Duckworth-Lewis-Stern explained: how rain rules work

A limited-overs match is a contract. Each side gets the same number of overs and the same ten wickets, and whoever uses that allocation better wins. Rain breaks the contract. Once one side has had fewer overs than the other, or has had its overs taken away at a different point in the innings, the two performances are no longer being compared on the same terms, and somebody has to decide what a fair target looks like.

The Duckworth-Lewis-Stern method is the answer international cricket settled on. It is named for the two statisticians who devised it, Frank Duckworth and Tony Lewis, and for Steven Stern, who took over as custodian and revised the model as scoring patterns changed. It is not a formula for predicting what would have happened. It is a formula for restating one side's performance in terms of the resources the other side actually had.

Why rain rules exist at all

The obvious solution, comparing run rates, fails badly, and understanding why is the fastest route into what DLS is doing.

Suppose a side bats fifty overs and scores at five an over. Rain then reduces the chase to twenty-five overs. Asking the chasing side to score at five an over for twenty-five overs sounds even-handed. It is not remotely even-handed, because a side batting twenty-five overs with ten wickets in hand can attack from the first ball in a way that a side pacing a fifty-over innings cannot. Run rate ignores wickets entirely, and wickets are precisely what a shortened innings frees a batting side to spend.

The refinements that came before DLS failed for the opposite reason. One widely used approach struck out the chasing side's least productive overs, on the logic that if you take five overs away, you should take away the five cheapest. That preserved the total the first side had scored but destroyed the shape of the chase, and it could leave a side needing a preposterous number of runs from a handful of deliveries, sometimes a target that could not be reached even by hitting every remaining ball for six. A method that can produce a mathematically impossible target is not a method.

What both failures have in common is that they modelled only one of the two things a batting side spends. DLS models both.

Resources: the central idea

At any moment in an innings, a batting side has two resources left: overs remaining and wickets in hand. DLS combines them into a single number, expressed as a percentage of a full, uninterrupted innings.

A side at the start of a fifty-over innings, ten wickets standing, has 100% of its resources. A side with no overs left, or no wickets left, has 0%. Everything else falls in between, and the whole method consists of a table that maps every combination of overs remaining and wickets lost to a resource percentage.

Three properties of that table are worth understanding, because almost every argument about DLS comes from misreading one of them.

It is two-dimensional. Resources depend on overs and wickets together. Ten overs remaining with nine wickets in hand and ten overs remaining with two wickets in hand are radically different positions, and the table treats them as such. A side that has lost wickets has lost resources even though not a single ball has been taken away.

It is not linear in overs. Overs at the start of an innings are worth less per over than the ones in the middle in terms of runs produced, and the last overs are worth more than any of them. Removing ten overs from the middle of an innings costs a side a different amount of resource than removing ten from the end. The table is built from the observed shape of scoring across innings, not from a straight-line assumption.

It is steeply non-linear in wickets. The first few wickets cost relatively little resource. The later ones cost a great deal, because a side eight down cannot use its remaining overs. This is the single most misunderstood feature of the method, and it drives most of the results people find counter-intuitive.

The resource table itself is a published reference, and the professional edition used in international cricket applies a computerised model rather than a printed grid, but the underlying idea does not change: every state of an innings has a resource value, and DLS works entirely in those values.

How a revised target is set

The calculation is a proportion. Call the resources available to the side batting first R1, and the resources available to the side batting second R2.

If the chasing side has fewer resources than the first innings did, the target is scaled down in proportion:

Revised target = Team 1's score × (R2 ÷ R1), then add one run

The extra run exists because the target is the score needed to win. Matching the scaled figure exactly is a tie.

Two examples make the shape of it clear. These are illustrative rather than drawn from any match.

Interruption before the chase begins. Team 1 bats its full fifty overs, so R1 is 100%. Rain then cuts the chase to thirty overs, and Team 2 starts knowing it has thirty overs and ten wickets. Suppose the table gives that position around three quarters of a full innings' resources. Team 2's target is roughly three quarters of Team 1's score, plus one. Note that this is a good deal more than three fifths, which is what a simple pro-rata of overs would suggest, precisely because Team 2 can attack from the first ball with all ten wickets intact.

