Duckworth-Lewis and PPC

In this post, I explore how a tool initially designed for improving fairness in cricket could have its place in reallocating resource in a retail environment.

Cricket

I’m not a keen follower of cricket. I last played at school, where it was a summer sport option. Rain never stopped play. If it was raining, we never started play, and went inside to play dodgeball instead. For this reason, I didn’t encounter Duckworth or Lewis until recently. Allow me to explain.

In 1 day cricket, each team bats an equal number of overs, typically between 20 and 50. The winning team is the side who scores the largest number of runs either before they lose 10 wickets or bat all of their overs.

All well and good, provided that each team can play all their overs. Suppose that team 1 bat first and score 260 runs in 50 overs, but then it rains for a few hours. A mix of safety concerns and the players not wanting to get wet stops play. Both teams take a longer lunch break and there is only enough time for team 2 to play 25 overs.  It would be unfair for them to need to score 261 runs for the win from fewer balls. How many runs should they have to score for a win?

An initial suggestion is to scale the runs required in accordance to the overs faced. If team 1 score 260 in 50 overs, then in 25 overs team 2 need to score more than 130 runs for the win. Simple, right? This is the ‘average run-rate’ approach and was in the rule book until the late 1990’s.

The most famous example of the injustice of this method was at the 1992 Cricket World Cup semi-final between England and South Africa. Rain stopped play, and South Africa were required to bat 22 with 1 ball – an impossibility! The ICC appealed to mathematicians to solve the problem of fairness and fix it!

South Africa faced an impossible task

The unfairness of this method stems from the lack of consideration of wickets. A team needing playing with more wickets in hand can play more aggressively than a team with only 1 wicket remaining. They can take more risks, aiming to hit more sixes and will score more runs as a result. Enter Duckworth-Lewis.

The Duckworth-Lewis Method is a mathematical formulation which, given the number of overs available for play together with the number of wickets in hand, helps calculates an adjusted run target. I’m interested in whether a similar model could help retailers adjust after periods of poor weather.

PPC Metaphor

This summer, the UK weather has seemingly swung wildly between heatwaves and thunderstorms. At homeware retailer Dunelm, it’s made comparing our actual sales to the forecast less of a science – variations could be explained away by the British weather which the finance teams couldn’t have accounted for.

Let’s consider the following scenario and explore how a variant of the Duckworth-Lewis model could be applied. Time for an analogy.

It’s a sunny day in Leicestershire. The doors to Dunelm are open and across the country, customers are shopping in their local store. The 50 day summer sale is underway and is comparable to a cricket match, each day representing an over. Instead of scoring runs, winning depends on revenue generated by the end of the sale. And instead of wickets, the teams have PPC advertising spend at their disposal. The more advertising spend, and the more days remaining of sale, the higher the revenue expected to be generated between now and sale end.

1 similarity between Cricket and Retail is that of timing. Just as a cricket team aim to lose their final wicket in the final ball of the final over, a store aims to sell their final sale products as the sale ends. Having lots of stock left to go into clearance or selling out of stock before sale ends are both inefficient, just as having lots of wickets left on the final ball or getting out with overs to spare is a sign of a poorly played innings.

So, who are the teams?

First to bat is the Dunelm forecast. Set by the finance team in advance, it plays the innings, assigning a target score for each day of sale. After the 50 days, it expects £x from £y of PPC spend. Sunshine throughout; the forecast’s overs are never stopped by rain.

Second to bat is the Dunelm PPC Team. They have a good first week and sales are ahead of forecast; garden furniture is flying! But then the rain starts, 2 weeks of poor sales occur before finally relenting. We’re behind forecast, but can we still generate enough revenue to win the match?

In cricketing terms, we’re asking if a team can score the same number of runs from fewer overs and with the same performance. Normally, the answer is no, which is why Duckworth-Lewis sets adjusted targets. But the Dunelm PPC team have a trick up their sleeve. In order the counteract the loss of ‘overs’ i.e. good days of sales, we can buy more wickets i.e. increase PPC spend. Just as this would allow a cricket team to bat more aggressively to score runs, we can bid more aggressively to increase sales. One thing to be noted is the idea of pent-up demand, where demand is simply delayed for a period before a period of higher demand. This is another tool on the PPC’s team side.

Now we understand the analogy, the question is whether Duckworth-Lewis can help determine how much additional PPC spend we would need to win the match.

Duckworth-Lewis in practice

Duckworth-Lewis is a tool to quantify resource remaining taking multiple factors into account, the principal output being the table displayed below. Values range from between 0 and 100%, an estimate of resource remaining. These percentages are multiplied by the score of team 1 to produce winning thresholds for team 2.

Parameters Zo and b are set from historic data points

The values are the result of an exponential decay relationship. U is the number of overs remaining, w is the number of wickets lost. Zo(w) and b are each extrapolated from the results of previous cricket matches. A nice application of mathematics to sport.

Every loss of overs can theoretically be equated to a loss of wickets

The red ovals highlight that (26 overs remaining, 0 wickets lost) has the same net resource loss as (38 overs remaining, 3 wickets lost). In theory, every loss of wickets can be equated to a loss of overs.

One difference in our PPC analogy that plays to our advantage is the continuity of PPC spend vs the discrete values taken by wickets lost. This would allow us to more accurately associate PPC spend to time lost to poor sales, alternatively to the amount behind forecast. From looking at historic PPC bidding data, we could produce an equivalent table for our problem. Then, if poor weather put us behind forecast, we could look up an estimate of how much additional spend would be needed to get back on target – and on whether it would be worth it. If the PPC costs were too high for profitability to remain, then it would be sensible to wait for clearance.

Certainly, more thought needs to go into this structure for it to be useful to a paid marketing team. This post aims to plant the seed, maybe I’ll revisit the idea again soon.

Until next time,

Scott