Play · a dartboard, a bell curve and one jump

Aim Where Your Misses Pay

Treble 20 pays best only for throwers whose darts land tightly around the aim: on the idealised board of a 2011 statistics paper, with a spread (a standard deviation) under about 16.9 mm. Past that, the best aim jumps once, to treble 19, then edges toward the 7 and slides through the 7, 16 and 8 to a spot just left of and below the centre. Drag the spread and watch the target move, estimate your own spread from nothing but your scores, then tilt it and watch the target move again.

What the slider means

Each dart lands where you aimed plus an error. In the model this page uses, from a 2011 paper by Ryan Tibshirani, Andrew Price and Jonathan Taylor, the error is a bell curve (a Gaussian) with the same spread, σ, across and up and down. σ is a standard deviation, not your typical miss. Tibshirani's magazine version of the paper describes it as the distance from the target within which 39% of his shots land: exactly 1 − e^−1/2, which is 39.35%. Half of all darts land within 1.177σ, and one in ten lands further out than 2.146σ.

One jump

For a thrower who almost never misses, the middle of treble 20 is the place: it is the only 60 on the board. By σ = 16.5 mm the best aim has crept up and slightly to the left: it is 1.9 mm left of the centre line and 106.2 mm up, against 103 mm for the middle of the treble. Then, just past σ = 16.9 mm, it leaves. It jumps across the board into treble 19, near the middle of the segment; by σ = 20 mm it sits on the outer edge of treble 19, and by 25 mm it leans toward the 7.

From there it slides inward and round: across the single 7 (at σ = 40 mm), the single 16 (50 mm) and the single 8 (60 to 80 mm), until at σ = 100 mm it is in the single bull, 10.7 mm left of the centre and 2.8 mm below it. In the paper's words, it curls into the center of the board, stopping a bit lower than and the left of the bullseye at σ = 100.

There is one jump. Stepping σ by 0.1 mm from 1 to 100 mm, the best aim never moves more than 3 mm from one step to the next except once, between 16.9 and 17.0 mm, when it moves 209 mm. At the switch itself, σ = 16.907 mm, the two aims are worth the same, 19.82 points per dart: the advice jumps, the score does not. At σ = 16.88 mm the treble 20 side is ahead by 0.003 points; at 16.94 mm the treble 19 side is ahead by 0.004.

At the other end, below about 2.5 mm, there is no single best point. Along an arc of treble 20, every aim scores within a millionth of a point of the best: at σ = 1 mm the arc is 11.9° of the ring (21.4 mm long), at 2.5 mm it is 2.7° (4.9 mm), and by 3 mm it is under half a degree. The board draws it as a band.

Two of the paper's three authors each threw 100 darts at the bull, and from the scores alone it estimated σ = 64.6 mm for author 1 and 26.9 mm for author 2. Its advice: author 1 should be aiming at the S8, close to the bullseye. Meanwhile, author 2 (who is a fairly skilled darts player) should be aiming at the T19. The exact integral agrees, and puts author 2's aim 4.55° from the border between treble 19 and treble 7.

Three printed cases, and where their numbers come from

Tibshirani's article in Significance works three cases. For σ = 25 mm the best aim earns him or her an average of 15.8 points per throw, and aiming at the triple 20 gives only 14.9; the exact integral gives 15.80 and 14.93. For σ = 60 mm, aiming low and left of the centre gives 12.4 points per throw where treble 20 gives 10.2; the exact integral gives 12.43 and 10.25. For σ = 5 mm, treble 20 gives an average of 42.9 points per throw; the exact integral gives 43.02 at the best point and 43.01 at the middle of treble 20.

Two of those printed figures are not the exact integral's: the 42.9, which it puts at 43.0, and the 10.2, which it puts at 10.3. Both are the grid's. The paper evaluates the score and the bell curve on a grid, say, on a millimeter scale across the dartboard, and Tibshirani's R package, darts 1.0, samples the score at each whole millimetre of a 681 by 681 grid and convolves it with a bell curve sampled at the same points. Rebuilt here from that description, the grid gives 42.94 at σ = 5 mm, and 10.24 at the middle of treble 20 at σ = 60 mm.

Where the printed 16.4 comes from

The paper puts the switch at a different σ: The path appears to be continuous except for a single jump at σ = 16.4. The rebuilt grid gives exactly that. At σ = 16.3 mm its best whole-millimetre aim is still in treble 20, at (−2, 106), ahead of the best near treble 19 by 0.0010 points per dart; at σ = 16.4 mm the treble 19 side, at (−35, −99), is ahead by 0.0103. Integrate the same board exactly and the switch moves to 16.9 mm.

Nothing that matters hangs on the difference. A treble ring is 8 mm across, and a sample at each whole millimetre stands for a square millimetre that may straddle a wire: enough to move a near-tie by half a millimetre of σ. Between σ = 16.4 and 16.9 mm the two aims are never more than 0.06 points per dart apart. Switch the board above to the paper's grid and drag through 16.4 mm to watch its jump.

