Binary, scalar and categorical contracts

The binary contract is the default and the one everything else is built from. The question has a yes-or-no answer, the contract settles at $1 if the answer is yes and $0 if it is no, and the price is the market’s probability. Buying YES at 68¢ and selling NO at 32¢ are the same position expressed two ways, which is why the two quotes are tied together: if they were not, holding one of each for less than $1 would be riskless profit.

The whole appeal of the shape is that it collapses a question into one number you can read without training. The cost of that simplicity is that it throws away magnitude. A binary on whether a threshold is crossed pays the same dollar whether it is crossed by a hair or by a mile, so a trader with a strong view about how far past the line the outcome lands has no way to express it here.

A categorical market asks which of several mutually exclusive outcomes occurs — which candidate, which city, which of five ranges a figure falls into. Mechanically it is usually a set of binaries sharing a resolution event, and exactly one of them settles at $1.

Because exactly one wins, the prices across the whole set should sum to about 100¢, and the amount by which they exceed that is the overround. Suppose five outcomes are quoted at 40¢, 30¢, 15¢, 10¢ and 8¢. Those sum to 103¢, so each price is inflated by the same 3% factor. Dividing each by 1.03 gives 38.8%, 29.1%, 14.6%, 9.7% and 7.8% — which now sum to 100.0%. That is the same proportional division you apply to a two-sided market as YES ÷ (YES + NO), generalised to more than two legs.

Two things to watch. First, if the venue implements the legs as independent binaries rather than as one linked market, nothing forces the sum to stay near 100¢, and a set summing to 118¢ is telling you that the legs are being priced by different people who are not looking at each other. Second, the set has to actually be exhaustive: a five-way market on a race with a sixth possible winner is not a probability distribution, and dividing by the sum will overstate every leg.

A scalar contract has a numeric range rather than a yes-or-no answer, and it settles somewhere inside that range. If the contract is written on a value between 3.0 and 5.0 and the published figure comes in at 4.2, the contract settles at (4.2 − 3.0) ÷ (5.0 − 3.0) = 0.6, so 60¢ per contract. A figure below the floor settles at $0 and one above the ceiling at $1; the range is clamped at both ends.

This changes what the price means. A scalar quoted at 60¢ is not saying there is a 60% chance of anything. It is saying the market’s expected settlement value is 0.6 of the way up the range — which is compatible with near-certainty about a middling outcome and equally compatible with a coin flip between the two extremes. The single number has lost the shape of the distribution behind it, and there is no way to recover it from the price alone.

Many venues sidestep scalar contracts entirely by cutting the range into buckets and listing a categorical market instead: “below 3.5”, “3.5 to 4.0”, “4.0 to 4.5”, and so on. That is more contracts to look at, but it gives you the distribution back — the prices across the ladder are the market’s probability mass, and you can see whether the expectation comes from confidence or from a split.

Before treating any quote as a forecast, establish which of the three you are looking at, because the same 60¢ means three different things. On a binary it is a 60% chance. On one leg of a categorical set it is a 60% chance only after you have checked what the whole set sums to. On a scalar it is not a chance at all.

The type also changes what a comparison across venues is worth. Two binaries on the same event are comparable once each is de-vigged. Two scalars are comparable only if the ranges are identical — the same real-world expectation maps to a completely different price under a different floor and ceiling. And a categorical leg on one venue against a binary on another is usually a mismatch hiding in plain sight, because the binary’s “no” bundles together every alternative the categorical market lists separately.