# A Red Rag To A Bull (The Duckworth-Lewis-Stern Method)

The original D/L model started by assuming that the number of runs that can still be scored (called $Z$), for a given number of overs remaining (called $u$) and wickets lost (called $w$), takes the following exponential decay relationship:

$Z(u,w)=Z_{0}(w)\left({1-e^{-b(w)u}}\right)$ where the constant $Z_{0}$ is the asymptotic average total score in unlimited overs (under one-day rules), and $b$ is the exponential decay constant. Both vary with $w$ (only). The values of these two parameters for each $w$ from 0 to 9 were estimated from scores from ‘hundreds of one-day internationals’ and ‘extensive research and experimentation’,

though were not disclosed due to ‘commercial confidentiality’.Wikipedia: Duckworth-Lewis-Stern Method; Mathematical Theory

Commercial confidentiality, you say? Pah! You can’t keep something as important as the cricket score secret! It just isn’t… well.

(I’m not going to go into the ins and outs of DLS here - it’s a way of deciding who won a rain-interrupted cricket match. Given that I live in England, that happens a lot.)

Anyway. Since my dander was well and truly up, I figured I could probably work out the constants $Z_0$ and $b$ for any given number of wickets, given that the tables are easy enough to find in several places.

What’s more, it’s easy enough to tidy up the tables in the spreadsheet program of your choice, then copy-and-paste into Desmos – although it seems to have a five-column limit. ((That’s not as big a problem as it might be. I can just do it in two blocks.))

And, using Desmos’s handy $\simeq$ functionality, I can fit the given model to the data and come up with very good estimates of $Z_0$ and $b$ for every $w$:

W | Z_0 | b |

0 | 134.1022939 | 0.027393558 |

1 | 118.5253606 | 0.030996527 |

2 | 101.9143572 | 0.036038757 |

3 | 84.45285677 | 0.043504654 |

4 | 66.99557009 | 0.054869035 |

5 | 50.28100499 | 0.073076761 |

6 | 35.11610741 | 0.104616858 |

7 | 21.98992361 | 0.167213062 |

8 | 11.90743731 | 0.309870559 |

9 | 4.700112328 | 0.763221136 |