Define "the end of the universe".

What happens in a trillion years. Maybe he thought I meant forever.

I just ran few some simulations of bankroll equation.

R=R0*(1+xb)^(np)*(1-x)^(nq)

x is the proportion of your roll you use per bet and n is total number of bets. Our aim is to make R as big as possible after n bets, if you do that you have also made the bankroll growth as big as possible, the x that does this is the kelly proportion.

maximizing my bankroll growth would be nice, one of my biggest difficulties is knowing how accurately ive predicted p.

Im only going to look at bets where the probability of winning is 0.5 and I get paid at odds of b to 1.

The roll equation becomes

R=R0*(1+xb)^(n0.5)*(1-x)^(n0.5)
If I want my roll to give non negative returns I need,
(1+xb)^(n0.5)*(1-x)^(n0.5)>=1
which gives,
x<=(b-1)/b

kelly proportion when p=0.5 with odds of b to 1 is

x=0.5*(b-1)/b

So if you bet more than double the kelly proportion you will get negative roll growth.

They'll all go broke, and a trillion years is plenty of time for that to happen. Adding a googolplex losses together does not equal a combined profit.

IRL, there is a minimum wager, so there is a RoR even when kelly-betting (not that this is the reason you're wrong, but it might help you see that you are). Define the ruin point as some small fraction y of your initial bankroll where you're no longer able to place another wager.

There is a formula for the probability of ever reaching y when betting x*f. When x=1, you're kelly-betting and the formula simplifies to p = 1/y.

The formula is P(reach y) = y^{(2/x - 1)}

Notice that when x=2, the probability is 100%. The formula returns >100% for x>2, which really means 100%.

So if each of the googolplex participants has a 100% chance of going broke, that means every single one will go broke.

But even if there were no minimum wager (meaning no such thing as "broke"), they'd all be approaching 0 forever, which still isn't profit. They'd reach y, then they'd reach y^2, and so on until the universe ended.

If you have a bet that is paying 1:1 odds and has a true win probability of 0.52. That would lead to a kelly bet of 4%. But if you incorrectly thought the win prob was 0.54, your incorrect kelly would be 8%. Incorrectly thinking true win prob is greater than 0.54 would lead to negative long run growth.

If you are handicapping NBA games that would be about 0.75pts or so away from the true line or about 2pts off in an AFL game. Tight margin!!!

So a punter sees an line at -8.5 paying even money, he runs his model and thinks line should be at -6.5, it turns out his model always overestimates win margins and the true line is -7.5. Genius model builder bets full kelly and is rewarded with a negatively growing bankroll. He cant figure how he is right so often but his bankroll is always shrinking.




If you have a bet that is paying 10:1 odds and has a true win probability of 0.125. That would lead to a kelly bet of 3.75%. But if you incorrectly thought the win prob was 0.16, your incorrect kelly would be 7.6%.

which mistake is harder to make;

thinking occurrence is 54% when it is actually 52%
or, thinking occurrence is 16% when it is actually 12.5%

you would need to be pretty confident in your modelling to go full kelly.

How many people out there would have developed a +ev handicapping system, which unfortunately consistently over estimates the chance of winning. They would be losing punters using a winning system with bet sizing chosen by kelly!!

Fmp.

Exactly, which is why one shouldn't bet full kelly in sports-betting or financial markets, where one never actually knows the true kelly.

Are lot of bettors would bet quarter Kelly where they they don't know the true probabbility for say a sports game, but would that still be considered Kelly? as its sort of like telling a chef who uses a full cup of a ingredient which is the "main ingredient" for a recipe, but to now use a quarter of that same ingredient.

It wouldnt be kelly as it doesnt maximise growth.

Their combined bankroll will be more than they started with even if they each made a bet a second and bet everything they had each time.

Ok I think you're right about that scenario.
Since it's only a trillion years, 1 in 2.5^(3.1536 E19) people won't go broke. They'll have hit the lottery and their wins will be enormous (the part I overlooked) - enough to compensate for all the other people's losses. This is another way of saying that the game is +EV, that 10^19 << googolplex, and that one trillion <<< infinity. Give them more time and they'll all go broke.

Remind me how this is supposed to relate to an individual managing his own bankroll? In one trillion years, an individual's chance of going broke would be 99.999...% with I can't even count how many 9's. The fact that if there were enough people doing the same thing, one of them wouldn't go broke, doesn't help Hero (who will go broke). If there were enough people or enough parallel universes, someone would jump off a bridge without falling; it's still correct to say, "If you jump, you will fall."

I stand by my statement that you will lose money by betting >= double-kelly. The only way not to is to essentially win a lottery. Why play a very expensive lottery (with possibly way worse chances than any actual lottery) when you can make bets that reliably grow your bankroll indefinitely?

Another thing to realize is that with geometric (fractional) betting, EG is the new EV because the geometric mean is what matters rather than the linear mean.

If you wanted to be super technical, you could maximize the growth for a finite number of wagers and the result would be slightly more aggressive than Kelly (but certainly not double-kelly unless you're making like two wagers ever).

I went through a very similar argument on this forum about ten years ago. The problem I had was that your words, until you corrected yourself, could be taken to mean, by others who are not following closely, that a basket of all good bets could somehow result in a basket whose average value was less than what you started with. I cannot allow such thoughts to enter the mind of a two plus twoer.

Fair enough, but I think it's also dangerous to refer to those bets as "good".

There are two things happening simultaneously: positive EV's are being added to form a bigger +EV, and negative EG's are being added to form a bigger -EG. The bets are "good" in that the first thing is happening, but bad in that the 2nd thing also is. For an individual person, the first is imaginary (they'll win imaginary S-bucks while losing real currency) while the 2nd is real.

