A million years ago, when you could still find video poker games with 100%+ theoretical return or poorly thought-out promotions offering enough cash-back to get you over 100%, we'd calculate the Kelly number for a given opportunity -- the bankroll necessary to ride out hills and valleys in favorable situations.
Spoiler: It's almost always 3-4x the value of a royal flush. So you needed $12-16k if you were playing a $1-per-coin game with a 1% edge at a pretty good clip.
And what do you earn with perfect play in that situation? The princely sum of around $30 an hour.
bdjsiqoocwk 51 minutes ago [-]
I would like to understand in detail what you just wrote.
"$1 per coin game" is this a game where you put in $1 to play and get paid either $2 or $0 with 50-50 probability (0 expected).
And the what does it mean %1 edge? Does it mean the probabilities are such that the expected payout is 1c per coin flip?
roenxi 2 hours ago [-]
A word that is good to know here is ergodic [0]. Which I must admit to not really understanding although it is something like the average system behaviour being equivalent to a typical point's behaviour. If a process is non-ergodic then E[X] is usually not as helpful as it seems in formulating a strategy.
An illustrative example to explain ergodicity. Consider the following game. Players start with $100. At every turn, a fair coin is flipped. If tails, the amount of player's money is increased by 50%. If heads, the amount of player's money is decreased by 40%. To play or not to play, that is the question.
diab0lic 45 minutes ago [-]
An example that may be useful to aid in understanding… Casinos are non ergodic.
A million players each placing a single bet will have an expectation of losing the house edge.
A single player placing a million bets has an expectation of $0.
30 minutes ago [-]
KK7NIL 1 hours ago [-]
Some math/finance nerds made a whole YouTube channel about ergodicity, which I've been really enjoying: https://youtu.be/VCb2AMN87cg
TL;DR: while a single investment may be ergodic, portfolio management (the math behind weighting successive and concurrent investments/bets) is not, as it has a strong dependence on all prior states.
uoaei 1 hours ago [-]
this comment may be confusing and I doubt this will help much but:
Ergodicity is less about memorylessness and more about the constraints on transitions into this or that state. A system is ergodic if "anything that can be an outcome, eventually will happen".
wenc 3 hours ago [-]
The Kelly criterion is almost never used as-is because it is very sensitive to probability of success, which is hard to know accurately and in many cases, dynamically changing. This is easy to see in an Excel spreadsheet. Changing the probability by even 0.01 percent can vastly shift the results. The article calls this out in the last paragraph.
The article mentions fractional Kelly is a hedge. But what fraction is optimal to use? That is also unknowable.
Finance folks, correct me if I’m wrong, but the Kelly Criterion is rarely used in financial models but is more a rule of thumb that says roughly if you have x $ and probability p, in a perfect world you should only bet y amount. But in reality y cannot be determined accurately because p is always changing or hard to measure.
intuitionist 30 minutes ago [-]
Yeah, but I think this misses the point a bit. The fact that your true edge isn’t knowable wouldn’t be so bad except that if you’re betting full-Kelly and overestimate your edge even a little bit, your probability of ruin in the long run goes to 1. Whereas if you underbet, you’ll compound wealth at a little lower rate but won’t risk ruin.
eftychis 3 hours ago [-]
I am not sure what you mean by "never used as is."
The Kelly criterion is an optimization of capital growth (its logarithm) method/guide. Not using it doesn't change its correctness.
But yes you need to know the advantage/the edge you have. Like with pricing methods eg for European options for Black Scholes you need to know the volatility and there is no way to know it, you estimate. This is where all the adjusting for bias and ML comes in.
wenc 1 hours ago [-]
But do you calibrate p (say through estimation) and then apply the Kelly criterion in your portfolio?
I don’t think it is used in this way. It swings too much with a given p.
uoaei 1 hours ago [-]
You calibrate for a reasonable distribution of p and use that to estimate (Monte Carlo, etc.) expected gain, optimizing your investment based on that. With this technique your estimate will probably end up somewhere around the common heuristics.
avidiax 2 hours ago [-]
Here's a link to a bigger graph for the Blackjack Scenario:
I think it shows that Blackjack is not even theoretically winnable over time if you have to pay for information on the count in the form on minimum bets. The ideal case it that you bet $0.49 for every $1,000 in your investment pool when the count is extraordinarily high.
Even if you hack the casino's cameras so you know the count without having to be at the table, your reward is a growth rate that is very low per hand.
quickquest 3 hours ago [-]
For the coin flipping scenario, what happens to the casino? Shouldn't they lose money in the long run as well? Or is it that they're under the kelly threshold with all the house cash?
headPoet 2 hours ago [-]
The casino will break even, but for the gamblers there will be a small number that win big, and a much larger number that lose out.
Consider two rounds, there's a 25% chance you 4x your money, a 50% chance you 0.75x your money and a 25% chance you 0.25x your money
Rendered at 02:00:42 GMT+0000 (Coordinated Universal Time) with Vercel.
Spoiler: It's almost always 3-4x the value of a royal flush. So you needed $12-16k if you were playing a $1-per-coin game with a 1% edge at a pretty good clip.
And what do you earn with perfect play in that situation? The princely sum of around $30 an hour.
"$1 per coin game" is this a game where you put in $1 to play and get paid either $2 or $0 with 50-50 probability (0 expected).
And the what does it mean %1 edge? Does it mean the probabilities are such that the expected payout is 1c per coin flip?
[0] https://en.wikipedia.org/wiki/Ergodic_process
A million players each placing a single bet will have an expectation of losing the house edge.
A single player placing a million bets has an expectation of $0.
Nassim Taleb also talks about this quite a lot: https://youtu.be/91IOwS0gf3g
TL;DR: while a single investment may be ergodic, portfolio management (the math behind weighting successive and concurrent investments/bets) is not, as it has a strong dependence on all prior states.
Ergodicity is less about memorylessness and more about the constraints on transitions into this or that state. A system is ergodic if "anything that can be an outcome, eventually will happen".
The article mentions fractional Kelly is a hedge. But what fraction is optimal to use? That is also unknowable.
Finance folks, correct me if I’m wrong, but the Kelly Criterion is rarely used in financial models but is more a rule of thumb that says roughly if you have x $ and probability p, in a perfect world you should only bet y amount. But in reality y cannot be determined accurately because p is always changing or hard to measure.
The Kelly criterion is an optimization of capital growth (its logarithm) method/guide. Not using it doesn't change its correctness.
But yes you need to know the advantage/the edge you have. Like with pricing methods eg for European options for Black Scholes you need to know the volatility and there is no way to know it, you estimate. This is where all the adjusting for bias and ML comes in.
I don’t think it is used in this way. It swings too much with a given p.
https://github.com/obrhubr/kelly-criterion-blackjack/blob/ma...
I think it shows that Blackjack is not even theoretically winnable over time if you have to pay for information on the count in the form on minimum bets. The ideal case it that you bet $0.49 for every $1,000 in your investment pool when the count is extraordinarily high.
Even if you hack the casino's cameras so you know the count without having to be at the table, your reward is a growth rate that is very low per hand.