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The infamous coin toss

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Re: The infamous coin toss

#151

Post by kgwgk »

Consider this alternative bet:

  heads +100%
  tails -75%
The expected value is positive: 12.5%

If you play five times in parallel you're as likely to make money as to lose money but when you gain you gain 45% on average and when you lose you lose 20% on average. Overall, the expected gain is 12.5%.

If you play five times in sequence you're unlikely to make money (less than 20% probability) but then you make a 300% gain (four heads) or even a 3100% gain (five heads). Overall, the expected gain is 12.5% compounded five times which is 80%.

As the number of coin flips grows it is less and less likely that you lose if you do it in parallel (the expected gain is 12.5% and the distribution of returns converges to that value) and more and more likely that you lose if you do it in sequence (the expected gain is again 12.5% compounded but the distribution of returns is an increasingly skewed lognormal).

Exercise for the reader: take this interesting but simple and well-known fact and make a career out of it from repeating it again, and again, and again, and again, and again, and again, and again...

Re: The infamous coin toss

#152

Post by Fredkin »

anonymoushn wrote: post #136
robertlagrant wrote: post #133

Earlier quoted context omitted.

You eat the rich and you're hungry again in an hour. And now there's no food.

Man Elon Musk is personally farming soooo much corn every day! Wow!

He may not be personally running a farm. But if much of his wealth is invested as is usually the case for the richest people (no way it's just sitting around as cash), then he's indirectly supporting many productive endeavors. Even if it was all just cash deposited in a bank, it's not doing nothing. Farmers and others need loans, banks need deposits (well ... before QE)

Re: The infamous coin toss

#153

Post by cubefox »

olooney wrote: post #7
didgeoridoo wrote: post #5

The +50% / -40% is cleverly chosen, because it seems like the bet is weighted toward the gambler if you’re just using a naïve expected value. However, if you were to make it “double your money” (+100%), it would become clear that the only fair downside would be “halve your money” (-50%). For these values, the “trick” becomes much more obvious: that increases in repeated games need to be far greater in percentage term…

Yes, this is called the Kelly Criterion[1] which tells us we should optimize the expected value of the log of wealth, not the expected value of wealth directly. On a log scale, it's clear that doubling and halving have the same magnitude but opposite signs so are perfectly balanced. [1]: https://en.m.wikipedia.org/wiki/Kelly_criterion

Yes. Though the logarithms aren't essential here. It really is about maximizing the "geometric expectation", the expected value calculated by using the geometric mean instead of the usual arithmetic mean.

Arithmetic mean: sum all values, then divide by the number of values.

Geometric mean: multiply all values, then take the nth root, where n is the number of values.

Re: The infamous coin toss

#154

Post by haktai »

jstanley wrote: post #8

I think the part about the misalignment between the individual and the collective is basically wrong. In the given game, the collective loses money just as the individual does. The collective wealth is the summation of the individuals' wealths, and both the collective and individual wealths drop over time. Write a simulation and try it if you don't believe me. I did, because I couldn't work out how the collective wea…

[dead]

Re: The infamous coin toss

#155

Post by n2d4 »

svnt wrote: post #119
hgsgm wrote: post #117

Earlier quoted context omitted.

Why do you agree with something plainly wrong than you disproved?

They are saying parent is correct that plus 50% and minus 50%, while appearing to be balanced, is a losing proposition for the individual. They are also saying parent is not correct in saying that it also loses on a population basis, showing that the average across a population is still winning.

Yes, that's what I meant. You probably don't want to take unnecessary risk by going +50%/-50% markets, but it's not for the reason the parent mentioned in the conclusion.

Re: The infamous coin toss

#156

Post by shove »

Fredkin wrote: post #152
anonymoushn wrote: post #136

Earlier quoted context omitted.

Man Elon Musk is personally farming soooo much corn every day! Wow!

He may not be personally running a farm. But if much of his wealth is invested as is usually the case for the richest people (no way it's just sitting around as cash), then he's indirectly supporting many productive endeavors. Even if it was all just cash deposited in a bank, it's not doing nothing. Farmers and others need loans, banks need deposits (well ... before QE)

Hahahahaha

deep breath

Hahahahaha

Re: The infamous coin toss

#157

Post by robertlagrant »

shove wrote: post #139
robertlagrant wrote: post #133

Earlier quoted context omitted.

You eat the rich and you're hungry again in an hour. And now there's no food.

Yes we rich-eaters have been surviving on a mono-culture of rich folk and will surely starve without the rich who sacrifice themselves selflessly for our sustenance.

Most people are rich because they own stock that's valuable, because they made or did something that's valuable.

Collapse that chain and you replace value creation as a form of mild power with political ability as a way to access direct and high levels of power and things start going wrong fast[0][1][2].

[0] https://www.britannica.com/topic/Stalinism [1] https://en.wikipedia.org/wiki/Great_Leap_Forward#Consequence... [2] https://en.wikipedia.org/wiki/Venezuela#Suspension_of_consti...

Re: The infamous coin toss

#158

Post by joncrocks »

ano-ther wrote: post #143
rerdavies wrote: post #64

Huh. So I wrote the code, and ran the simulation. Now I get it. Investors: 100,000 Iterations: 100 Average worth after 100 iterations: $83.923 Average net worth increases. However the distribution of wealth is skewed dramatically. Winners: 13,704 (net worth of more than $1 at the end) Investors worth What That Guy was worth (the investor who made the most money): $1,171,830.00 He flipped 71 heads and 29 tails. Median…

> Average wealth increases; but average log(wealth) decreases What’s the mathematical rationale (or intuition) for looking at the log values? I don’t quite understand what they represent.

It's reducing the impact of the outliers.

e.g.

x = 60, 10, 10, 10, 10

avg. x = 20

avg log(x) = 1.1556 (which is 10^1.1556 = 14.3)

Re: The infamous coin toss

#159

Post by robertlagrant »

Fredkin wrote: post #152
anonymoushn wrote: post #136

Earlier quoted context omitted.

Man Elon Musk is personally farming soooo much corn every day! Wow!

He may not be personally running a farm. But if much of his wealth is invested as is usually the case for the richest people (no way it's just sitting around as cash), then he's indirectly supporting many productive endeavors. Even if it was all just cash deposited in a bank, it's not doing nothing. Farmers and others need loans, banks need deposits (well ... before QE)

Almost all of his wealth is in stock. Start taking that and it becomes valueless, because things that are legal to steal at any moment are not worth buying.

Re: The infamous coin toss

#160

Post by waldothedog »

ta1243 wrote: post #71
rerdavies wrote: post #64

Huh. So I wrote the code, and ran the simulation. Now I get it. Investors: 100,000 Iterations: 100 Average worth after 100 iterations: $83.923 Average net worth increases. However the distribution of wealth is skewed dramatically. Winners: 13,704 (net worth of more than $1 at the end) Investors worth What That Guy was worth (the investor who made the most money): $1,171,830.00 He flipped 71 heads and 29 tails. Median…

Or you could redistribute wealth to even out the winner-takes-all part while still encouraging merit of some sort by having winner-takes-some

Like a large(r) tax on upper incomes and capital gains??
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