Recursion, DP, and the Duckworth–Lewis–Stern method - 6/7

Part - 5 is here.

Introduction

The Duckworth–Lewis–Stern method (DLS) is designed to calculate the target score for the team batting second when a limited-overs match is marred by an interruption (rain, etc.). The basic principle is that each team in a limited-overs game has two types of resources available using which they can score runs.?Those resources are:

1. Overs left to play; and,
2. Wickets remaining.

The target for the second team is adjusted proportionally to the change in the combination of these two resources.

The unique selling proposition of DLS is how it interprets the game, resulting in the target score calculation being more realistic. It computes the percentage of resources left with a team when some overs are lost due to interruption. The superiority of the DLS method over the other techniques for target calculation is that it closely tries to follow the path that the game would have taken if it wasn't interrupted.

Mathematical Theory of original D/L method

The original D/L model started by assuming that the number of runs that can be scored for a given number of overs remaining and wickets lost, followed an exponential decay model.

The exponential decay model implies that the rate of decay of a thing is proportional to its current value. Its equation is generally written as below:

Before its application to the game of cricket, this model was already seen in chemical reactions, electrostatics, geophysics, heat transfer, radioactivity, and finance, to name a few domains.

The plot of the exponential decay curve in the D/L model looks like this:

Using the exponential decay equation, the resources for each {wicket, over} combination would be:

Improvements to the original method

1. The original method is also called the Standard Edition. This method did not use the score of the first innings at all. Therefore, the resource percentage numbers were static. The reason was that computers would not be guaranteed to be present. This approach started cracking for high-scoring matches. This problem of high-score games caught the attention of Duckworth and Lewis during the 2003 World Cup final match between India and Australia.
2. An additional parameter was added to the equation that would be determined based on the first inning of the game. This parameter ensured increased fairness in target calculation. However, this parameter could not be calculated manually. In 2003, ICC decided that computers are ubiquitous and adopted the more complex method, which would be called the Professional Edition.
3. In 2009 the D/L method was reviewed for the T20 format. Programmatically speaking, the technique developed for the 50-over game did not fit the 20-over match. Teams approached an ODI differently than a T20.
4. For the 2015 World Cup, the ICC used the Duckworth-Lewis-Stern method. Professor Steven Stern modified the formula and became the new custodian of the technique. If you remember earlier matches, the teams tried to keep wickets in hand to get an advantage in the D/L method. Professor Stern's changes incorporated that teams should start with a higher scoring rate when chasing targets instead of trying to keep wickets in hand. Below is the graphical representation of Stern's changes. The thick line is the corrected curve that's followed now. The thin lines are from the original method.

Part - 7 is here.