Hi rtabell!
I'll first introduce the notation needed to use the formula I've found.
Let's call X1 and X2 to the values of both bets to win. Also, let's
say you define X1 so that it is smaller than X2. For example, in the
case you describe, X1 shall be $25 and X2 shall be $26, so you will
buy the bet at X1 and sell it at X2.
Once you've defined X1 and X2, let's call t1 and t2 respectively to
the profit share you get to keep in platform 1 (the one with the
"cheap" bet), and the profit share you get to keep in platform 2 (the
one with "expensive" bet). In your case, the "cheap" bet is in
platform A. Therefore, since in that platform you get to keep 100% of
the profits, then t1 will be equal to 1; and since in the other one
you keep 95%, then t2 will be equal to 0.95.
Note that I decided to include the possibility that it's the "cheap"
platform the one that charges you fee on profits. This is realistic,
as you wouldn't expect the same platform to always be the cheap one.
So, here's the formula:
(100 - X1)*t1 + X1
Q = --------------------
100 - X2*(1 - t2)
The value Q will represent how many times higher will the bet in the
expensive platform than in the cheap platform. All this will become
clear once we plug in the values from the example you give.
In your example, we said that X1 = 25, X2 = 26, t1 = 1, and t2 = 0.95.
Plugging this values into the above formula, we get that Q =
1.0131712... This means that for every 1 unit you buy of the bet at
platform A, you should sell 1.0131... units at platform B. Let's check
that this value gives you the same profit no matter the outcome of the
game.
You buy 1 unit of the bet to win at Platform A, and sell 1.0131 units
of the bet to win at Platform B. [Note that that the profit will not
be exactly equal because I've rounded Q to 4 decimals - to obtain the
exact same profit you should use the exact value of Q, which might
prove difficult depending on your bet sizes]
If Miami Heat wins, each unit at platform A had a profit of $75, while
each unit at platform B had a loss of $74. Since you bought 1 unit at
A and sold 1.0131 at B, your net profit is 75 - 1.0131*74 = $0.0306
If Miami Heat loses, each unit at platform A lost $25, while each unit
at platform B had a profit of $26*0.95. Since you bought 1 unit at A
and sold 1.0131 at B, your net profit is 26*0.95*1.0131 - 25 =
$0.0235.
As I said, the profit is not the same due to rounding. If you do these
calculations with the precise value of Q, you'll get the same profit
in both outcomes.
It's also very important to notice that, depending on the values of
x1, x2, t1 and t2, the bet size that produces the same result
independently of the outcome of the game might very well produce
guaranteed LOSS. Think of extreme cases where both companies charge
very high profit fees and the difference between x1 and x2 is very
small. In these cases, the Q from the formula will surely be such that
you lose (the same amount) no matter the outcome of the game. So
before applying the formula in real life, be sure to check that there
is a guaranteed profit rather than a guaranteed loss to be made from
the bet.
I hope this helps! Please let me know through a clarification request
if you have any trouble using the formula or if it's not what you were
looking for, and I'll keep working on it.
Best wishes,
elmarto |