to P1 category | (Individual batting resource × 100/team’s total batting resource)% of team’s total batting resources = {100 × 100/(100 + 100 + 100 + 100 + 100 + 100 + 80 + 90 + 60 + 10 + 80)}% of team’s total batting resource = 10.87% of team’s total batting resources ~11% team’s total batting resources So any batsman of West Indies who belongs to P1 category possesses 11 units of total team’s batting resource |
A batsman of West Indies who belongs to P3 category | (Individual batting resource × 100/team’s total batting resource)% of team’s total batting resources = {80 × 100/(100 + 100 + 100 + 100 + 100 + 100 + 80 + 90 + 60 + 10 + 80)}% of team’s total batting resources = 8.7% of team’s total batting resources ~9% team’s total batting resources So any batsman of West Indies who belongs to P3 category possesses 9 units of total team’s batting resource |
South African Bowlers | Individual bowling Resource allocation computation in team’s total bowling resource |
A bowler of South Africa who belongs to P2 category | (Individual bowling resource × 100/team’s total bowling resource)% of team’s total bowling resources = 90 × 100/(10 + 10 + 10 + 30 + 20 + 10 + 80 + 80 + 10 + 70 + 90)% of team’s total bowling resources = 21.95% of team’s total bowling resources ~22% of team bowling resources So any bowler of South Africa who belongs to P2 category possesses 22 units of total team’s bowling resource. |
A bowler of South Africa who belongs to P3 category | (Individual bowling resource × 100/team’s total bowling resource)% of team’s total bowling resources = 80 × 100/(10 + 10 + 10 + 30 + 20 + 10 + 80 + 80 + 10 + 70 + 90)% of team’s total bowling resource = 19.51% of team’s total bowling resources ~20% of team bowling resources So any bowler of South Africa who belongs to P3 category possesses 20 units of total team’s bowling resource |
According to
Serial No. of matches | Second batting team | Second fielding team | Remaining batting resources of the second batting team | Remaining bowling resources of the second fielding team | Net resource factor of the second batting team |
---|---|---|---|---|---|
1 | West Indies | Pakistan | 0.52 | 0.54 | 0.99 |
2 | England | India | 0.49 | 0.54 | 0.98 |
3 | South Africa | Sri Lanka | 0.40 | 0.43 | 0.99 |
4 | New Zealand | Sri Lanka | 0.67 | 0.62 | 1.02 |
As per
Serial No. of matches | Remaining batting resources as per proposed method (till the point of interruption) | Remaining batting resources as per D/L Standard edition (till the point of interruption) |
---|---|---|
1 | 0.520 | 0.448 |
2 | 0.490 | 0.616 |
3 | 0.400 | 0.347 |
4 | 0.670 | 0.584 |
Here, we present an excerpt of net resource factor table which facilitates the computation of net resource factor for different combination of remaining batting resource of the second batting team and remaining bowling resource of the second fielding team. Here, we present a match scenario where 10 overs are left to finish off the revised over limit of the second innings (in case of temporary interruption) or 10 overs would have been bowled (in case of permanent interruption) and length of the first innings, i.e. 30 overs. In
Remaining batting resources of the team batting second. | Remaining bowling resources of the team bowling second | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.00 | |
0.10 | 1.00 | 0.97 | 0.94 | 0.91 | 0.88 | 0.85 | 0.82 | 0.80 | 0.77 | 0.75 |
0.20 | 1.03 | 1.00 | 0.97 | 0.94 | 0.91 | 0.88 | 0.86 | 0.83 | 0.81 | 0.78 |
0.30 | 1.06 | 1.03 | 1.00 | 0.97 | 0.94 | 0.91 | 0.89 | 0.86 | 0.84 | 0.81 |
0.40 | 1.09 | 1.06 | 1.03 | 1.00 | 0.97 | 0.94 | 0.92 | 0.89 | 0.87 | 0.84 |
0.50 | 1.12 | 1.09 | 1.06 | 1.03 | 1.00 | 0.97 | 0.95 | 0.92 | 0.89 | 0.87 |
0.60 | 1.15 | 1.12 | 1.09 | 1.06 | 1.03 | 1.00 | 0.97 | 0.95 | 0.92 | 0.90 |
0.70 | 1.18 | 1.14 | 1.11 | 1.08 | 1.05 | 1.03 | 1.00 | 0.97 | 0.95 | 0.92 |
0.80 | 1.20 | 1.17 | 1.14 | 1.11 | 1.08 | 1.05 | 1.03 | 1.00 | 0.97 | 0.95 |
0.90 | 1.23 | 1.19 | 1.16 | 1.13 | 1.11 | 1.08 | 1.05 | 1.03 | 1.00 | 0.98 |
1.00 | 1.25 | 1.22 | 1.19 | 1.16 | 1.13 | 1.10 | 1.08 | 1.05 | 1.02 | 1.00 |