Kelly Can't Fail

Note that you need to be able to infinitely divide your stake for this to work out for you all the time.

For example, if the deck has 26 red cards on top, you'd end up dwindling your initial $1.00 stake to 0.000000134 before riding it back up to 9.08

Very good point. I did some experiments and the system is very sensitive to any sort of quantization or rounding of bets. You get the expected value about the right place, but the variance goes up quickly. So in addition to your important case, things are a bit dicey in general.

If you start out with a $1e12 stake, you're able to avoid catastrophic rounding errors even in the worst case. There's probably a life lesson here.

Is the lesson: choose to be born to wealthy parents?

Or is it to choose appropriate betting amounts based on your capacity for risk

IEEE-754 isn't precise enough for my capacity :( I too need rich parents.

>>Note that you need to be able to infinitely divide your stake for this to work out for you all the time.

This is what most people discover, you need to play like every toss of the coin(i.e tosses over a very long periods of time). In series, like the whole strategy for it to work as is. You can't miss a toss. If you do you basically are missing out on either series of profitable tosses, or that one toss where you make a good return. If you draw the price vs time chart, like a renko chart you pretty much see a how any chart for any instrument would look.

Here is the catch. In the real world stock/crypto/forex trading scenario that means you basically have to take nearly trade. Other wise the strategy doesn't work as good.

The deal about tossing coins to conduct this experiment is you don't change the coin during the experiment. You don't skip tosses, you don't change anything at all. While you are trading all this means- You can't change the stock that you are trading(Else you would be missing those phases where the instruments perform well, and will likely keep landing into situations with other instruments where its performing bad), you can't miss trades, and of course you have to keep at these for very long periods of time to work.

Needless to say this is not for insanely consistent. Doing this day after day can also be draining on your mental and physical health, where there is money there is stress. You can't do this for long basically.

I guess it then does follow that having rich parents does expand your capacity for risk!

Yup, the dual would be saying Martingale can't fail with infinite money.

My simulation shows that with a 52 card deck, if you round the bet to the nearest $.01 you will need to start with $35,522.08 to win a total of $293,601.28.

If you start with $35,522.07 or less, you will lose it all after 26 incorrect cards.

The lesson I'm taking away is "learn math and how to use it."

While I don't agree on nearly anything you stated, I enjoyed your prose: I suppose you left out words here and there as a metaphorical proof of your claim that you can't miss a single toss, didn't you?

Nearest rounding does seem like a mistake here. Rounding down is quite safe: rather than lose it all, you end up with at least 2^26 pennies.

Learn math and discover poignantly all the situations where it is effectively useless.

If you assume coin tosses are independent, it shouldn’t matter if you miss coin tosses.

It’s easier to make money if you already habe money

Applying math to a more practical betting situation, like poker, is a lot harder. You'd have to be able to calculate your exact odds of winning given only a small amount of information, without a calculator and without it taking so long that the other players notice, and then also factor in the odds that the other players are bluffing and the advantages that you might have from (not) bluffing.

Finally a real world use case for bitcoin!

Coin tosses are not independent. Unless the premise is coins toss themselves.

A person tosses a coin, so tosses are are connected to each other.

Ask yourself this question- Would your thumb hurt if you toss a coin 5000 times? If so, would that change the results?

>>I suppose you left out words here and there as a metaphorical proof of your claim that you can't miss a single toss, didn't you?

You must always practice in real world conditions. Notice in the experiments conducted in programs, you are taking series of tosses as they come, even if they are in thousands in numbers, one after the other, without missing a single one. Unless you can repeat this in a live scenario. This is not a very useful strategy.

Kelly criterion is for people who are planning to take large number of trades over a long period of time, hence the idea is to ensure failures are not fatal(this is what ensures you can play for long). As it turns out if you play for really long, even with a small edge, small wins/profits tend to add to something big.

If you remove all the math behind it, its just this. If you have a small edge to win in a game of bets, find how much you can bet such that you don't lose your capital. If you play this game for long, like really really long, you are likely to make big wins.

It's a good point. I think it affects the realism of the model. When the stake is very low, finding a penny on the street gives an astronomical improvement in the end results. At the high end, it's possible the counterparty might run out of money.

