You are ignoring half of the possible outcomes. If you get HH or TT, that's $2.25 and 0.36, which averages to $1.25.
50% of ($1.25 + $.90) for a $1 bet seems like a great game to play.
Re:What is ergodicity? (2016)
by yellowstuff on Friday June 28, 2019 @12:13PM
(#20305840)
This is actually an awesome illustration of why a lot of famous results in academic finance were wrong in the 70s and 80s. Returns don't average, they multiply. If you get HHTT or TTHH you're in terrible shape:
2.25 * .36 = .36 * 2.25 = .81
You started with $1 and now have 81 cents. I recently learned that a lot of early computer-era academic finance actually made the same error of averaging returns, as described in Finding Alpha:
http://falkenblog.blogspot.com/2016/08/finding-alpha-pdf.htm...
After many rounds, you expect to see half heads and half tails. In that case your outcome at n rounds would be around 1.5^n/2 * 0.6^n/2 = 0.9^n/2 which trends towards zero.
You would need around 26% more heads than tails just break even (0.6*(1.5^1.26)=1). But even if you were lucky at first, you will always "regress toward the means" in terms of head to tails ratio if you kept playing.
> If you bet more than that in the long run you will go broke.
The optimal bet is not the same as the critical bet separating the positive growth and negative growth regions.
Re:What is ergodicity? (2016)
by yellowstuff on Thursday June 27, 2019 @04:08PM
(#20298142)
You're correct. Betting more than double the Kelly amount results in negative expected growth:
($1 * 1.5) * .6 = $.90
What if you get a tails then a heads?
($1 * .6) * 1.5 = $.90
Doesn't seem like such a great game now, does it?
The Kelly Criterion sets a limit for how much of your total wealth you should bet when the odds favor you. If you bet more than that limit you increase your odds of losing everything without improving your expected return.
So imagine a better game where you could bet any amount, and you still got paid $5 for every $4 you risked and had a 50% chance of winning. The correct amount to bet is 10% of your bankroll. If you bet more than that in the long run you will go broke.