Expected value, done properly
The usual EV formula assumes a bet either wins or loses. Asian handicaps, quarter lines and draw-no-bet also push, half-win and half-lose, and the standard formula silently misprices all of them. Settlement-aware EV, with worked examples.
Expected value is the average result of a bet if it could be repeated forever at the same price. The standard formula, EV = p × (odds − 1) − (1 − p), is correct only when exactly two things can happen. A large share of the betting market has three, four or five settlement states, and applying the binary formula to those markets produces confidently wrong numbers.
The standard formula and its exact scope
EV = p × (odds − 1) − (1 − p)p = your probability of winning; odds = decimal price. Result is in units per unit staked.Worked example: you estimate 55%, the price is 2.00. EV = 0.55 × 1.00 − 0.45 = +0.10, or ten percent of stake per bet on average.
This is valid for a market where the only outcomes are "win" and "lose": a two-way moneyline with no draw, both teams to score, over/under on a half-line such as 2.5. It is not valid for anything else.
The five settlement states
Real markets settle in more than two ways, and each state has a different payout multiplier:
| State | Returns | Where it occurs |
|---|---|---|
| Full win | stake × odds | everywhere |
| Half win | half at odds, half returned | quarter lines: −0.25, +0.75 |
| Push | stake returned | whole totals (3.0), level handicap (0), draw-no-bet on a draw |
| Half loss | half lost, half returned | quarter lines: −0.75, +0.25 |
| Full loss | nothing | everywhere |
The general form that covers all five:
EV = p_win×(d−1) + p_halfwin×(d−1)/2 − p_halfloss×0.5 − p_lossPush contributes exactly zero: it neither gains nor loses.Why is a push not a loss?
On a total of exactly 3.0, a game ending 2–1 returns the stake. Treating that as a loss inflates the loss probability and understates EV; treating it as a win does the reverse. Both errors are large in markets where pushes are common: level handicaps in evenly matched fixtures push often.
Draw-no-bet is the cleanest illustration. Three outcomes exist (win, draw, lose), but only two of them are win and loss. The draw is a push. The correct probability to compare against the price is not the raw win probability but the win probability conditional on the bet being live.
The quarter-line trap
A handicap of −0.25 splits the stake: half on 0, half on −0.5. A draw makes the first half push and the second half lose: a half loss overall. This is where the binary formula fails most visibly, and the failure has a signature you can test for.
A bug we shipped and caught. Our first implementation counted half-states at full weight in the denominator when converting settlement probabilities back to a model probability. The symptom was diagnostic: on a quarter line, both sides of the same market showed a negative edge, which is impossible: de-vigged, they must sum to one. The fix was to weight half-states at 0.5:
p_model = (p_fullwin + 0.5 × p_halfwin) / (p_fullwin + 0.5×p_halfwin + 0.5×p_halfloss + p_fullloss)There is now a regression test asserting that both sides of a quarter line sum to one. A test that fires on a symmetry violation catches this class of bug far better than a test asserting a particular number, because it does not need to know the right answer to detect a wrong one.
EV is only as good as the probability behind it
Two inputs go into EV: the price, which is observable, and the probability, which is a belief. The formula cannot improve a bad belief; it launders one into an official-looking number.
This is why the sequence matters: de-vig the market correctly to get a fair market probability, produce an independent model probability, and only then compute EV against the actual price. Reversing the order (deriving a "model" probability from the price and then computing EV against the same price) produces a number that always looks reasonable and means nothing. See de-vigging methods.
How does CONSENSUS compute expected value?
Every market carries a settlement schema (binary, three-way, push-aware, quarter line, draw-no-bet or combo), and EV is computed from the settlement-state probabilities rather than a single win probability. Combination markets are priced from the joint distribution over the score matrix rather than by multiplying marginals, because multiplying marginals assumes independence that combination markets specifically violate.
Check any of this against our record
Every signal CONSENSUS publishes carries the bookmaker odds fixed before the event starts and the settled result afterwards — including the drawdowns and the losing runs. The running total is on the front page and every entry is in the log.