You are conflating 2 concepts: a) that the reality converges to what the theory predicts only after a great number of samples; b) that if you skip some events the results will vary.

Now, b) is false. You can change the code to extract 3 random numbers each time, discard the first 2 and only consider the third one, the results won't change.

Instead a) is generally true. In this case, the Kelly strategy is the best strategy to play a great number of repeated games. You could play some games with another strategy and win more money, but you'll find that you can't beat Kelly in the long term, ideally when the repetitions approach infinity.

>>Now, b) is false. You can change the code to extract 3 random numbers each time, discard the first 2 and only consider the third one, the results won't change.

Might be in theory. In practice, this is rarely true.

Take for example in trading. What happens(is about to happen), depends on what just happened. A stock could over bought/over sold, range bound, moving in a specific direction etc. This decides whats about to happen next. Reality is rarely ever random.

Im sure if you study a coin toss for example, you can find similar patterns, for eg- if you have tired thumb, Im pretty sure it effects the height of the toss, effecting results.

>>Instead a) is generally true. In this case, the Kelly strategy is the best strategy to play a great number of repeated games.

Indeed. But do make it a point to repeat exact sequences of events you practiced.

Or is it to choose appropriate betting amounts based on your parents?

It would really help if your parents know someone who can and will take the other side in this game.

A popular view is that having wealthy parents gives one a great advantage. Another popular view is that working extraordinarily hard for money is a waste of one’s life even if one gets the money. But the two are only consistent if one believes that one’s own life is the optimization target. If I live a life of misery so that my children live a life of prosperity that would strike me as a phenomenal result.

So another reading is “choose to give your children wealthy parents”.

Naturally tossed coins tend to land on the same side they started with 0.51 of the time, see

https://www.stat.berkeley.edu/~aldous/157/Papers/diaconis_co...

It's not because there is a finite amount of money at which this can't fail, which is never the case for martingale. Martingale is actually likely to bankrupt you against a casino that is much more well staked than you even if you have a small advantage.

Linked paper does not state that; it states that tossed coins tend to be caught on the same side they stared with slightly more than half the time. The results explicitly exclude any bouncing (which will happen if a coin lands on a hard surface).

The paper does discuss coins allowed to land on a hard surface; it is clear that this will affect the randomness, but not clear if it increases or decreases randomness, and suggests further research is needed.

This is a truism

This sounds similar to the Martingale system.

https://en.wikipedia.org/wiki/Martingale_(betting_system)

Bitcoin isn't infinitely divisible, you can't do smaller than one satoshi.

Practically if you live life fearing a 26 red cards event something like e-15, you'd never leave the house and leave your money under your bed

Nice try, but-

The machine they use to toss the coin has a spring, and Im sure the spring tension varies through time effecting results.

Are you sure about those numbers? I get that the smallest fraction of original stake we hit is around 1.35E-7:

          ⊢min←⌊⌿ratio←×⍀{1-d×0⍪¯1↓(+⍀d←¯1+2×⍵)÷⌽⍳≢⍵}52↑26⍴1
    1.353223554704754E¯7

In which case we need to start with $73,897.62

          ⌈÷min
    7389762

For a total payout of $671,088.64

          ⌊(⌈÷min)×⊃⌽ratio
    67108864

Thanks for getting me to actually check this!

Note: above code is Dyalog APL.

I cannot decipher the runes you write, but your magic looks to be more powerful than mine so you are probably right. (i.e., my Python script may have errors; it was pretty hacked together)

I've added a follow-up note on the case of discrete stakes here: https://win-vector.com/2024/12/21/kelly-betting-with-discret... . It is known there is a dynamic programming strategy that guarantees a return of $8.08 on a $1 bet. Simple rounding of the Kelly strategy does not achieve this.

When Poland first introduced the capital gains tax in 2002, banks were quick to notice the tax law still generally required tax amounts to be rounded to nearest full złoty when accrued, so they started offering financial products with daily capitalization, which were effectively exempt from the new capital gains tax, as for most customers, the daily gain would be low enough that the tax on it always rounded down to zero. This only got corrected 10 years later.

I find it fascinating that we could have a whole class of financial products hinging on something seemingly so trivial as a rounding strategy.

I think the word you need to use in this conversation is iid.