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The Player Line / The correlation
One player is not independent of the team
Why a single-player price is not independent of the team
A player does not perform in a vacuum. He scores more often in the matches his team wins, and the market on him carries that fact whether the page says so or not. The single-player price already contains a view about the team, which is why the two markets move together and why pricing one without the other is a mistake.
Desk spec
- scores in wins
- 30.0%
- scores in losses
- 5.0%
- overall
- 21.25%
- fair price
- 4.71
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 answerA single-player price is conditional on the team's performance, not independent of it. The sample player scores in 30.0% of the matches his team wins, 10.0% of its draws and 5.0% of its losses - so his overall chance of scoring, 21.25%, is an average over outcomes the team market is pricing at the same time. The fair price follows: 4.71.
The conditional rates are the whole story
Stated as a single number, a player's chance of scoring looks like a property of the player. Stated as three numbers, it is a property of the player and the match together. The sample player's rate is six times higher in a win than in a loss, which means a reader who thinks his price is too long is, whether he notices or not, also taking a view that the team will win. The two markets are not separate products; they are the same match read twice.
The correlation runs the other way as well. A team that is priced short to win is, on the samples, more likely to have a forward on its books who is priced short to score, so the two markets often move in the same direction without either having changed its own inputs. The desk's point is narrow: a single-player price is an estimate of one person's output conditional on a set of team outcomes, and the words "conditional on" do real work.
What the samples show
Sample D - the sample player's scoring rate inside each team outcome
| Team outcome | How often | Player scores | Joint chance |
|---|
| The team wins | 60.0% | 30.0% | 18.00% |
| The team draws | 25.0% | 10.0% | 2.50% |
| The team loses | 15.0% | 5.0% | 0.75% |
| The player scores, whatever the team does | 100.0% | 21.25% | fair price 4.71 |
sample D - the player's overall rate, and where it comes from
the player scores in 30.0% of wins, 10.0% of draws and 5.0% of losses
the team wins 60.0%, draws 25.0% and loses 15.0% of its matches
overall chance of scoring
= 0.30 x 0.60 + 0.10 x 0.25 + 0.05 x 0.15
= 0.1800 + 0.0250 + 0.0075 = 21.25%
fair price = 1 / 0.2125 = 4.71
if the rate were the same in every outcome - say 21.25% flat -
the overall figure would be unchanged, but the market would be
wrong about every individual match: too long in a win, too short in a loss.
the price is the average; the bet is settled in one of the three cases.
sample D - why the two markets move together
chance the player scores and the team wins = 0.30 x 0.60 = 18.00%
if the two were independent:
0.60 x 0.2125 = 0.1275 = 12.75%
so the pair happens 18.00% of the time against 12.75% if independence held:
a ratio of 1.41, which is the correlation the player market carries.
the fair price of "team wins and player scores", from the conditional rates
= 1 / 0.1800 = 5.56.
the fair price of the same pair priced as the two separate markets multiplied
= (1 / 0.60) x (1 / 0.2125) = 1.67 x 4.71 = 7.86.
so a reader who composes the two markets gets a price 2.30 of price longer
than the conditional facts allow, 2.30 / 7.86 = 29.3%
- which is the correlation the teams-and-players markets share.
the single-player market already embeds this; the reader usually does not.
Five checks on a single-player market against its team
- Whether the player's rate is published as one number or as separate conditional rates.
- Whether the player market is priced before or after the team market, and on the same model.
- Whether a heavy favourite's player markets are all shorter for the same reason.
- Whether the correlation is disclosed anywhere, or only shows up in the price.
- Whether a bet on the player and a bet on the team are treated as one position or two.
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