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The Player Line / The projection
Expected minutes against a per-90 rate
How the projection behind a single-player line is built
The line a single-player market posts is not an opinion about the player in the abstract. It is the output of two numbers that are estimated separately - how long he will play, and what he does per minute - multiplied together and then rounded to a bet a reader can take.
Desk spec
- expected minutes
- 78.0
- per-90 rate
- 1.40
- expected output
- 1.21
- rounding
- to the line
the projectionWhat a player is expected to do: an expected-minutes figure multiplied by a per-90 rate. The sample player is expected to play 78.0 minutes at 1.40 shots on target per 90, so the model expects 1.21 of them rather than the 1.40 a full match would imply.
the participation definitionWhat the operator's rules mean by "to play". Across 400 sampled single-player markets the same bets stand 308 times under "named in the starting XI" and 369 times under "takes the field", so the definition decides whether the bet is settled at all.
the confirmationThe source that counts the stat after the match and the audit behind it. 14 of 1,200 sampled markets were corrected after first publication, 5 of them more than an hour later, and 3 after the payout had already been made.
Direct answerThe projection is expected minutes multiplied by a per-90 rate and divided by ninety. The sample player is expected to play 78.0 minutes at 1.40 shots on target per 90, so the model expects 1.21; that figure is not a bet, and it is rounded to the nearest half-line, 1.5, whose probability of 2 or more is 34.10% and whose fair price is 2.93.
The two numbers are estimated separately
The minutes figure and the rate figure come from different places and are uncertain in different ways. Minutes depend on the team sheet, the role, the opponent, the score a manager expects and the risk of an early injury; the rate is a far more stable quantity that can still be skewed by a small sample of past matches. A projection that states 78.0 minutes and 1.40 per 90 is presenting one number that is routinely wrong by a quarter of an hour and one that is usually right to a decimal place, and the market's price rests on both.
That is why the honest output of the projection is a band rather than a figure. Sample J moves the minutes estimate twelve either way: at 66 minutes the expected output is 1.03 and the fair price is 3.63; at the full 90 minutes it is 1.40 and the fair price is 2.45. The same player, on the same day, is honestly quotable anywhere across a 1.18 range of price.
What the samples show
Sample A and sample J - one projection, read across the minutes estimate
| Minutes assumed | Expected output | Chance of 2+ | Fair price |
|---|
| 66.0 (the low estimate) | 1.03 | 27.53% | 3.63 |
| 78.0 (the central estimate) | 1.21 | 34.10% | 2.93 |
| 90.0 (a full match) | 1.40 | 40.82% | 2.45 |
| The width of the whole answer | 1.03 to 1.40 | 27.53% to 40.82% | 1.18 of price |
sample J - how wide the minutes estimate is
central estimate = 78.0 minutes, output 1.21, fair price 2.93
uncertainty on the minutes = +/- 12.0 minutes -> 66.0 to 90.0
at 66.0 minutes: output = 1.40 x 66 / 90 = 1.0267 -> 1.03
P(0) = e^-1.03 = 0.35701; P(1) = 1.03 x 0.35701 = 0.36772
P(2 or more) = 0.27527 -> fair price = 1 / 0.27527 = 3.63
at 90.0 minutes: output = 1.40
P(0) = e^-1.40 = 0.24660; P(1) = 1.40 x 0.24660 = 0.34524
P(2 or more) = 0.40816 -> fair price = 2.45
the band is 3.63 - 2.45 = 1.18 of price
= 1.18 / 2.93 = 40.3% of the central price.
the operator publishes one number from inside that band.
sample A - what the rounding to the line costs
model mean = 1.21
a line must be a whole number, so the market offers 1.5 (2 or more)
chance at the line of 1.5 = 34.10%
chance if the line could be 1.21 = 1 - P(0) - P(1) at 1.21 = 34.10%
so rounding 1.21 up to 1.5 leaves the price essentially unchanged
at this particular mean, but not in general:
at a mean of 1.50 the chance of 2 or more is 44.25% (fair 2.26),
while the chance of 1 or more is 77.69% (fair 1.29).
the line is where a projection that is a decimal becomes a bet
that is whole, and the two do not always sit together.
Five checks on a projection you are shown
- Whether the page states the expected minutes at all, or publishes only the rate as if a full match were certain.
- Whether the rate is per 90 minutes or per appearance, which the two are not the same number.
- How far back the sample behind the rate goes, and whether it is one hundred matches or six.
- Whether the projection is published as a band or as the single decimal the price was taken from.
- Whether the line was rounded up or down from the model's mean, and which way that pushes the price.
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