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.

Update

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.