Interruption during the chase. Team 2 is fifteen overs into a fifty-over chase, four wickets down, when rain arrives and the match is reduced. R2 is now the resources used before the break plus the resources available after it, and the two are added together. The revised target scales Team 1's score by that combined figure. Because Team 2 has already spent four wickets, the resources it can still use are lower than they would be for a side with wickets in hand, and the target falls by less.

If the chasing side ends up with more resources than the first innings had, which happens when the first innings is cut short mid-stream and both sides then get the same reduced allocation, the target cannot simply be scaled up in proportion. Multiplying a score by a number greater than one would assume the first side would have kept scoring at the same rate through overs it never got, which is not how innings behave. Instead the method adds a fixed quantity for the excess resource, based on an average total for the format and level. The practical consequence surprises people: a chasing side can be set a target higher than the score the first side actually made. That is not a glitch. The chasing side has been handed the advantage of knowing the shorter innings length from the first ball, and it is being charged for it.

The par score

During an interrupted chase, broadcasters display a par score. It is the same calculation run continuously.

At any point in Team 2's innings, the par score is Team 1's score scaled by the resources Team 2 has consumed so far, relative to R1. If Team 2 is level with par, the match is exactly even. If it is ahead of par, it is ahead of the game.

Par is not the target. The target is the final figure needed to win over the full revised innings. Par is a running measure of whether the chase is on track at this instant, and it exists for one specific reason: if play is abandoned at that moment and the minimum overs have been bowled, the side ahead of par wins.

This produces the situation that most reliably confuses viewers. A side can be comfortably ahead of the required run rate and simultaneously behind par, because par accounts for wickets and the run rate does not. A side four down is not entitled to the same allowance as a side one down, and par says so.

The par figure moves with every ball, and it moves in two directions at once. Runs scored push the side above par. Wickets lost push it below, sometimes sharply, because losing a wicket reduces the resources still available and therefore raises the score the side should have reached by now.

Why DLS is less generous to the chasing side than people assume

There is a widespread belief that rain favours the side batting second, and that DLS hands out easy targets. The method is built to do the opposite, and there are four distinct reasons it is stingier than expected.

It charges for certainty. A side chasing in a shortened innings knows exactly how many overs it has. That knowledge is worth a great deal: it removes the need to pace an innings, allows aggressive batting from the start, and lets the side attack with wickets in hand. DLS prices that advantage into R2, which is why the target for a shortened chase is proportionally higher than a simple overs-based reduction would give.

Wickets dominate the arithmetic. A chasing side that has lost wickets before an interruption gets much less relief than one that has not. Two sides on the same score at the same point, one two down and one six down, receive very different revised targets, and the side six down may find the target barely moves at all. Losing wickets is the fastest way to make a rain break unhelpful.

Resources already spent are not refunded. The overs a chasing side has already used, and the wickets it has already lost, are consumed. DLS scales by resources used plus available, not by what remains. A side that batted slowly through its first fifteen overs cannot recover that expenditure through an interruption.

Late interruptions barely move the number. If rain arrives with three overs left, almost all of the innings' resources have already been consumed. Removing the last few overs takes away a small percentage, and the revised target falls by a correspondingly small amount. The intuition that a late stoppage produces a dramatic reduction is simply wrong.

The corollary matters too. The side batting first is not defenceless. If the first innings is interrupted, the method credits that side for the resources it lost, and the chasing team is asked for a target that reflects what the first side could reasonably have done with the overs it never received.

Common misunderstandings

"DLS predicts what would have happened." It does not. It contains no model of the specific match, the pitch, the bowlers, the batters at the crease or the form of either side. It restates one innings in terms of the other's resources using patterns drawn from a large body of scoring data. It answers "what is a fair equivalent?", not "what would the score have been?"

