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Q: Find Math Formula for Optimal Bet Size ( Answered 5 out of 5 stars,   1 Comment )
Subject: Find Math Formula for Optimal Bet Size
Category: Science > Math
Asked by: rtabell-ga
List Price: $10.00
Posted: 11 May 2005 05:47 PDT
Expires: 10 Jun 2005 05:47 PDT
Question ID: 520399
Here is an easy problem for any Mathematician.

I want a formula, which would determine optimal bet size across two
platforms if one platform charges fees on profits.

For example:

I can buy Miami Heat to win the NBA Championship $25 to pay $100 on Platform A.

I can sell Miami Heat to win the NBA Championship $26 on Platform B.

If there are no fees, my profit from the transaction is a
guaranteed $1. If there are no fees and the Heat win, I make $75 on
Platform A, and lose $74 on Platform B; and if the Heat lose I make
$26 on Platform B and lose $25 on Platform A. Either outcome with no
fees I make $1.

However let us assume Platform B charges a 5% fee on profits. 

As a result if Miami Heat win, I still get a $75 profit in Platform A,
and a $74 loss on platform B; but if they lose I lose $25 on Platform
A and I make (0.95*26) = 24.7 on Platform B for a net loss of $0.3.

Give me a formula which will determine the optimal Bet Size for each
platform which will make me indifferent as to which team wins/loses..
ie. No matter which team wins I want an equivalent profit for each

I don't need a mathematical proof, just the formula will do; if it
works I will know it works.
Subject: Re: Find Math Formula for Optimal Bet Size
Answered By: elmarto-ga on 11 May 2005 11:35 PDT
Rated:5 out of 5 stars
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

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 =

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,
rtabell-ga rated this answer:5 out of 5 stars and gave an additional tip of: $1.00
Thank you very much. You have answered the question correctly and
provided even more useful info. You deserve an extra dollar. I will
submit another question if I need a further formula.

Subject: Re: Find Math Formula for Optimal Bet Size
From: elmarto-ga on 12 May 2005 12:51 PDT
Thank you for the rating and tip!

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