The short answer
Bookmakers build a margin into every price, so the implied probabilities across a market always add up to more than 100 per cent. Devigging removes that margin and rescales the probabilities so they sum to 100.
The result is a fair price, sometimes called a no vig price or true odds. It is an estimate of what the market believes, with the bookmaker's cut taken out.
You need this because you cannot sensibly compare your own probability estimate against a number that has a business's profit margin baked into it.
The simplest method, and a worked example
The most common approach is proportional, sometimes called the multiplicative method. Convert each price to an implied probability, add them up, then divide each one by that total.
Take a market priced at 1.90 and 1.90. Each implies 52.63 per cent and the total is 105.26 per cent. Divide 52.63 by 105.26 and you get 50 per cent for each side. The fair price for both is 2.00.
A less symmetrical example: prices of 1.55 and 2.55 imply 64.5 and 39.2 per cent, totalling 103.7. Dividing each by 103.7 gives 62.2 per cent and 37.8 per cent. The fair prices are about 1.61 and 2.65.
The no vig fair odds calculator on this site does this for you if you would rather enter two prices than run the arithmetic.
Why the simple method is not always right
Proportional devigging assumes the margin is spread evenly across the outcomes. In many markets it is not.
Bookmakers commonly load more margin onto longshots than onto favourites, a pattern related to the well documented favourite longshot bias. When that happens, dividing everything by the same total understates the favourite's true chance and overstates the outsider's.
Two alternatives exist for this reason. Additive devigging subtracts an equal share of the margin from each outcome rather than scaling proportionally. Shin's method and the power method both try to model the uneven loading explicitly. Each makes different assumptions, and none of them is correct in every market.
For two way markets with small margins, the three approaches produce answers close enough that the choice rarely decides anything. On long multi outcome markets with fat margins, such as a race or an outright, the difference matters a great deal.
Which price to devig
This matters more than the method. Devigging a price from a book with a wide margin and little volume gives you a fair estimate of a poorly informed opinion.
The usual practice is to devig the sharpest price available, on the reasoning that the book with the most volume and the tightest margin is the one carrying the most information. A price from a book that moves last and follows others is worth less as a signal.
Devigging the same book you are about to bet with is close to circular. If you strip the margin from a price and then compare it against that same price, you have proved that the price contains a margin, which you already knew.
How to use it in practice
Devig a sharp price to get a fair probability. Compare that fair probability against the price you can actually get somewhere else. If the price you can get implies a lower chance than the fair estimate, that is where positive expected value comes from.
Treat the output as an estimate with error bars, not a measurement. The method makes assumptions, the source price carries opinion, and both can be wrong.
PandaBet's software detects positive expected value against modelled true odds, which is this idea applied continuously across eight Australian bookmakers rather than one market at a time by hand.