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Arbitrage and middles

How arbs and middles work, the math behind each, and what can go wrong.

TommyLocks, Nov 3, 4 min read

Most bets are a prediction: you think a side wins more often than its price says. Arbitrage and middles work differently. They come from two sportsbooks disagreeing with each other, and the bettor takes both sides at different books.

Arbitrage

An arb happens when the best price on each side of a bet, taken from different books, adds up to less than 100% implied probability. Betting both sides then returns a profit whichever side wins.

Example:

• Team A +110 at one book: implied probability 47.6%
• Team B +105 at another: implied probability 48.8%
• Together: 96.4%, under 100%

Split $100 so both sides pay the same:

• $49.40 on Team A at +110 returns $103.74
• $50.60 on Team B at +105 returns $103.73

Either way you get back about $103.73: a $3.73 profit, or 3.7% of the total stake. The stake on each side is its implied probability divided by the total (47.6 ÷ 96.4 for Team A), and the return is the stake divided by the total (100 ÷ 0.964).

Most arbs are much thinner than this, often 1–2%.

Middles

A middle happens when two books have different lines on the same bet. You take the over at the lower number and the under at the higher one, and if the result lands between them, both bets win.

Example (game total):

• Over 227.5 at -110 at one book
• Under 229.5 at -110 at another
• $110 on each side

If the total lands on 228 or 229, you win $200. Anywhere else, you lose the $10 of vig. To break even, the middle has to hit about 1 time in 21 (4.8%), because 1 hit at +$200 has to cover 20 misses at -$10.

The same works on spreads (Team A -2.5 at one book, Team B +3.5 at another, which both win if Team A wins by exactly 3) and on player props (Over 22.5 points at one book, Under 24.5 at another, which both win on 23 or 24).

How wide the window is matters most. Each extra number inside the middle adds to how often it hits. Where the window sits matters too: in the NFL, many games end on margins of 3 and 7, so a middle that covers one of those numbers hits more often than one that doesn’t. NBA scores spread out more evenly, so each number in the window adds less.

Middles are not guaranteed. Unlike an arb, a middle usually loses a small amount. It pays off only when the hit rate is high enough to cover those small losses over many tries.

What can go wrong

• Prices move. Arb and middle windows are short. If one side moves before you place the second bet, the arb can turn into an ordinary bet at a worse price. Many bettors place the side most likely to move first.
• Bet limits. A book may only take a small amount at its price, which caps the size of the arb.
• Different rules. Books handle voids differently. If a player doesn’t play, one book may void a prop while the other grades it a loss, leaving one side exposed.
• Mistakes in the posted price. When a price is obviously wrong, a book may void bets made at it, again leaving one side exposed.
• Account limits. Sportsbooks watch for arb and middle betting and may cut the maximum stake on accounts that do it often.

The overall point

Arbs and middles come from the market disagreeing with itself, not from knowing who wins. An arb locks in a small, certain return. A middle takes a small, likely loss for a chance at a larger win. Both depend on finding books that disagree and acting before the prices line up again, and both pay off through volume more than any single bet.

Tools for finding them

• The arbs & middles tab in Educated Wagers checks every book’s prices and lines for current arbs and middles, with stakes split so both sides pay the same. It also shows near-arbs within 2%.
• The odds screen shows every book’s price side by side. Sort by % diff to see where books disagree most.
• The odds discrepancies tab lists the biggest price differences across books.
• Odds discrepancies between sportsbooks explains why books disagree in the first place.