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Utility of Duckworth–Lewis method in cricket

My answer to: In cricket, why doesn’t the run rate take into account the stage of the innings? (To vote and comment on Quora, visit the link here.)

The progress of an innings (number of runs scored) is not a linear function. Normally, teams prefer to score slowly initially while they save wickets, and they take risks towards the end. When they take risks (as the innings approaches finish), they score more runs but are also likely to lose more wickets. When they don’t take risks (when the innings is fresh), they score less runs and are also likely to have wickets intact. And that is why the meaning of the phrase “10 runs per over required to win” holds different meanings when the number of overs is 20 and when it is 2.

Mathematically speaking, the run rate is simply the “average number of runs being scored per over”. It does not present the instantaneous rate of scoring.

Run rate = Number of runs scored / number of overs bowled

The answer to your question is that the run rate doesn’t intend to give the information that you are seeking. What you seek, has been provided by Duckworth and Lewis. They have said that a team’s scoring rate is linearly related to the percentage of resources they have utilized, not the number of overs they have played. And resources are a combination of overs and wickets.

This is a short version of the D/L table for a 50-overs-a-side ODI. When a team starts its innings (50 overs are left), it has 100% resources left. When a team has 40 overs left, and all 10 wickets still in hand, it has 89.3% resources left. When a team has 10 overs left, and 4 wickets in hand, it still has 22.8% resources left. When a team has any number of overs left but no wickets left, it has zero resources left. And so on.

Basically, D/L tables have stated that an innings’ progress is based on a function involving both overs and wickets left.

So, a much more accurate statistic than runs per over would be runs per percentage point of resources.

Worked Examples from this above table:

1) If Team B is chasing 251 and is 182/6 after 40 overs:

Current run rate = 182/40 = 4.55 runs per over
Required run rate = 69/10 = 6.90 runs per over
Resources left = 10 overs and 4 wickets  ≡ 22.8%
Resources utilized = (100 – 22.8) % = 77.2%
Runs per percentage point of resources scored by Team A = 250/100 = 2.50 runs per hundredth of resource
Runs per percentage point of resources scored by Team B = 182/77.2 = 2.36 runs per hundredth of resource

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So, we can state that at this stage, Team A is leading, and to say so, we used a linear model. If rain were to finish the match now, Team A would win by D/L method.

2) Imagine Team B being 182/4 after 40 overs. Now,

current run rate and required run rate are the same as before (4.55 and 6.90).
Resources left = 10 overs and 6 wickets  ≡ 28.3%
Resources utilized = 71.7%
Runs per percentage point of resources scored by Team A = 250/100 = 2.50 runs per hundredth of resource
Runs per percentage point of resources scored by Team B = 182/71.7 = 2.53 runs per hundredth of resource

Here, team B is leading. And understandably so, to get 69 runs in 10 overs with 6 wickets in hand is far more probable than with 4 wickets in hand.

What we conclude is that run rates are not even equipped to tell anything about the current scoring pattern or the state of the match. Run rate is just an average. To have an accurate idea, you need to use a D/L table and define a quantity such as “Runs scored per percentage point of resources”. This quantity offers a linear model, and hence it provides easier predictions.

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