This twatter thread blew my mind



Consider a coin flip bet: Heads you get 1.5 times your current wealth. Tails you get 0.6. Should you take it?

Expected value is 1/2*0.6 + 1/2*1.5 = 1.05 times your wealth. So take it right?


Averaging over many people, it's true that the expected value of wealth increases. But if you watch one person's wealth over time, it decreases!

If you work it out, you get sqrt(0.6*1.5) ~= 0.95 times as much wealth at each time step.

How can this be? Turns out, a few people get very rich taking this bet repeatedly, while most go broke.

In other words, the classic "expected value" framework has a built-in hidden assumption about how to count the outcomes of multiple people.

Google Kelly Criterion and read. Should clear things up for you.

To clarify, it won't clear up why someone would write such a tweet.

it's about brm, you're wagering far too much of your roll on a single bet

furthermore, given how much wealth is multiplied via traditional investing it's far worse to lose 40% than it is to gain 50%

If have 2 guys who all invest a flat rate of 6% annual returns
A starts with $60
B starts with $100

It takes 9 years for A to recoup losses to hit $100
It takes 7 years for B to hit $150

So in terms of traditional investing, it's losing 9 years vs gaining 7 - horrible deal

indeed, thanks

something clicked for me, and there are indeed economical and ethical implications.

yeah, it's brm indeed, i never thought about it that way though. Also, i didn't realize that if you wager everything and run average, you actually go broke pretty quickly.

There are two expected values for the individual to consider if he makes this bet repeatedly. One is the expected value of his bankroll. Since the EV of each bet is positive, this expected value must be positive. If it wasn't it would open the possibility of betting systems that turn negative EV bets into positive ones. There are none.

The other EV is the expected value of the "Growth Rate" of the bank roll. The Growth Rate of the bankroll is a random variable, say Rn, where n is the number of iterations of the bet. Since in this case you're betting the entire bankroll each time, Rn is calculated as the solution to the equation:

Bn = B(1+Rn)^n

where B is the original bankroll and Bn (a random variable) is the bankroll after n bets.

You can think of Rn as the constant rate of interest you would have to earn on your bankroll, compounded n times, to get the random result of n bets. So Rn is random depending on the outcome of n bets.

Since Rn is a random variable you can calculate its expected value. In this case EV[Rn] is negative. That means that after n bets, for n large, you can expect the highly likely outcome of your n bets to produce a bankroll that behaves as if it was being compounded down by a negative interest rate bet after bet.

The disparity between the two EV's is in the difference between the highly likely outcomes and the progressively less and less likely outcomes. If you look to do this a predetermined number of numerous times N it's like spending your entire bankroll on lottery tickets for Megabucks where the Megabucks running jackpot has grown so huge as to make buying the tickets positive EV. You have a small chance of getting very very rich and a large chance of going broke.

This is why the Kelly Criteria was invented. Kelly realized that the important thing for a professional gambler is to maximize the EV of Rn rather than Bn.


PairTheBoard

...isn't that obvious?

Post this as a prop bet in a strategy forum where you can find many people who claim to be willing to accept any amount of variance for positive EV. Add in a requirement that the bet must be repeated a large number of times. Maybe you can find some takers.

I'm not an economist and I'm obv not too well-versed in probability theory.

Is the overall economy comparable to a mtt staking stable where the +5% EV stakees are forced to play until they are out of makeup? This suddenly seems like a really exploitative scenario.

If so, the math seems to make a case for higher taxation of the super-wealthy because a lot of them have been on the receiving end of EV-based economic models, right?

oh man... please don't politicize this

and for the record, higher tax rates on the wealthy for the sole reason that they have more is just tyranny spouting from sour grapes

there are many viable arguments for having higher rates of taxation on the wealthy but "because you have more money than me" isn't one of them

This bet is good if your total "wealth" is like 1-2-5k and you make 5-10k per month anyway.

It is not at all good if you repeat it many times or if you have substantial wealth.

