July 25, 2021

The Cost of Online Betting

What is the cost of online betting? What part of the stakes is paid out again by the betting company, how high is the “betting fee” the betting company is keeping for themselves? How many bets do you have to do before the betting fee eats up your stake?

Let us take as example the upcoming semi-final football matches in the Champions League. The betting company bwin displays the following return rates on April 30, 2016 (rates can and will change over time).

Semi-Final 1 / May 3, 2016 / Bayern Munich – Atletico Madrid

Home win:         r1 = 1.53

Draw:                  r2 = 3.90

Away win:          r3 = 5.25

Semi-Final 2 / May 4, 2016 / Real Madrid – Manchester City

Home win:         r1 = 1.53

Draw:                  r2 = 3.75

Away win:          r3 = 5.50

The rate r1 means, when betting 1 you will get paid 1.53 for a home win.

The return rate includes the stake in this case i.e. the stake of 1 is returned plus the gain of 0.53.

Return rates are usually calculated based on the ratio of people betting per outcome. When more people bet on a certain outcome, the potential return will be smaller. When less people bet on the outcome, the return will be higher.

In general, a more probable outcome decreases the potential win; a less probable outcome increases it. Bettors might have different strategies so the betting ration might or might not depend on the probability of outcome, but this is not relevant for this analysis.

When n1 people are betting on outcome 1 (home win),

r1/r2 * n1 people will have bet on outcome 2 (draw) and

r1/r3 * n1 people on outcome 3 (away win).

Therefore, the total number of bets N is:

N = (1 + r1/r2 + r1/r3) * n1

Assuming s is the stake per bet, the sum of all stakes and the total income I of the betting company will be:

I = N * s

I = (1 + r1/r2 + r1/r3) * n1 * s

The betting company will pay out P < I after the event.

In case outcome 1 wins:

P = r1 * n1 * s

In case outcome 2 wins:

P = r2 * r1/r2 * n1 * s

P = r1 * n1 * s

In case outcome 3 wins:

P = r3 * r1/r3 * n1 * s

P = r1 * n1 * s

This shows that the total payout P will be the same whatever the outcome. This is what one would expect as the ground rule of any bet.

Comparing the betting company’s income I with the total payout P shows that the calculations are independent of how many people are betting. Calculating the ratio will eliminate n1.

What matters is the relative cost of betting for the bettor i. e. the share of the stakes the betting company keeps for themselves. This is sort of a “betting fee” BF = I – P, or in percents:

BF = 1 – P/I

BF = 1 – (r1 * n1 * s) / [(1 + r1/r2 + r1/r3) * n1 * s]

BF = 1 – 1 / (1 + r1/r2 + r1/r3)

First semi-final:               BF1 = 9.13%

Second semi-final:         BF2 = 9.26%

The difference between both numbers can be due to rounding betting companies applied to the return rates to make them more readable.

The betting fee is:

  1.  the percentage of all stakes that that is not paid out. It is therefore the total loss of all participants.
  2. the average expected loss of a bettor with every bet.

The law of large numbers (LLN) is applicable here. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

The following table shows how quickly losses can be accumulated if the betting fee is 9.20% (assuming constant stakes):

loss_table_920

We can observe the following:

1. The expected average loss per bet is 9.20%.

2. Half value time after 6 bets

After 6 bets, the accumulated average loss is expected to be more than 50%. In other words, the bettor must expect after 6 bets to have lost more than half of a stake.

3. Total write-off after 11 bets

After 11 bets, the accumulated average loss is expected to exceed 100%. The participant must therefore expect after 11 bets to have lost more than one stake.

Obviously, online betting companies need to make money but the fees appear to be quite high. A fee of below 0.5% would be desirable to flatten the loss curve.

The question is how many bettors don’t know that fees and loss probabilities are that high, and chances to win such reduced.