Originally Posted by
wack
Just quickly, how come $205k or more?
There's no simple logical answer to this ... it's just how the math happens to work itself out.
But just for fun (I had a late night) here's how it goes:
Originally Posted by
Original Problem
A given event has 3 mutually exclusive outcomes, call them i, ii, and iii.
These occur with probabilities:
- 0.46080
- 0.31698
- 0.22222
With bets offered at decimal odds of:
- 2.1368
- 3.3703
- 5.9390
There's no maximum bet on outcome i, but the maximum bets on outcomes ii and iii are $1,760 and $1,000, respectively.
Q: How large would the player's total bankroll need to be such that the expected growth maximizing allocation for bet i would be $0?
Solution:
We have the objective function (assuming the $1,760 and $1,000 constraints bind):
Code:
E(U) =
0.46080 * ln($B + 1.1368 * $x - $1,760 - $1,000) +
0.31698 * ln($B - $x + 2.3703 * $1,760 - $1,000) +
0.22222 * ln($B - $x - $1,760 + 4.9390 * $1,000)
where $B is total bankroll and $x is the amount bet on outcome i.
So we maximize E(U) with respect to $x, subject to $x ≥ $0 and $x + $1,760 + $1,000 ≤ $B (and of course we know that the latter constraint won't bind for moderate levels of $B -- "moderate" being around $5,319 -- but that's another post).
Setting dE(U)/d$x to zero and solving for $x (the 2nd derivative is everywhere negative) we come up with
Code:
$x =
5.62984 × 10-11 *
( 8.76123 × 109 ± 9.00127 × 109 *
sqrt (B2 + 797.194 B + 158902)
+ 5.28259 × 1013 )
for $x ≥ $0
Setting $x = 0 and solving for $B we come up with a value of $205,124.2305 which I'm slightly surprised to discover is so close to my $205,000 figure.
This means that for bankrolls of about $205,124.2305 or greater the expected growth maximizing allocation for bet i would be $0.