If you repeated many time you are typically going to end up (1.5*0.96)^n=0.96^n times your initial wealth where n is the number of times you did it. So typically you lose your wealth that way.

But imagine you have 200k in liquid assets and you wanted to go to 300k or drop to 120k in a 50-50 bet. Obviously there is such a thing as utility of money (and it is not a linear function of money) and the move from 200 to 300 is not as significant in its consequences as the move from 200 to 120.

That is only a good idea if you replenish your wealth constantly because it is so low. Like i would gladly take that risk with all the money i make every month for a few months (if i could hide my true wealth and appear bankrupt) but i am not going to be excited about it either to be fair because over a year you still stand a decent chance to be down by 1-2 monthly wages and are only expected to make on avg 0.5 wages per year.

Imagine if you hid your money and claimed to be worth only 5k and you make 5k per month.

See it that way too. Imagine you did that every month of the year and you started with 5k and are saving 1k per month after expenses.

At the end of 10 years the chance to not have lost any money is only 27% by simulations. Now that is ridiculous management. Sure the chance to have 50x the money saved and initial capital this time is like 2.5% which is why the EV is still positive.

You do not want to invest in your life in such a way that the majority of the time you are broke only so that some small fraction of the time you are millionaire.

As a one time thing its ok only if it opens the door to a lucrative opportunity like say something happens at 300k like the down payment to an amazing real estate deal or the money needed to save your small company from default knowing that without it you end up winning in a few years substantial profits typically.


The other reason one may want to do that is because it offers a chance that is say 0.2% to end the year if done every month with 1.5^12~130 times their money. So if one wanted to reach a very advanced position quickly it gives them a way to do it that is plus EV. So you start with 10k and you end with 1.3 mil. Now you can do things that you would never do the slow way. So in that sense if one is near bankrupt and earns money every month they are ok to do it if they want to reach something remarkable and they have only a small chance to get there but its not horrible and still the rest of the time they are ahead often ie say if you did it for 12 months you would have often 8wins vs 4losses results and be up 3.3x ie a good volatility opportunity.


The correct way to do it however is not with all your money.

Look how to optimize this;

You start with K capital and you risk x fraction of your money in a 50-50 1.5 vs 0.6 bet.

You do it many times. You will have typically after n attempts where n is very big K*(1+x*0.5)^(n/2)*(1-x*0.4)^(n/2).

So you want to optimize (1+x*0.5)*(1-x*0.4)=-0.2x^2+0.1x+1 which obviously is optimized (see derivatives or vertex of a parabola) at x=-0.1/2/-0.2=0.00227 yielding an avg growth rate per event of 1.000225969.

So if you risked only 2.27% of your wealth in this bet you would get optimal avg growth rate of 0.025% per trial.

If you can do that every day you would have after a year 8.6% return on your money slightly beating the avg stock market index or real estate ideas. But if you have to do it less frequently you have better usage for your money elsewhere. Compare with doing it for all your money every day of a year which in order to break even requires you to win near 56% of these 365 bets when expected is 50% or 2.3 standard deviations more than expected.

Made a stupid calculator typo here that propagated as wrong number ie

the maximum is at x=-0.1/2/-0.2=0.25 yielding an avg growth rate per event of 1.25%

So if you risked only 25% of your wealth in this bet you would get optimal avg growth rate of 1.25% per trial.

If you can do that every day you would have after a year 93.2 times your money or if you did it once a month 16% growth per year beating the avg stock market index or real estate ideas. Compare with doing it for all your money every day of a year which in order to break even requires you to win near 56% of these 365 bets when expected is 50% or 2.3 standard deviations more than expected.


