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For the vast majority of betting markets, most bookmakers offer similar betting prices, with only a fairly narrow spread of odds about an average. For the France v England example from the last lesson, the best price for France (2.50) was only marginally better than the average (2.45). Consequently no matter how we tried to back all three possible outcomes (assuming of course that we wanted to) we could never return a profit by doing so. Sometimes, however, it is possible and such an opportunity is known as an arbitrage or “arb” for short, sometime called a “surebet”. Consider the following example for the 1X2 market for Liverpool v Chelsea.

Liverpool Draw Chelsea Overround Payout
Average Prices 2.61 3.3 2.66 106.21% 94.15%
Best Prices 3.2 3.4 2.8 96.38% 103.76%

The value for the payout is simply the inverse of the overround. A payout figure of 103.76% means that, whatever the result between Liverpool and Chelsea, we can guarantee a profit of 3.76% on turnover. The stakes for each outcome are simply calculated by the inverting the betting odds and multiplied by some constant to ensure we are betting the sizes appropriate to our risk preferences. In the example below, this is £100.

Liverpool Draw Chelsea Total
1/odds 0.3125 0.2941 0.3571 0.9638
Stakes £31.25 £29.41 £35.71 £96.38

The final table then shows the computed returns for each possible outcome: Liverpool win, draw and Chelsea win. Whatever the result, we will make £3.62, which is equivalent to 3.76% of the total turnover of £96.38.

Home Draw Away Total Profit over turnover
Liverpool wins £68.75 -£29.41 -£35.71 £3.62 3.76%
Draw -£31.25 £70.59 -£35.71 £3.62 3.76%
Chelsea wins -£31.25 -£29.41 £64.29 £3.62 3.76%

Because of the huge number of betting markets available, numerous arbitrage opportunities exist at the bookmakers every day, and they can be found simply by means of an odds comparison. For a fee some arbitrage services will also find these for you. Often, one finds them describing such opportunities as risk-free. Of course, there is no such thing, since even arbitrage bets can, and do, go wrong. Firstly, the punter must ensure that he can obtain all the necessary odds to make the arbitrage work. In the example above, in the time it takes to place bets on Liverpool and Chelsea to draw and Chelsea to win, the 3.20 for Liverpool may have dropped below the value necessary to make this arbitrage work. Given that bets can usually be placed in seconds, this should not normally present a problem, but it does occasionally happen. Possibly more of a concern would be the impact of stake limitation. What happens if the bettor has his £31.25 for Liverpool and his £35.71 for Chelsea accepted only to find that the bookmaker offering 3.40 for the draw will not accept a stake of £29.41? Of course,  that’s unlikely at these stake sizes (unless the bookmarker suspects you of arbing, and most bookmakers don’t like this sort of thing at all), but for 3 or 4-figure stake sizes required to return some sort of meaningful profit, such limitations become a regular headache.

Other practical issues besides price drops and stake acceptance also exist. For some sports, different bookmakers may use different rules regarding the settlement of bets for events which do not complete. This is particularly common in tennis where player retirements frequently occur. When this happens some bookmakers will pay out provided at least one ball has been struck in the match, whilst others will simply void bets, requiring the whole match to be completed to settle wagers. Still others will settle wagers only if one or occasionally two sets of tennis have been completed. If the arbitrage hunter has placed his stakes with bookmakers using different rules, a retirement may leave him with a lost stake on one player and a void bet on the other. Furthermore, to take advantage of as many arbitrage opportunities as possible, the punter will be required to have a large number of bookmaker accounts, many with less well established reputations for customer service and betting account management, and a considerable amount of funds potentially residing with these bookmakers at any one time. Even the best arbitrages amount to only a few percent. To make £50 from a 5% arb requires £1,000 to be turned over in just one go.

Perhaps the biggest drawback with arbitrage betting, however, is the fact that the majority of bookmakers, with the exception of the most dynamic odds setters like Pinnacle Sports, don’t like it. Catch you at it and they will likely limit your stakes or simply close your account. The problem is, given the precise staking required to make arbitrage work properly, it’s fairly obvious when you’re doing it. To get round this difficulty we could round the stakes to make them look less obvious, for example in the Liverpool v Chelsea arbitrage above, to £30, £30 and £35. We would still be guaranteed a payout, but this time only £1 if Liverpool wins, whereas as much as £7 if the match is drawn (and £3 if Chelsea wins). Bookmakers, however, have very sophisticated means of tracking all your betting activity with them and despite such attempts at being more inconspicuous, there is no guarantee that you will still not be spotted.

To reduce the risk of the bookmakers rumbling our arbitrage activity, we can instead back just one of the possible outcomes for an underround book, instead of all of them. Of course, this time not hedging for the other results will mean there is no profit guaranteed on a bet-by-bet basis. Nevertheless, if the small advantage is spread proportionally across all possible results, the betting odds for each of them should, on their own, offer theoretical value and positive profit expectancy. Over the long term, if the bookmakers’ assessments of result expectancies are broadly correct, we should make a profit. Such an approach might be described as arithmetic value betting.

The advantage over arbitrage of arithmetic value betting is that we need not go to all the lengths of precisely calculating stake sizes. Instead, we can just bet the stakes we want; whatever the size, the profit expectancies as a percentage will remain the same, and they won’t stand out as much with the bookmakers. Furthermore, we need not have such a large number of bookmaker accounts and funds deposited in them, instead just limiting ourselves to our preferred brands. Unfortunately, bookmakers are also very good at spotting particular betting patterns. If they see a punter repeatedly targeting best prices, it won’t be long before the majority of them slap some sort of limitation on him. One way to reduce that risk would be to throw in a few “duff” wagers now and again (for example some randomly selected racing bets) where the bookmaker’s odds we are backing are shorter than the market best price.

Perhaps the biggest difficulty facing arithmetic value betters, which arbers do not have to worry about, is the precise location of the betting value. For the Liverpool v Chelsea match, the full market value is still relatively small at 3.76%. There is nothing to say that each possible outcome – a Liverpool win, a draw or a Chelsea win – will have exactly 3.76% value within the odds. It would only take a relatively minor shift in the value of one or more outcomes to eliminate the value for the other(s). Suppose the “true” probability of both a Liverpool win and the draw was 35% (fair odds 2.86). This would mean that the actual betting odds for Chelsea, 2.80, did not offer any value at all. Indeed, with a fair price of 3.33, we would be facing a negative profit expectancy of -0.16. Unless the actual betting odds closely mirror the “true” probabilities of each outcome, it is very likely that at least one of the possible outcomes will lack value for all but the largest of underround books. Fortunately, it is fairly well known that the favourites in a betting market are more likely to hold the value whilst the longshots are not. Why this should be so will be the focus of the next lesson on the favourite–longshot bias.

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