An Aggressive Approach to Bankroll Management

[alon.albert]

I found this article to be very interesting but I would really like to get a clarification from the author.

In order to understand the math, I tried to plug in numbers from a trivial theoretical case. I tried to apply the math to a Heads Up game with no fee and a 50% win rate. Since there is no expected profit from this situation I would expect the math to calculate F* as 0. Here's how the math turned out:

Assume 1000 games, 500 wins.
Payout is 2 for 1'st place.

<font class="small">Code:</font><hr /><pre>b = 2 * 500 / 500 = 2
p = 0.5
q = 0.5

F* = (2 * 0.5 - 0.5) / 2 = 0.25</pre><hr />

This is obviously wrong.

I believe the problem is that b needs to be calculated differently. In the article, b is calculated in terms of earning rather than profit. I believe b should be calculated like this:

<font class="small">Code:</font><hr /><pre>b = (2 - 1) * 500 / 500 = 1<font class="small">Code:</font><hr /><pre>

The "-1" subtracts the tourney cost from the earnings resulting in profit.

Now we get:

<font class="small">Code:</font><hr /><pre>F* = (1 * 0.5 - 0.5) / 1 = 0</pre><hr />

Which is what I would expect.

If I use this approach on the authors example:

<font class="small">Code:</font><hr /><pre>(210 * (4.2 - 1)) + (238 * (2.5 - 1)) + (152 * (1.68 - 1)) / 600 = 1.89</pre><hr />

This will in turn result in:
<font class="small">Code:</font><hr /><pre>F* = (1.89 * .40) - .60 / 1.89 = 8%</pre><hr />

Am I missing something?