Post
Topic
Board Service Announcements (Altcoins)
Re: Kelly Criterion
by
dooglus
on 18/06/2014, 19:32:49 UTC
When a player risks 1 unit at 33% for a 3x payout for example, the house is risking 2 units for a 1.5x payout with a 67% chance of winning. If you apply the Kelly Criterion on those numbers you'll see that it's still optimal to risk 1% of the bank.

f = (bp - q) / b

where f is the fraction to risk,
b is the payout multiplier minus 1
p is the probability of the house winning
q is the probability of the house losing

So in the 49.5% bet, where the payout is 2x,
b = 2-1 = 1
p = 0.505
q = 0.495
f = (1*0.505 - 0.495) / 1 = 0.505 - 0.495 = 0.01 = 1%

And in the 33% bet, where the payout (for the house) is 1.5x,
b = 1.5-1 = 0.5
p = 0.67
q = 0.33
f = (0.5*0.67 - 0.33) / 0.5 = (0.335 - 0.33) * 2 = 0.01 = 1%

And it's the same for every other bet:

If the player has probability q of winning, the payout to the player is 0.99/q
When the player risks 1 unit, the house risks 0.99/q - 1 unit
That's a payout multiplier for the house of 1 / (0.99/q - 1) + 1

b = 1 / (0.99/q - 1) + 1 - 1
f = (b*p - q) / b
  = (p/(0.99/q - 1) - q) * (0.99/q - 1)
  = (p*q/(0.99 - q) - q) * (0.99 - q)/q
  = (p/(p - 0.01) - 1) * (p - 0.01)
  = p - (p - 0.01)
  = 0.01
  = 1%