[Data]

I just realized that now is not just the end of the year but the end of the decade. I think now is the perfect time to give credits where credits are due.

I nominate this post:
http://www.sportsbookreview.com/forum/handicappe...ml#post4977571

I've for some time been a bit embarrassed by my rather egregiousness overcomplication of the mathematical curiosity that is the unconstrained simultaneous independent event Kelly solution, which I detailed over 3 years ago in this thread.

So to rectify in brief:

Given N independent events, x₁, x₂, ..., x_N, with corresponding single bet Kelly stakes of κ₁, κ₂, ..., κ_N, the unconstrained Kelly solution (for any Kelly multiplier > 0) consists of the 2^N-1 parlays such that the wager on a given parlay comprised of all events in set S would be:

[nbtable] [tr] [td] [/td] [td] κ_i[/td] [td]   [/td] [td]   [/td] [td] × [/td] [td]   [/td] [td] [/td] [td] (1-κ_i) [/td] [/tr] [/nbtable]

So given, for example, events A, B, C, D, and E, with corresponding single-bet Kelly stakes of κ_A, κ_B, κ_C, κ_D, and κ_E, then the Kelly stake for the 1-team parlay consisting of only bet A would be:

κ_A * (1-κ_B) * (1-κ_C) * (1-κ_D) * (1-κ_E)

While the Kelly stake for the 3-team parlay consisting of bets A, B, and C would be:

κ_A * κ_B * κ_C * (1-κ_D) * (1-κ_E)

Much simpler, no?

Now if only I had used this logic in my old-school JavaScript Kelly calculator, it would run a hell of a lot faster. Well, c'est la vie.

If anyone's interested in the C-code for this (and or/a DLL linkable from Excel) let me know and I'll post it here. (Although as I've said, this is really more of a curiosity than anything else.)

[bztips]

I haven't reviewed ALL of Ganchrow's posts, but no doubt he is head and shoulders above everyone else here. Based on what I have seen however, I would agree 100% with your nomination. Only reason the post itself is not perfect is that I believe Ganch (probably intentionally) undermines it by saying the formula is mainly a "curiousity" . I actually use it ALL THE TIME when I'm betting multiple MLB games at the same time. (Full disclosure -- fractional Kelly of course, I don't have enough confidence in my EV estimates to bet the full boat.)

[Indecent]

Absolutely. The solution is as elegant as it is simple. I'm surprised I didn't notice this post when he made his return, but I'm glad you posted a reminder Data.

[Dark Horse]

Ganch is going Zen. I hope he'll write a book with pages such as the one quoted above. I'll read it in a quiet place.

[subs]

+1

but also appears to be a gentleman.

[Pokerjoe]

I like Ganch, and appreciate his contributions here, and the elegance of his reformulation, but the post, tho fun, isn't actually helping anyone make money.

The Kelly IDEA is extremely useful and important to know. You will not succeed in this game without it.
(Cue Trix Trix's response, something along the lines of "You're wrong, Pokerjoe, because the Kelly IDEA is extremely useful and important to know and you will not succeed in this game without it!, followed by Data telling us, yeah, also, Sacramento is the capital of California, so there!)

You might call this the Pokerjoe Uncertainty Principle: The Kelly FORMULA, apart from modeling the idea, is not very useful because no one ever knows their EV. You can at times estimate EV improvements, and Kelly can help you figure out how much to change your bet, generally, but it's a trap to think you have real precision.

Almost no one is dumb enough to think their EV calcs are accurate (actually most bettors aren't thinking in terms of EV, so I mean, of those who are), so they use a Kelly fraction. But as soon as you pull a fraction out of your ass, you're using the Kelly IDEA, not the Kelly FORMULA, even if you don't admit it to yourself. The formula, for fraction users, becomes a way to consistently articulate the Kelly idea, but don't kid yourself that you are being precise, or that you are gaining edge by using it.

Here's the reality:
You're standing in Vegas, admiring the neon signs (Data is; the rest of us, when in Vegas, admire the cocktail waitresses).

Guy steps up, sizes you up for being a typically not-nearly-as-bright-as-they-think math guy and says "Hey, got a proposition for you. How about we flip this coin, and if you get it right I give you 3-1."

Data goes "Yee, haw! How much can I bet?"

"As much as you want, my fine fellow."

Data laughs and says "Fool, I know Kelly!" He breaks out his calculator, punches in the numbers, and shows the Vegas guy the Kelly calculation. "This much?" he asks.

"Sure, buddy!"

Faux-sharps think they know their EV in that spot and use Kelly to figure out how much to bet. Real sharps say GTFO and go back to flirting with the cocktail waitress.

