Arbitrage Betting – 1: Introduction & Theory

Arbitrage betting involves placing bets on each possible outcome of an event to make a guaranteed profit, regardless of the event outcome.

Most people associate sports betting arbitrage with opportunities where different agencies offer sufficiently different odds to make a guaranteed profit. However, the majority of my own opportunities have involved varying odds over time rather than inter-agency discrepancies. For example, the odds for Australia to beat South Africa in a test match will change each day depending on the previous day’s results. This also applies to intra-game betting, like soccer, where the odds drastically change with each goal scored. While there are many people who do engage in disparate odds arbitrage, you typically need accounts with 25+ betting agencies to take advantage of it on a regular basis.

This is the first in a series of posts which will cover arbitrage theory, opportunities to look out for and how to best take advantage of them. Please keep in mind that I am no expert on sports betting and these posts are intended for your amusement only. Please don’t rely on my mathematics and logic! Verify everything for yourself. You can view the Wikipedia article on arbitrage betting here.


Part 2: Three-outcome Betting is now available

Part 3: Arbitrage Opportunities is now available

Arbitrage Theory

Arbitrage betting involves placing multiple bets on the same event that combine to provide a guaranteed profit.

Arbitrage with Bookmakers

When an agency offers betting odds for an event, the sum of the inverses of the odds will always sum to greater than one. So if an agency offers the following odds for a tennis match:

Andy Murray    1.68
Andy Roddick    2.20

The sum of the inverses of the odds is 1/1.68 + 1/2.20 = 1.05. This means the betting agency will earn 5% on all bets for this game. It is worth remembering this, as it enables you to see which bookmakers offer better rates than others. The higher the figure, the greater the profit the bookmaker is taking.

If you placed equitable bets (providing the same profit) on both Murray and Roddick, you would be guaranteed a loss due to the agency’s margin, but if two agencies offered different odds, you may be able to bet on Murray with one agency and Roddick with another. Suppose two agencies offered the following odds:

     Agency 1    Agency 2
Andy Murray    1.68    2.20
Andy Roddick    1.40    2.98

When you sum the inverses of agency 1’s odds for Murray with agency 2’s odds for Roddick, the result is less than one, which means an arbitrage opportunity does exist.

1/1.68 + 1/2.98 = 0.931

If you placed a $100 bet on Murray with agency 1 and a $100 x (1.68/2.98) = $56.38 bet on Roddick with agency 2, you would receive a guaranteed profit of $11.62 regardless of the result. If you were confident Andy Murray would win, you could instead bet $100 on Murray with agency 1 and $100/(2.98-1) = $50.51 on Roddick with agency 2. This would provide a profit of $17.49 if Murray won with no profit or loss if Roddick won.

In general terms, if you set the second bet equal the first bet multiplied by the ratio of the odds, you will get an equal payout regardless of the result. If you are confident of a particular result but would like protection from being wrong, you can set the second bet equal to the first bet divided by one less than the second bet’s odds. I call the first type unbiased arbitrage and the second type biased arbitrage.

b1 = Bet amount on outcome 1
b2 = Bet amount on outcome 2

o1 = Odds for outcome 1
o2 = Odds for outcome 2

Unbiased Arbitrage

b2 = b1 x (o1 / o2)

Biased Arbitrage

b2 = b1 / (o2 – 1)

If you had a total bet amount in mind and wanted to calculate your two bets, you can use these formulas.

B = b1 + b2 = combined bet amount

Unbiased Arbitrage

b1 = B / (o1/o2 + 1)
b2 = B / (o2/o1 + 1)

Biased Arbitrage – where you predict b1 will be correct

b2 = B / o2
b1 = B – b2

Using whole dollar bets to reflect most betting agency rules, the possible outcomes for the Murray vs. Roddick example are displayed below.

    |   No Arbitrage   |   Unbiased Arbitrage   |   Biased Arbitrage
    |       |       |    
Murray Bet   |   $100.00   |   $100.00   |   $100.00
    |       |       |    
Roddick Bet   |   $0.00   |   $57.00   |   $51.00
    |       |       |    
Profit if Murray Wins   |   $168.00   |   $11.00   |   $17.00
    |       |       |    
Profit if Roddick Wins   |   $-100.00   |   $12.86   |   $0.98

I suggest you set up a spreadsheet to regularly test for arbitrage opportunities. Also, I have created an arbitrage calculator which is available in the tools section.

My next post will provide theory on arbitrage betting with three possible outcomes: win, draw and lose. This is important for sports like soccer and test cricket.

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5 Responses to "Arbitrage Betting – 1: Introduction & Theory"

  1. can you tell me the wager amounts required to make arbitrage back/lay bets with odds of
    back 3.55
    lay 3.45
    so i would like to make a biased bet on 1 above and a biased bet on 2 above so that one of the wagers must win and the others will be no win ,no loss

    to include betfairs 5% or alternatively can you tell me where a calc to do these types of wagers can be found, thanks for your time regards don rees

  2. i think i have made a mess of submitting my question so i will try again.assume i have back odds of $3.55 and lay odds of $3.45 ( for say U&O 2.5 goals)
    i would like to know the wager amounts required for biased bets on 1 and biased bets on 2, so that one of these must be a winning wager and the others no win , no loss and the betfair commission to me is 5%, or a calc that will do these calculations for me, thanks don rees

    1. Hi Don, Betfair calculates the commission based on your net profit in the market. With biased arbitrage no commission would apply in a scenario with no profit.

      If you have made a back wager of $100 at 3.55, for biased arbitrage towards the back bet you would lay $100 at 3.45. If the back bet wins you net $255 – $245 = $10, minus commission. If the lay bet wins you net $100 – $100 = $0.

      If you have made a back wager of $100 at 3.55, for biased arbitrage towards the lay bet you would lay $100*(3.55 – 1)/(3.45 – 1) = $104.08. If the back bet wins you net $255 – $255 = $0. If the lay bet wins you net $104.08 – $100 = $4.08, before commission.

      Let me know if you have further questions.

      1. hi i would like to have a calculator that would tell me the wager amounts required when i wanted to make say a bet of $100.00 on team A at odds of say $1.57 and a “saver” bet on team B at odds of $ 4.22 , so that if TA loses i have sufficient money on TB to cover my stakes for the two is a free bet ??
        this is an arbitrage bet so the odds are right to make a profitable biased bet.
        i would then like to know the wager amounts required to make a $100.00 biased bet on team B and a ” saver” bet on team A.
        i regret to admit that at 79 y/o i have forgetten all of my algebra learnings ,so if you could make your answers as simple as possible please,
        if i could have a go myself i believe that in the first instance the wager amount required on T B would be $ 33.00 resulting in a win of $24.00 and the wager amount required on TA in the second instance would be $125.00 resulting in a win of $197.00, are these correct, ? thank you don


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