Maths For Lay Dutching To A Fixed Liability

This evening I 'Googled' for the maths behind lay dutching to a fixed max liability and found loads of links to on-line dutching calculators and little else. There is of course lots of trading/betting software out there that also has dutching calculators, but as I needed the maths for a bot I'm developing, I wound up working it out for myself.

Requirement: Calculate the individual stakes required based on a fixed overall maximum liability.

Looking at the correct score market of the Arsenal v AC Milan match to be held tomorrow night (6th Mar 2012) we see the lay odds for the 0-1, 1-1, 2-1, 3-0 and 3-1 scores are as listed in the table below. Note, those scorelines have been picked at random for illustration purposes and do not indicate any thoughts I may have on the game.

From those odds, the implied probabilities can be calculated by dividing 1 by the odds. The table summarises those calculations so far:

Score	Odds	Win Prob
0-1	16.0	0.0625
1-1	8.4	0.1190
2-1	10.5	0.0952
3-0	29.0	0.0345
3-1	20.0	0.0500

So, the total probability of one of the chosen results being the full time score is simply the total of the last column, i.e. 0.3612, or 36.12% if you prefer. However, we don't want those picks to be the winning score since we wish to lay them all. So the overall probability of the final score being something other than one of our five picks is simply:

1 - 0.3612 = 0.6388

This is equivalent to decimal odds of:

1/0.6388 = 1.5654

Now, we need to calculate the total stake to be used when laying these picks so that the overall liability is fixed at £100 say. This is simply:

Total Stake = Fixed Liability * (Overall Odds - 1)

=> Total Stake = 100 * (1.5654 - 1) = 56.54

Now, all we have to do is to split this total stake across the five picks in proportion to their odds. This is done using the formula:

Stake = Total Stake * Win Prob / Total Win Prob

So, in the case of the 0-1 scoreline, the Stake is:

56.64 * 0.0625 / 0.3612 = 9.80

The following table summarises the results for all the chosen scorelines:

Score	Odds	Stake
0-1	16.0	9.80
1-1	8.4	18.66
2-1	10.5	14.93
3-0	29.0	5.41
3-1	20.0	7.84

So we now have the stakes required for each selection ensuring that the liabilty should any of the selections end up winning is fixed at 100. If we decided to add another scoreline to our picks, the overall liability would remain the same, but all the individual stakes would change. 

Here's a screen shot from the Fairbot dutching calculator to confirm the figures, albeit with a few pennies different due to rounding differences.