Data, you like Ganch's post because you dream of finding EV and plugging it into Kelly. So, Ganch's post is your porn. But you're putting the cart before the horse with this math. Find yourself some EV first.

And here's the real point of my story about the Vegas coin flipper: the greater your perceived edge, the greater the chance you're wrong.

If nothing else, if you are so numerically anal (I sometimes am) that you insist on using math where none is really needed, telescope your EV estimates by using the square root, not a fraction, such that an est. EV of 9% becomes 3% in your Kelly calcs, and so on.

I'm not even going to look back at this thread, LOL, because I can smell the emotionalism and non-reading coming. I'm still stunned by the number of people who read "The misunderstanding of BTCL" to mean "The error of BTCL." And then continued to use the phrase BTCL without understanding it.

But I do trust that Ganchrow is not the least bit offended by this post. He shouldn't be. It's not about him. Props to him for the insight that lead to the work that led to the post that led to this thread.

[TomG]

How >99% of the readers view the "best post of the decade"

[dippeli]

what is 1-team parlay?

[Socrates]

Good stuff.

[dvsbmx]

what is 1-team parlay?

^^^Post of the decade.

[trixtrix]

You might call this the Pokerjoe Uncertainty Principle: The Kelly FORMULA, apart from modeling the idea, is not very useful because no one ever knows their EV. .

congratulations on reinventing baye's theorem and placing your name on top of it.

[dippeli]

^^^Post of the decade.

lol

sry, we are not all experts. I just always thought parlay is >= 2 bets in one. Well, I play singles anyway so never mind.

[roasthawg]

I bet full Kelly on multiple events... never understood the logic (or math) behind lowering your bet for simultaneous events although I suppose it's intended to lower your risk of ruin. This post does little to clear it up for me at first glance although I haven't taken the time to try and fully comprehend the math behind it.

[bztips]

I bet full Kelly on multiple events... never understood the logic (or math) behind lowering your bet for simultaneous events although I suppose it's intended to lower your risk of ruin. This post does little to clear it up for me at first glance although I haven't taken the time to try and fully comprehend the math behind it.

The logic is straightforward: Just as the familiar Kelly formula is "optimal" for a single bet, the revised formula above is "optimal" for simultaneous events. "Optimal" means that it maximizes the expected growth in your bankroll.

[donjuan]

I bet full Kelly on multiple events... never understood the logic (or math) behind lowering your bet for simultaneous events although I suppose it's intended to lower your risk of ruin. This post does little to clear it up for me at first glance although I haven't taken the time to try and fully comprehend the math behind it.

So what happens when you have 40 simultaneous events with full kelly of 3% each?

[RickySteve]

So what happens when you have 40 simultaneous events with full kelly of 3% each?

3rd mortgage on the trailer, obv.

[Justin7]

I bet full Kelly on multiple events... never understood the logic (or math) behind lowering your bet for simultaneous events although I suppose it's intended to lower your risk of ruin. This post does little to clear it up for me at first glance although I haven't taken the time to try and fully comprehend the math behind it.

Pretend you find two amazing opportunities. A fair coin will be flipped twice. If it is Heads both times, you lose. Otherwise you win. The bet pays at even money. 50% EV, Kelly stake is 50%.

This event is being run at the same time. Before the first flip of the two events, you have to decide how much to bet on both. If you risk 50% on both, you have a 1/16 chance of losing your whole bankroll. If you do this every week for 16 weeks, you're likely to bust out.

If you're following Kelly staking diligently, there should never be any chance of busting out. Your strategy fails in this scenario. In more realistic scenarios, you are still overbetting, which will lower your long-term growth.

[Data]

3rd mortgage on the trailer, obv.

[roasthawg]

Pretend you find two amazing opportunities. A fair coin will be flipped twice. If it is Heads both times, you lose. Otherwise you win. The bet pays at even money. 50% EV, Kelly stake is 50%.

This event is being run at the same time. Before the first flip of the two events, you have to decide how much to bet on both. If you risk 50% on both, you have a 1/16 chance of losing your whole bankroll. If you do this every week for 16 weeks, you're likely to bust out.

If you're following Kelly staking diligently, there should never be any chance of busting out. Your strategy fails in this scenario. In more realistic scenarios, you are still overbetting, which will lower your long-term growth.

Ah, this was a good explanation... without diving into the math behind it this makes sense to me at least.

In actual practice I make all of my bets for the day first thing in the morning. Normally I don't have my entire roll bet out so it's tough for me to rationalize lowering my stake simply because I have two plays one day as opposed to one on another. Even when I do have my entire roll at stake it's spread over so many bets that the risk of ruin is essentially nil.