Parlay optimization: beating compounded vig and correlation

A parlay is a chain of bets that all have to win. The payout multiplies, which is what makes a $10 ticket paying $600 so appealing. What multiplies with it is the sportsbook's margin: every leg carries a small overcharge, and when you chain three of them, you are paying all three overcharges on one ticket.

Think of three stores that each add a 5% fee. Buy one item from one store and you lose 5%. Buy a bundle that only counts if all three purchases go through, and you have paid about 15% in fees for the privilege. That bundle is a normal parlay.

Some legs are connected. If a quarterback throws for 300 yards, his top receiver probably had a big game too, so a parlay on both is more likely to hit than two unrelated bets would be. Sportsbooks know this. On a same-game parlay they shorten the payout to account for the connection, which is why the number on the slip is lower than the legs multiplied out.

• Positive correlation — the legs tend to win together (a team's win and its star's big game). The book pays less than the plain multiplication.
• Negative correlation — the legs work against each other (the Under and a player's big yardage day). The book sometimes pays more.
• Independent — legs from different games barely affect each other, and the payout is close to the legs multiplied.

The trick is not finding a magic combination of teams. It is choosing legs that are each already a good price on their own, so each leg adds edge instead of fee, and then placing the ticket at the sportsbook that is paying the most for that exact combination. Done properly, the parlay is better than any one of its legs.

Same-game legs are rarely independent, so the plain product overstates or understates the real probability. Sportsbooks handle this by quoting a same-game parlay at a price that is not the product of the legs. A simple way to express the adjustment is a single factor applied to the product.

To value a same-game parlay you need a correlation estimate that does not come from the book you are about to bet with. Otherwise the payout cancels out of the math entirely: a book that shortens the price heavily looks exactly as good as one that barely does.

• Fewer legs, more edge per leg — the ticket's EV is driven by the weakest leg; one −EV leg can erase the others.
• Shop the ticket, not just the legs — books disagree more on parlay prices than on single legs. The same slip can pay +620 at one book and +780 at another.
• Mind the double count — two legs on the same stat for one player (total bases and a double, say) are not two opinions, they are one.
• Beware a number that looks too good — a card showing +100% on four legs is nearly always a pricing or devig error, not an edge.

• Devigged, no fallback — in EV mode legs are scored with the Positive EV strategy (worst-case devig, median across benchmark books). A row the pipeline could not devig has no fair value and no leg, which drops one-sided markets. Falling back to the raw one-sided average once shipped a +141.8% card whose four legs all took that path and priced the parlay at +4044 against FanDuel's +9381.
• At least four paired books — a leg needs fairBookCount ≥ 4. One paired book is not a market: in deep alt ladders, the lone book quoting a 'No' side posted a boilerplate -10645, which handed every player in the league the same fair probability and scored eight unrelated longshots at exactly +10.3%.
• Gated lines never bet — any line carrying an exclusion reason (consensus outlier, low liquidity, swapped sides) is excluded from the pool. The pool is pregame only, both sides, with a leg value floor of −25%.
• Conflicts — two legs cannot share a market group (both sides, or two rungs of one alt ladder), and two legs on the same player in the same event cannot overlap in the stats they count (total bases already counts a double). Stat overlap is declared per sport in a stat-group registry.

A book's quote is never valued against its own factor. If it were, the payout would drop out of the EV entirely, and two books offering the identical ticket at +750 and +894 would score the same. Instead every slip gets one consensus factor, and each book's price is measured against it.

• Consensus factor — the geometric mean of the quoting books' factors (votes in log space), with exchanges excluded and a price family averaged into one vote, shrunk toward a prior. The shrinkage weight is between-slip variance ÷ (between-slip variance + within-slip variance ÷ n), with measured spreads of 0.235 between slips and 0.126 within: one vote earns 0.78 of the weight, three votes 0.91.
• The prior — 0.958 per leg beyond the first, so 0.958^(n−1). Measured slip medians were 0.916 at three legs and 0.886 at four, within 0.3% of that curve. Cross-game parlays use a prior of 1.0, and a slip with one vote or fewer is marked low confidence.
• Why shrinkage rather than a cutoff — only 44% of slips see three votes and 12% see four. A minimum-books rule would have thrown away most of the board.

Behind those constants: on 2026-09-17, 132 slips produced 426 priced quotes across 16 NFL games plus MLB. Only DraftKings and FanDuel return a correlation field (81 of 81 of their quotes, 0 of the other 345), and the derived quoted ÷ naive factor matched the reported one at a median of 0.00%. DraftKings and FanDuel disagreed on identical legs by +4.1% on average with a 15.6% spread, agreeing on the sign only 60% of the time. Leave-one-out, the geometric mean predicts a held-out book's factor with −0.0% bias and a 6.3% median error; taking the maximum instead is +9.2% biased. 71% of quotes shortened the price, with a median factor of 0.914.

• SGP tokens — a book can quote a same-game parlay only when it has issued an SGP token for every leg. The headline price on a card must come from a book that can sell the ticket. Ranking on the best naive product across all books put Bet365, which prices legs longest and issues no tokens at all, at the head of every card: slips selected at +67% to +79% that the books able to price them were paying −13% to −22% on.
• Estimate fallback — if no placeable book quotes the slip, the card shows naive product × consensus factor and is flagged as an estimate.
• Cross-game slips — priced through the same endpoint, no token required, and reported at about zero correlation.
• Min books — a promise about the price table: how many books returned a quote for the parlay, with a price family counted once, across the whole board rather than your book filter. It is only knowable after the books answer, so above 1 the search prices up to three slices down the ranking until the pack fills; a pack can still come back short.

• Per-bucket budgets — up to 40 pool legs per game (10 per game and 60 overall when the slate pool is on), a combination walk capped at 150,000 total and no fewer than 2,500 per game, at most 8 legs, and up to 12 revealed slips. Each game's slice keeps a quota per price band (+150 and +600 floors) so the pool is not all longshots. A full generation takes about nine seconds, nearly all of it the books' own SGP latency.
• EV mode — the default. Devigged median legs, correlation applied.
• Mispriced mode — scores the board with the one-sided Prop Outliers average, so one-sided markets enter the pool, and applies no correlation correction. The number is Value % compounded across legs and deliberately keeps the vig; it reads far higher than EV and is labelled accordingly.
• Promo valuation — EV = p × profit on win − (1 − p) × (at risk − refund). A profit boost multiplies the profit by (1 + boost); an odds boost adds American points to the final price; a bonus bet puts nothing at risk; a no-sweat bet refunds stake × conversion rate on a loss. The Promo sort ranks by promo EV minus straight EV.