Real boards differ from the paper's board too. The paper takes its measurements from the BDO's playing rules, citing a 2006 web page: the treble from 99 to 107 mm from the centre, the double from 162 to 170 mm, the bulls out to 6.35 and 15.9 mm, all with wires of no width. The World Darts Federation's current rules (the 20th edition, of 28 February 2018) give 107.4 mm to the outside edge of the treble wire, 170.0 mm to the outside edge of the double wire, rings 8.0 mm wide inside to inside, bulls 12.7 and 31.8 mm across inside, and wire from 1.27 to 1.85 mm thick; a dart scores on the side of the wire it enters. Read the wires as having no width on their outside edges and the switch comes at 16.98 mm. Put the scoring line down the middle of a 1.27 mm wire and it comes at 16.68 mm; of a 1.85 mm wire, 16.56 mm. For a conventional wire board at its nominal dimensions, then, about 16.6 to 17.0 mm. (The federation's rules of 1995 gave the paper's 107.0 mm for the treble.)

Not a plan for 501

All of this maximises points per dart, and nothing else. The paper says so: It is important to note that we are not proposing an optimal strategy for a specific darts game. In 501, as the magazine article puts it, players count down from 501 points to zero and are penalised for going negative, and under the federation's rules a leg is won only by finishing on a double (or the bull). Points per dart is the right target only while every point is worth the same to you; on the way out it is the wrong one. The order of the twenty numbers, which decides how much a sideways miss costs, belongs to another layer: the order punishes the wobble, and this one asks what the wobble should make you do.

But nobody knows their σ, and nobody throws in circles

That is the fair objection to everything above: a round bell curve of known size is a toy. The paper has an answer to each half, and both can be worked here.

Your σ from your scores alone

You could measure where 100 darts land, but, as the magazine article says of that job, This is a pretty dull task! The paper uses only the scores of darts all aimed at the double bull. A score of 20 says the dart is somewhere in one of three pieces of board (the single 20 inside or outside the treble, or the double 10); a 25 or a 50 says it is close; a treble or a double says it is far. The EM algorithm takes a guess at σ, works out how far from the centre each dart probably was given its score, re-estimates σ from those distances, and repeats until σ stops moving. With the aim at the centre each step has a closed form (the paper's Appendix A.2), and the page runs it in your browser.

Does it work? For each σ below, ten sessions of 50 simulated darts (seeded, so the same every time), with σ estimated from where the darts landed and from their scores alone. As in the paper's own test, the two are close.

Tilt the scatter

The second answer lets the scatter be an ellipse. The paper suggests that it is common for most players to have a smaller variance in the horizontal direction than in the vertical one, and adds that a right-handed player may possess a different “tilt” to his or her error distribution (defined by the sign of the correlation) than a left-handed player. From author 2's 100 scores it fits σx = 17.9 and σy = 39.1. Its printed covariance also carries a correlation of −0.22: an ellipse with deviations of 39.4 mm along its long axis and 17.4 mm along its short one, the long axis 7.1° from vertical with its top to the left. For that ellipse the best aim is just above treble 20, 1.8 mm left of the centre line and 110.6 mm up, worth 16.05 points per dart; in the paper's words, the new heatmap tells him to aim slightly above the T20. The round model of the same thrower had said treble 19.

Now mirror the ellipse, top to the right, and change nothing else. The best aim goes back to treble 19's outer edge, 16.185 points per dart there against 16.112 for the best spot near treble 20. The elongation alone does not settle where to aim; the sign of one fitted correlation does. Turn the ellipse and the target moves again.

Turned a degree at a time, the ellipse keeps the best aim in the 20's segment, just above the treble, only between 5° with its top to the right and 16° with its top to the left. Lean it further to the left and the aim moves to the 19's segment (17° to 20°; at 20° it is right on the wire it shares with the 7), then the 7's (21° to 79°); lean it to the right and it moves to the 19's (6° to 51°), then the 16's (52° to 74°); lying nearly flat, from 75° to the right through to 80° to the left, it is in the 19's again.

How sure is the sign?

The paper gives no uncertainty for its fitted ellipse. One way to see what 100 scores can tell: suppose author 2 really does throw with exactly that ellipse, simulate 200 sessions of 100 darts at the bull, and fit an ellipse to each session's scores alone (the same EM; this page works out its averages over each scored region by quadrature, where the paper used importance sampling). In 10 of the 200 sessions the fitted correlation came out positive, so 100 scores mostly do settle which way the ellipse leans. They do not settle the advice. The fitted ellipse's best aim stayed in the 20's segment in 133 sessions, and went to treble 19's segment in 43, the 8's in 12, the 11's in 8 and the 7's in 4: one session in three would have sent author 2 somewhere else. The paper's verdict for him is a point estimate, and not a firm one.

What the model leaves out

• One bell curve for every target. Fitting professionals' darts, Haugh and Wang write that A simple model such as (1) will not be sufficiently rich for modeling the skills of professional darts players, because these players tend to focus on (and practice throwing at) specific parts of the darts board.
• Independent darts, thrown by a player already warmed up: the paper's section 4.2 raises both.
• A scatter as likely to fall low as high. The paper also fits a skewed version for both of its throwers and reports that the size of the skew is not large enough to produce heatmaps that differ significantly from Figure 4.
• How common an elongated scatter is. The paper argues it from the throwing motion; its evidence is the fits for its two throwers.
• Darts that bounce out, and the wires themselves: on the paper's board every dart scores the region its point is in.

curl -sO https://artwaste.land/checks/verify-aim-where-your-misses-pay.mjs
node verify-aim-where-your-misses-pay.mjs