How is potential profit from an arbitrage bet (a.k.a. surebet) calculated?

Question

Arbitrage bet or surebet means betting on all possible outcomes in which you make profit independently on the result of the event.

How exactly can I calculate whether given odds form an arbitrage and what the potential profit could be?

Answer 1

Complementary bets

First let us look at the situation where we have several possible outcomes, which are complementary, i.e., exactly one of these results can happen. There can be various number of complementary outcomes; we can have 2 of them (two players in a tennis match, over/under in a soccer match, home/away in ice hockey), 3 results (home/draw/away in soccer, or regulation time in ice hockey) or even more results (for example, betting on exact score 0-0, 1-0, 0-1, and complementing this with over 1.5). Let us try the case of 3 possible outcomes; the computations are similar in the case of other number of results.

We may have the following outcomes and odds:

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|   | odds | stake | winnings |
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| 1 |  o1  |  b1   |   o1*b1  |
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| X |  o2  |  b2   |   o2*b2  |
-------------------------------
| 2 |  o3  |  b3   |   o3*b3  |
-------------------------------

We would like to maximize the winning in the case of any outcome, hence W=o1b1=o2b2=o3b3 which yields b1=W/o1 b2=W/o2 b3=W/o3

If we bet the amount of B altogether, then
B=b_1+b_2+b_3=W(1/o1+1/o2+1/o3).
To obtain an arbitrage, we need W>B, i.e.,
1/o1+1/o2+1/o3<1. And the fraction
W/B=1/(1/o1+1/o2+1/o3)
represents how much is the winning larger than our stake. We can also calculate from the above equations b1,b2,b3; i.e., the percentage of our stake to put on various outcomes.

We can have a look on a particular example. Suppose we find the following odds at various bookmakers

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| 1 |   6   |
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| X |   5   |
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| 2 | 1.746 |
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Since 1/6+1/5+1/1.746=0.9394<1 this is an arbitrage, and the profit is 1/0.9394=1.0645; which means the potential profit is 6.45%.

Arbitrage of type pk/X/2

Let us have a look at one more type of arbitrage.

Many bookmakers offer bet called pk (standing for pick-em) or handicap 0 which works as follows: If the team wins, the bettors obtains the winning; in the case of draw the bettor gets his stake back. If the team loses the match, the bettor loses his stake.

Let us have a look again what happens if the bettor bets on all three possible outcomes. (They are, so to say, partially complementary.)

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|   | odds | stake | winnings |
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| pk|  o1  |  b1   |   o1*b1  |
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| X |  o2  |  b2   | b1+o2*b2 |
-------------------------------
| 2 |  o3  |  b3   |   o3*b3  |
-------------------------------

We try to optimize the wagers in a such way that the winning in all cases is maximal possible. This yields the winning
W=o1b1=o3b3=b1+o2b2

The first equality yields
b1=W/o1
b3=W/o3
Now we can use b1 to compute b2.
b_2=(W-b1)/o2=W/o2-W/(o1o2)

Using b1+b2+b3=1 (i.e., assuming b1,b2,b3 represent the percentage of the stake) we get:
W/o1+W/o2-W/(o1o2)+W/o3=1
1/o1+1/o2-1/(o1o2)+1/o3=1/W
1/(1/o1+1/o2-1/(o1o2)+1/o3)=W
Hence we can obtain an arbitrage if
1/o1+1/o2-1/(o1o2)+1/o3<1 (which is equivalent to W>1.)

Again, we can compute b1, b2, b3 from the above equations. We can have a look at an example again. Suppose we have match with the following odds:

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| home, pk | 2.24 |
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|   home   | 2.98 |
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|   draw   | 4.1  |
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|   away   | 2.45 |
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The usual 1/X/2 arbitrage with the odds 2.98/4.1/2.45 gives profit 1.252%. Using pk for the home team yields 1.05% (Hence, in this case, using this type of arbitrage bet did not bring an advantage.)

Risks

Perhaps it is important several risks connected with this types of betting. Many of them are mentioned in the Wikipedia article about arbitrage betting, too. (Here is link to the revision at the time when this answer was posted and to the current revision.)

• You have to be careful about the limits you are allowed to stake.
• You need to be careful about the bookmarkers rules. For example, in tennis some bookmakers make the stake void if one of players withdraws during the match; some consider the other player the winner. (And some have even more complicated rules depending on number of sets that have already been played.) Similarly in baseball, some bookmakers make the bet void unless the listed pitchers started.
• There are bookmakers that have different rules for pk (for example, they return only 90% instead of the whole bet).