Team totals and the ladder behind them

A team total is an over/under on one side's score, settled without reference to the opponent's. It is the only market that lets a view about one team's production be bet without any exposure to who wins. The rungs above and below the posted line form a nested ladder, and their probabilities must fall monotonically as the number rises.

Because the rungs are nested, the difference between two adjacent ones is the probability of landing in between them. Over 0.5 minus over 1.5 is the chance of exactly one; over 1.5 minus over 2.5 is the chance of exactly two. The ladder is a distribution written out as prices, and subtraction reads it.

Those differences still contain margin, since each rung was quoted with its own cut in it, so the exact-score numbers they produce are inflated. De-vigging each rung against its own opposite price first gives a considerably more honest set of differences.

Which is where it goes wrong. A team total reads as a bet on one team and is settled by a match the opponent is also playing. A side three goals up stops attacking, and a side three goals down starts chasing, so the game state the total needs is frequently the opposite of the game state that would make it happen.

What to do with it. Pull the whole rung ladder for the team, difference it into exact-score probabilities, and check the single rung you fancy against that shape.