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The Composed Market / The model
A joint chance, built out of pairs, with a band around it

How the joint chance behind a composed price is estimated

The number that decides a composed price is one probability: that all the legs happen together. It is not measured, because the combination has never happened; it is estimated from the legs, the relationships between them, and a model that has to answer even when the answer is uncertain.

Desk spec
pairs at 3 legs
3
pairs at 12 legs
66
joint states at 12
4,096
price band
4.48 to 5.17
the composed priceThe number returned for a set of outcomes the customer named. It is not the product of the legs: on the samples 5.60 multiplies and 4.80 is offered, because the legs happen together more often than independence implies.
the requestA customer asking an operator to invent a market. 500 arrived in the sample month, 218 produced a price, 131 could not be priced at all, and the median answer took 3 hours 40 minutes.
the void legA leg that never happens, repaired by re-pricing the combination from its remaining legs. Dividing the price by the void leg gives 4.50 where the real three-leg price is 4.80 - a 6.7% error.
Direct answer

The joint chance of a combination is estimated from each leg's own chance and the pairwise relationships between them. Three legs need 3 pairs; twelve legs need 66, and the model must resolve 4,096 joint states. Sample J shows the answer is a band, not a number: a 1.5-point uncertainty in a 20.83% joint chance moves the price from 4.48 to 5.17.

What goes in

Three inputs, and only the first is visible to the reader. Each leg has a chance of its own, which the published price already states. Each pair of legs has a relationship, which nothing published states. And the model has a way of turning those into one joint figure - a way that also decides how the answer behaves at the edges, when a leg is almost certain or almost impossible.

The count is what surprises readers. Three legs need 3 pairs. Twelve legs need 66 - C(12,2) - and the number of states the combination can end in rises as 2^12 = 4,096. Every one of those 4,096 outcomes has to be resolved for the combined market to exist, which is why a covered market is not always a market the model can price: sample B records 131 of 500 requests that were never priced for exactly this reason.

What comes out, and how wide it is

Sample J - one joint chance, priced at the edge of its own band
Estimate for the joint chancePrice it impliesAgainst 4.80
The model's central figure, 20.83%4.80-
The low edge, 19.33%5.17+0.37
The high edge, 22.33%4.48-0.32
The band around one price0.69 wide14.4% of 4.80
sample J - one probability, two prices model joint chance = 0.2083 uncertainty on that estimate = +/- 1.50 points low estimate 0.2083 - 0.0150 = 0.1933 -> price = 1 / 0.1933 = 5.17 high estimate 0.2083 + 0.0150 = 0.2233 -> price = 1 / 0.2233 = 4.48 the band is 5.17 - 4.48 = 0.69 of price = 0.69 / 4.80 = 14.4% so the same three legs can honestly be quoted at 4.48 or 5.17, depending on where in its own uncertainty the model is read. nothing on the page the customer sees says which of the two they were given.
sample E - what a long combination costs to price legs in a combination min 2, max 12 single chances needed = 12 pairs needed = C(12,2) = 12 x 11 / 2 = 66 joint states the model resolves = 2^12 = 4,096 compare 3 legs: 3 pairs and 8 joint states a 12-leg request therefore asks for 22 times the pairs and 512 times the states of a 3-leg request, for one price. that is the arithmetic behind the maximum leg count: beyond twelve, the model is answering about a market nobody is really asking for.

Where the model meets the reader

A model that returns a band rather than a number produces a commercial choice: the operator picks the point in the band at which to quote. Nothing in the samples says the choice is dishonest, and nothing lets a reader see it either - the composed price arrives as a single figure, and the uncertainty behind it is not published. The practical consequence for a reader is simple to state and hard to act on: two requests that look identical can come back at different prices, and the second figure is not a better price because the market moved, but because the estimate did.

Five questions a model raises
  • Does the page anywhere state that the composed price comes from an estimated joint chance rather than from the legs?
  • Is the minimum accepted price published, so a reader can tell a refusal from a quiet re-quote?
  • Does the operator publish how often a combination it priced came back at a different figure the next day?
  • Is the same model used for the published version of the market once the combination is posted to everyone?
  • If two legs are near-certain, does the model still treat the pair as independent - and does the price change materially if it does not?

Read next

The price of legsThe gap correlation explainsThe requestsThe queue the model sits behindThe limitsWhere the model is refusedThe refusalsThe four reasons a request fails