I think the issue is the bankroll equation gives you the most likely outcome, it doesnt give you the EV of the series of bets.

if you take two bets each 50% win chance and each paying 1.1:1 odds

the most likely outcome would be one win and one loss

it gives a payoff as per the bankroll equation

B=B0*(1+k*1.1)*(1-k), this is made as big as possible by choosing k to be Kelly, 1/22 of your roll
So your payoff is most likely to be, B=B0*441/440=B0*1.00227

The EV of such a bet would be

EV=sum prob(X)*X

EV = 1/4*(B0*(1+x*1.1)^2 + 2*B0*(1+x*1.1)(1-x) + B0*(1-x)^2), this is made as big as possible by choosing x to be as big as possible.

So to maximise the return of the most likely sequence of outcomes, you use Kelly. To optimise your EV you choose bet proportions size to be as big as possible.


If you extend it to 100 bets, using Kelly, there is a 38% chance of getting negative returns, that is with 48 or less wins. With an EV of 1.25*B0

if you bet double kelly 46% of the time you will experience negative returns, but your EV will be 1.57*B0.

if you bet 50% of your roll on each bet, you experience negative return 99% of the time, but your EV would be 11.81*B0



If you are the banker of a game where the player wins 50% of the time and gets paid at 1.1:1 odds over a large number of games

Would you allow the player to bet
a) 0% of his roll
b)over double kelly of his roll?

I wonder if multiplying a series of growth factors together, where the growth factors come from a normal distribution with a negative mean, Could lead to +EV returns.

I guess they can, thats why betting over 2*kelly gives positive EV and negative growth.

If the banker is infinitely rolled, then (b) because it's +EV for the banker. If the player bets 100% kelly, the banker is martingaling, which is +EV with an infinite BR (putting aside that you can't get any richer than infinit-aire). If the player bets something like triple-kelly, then it's the same concept just to a lesser extreme.

So actually I wasn't precise enough in my last post. In David's scenario, the bets are +EV only because the time is finite. With infinite time, both the EG and EV are negative.

Oh but if the banker has a finite bankroll, I'd have to say (a), which is interesting because somehow neither party should want to play the zero-sum game. Weird! I may have to ponder that a bit.

Another situation is infinite bankroll but accepting wagers against an endless stream of players, which is the same as taking bets from one infinitely rolled player. That would be another case for (a), since the players as a whole would not be over-betting.

No, betting an infinite number of bets where you are betting more than kelly would lead to both negative growth and positive EV.

Negative growth is obvious.

if you bet say 2 bets, 50% win , paying $1.1:1, betting half your roll each time.

you win 2 1/4, end at from a $1 roll (make $1.4025)

win 1 lose 1 1/2 (lose 0.225).

lose all 2 1/4 (lose 0.75 )

Clearly + ev, but as kelly optimises growth you lose when you obtian the expected outcome of 1 loss one win

if you placed three bets instead of two, the above would still hold. As you continue to increase the number of bets the EV will never become negative.

If you placed a large number of 50/50 bets paying 1.1:1 betting 50% of your roll, you getting them all right will magnify your roll many times over, you would have an equal chance of losing them all and you would never even lose your entire roll.

That's true until you "reach" infinity. This was discussed before on here ad nauseam with regards to Martingaling, and the banker here is reverse-martingaling. Something something measure theory, but basically, with infinite time, the banker has a 100% chance of getting all the player's money. If the player is infinitely greedy, they have a 100% chance of losing their BR and a 0% chance of winning anything, which is a -EV proposition.

What were you guys discussing previously?

Proportional betting? Or proportional bet exceeding double kelly gives both neg growth and -ev?

Still a bit unhappy with your wording. The bets that are somewhat bigger than what you would call "good" are fine unless they are available to you indefinitely, you have no other source of income, and you have signed a contract that once you specify their proportional size you can never back down.

I don't know enough about transfinite arithmetic to know if this is correct or not. But there was available to you very strong evidence that the answer depended on finite vs infinite. How did you miss it? I speak of course of Tom Cowley's post. He wanted to be able to say that the bad guy was wrong but was going to let the good guy slide.

A -EG bet portfolio isn't a good bet portfolio. Heehaww's statement wasn't exactly right (a -EG bet can be fine if you have a strong enough expectation of being able to hedge after you make it), but it's close enough for the subject at hand.

I don't know what your finite/transfinite stuff is about though. Pick an e as small as you want, and there's a corresponding finite number of bets after which 10^10^100 people will be down money collectively with probability > (1-e). It's far larger than 10^19 bets apiece though obviously.

That martingaling on -EV wagers becomes +EV when both your bankroll and the betting limit are infinite. This means that for the house, reverse-martingaling on +EV wagers becomes -EV under those conditions. That's not the same thing you asked about, but I strongly suspect the same thing happens.

It's not transfinite arithmetic, it's a post-grad topic of calculus called measure theory, which I don't know about either.

I knew each player would be almost guaranteed to go broke, but neglected to consider whether any wouldn't win the lottery, given a googolplex people. Once I looked at the orders of magnitude involved, it was obvious why you chose a googolplex. In your first post, I read "end of universe" as "infinite time". When you specified a trillion years, I figured for an individual person, that's about as good as infinite time (which it is), and didn't think further. Shame shame, I know.

Good point

A lot of people take measure theory as undergrads (I didn't). Transfinite arithmetic seems to be closely related to measure theory if not the same thing (somehow I have never heard the term before, but a quick search tells me that Cantor coined the term leading me to believe it is closely related to measure theory).

https://clio.columbia.edu/catalog/4308919?counter=2

Ah I can be wrong, I didn't think comparing cardinalities had much to do with, say, calculating a Lebesgue integral. But measure theory covers much more than problems like this.