## Saturday, December 7, 2013

### The Duckworth-Lewis Method.

Cricket World Cup 1992 was the best World Cup ever played according to most of the cricketers, it was also the most controversial ever thanks to the Semi Final between South Africa and England. After the first Innings England made 252 in 45 overs for the loss of  6 wickets. In reply South Africa were 231/6 in 42.5 overs when rain interrupted. Match was resumed after 12 minutes and two overs of South Africa were discarded and they required 21 runs of 1 ball when the match began according to the Highest Overs Scoring Method.

After this match everyone started criticizing the method and called for an alternate method that would be less controversial. Highest Over Scoring Method was actually as such. If it rains and overs are deducted because of time wastage then a team batting second will have to score runs in the overs that were most scored by the other team. In this match South Africa had bowled two maiden overs so while batting their two overs were deducted.

Famous Cricket Journalist and Commentator, Christopher Martin said after the match that for sure someone else with a better formula should come forward and save cricket, like thousand of other cricket fans, famous mathematician Frank Duckworth was listening to the commentary as well. He gave it a thought and conceived that this is a mathematical problem and it can be solved. So with the help of Tony Lewis he devised such a method for rain affected methods that we all see in Cricket Matches.

 Frank Duckworth and Tony Lewis
This method is based on two things in a match, overs and wickets because these are the two most important things for a Cricket team during a match. Obviously a team that is batting scores runs according to the overs remaining and the wickets in hands.

Frank Duckworth and Tony Lewis devised a P(u,w) function which tells about the average overs and wickets  (resources) during a match. Here overs are u and remaining wickets are w. The D/L method works using the notion that teams have two resources with which to make as many runs as they can - these are the number of overs they have still to receive and the number of wickets they have in hand. From any stage in their innings, their further run-scoring capability depends on both these two resources in combination. The single table gives the percentage of these combined resources that remain for any number of overs left and wickets lost. An extract of the over-by-over table is given in Table 1. (A ball-by-ball version of the table has also been produced to enable scorers to deal with instances when play is interrupted mid-over.)

When a match is shortened after it has begun, the resources of one or both teams are depleted and the two teams usually have different amounts of resource for their innings. In this case a revised target must be set. The D/L method does this in accordance with the relative run-scoring resources available to the two teams. If stoppages cause the team batting second (referred to here as Team 2) to have less resources available, as is more often than not the case, then their target will be revised downwards. If, on the other hand, as often happens when Team 1's innings has been interrupted, the stoppages result in Team 2 having more resources available, then their target is revised upwards to compensate for the extra resources they have at their disposal.

Example: Premature curtailment of Team 2's innings
Team 1 have scored 250 runs from their 50 available overs and Team 2 lose 5 wickets in scoring 199 runs in 40 overs. Play is then stopped by the weather, the rain refuses to relent and the match is abandoned. A decision on the winner is required.

Team 1's innings: this was uninterrupted, so the resource percentage available is 100%.
Team 2's innings: resource % available at start of innings = 100%
After 40 overs Team 2 have 10 overs left and have lost 5 wickets.
From table, resource % left at suspension of play = 27.5%
As play is abandoned all this remaining resource is lost.
Hence resource % available for Team 2's innings = 100 - 27.5 = 72.5%
Team 2 had less resource available than Team 1 so their target must be scaled down by the ratio of resources, 72.5/100
Team 1 scored 250, so Team 2's 'target' is 250 x 72.5/100 = 181.25
As there is to be no further play, the winner is decided according to whether or not this target has been exceeded. With 199 runs on the board, they have exceeded their required target by 17.75 and so are declared the winners by 18 runs.

Note : The above result is quite fair as Team 2 were clearly in a strong position when play was stopped and would very likely have gone on to win the match if it hadn't rained. Most other methods of target revision in use would, unfairly, make Team 1 the winners. The average run rate method gives 201 to win, the Current ICC method gives 227 and the parabola method gives 226. [Setting the target by the method of Discounted Total Runs - the Australian rain-rule - requires knowledge of the runs made by Team 1 from their most productive overs but the target would almost certainly be no lower than that required under average run rate and would probably be much higher so that Team 2 would very probably lose by this method as well.

Example 2: Interruption to Team 2's innings

In an ECB Axa Life (Sunday) League match Team 1 have scored 200 runs from their 40 available overs and Team 2 lose 5 wickets in scoring 140 runs in 30 overs. Play is then suspended and 5 overs are lost. What is Team 2's revised target?

Team 1's innings: At the start of 40 over innings resource percentage available = 90.3%
Team 2's innings: resource % available at start of 40 over innings = 90.3%
After 30 overs Team 2 have 10 overs left and have lost 5 wickets.
From table, resource % left at start of suspension = 27.5%
5 overs are lost, so when play is resumed 5 overs are left.
From table, resource % left at resumption of play = 16.4%
Hence resource % lost = 27.5 - 16.4 = 11.1%
so resource % available for Team 2's innings = 90.3 - 11.1 = 79.2%
Team 2 had less resource available than Team 1 so their target must be scaled down by the ratio of resources, 79.2/90.3
Team 1 scored 200, so Team 2's target is 200 x 79.2/90.3 =175.42, or 176 to win, and they require a further 36 runs from 5 overs with 5 wickets in hand.

Example 3: Interruption to Team 1's innings

In an ODI, Team 1 have lost 2 wickets in scoring 100 runs in 25 overs from an expected 50 when extended rain leads to Team 1's innings being terminated and Team 2's innings is also restricted to 25 overs. What is the target score for Team 2?

Because of the different stages of the teams' innings that their 25 overs are lost, they represent different losses of resource. Team 1 have lost 2 wickets and had 25 overs left when the rain arrived and so from the table you will see that the premature termination of their innings has deprived them of the 61.8% resource percentage they had remaining. Having started with 100% they have used 100 - 61.8 = 38.2%; in other words they have had only 38.2% resources available for their innings.

Team 2 will also receive 25 overs. With 25 overs left and no wicket lost you will see from the table that the resource percentage which they have available (compared to a full 50 over innings) is 68.7%. Team 2 thus have 68.7 - 38.2 = 30.5% greater resource than had Team 1 and so they are set a target which is 30.5% of 225, or 68.63, more runs than Team 1 scored. [225 is the average in 50 overs for ODIs]

Team 2's revised target is therefore set at 168.63, or 169 to win in 25 overs, and the advantage to Team 2 from knowing in advance of the reduction in their overs is neutralised.

Note: Most of the other target resetting methods in use make no allowance for this interruption. They set the target of 101 to win simply because both teams are to receive the same number of overs. This is clearly an injustice to Team 1 who were pacing their innings to last 50 overs when it was curtailed, whereas Team 2 knew in advance of the reduction of their innings to 25 overs and have been handed an unfair advantage. D/L allows for this by setting Team 2 a higher target than the number of runs Team 1 actually scored, as described above.