Split an Arbitrage Into Legs and Read the Terms Behind Each One

What does Pinnacle's arbitrage article actually measure?

The figures most often repeated about Pinnacle and arbitrage come from one article by Joseph Buchdahl, a betting analyst, carried on Pinnacle's own site under the date of 7 May 2019. It covers 50,210 European league football matches played between 27 July 2012 and 5 March 2019, using the home, draw and away prices. In 16,861 of them (33.6 percent, a division the article leaves to the reader) the best price on each outcome across bookmakers added up to an implied probability under 100 percent before the market closed. Those are opportunities on paper: the count ignores limits, refusals, voids and fees.

That average means 0.81 percent on the stake, in the article's own words. The split by leg is the useful part. The Pinnacle leg's 95.5 percent is a 4.5 percent loss on the turnover placed on it, while the legs at other bookmakers returned 102.7 percent. In this sample the value of the arbitrage sat with the other books and the Pinnacle price was the leg that paid for it. Pinnacle's average margin on these matches would have implied 97.6 percent (the article gives the implied return, never the margin itself), and its other two prices in the same matches returned 98.3 percent.

The gap between 95.5 and 98.3 is what the author tested, and it is the figure most often misread. A one-tailed test returned a p-value of 8 percent: a gap that large would appear by chance 8 percent of the time if Pinnacle's prices in these matches were no different, against the 5 percent he says he needs. That is a weaker statement than an 8 percent chance of being wrong, and the article stops short of saying Pinnacle raises anything on purpose. His explanation is the favourite and longshot bias. The average Pinnacle price in the arbitrages was 4.74 against 3.61 for the other two options, and 79 percent of the Pinnacle legs were draws or away wins. At the close the same legs averaged 4.73 and returned 95.0 percent.

The last line, that Pinnacle "will always allow you to practice arbitrage", is the author's conclusion about its pricing, not a clause in any terms. The figures deserve a careful reading, because the data covers football only, stops in 2019, sits on the site of the operator it describes, and the article does not publish its data. For how the wider sharp-book view of prices fits together, see the sharp betting hub.

How do the legs add up to the result?

Start with 2.10 on one outcome and 1.95 on the other, total stake 100. The inverse odds add up to 0.989011, so the return is 101.11 percent. Stakes of 48.15 and 51.85 pay 101.11 on either outcome, which leaves a result of 1.11 if both legs are placed at these prices.

Now split that by leg against a reference with no margin, 50/50 on the two outcomes. The 2.10 leg has an expected value of +5.00 percent of its stake, +2.41 in money. The 1.95 leg sits at -2.50 percent, -1.30 in money. They sum to 1.11, and they always will, because every leg pays the same amount, so the stakes weighted by the reference probabilities add back up to the result.

The split is what moves. Against a book quoting 2.04 and 1.89 (its inverse odds add up to 1.0193, so it carries its own margin) both legs turn positive: +0.48 and +0.63 with the proportional method, +0.42 and +0.69 with the power method, and still 1.11 in total. The power method takes more margin off long prices, which is why the two disagree. With three outcomes the sign itself can flip. Prices of 2.20, 3.60 and 3.85 against a reference of 2.10, 3.40 and 3.60 return 100.80 percent and a result of 0.80 on 100 (stakes of 45.82, 28.00 and 26.18), yet the favourite leg reads -0.02 under one method and +0.57 under the other. A decomposition is conditional on its reference, and how a margin-free closing price is built explains where a reference can come from.

Decimal odds shown to two places are rounded, so a 2.10 on screen can stand for a slightly different price underneath. A rounding of 0.005 moves that leg's payout by about 0.24 percent, roughly a fifth of a 1.11 percent result.

What does one orphaned leg cost?

The result of 1.11 exists only while both legs are placed at the quoted prices. A leg that is refused, repriced or voided leaves the other as an ordinary bet sized by its stake. Keep only the 2.10 leg, stake 48.15, and you win 52.96 or lose 48.15, which is 43.3 times the result. Keep only the 1.95 leg, stake 51.85, and you win 49.26 or lose 51.85, 46.7 times the result. Against the 50/50 reference the orphaned 2.10 leg is worth +2.41 on average and the orphaned 1.95 leg -1.30, so which leg disappears matters as much as whether one does, and another reference changes the signs.

Repricing works the same way. If the 1.95 leg falls to 1.90, a recalculated stake of 100 at 2.10 and 1.90 returns 99.75 percent, a result of -0.25. With 2.10 fixed the other leg must quote at least 1.909, and with 1.95 fixed the 2.10 leg must quote at least 2.053. A mistake is sized by the stake, because a leg left alone behaves like a normal bet of that size. Which leg goes first is not something any published figure shows. Whether a run of such results pays over a season is a separate question, taken up in the page on whether betting is profitable, and the price-comparison tools by route are listed in the betting tools overview.

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