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"MvG will NEVER dominate again!"
- The Rocky of Darts
Exploring avant garde and out of the box concepts in professional darts, from a mathematician's perspective.
Ask the Orakel
Darts. Know it All!
By: Greg Heim
THE CONCEPT OF CONTINUATION EQUITY AND QUADRANT/SECTIONAL HEDGING
Though this concept can be applied to at any point of a leg, I am selecting an arbitrary starting point where both players are on ANY checkout. It is important this is always kept in mind during this analysis.
Continuation Equity places a theoretical value to the score which will remain if they are first to checkout and are unsuccessful. There are many factors which are important to this concept which will be discussed later.
CONSIDER THE FOLLOWING
This well-discussed example, which I and perhaps a few others refer to as the “Danny Baggish Spirit of ‘76” is a perfect example to demonstrate multiple concepts. As discussed in my previous article, videos, and posts – it is a gaming mathematician’s dream – and as American as a slice of apple to boot!
As stated by the arrow, Danny Baggish has the first opportunity to checkout 76, not just for the leg – BUT FOR THE SET and a commanding 2-0 set lead. His opponent, Damon Heta has three darts in-hand to take out 32.
If you have been following my material previously, the win probability for both players is an extrapolation of almost every scenario (numerical AND non-numerical) going WELL beyond Danny’s win probability to take out the 76 on his initial opportunity. Thanks to the compilation and next generation analyses of data, spearheaded by our friends at Darts Orakel, that one-time success probability is much easier to obtain.
In this example, the continuation equity for Danny Baggish is either close, or right at the lowest value possible with Damon Heta – who is sitting on D16 with three darts in hand ON THAT TURN ALONE.
Therefore, the player is going to focus and assume the highest, acceptable level of risk on the checkout path, and place the least (not zero) level of concern on what s/he leaves if their checkout is unsuccessful. In most cases, the breaking point of increased risk assumption starts when the player having first crack at a checkout has less than a 50% chance of getting at least one more visit if they are unsuccessful. Three darts at D16 exceeds that threshold by the widest of margins.
Danny Baggish eschewed the T20 D8 combination and checked out by hitting the T16 D14 combination on his initial opportunity. This gave Danny a two set to zero set lead. He would need as he would go on to defeat Damon Heta three sets to two. Despite this success, was this the correct decision (it was not a miscount)? Danny’s approach was the perfect example, of quadrant or sectional hedging. This all but guaranteed Danny the use of all three darts for the initial checkout attempt. To illustrate this concept, let us look at the board.
There are EIGHT contiguous sections, 12 to 19 inclusive which automatically guarantees that it will not be necessary to hit a treble on the second dart to complete the checkout. You can confirm this by doing the math on your own time.
If Danny goes for T20, that plan has a more-measurable chance at going awry if the dart gets snatched and lands in the Single 5, and each segment in the ONES. Given the importance of the leg, he was not willing to assume that risk, though the continuation was near its bottom value. It is also not a sign of disrespect to Danny, but his skill level is not in the upper echelon of more-experienced pros – and he was smart enough to check his ego at the door to realize this PRIOR to this first attempt at the 76 checkout.
WHAT IF THE CONTINUATION EQUITY IS HIGHER?
In the First Round of the 2021 PDC World Championship between Luke Woodhouse and Jamie Lewis: Luke Woodhouse has a 2-1 set lead and is behind 1-0 in the third set. Luke has the first shot at his 121 checkout. Jamie Lewis is sitting on 295.
There are a host of reasons why the continuation equity is not only higher, but significantly higher than the previous example.
The two main reasons are:
* The importance of the score left behind if the 121 is not checked out.
* Luke Woodhouse has at least six darts to win the leg since Jamie Lewis is not on a checkout.
When placing importance on the remaining score if the checkout is not successful, an overly aggressive path could result in leaving a low and/or undesirable double where options are fewer – referred to as The Madhouse.
Getting into The Madhouse when your continuation equity is quite high decreases your win probability not just for the leg, but for the match. That is because another concept called psychological equity dramatically shifts to your opponent if he wins or pinches a leg where you an overwhelming favorite.
WHAT DID LUKE WOODHOUSE DO?
He ignored the typical combination for T20 T15 D8 and did so successfully to tie the 4th set at 1-1.
This combination is something I have referred to as a flexible or flex checkout. The flexibility of this T20 T15 D8 is that even if he leaves a D4, he still has three darts at a double when he comes back to the board. If he misses the D8 on his last dart, he has four darts at a double on this next throw – which in turn maintains an acceptable level of continuation equity combined with an acceptable, but slightly higher level of risk.
When I say “slightly higher level of risk, this is under the assumption that the T20 and T15 are scored. If you analyze the possible and plausible scenarios if The T20 is or is not successful, and/or the T15 is or is not successful, the risk threshold is lower, but not to the point of risk-aversion. This bolsters the magnitude and frequency of the continuation equity of the remaining score left on an unsuccessful checkout.
In this case, the elevated risk was warranted in this instance because Luke has lost momentum and needed to make a statement. Taking everything into consideration, I really liked the move. It is another case where even the best raw data does not tell the whole story.
CONCLUSION AND AN IMPORTANT ANNOUNCEMENT
This article took far too long to compose for a wide variety of reasons. I made a promise that I would provide something more substantial than the teasers and smaller selections I have provided. There are many other projects and interests I have beside darts, though darts is in the upper part of that list.
In addition to my varied Tweets, I will be going in a new reaction of a “reactive blog” for darts. I think you shall find the material fresher with substance, and a bit more concise.
Looking forward to the World Matchplay in Blackpool. See you all on Twitter.
2-to-1 you'll LOVE it!