Let me use this stupid typo as opportunity to make something important noticed here. Lots of day traders that generally pick good trades lose money this way typically, although they are supposed to be correct in choices because they go on margin for all their positions on any new idea and put stop limits. So instead something smaller like 20% or 25% here or in options if you play to double or go bust something like 2p-1 where p is your win probability (if p is not 0.5 and its an even +100%win -100% loss risk) is way better. So if you have winning trades 60% of the time in the +100% or -100% model you must risk only 2p-1=20% of your position.

good stuff and humbling to read, i honestly feel very dumb and uneducated reading it

No. I mean, you technically can compare them in the same way as you can compare a lizard and a cheeseburger.

People naturally recognize this idea when they knowingly make -EV bets by paying insurance premiums to protect from catastrophic injury to their wealth. The insurance company can take these bets because its bankroll is huge compared to the payoffs they will have to make.

However, there's a problem for the insurance company if they overlook the chance the bets are correlated in the case of a black swan event. This is what happened in 2008 when AIG and others sold Mortgage insurance. Turns out they were picking up nickels in front of a bulldozer.

PairTheBoard

You could, instead, use the Kelly Criterion. *It maximizes growth rate. For OP’s example, it gives a bet size of 10% of current bankroll and bankroll geometric growth rate of 0.39920318409% per bet.

This, over 365 bets, this takes a bankroll (on average) from $1000 to $4281.01 while completely eliminating all possibility of busting. *This is quite a bit better than the stock market.

If you consider that insurance companies don't have agency (and therefore have no problems) and executives and investors and (some) workers have agency, then it makes a bit more sense.

pairtheboard is very insecure people won't be sure he's the one who wrote the stuff down

I sometimes sign "-adam" to the end of my posts if i want to personalize it a little or come off as a little more genuine but just don't see the EV is signing with your random and anonymous handle

What i did is the essence of Kelly criterion.

https://en.wikipedia.org/wiki/Kelly_criterion

The result Kelly gives is identical to the result i got not 10% but 25%, its higher than what you have there.

P/a-q/b or here .5/.4-0.5/0.5=0.25 same as i got the way one understands what Kelly is all about. I just wanted to show how to get there.


In a random small sample of bets you will not get the expected result of course in any dependable way but over time the fluctuation around it will not be significant making the avg result the target for maximization because you will be very close to it the vast majority of time.

I did exactly the Kelly thing. How did you get 10%?

There's a thread about this in ATF. I've been doing it since 2003. It used to be more popular. Mason still does it.


PairTheBoard

can you link to it, in all seriousness i'd be interested to know the science behind it

Read the second example in the "statement" section. It is the special case that is the same as OP's example.

0.5 - ((0.5 * 0.4) / 0.5) = 0.1

Look a little lower just before Bernoulli in the proof section.

P/a-q/b=0.5/0.4-0.5/0.5=0.25

https://forumserver.twoplustwo.com/5...luded-1189913/

PairTheBoard

According to this Kelly Criterion Calculator;

https://www.albionresearch.com/kelly/default.php

If you're given 5-4 odds on a bet with 50% chance of winning you should bet just under 10% of your bankroll ($99 on a $1000 bankroll). "Your fortune will grow, on average, by about 0.62% on each bet."

The reason it's not exactly 10% may be because you're limited by a minimum bet of $1. I don't know.

I'm puzzled by the following statement on that linked page:

"Assuming that your criterion is the same as Kelly's criterion — maximizing the long term growth rate of your fortune — the answer Kelly gives is to stake the fraction of your gambling or investment bankroll which exactly equals your advantage."

I believe that is just a rule of thumb approximation. In this case the advantage, or edge, is 12.5% if my great math abilities have not fouled up this simple calculation.

For a $100 bet at 5-4 odds your expected result is to win $125 half the time and lose $100 half the time. So you expect to win $12.50 per $100 bet on average. That's a 12.5% edge (or "advantage).

However, if you bet the rule of thumb 12.5% of your bankroll each time you will be over betting Kelly and reducing your expected growth rate to less than its maximum - but not necessarily negative. If I recall correctly the expected growth rate turns negative at twice the proper max Kelly percentage.


PairTheBoard