# Arbitrage Betting – 2: Three-outcome Betting

In my first arbitrage post I laid the groundwork for arbitrage betting with two possible outcomes: win and lose. This is case for sports where draws aren’t possible. Some examples are tennis, baseball, and basketball (with extra time included). For sports like test cricket and soccer, however, draws are not only a possibility, but occur frequently. This post, the second in my arbitrage series, covers arbitrage betting with three possible outcomes.

### Update

Part 3: Arbitrage Opportunities is now available

If you haven’t done so already, be sure to check out the three-outcome arbitrage calculator.

### Three-Outcome Arbitrage Theory

As with two-outcome betting, you can obtain a betting agency’s margin on their odds by summing the inverses of the odds. So if an agency offers the following odds for a soccer match:

 Everton 1.8 Draw 3.35 Tottenham 4.4

The sum of the inverses of the odds is 1/1.80 + 1/3.35 + 1/4.40 = 1.081. The larger this figure, the greater the margin that the bookmaker is taking. Because the sum is greater than 1, if you placed equitable bets (i.e. providing the same profit) on all three outcomes, you would be guaranteed a loss due to this margin.

If you can find differing sets of odds for the same event, you may be able to come up with a combination of win, draw and loss bets that guarantees a profit, regardless of the event outcome. The different betting odds could be provided by different bookmakers, as in the example below, or it may be due to odds changing over time as the game or series progresses. Arbitrage opportunities will be discussed in more detail in upcoming posts.

Suppose two agencies offered the following odds:

 Agency 1 Agency 2 Everton 1.80 2.30 Draw 3.35 3.25 Tottenham 4.40 2.95

If you sum the inverses of agency 2’s odds for an Everton win along with agency 1’s odds for a draw and an Everton loss, you get 1/2.30 + 1/3.35 + 1/4.40 = 0.961. This figure is below 1, so an arbitrage does opportunity exist. Note that a number of cross-combinations can be tested. To find the best combination, take the largest odds for each possible outcome.

To calculate the amount to bet on each outcome, determine the total amount you would like to bet. Then calculate the amounts to bet on each particular outcome as follows:

Definitions:
b1 = bet (in dollars) on outcome 1 (Everton win).
b2 = bet (in dollars) on outcome 2 (draw)
b3 = bet (in dollars) on outcome 3 (Everton loss)
B = b1 + b2 + b3 = combined bet amount
o1 = odds for outcome 1 (Everton win).
o2 = odds for outcome 2 (draw)
o3 = odds for outcome 3 (Everton loss)

In this example I will bet \$1,000 in total. Calculate the bets for each outcome as follows:

b1 = B / (1 + o1/o2 + o1/o3)
b2 = B / (1 + o2/o1 + o2/o3)
b3 = B / (1 + o3/o1 + o3/o2)

b1 = \$1000 / (1 + 2.30/3.35 + 2.30/4.40) = \$452.63
b2 = \$1000 / (1 + 3.35/2.30 + 3.35/4.40) = \$310.76
b3= \$1000 / (1 + 4.40/2.30 + 4.40/3.35) = \$236.60

Each bet equals the total bet amount divided by 1 plus the sum of the ratios of that outcome’s odds to the other outcomes’ odds

Calculate the guaranteed profit as b1o1 – B (or as b2o2 – B, etc)

\$452.63 x 2.30 – \$1000 = \$41.06
\$310.76 x 3.35 – \$1000 = \$41.06
\$236.60 x 4.40 – \$1000 = \$41.06

Betting \$452.63 on an Everton win, \$310.76 on a draw, and \$236.60 on an Everton loss would guarantee a profit of \$41.06, which is a 4.1% return on the total bets.

If you were confident of a particular result, you could employ a biased arbitrage strategy and make a larger profit if the pick is correct, with no loss if it isn’t.

In the following example let’s predict Everton will win.

Calculate the bets for a draw and an Everton loss as follows:

b2 = B / o2
b3 = B / o3

The bet on an Everton win will equal the total bet amount minus the bets for the other two outcomes:

b1 = B – b2 – b3

b2 = \$1000 / 3.35 = \$298.51
b3 = \$1000 / 4.40 = \$227.27
b1 = \$1000 – \$298.51 – \$227.27 = \$474.22

If a draw or Everton loss occurs there is no profit or loss. If Everton wins the result is a profit of \$474.22 x 2.30 – \$1,000 = \$90.71, which represents a 9.1% return as opposed to a 4.1% return for an unbiased arbitrage bet.

Using whole dollar bets to reflect most betting agency rules, the possible outcomes for this example are displayed below. The no arbitrage bet involves a simple bet of \$1,000 for Everton to win.

 | No Arbitrage | Unbiased Arbitrage | Biased Arbitrage Everton Win Bet | \$1,000 | \$453 | \$474 Everton Draw Bet | \$0 | \$311 | \$299 Everton Loss Bet | \$0 | \$237 | \$227 Profit if Everton Wins | \$1,300 | \$41.90 | \$90.20 Profit if Everton Draws | -\$1,000 | \$41.85 | \$1.65 Profit if Everton Loses | -\$1,000 | \$42.80 | -\$1.20

The unbiased arbitrage strategy provides a profit of between \$41.85 and \$42.80 depending on the result, whereas the biased arbitrage strategy provides a \$90.20 profit if Everton wins, with minimal profit or loss for the other results.

With the basics of arbitrage theory out of the way, my next post will cover some practical arbitrage betting opportunities, with references to actual previous bets I have made.

## 6 Responses to "Arbitrage Betting – 2: Three-outcome Betting"

I am sure that i will come back to your site soon. Keep us posting

2. I was looking for a long time for something good, I think I found it and owe it to you mister.I will follow your instuctions. Thank you. Greetings from Greece and to all Australian-Greeks. christos.

3. Bt wen you multiply with the original odd you get less than the sum you staked

4. Dmitry

Hi, thank you for all the info. I just wanted to find out where you got the 2.2 odds in your biased arb calculation please? It’s not in Agency A or B and doesn’t give a profit in your calculator or in my manual calculations either.
Thank you.

1. Dmitry

Correction – and the 1.8 odds in Agency A (whose draw and loss odds were used) doesn’t give a profit.

2. Hi Dmitry, the two references to 2.20 odds should state 2.30. I have amended the article to fix this.

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