"It's just a run-rate calculation." It is explicitly not. The whole reason the method exists is that run rate ignores wickets. Any explanation of DLS that never mentions wickets in hand is describing something else.

"The team ahead when rain comes should win." Only if that side is ahead of par, and being ahead of the required rate is not the same thing. A side well ahead on runs but seven wickets down is often behind par.

"Rain always helps the chasing side." Sometimes it does, sometimes it does not, and the direction depends almost entirely on wickets in hand at the moment of the stoppage. A team cruising with wickets intact usually benefits. A team scoring at the same rate but several wickets down usually does not.

"A minimum number of overs guarantees a result." The playing conditions require a minimum before a result can be declared, commonly twenty overs a side in the fifty-over format and five in Twenty20, but that is a floor, not a guarantee. If the minimum is not reached, the match is a no result regardless of how commanding one side's position looked.

"The target should never exceed the first innings score." As set out above, it can, when the chasing side ends up with more resources than the first innings had. This is the correct outcome, not an error.

"There is only one version of the method." There are two implementations. The standard edition uses a published table and is designed to be workable with a calculator, which suits club and lower-level cricket. The professional edition uses a computerised model with an adjustment for high-scoring innings, and it is what is used at international level. They can give different numbers for the same situation, and the professional edition handles very large first-innings totals considerably better.

"DLS is the only rain method." It is the standard in international cricket, but alternatives have been proposed and used in domestic competitions, most notably the VJD method developed in India, which models the innings in a different way. Different competitions may specify different methods in their playing conditions.

"DLS applies to Test cricket." It does not. There is no target to revise in a format with unlimited overs; lost time simply reduces the number of overs available to force a result, and a match runs out of time as a draw.

What the method does not and cannot know

DLS is a general model applied to a specific match, and the gap between the two is where every complaint about it lives.

It does not know the pitch. A surface that is deteriorating, or a pitch under lights where the ball begins to move, changes the difficulty of batting in a way no resource table captures.

It does not know about dew. In many parts of the world an evening interruption changes the game materially, because a wet ball is harder to grip and the side batting second gains an advantage that has nothing to do with overs or wickets.

It does not know who is batting. The resources of a side with a deep batting order and the resources of one with a long tail are treated identically at the same wickets-lost figure, even though the second side's remaining wickets are worth far less.

It does not know the match situation beyond runs and wickets. It has no view on which bowlers have overs left, whether the powerplay has been taken, or whether the best batter is at the crease.

It cannot resolve the deeper problem that a shortened match is a different match. Twenty overs a side is a different sport from fifty, and no scaling exercise can make the result of one a fair verdict on a contest that began as the other. The method's claim is narrower and more defensible: given that the game has been shortened, this is the fairest available restatement of what has already happened.

Reading a DLS situation as it unfolds

A few habits make an interrupted match much easier to follow.

• Watch the par score, not the required rate. Par is the number that decides the match if play stops.
• Watch the wickets column. Every wicket raises par. In a rain-threatened chase, wickets in hand are worth more than they look on the scoreboard.
• Expect the target to fall less than the overs do. A chase cut by forty per cent of its overs will not have its target cut by forty per cent.
• Note when the interruption happens. Early stoppages move the numbers a lot. Late ones barely move them.
• Check whether the minimum overs have been reached. Until they have, no par score matters, because there is no result to be had.
• Do not expect the first innings score to be a ceiling. If the chasing side has more resources, it will be asked for more runs.

The method's reputation for opacity comes almost entirely from the resource table being invisible to the viewer. The idea underneath it is not complicated: count what a batting side has left, count it in two dimensions rather than one, and compare like with like.

If the underlying scoreboard conventions are unfamiliar, how to read a cricket scorecard covers the columns a par score is calculated from, and the cricket statistics glossary defines run rate, required rate and the rest of the terminology used here. The differences between the formats DLS applies to are set out in Test, ODI and T20 explained.