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The infamous coin toss

ergodicityeconomics.com

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Re: The infamous coin toss

#171

Post by sobriquet9 »

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…

> The only solution: eat the rich.

The only? Maybe not bet 100% of your wealth every time? That's what Kelly criterion is about.

Re: The infamous coin toss

#172

Post by kgwgk »

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.

Multiplication becomes addition after you take logs. That makes many things easier to understand and manipulate.

Exponential trends become linear.

The geometric random walk where you multiply by X or divide by Y with equal probability becomes an additive random walk where you add log(X) or substract log(Y) and you can represent that as a standard random walk (where the up and down steps have the same magnitude and probability) plus a trend.

Re: The infamous coin toss

#173

Post by cubefox »

hgsgm wrote: post #114
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…

No, the trick is less obvious. It's even more shocking that a hugely higher expectedly value upside (+100% , -60%) ((2x+.4x)/2 = 1.2x) is still losing long-term.

We need the geometric expectation here: sqrt(2x * .4x)=0.89x

The arithmetic expectation is wrong in this case.

Re: The infamous coin toss

#174

Post by n2d4 »

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.

The other people gave good intuition on why the logarithm is good but not on why the logarithm is the best choice. For example, max(sqrt(EV)) would also reduce outliers.

There is a more mathematical way akin to the original derivation of the Kelly criterion [1]. Let's compare two strategies, max(EV) and max(log(EV)). Our game is coin-toss with heads=+50%/tails=-40% payouts and infinite repetitions; max(EV) would always go all-in, while max(log(EV)) would only bet a fraction of its current wealth at each turn.

- After the first turn, there are two cases; if heads, max(EV) has a higher payout, if tails, max(log(EV)) has a higher payout. On average, max(EV) has a higher payout because it went all-in, and there's a positive EV per toss.

- After two turns, there are four cases; in one of them (heads-heads), max(EV) has a higher payout, and in the other three, max(log(EV)) does. The latter is hence the "safer" choice. However, in the 25% chance that max(EV) wins, it has significantly more money than max(log(EV)), so the average (not median) payout is higher for max(EV).

- This repeats, and after more and more repetitions, the chance that max(log(EV)) has a higher payout converges towards 1. Note that the payout for max(EV) in the (very!) rare case that it gets lucky skyrockets at an even faster speed, so on average, max(EV) still has a higher payout.

- Turns out this is not just true for max(EV) and max(log(EV)), but in fact max(log(EV)) always maximizes the chance of higher payouts, given enough repetitions.

So, in other words, max(EV) maximizes the average payout, while max(log(EV)) maximizes the chance of having a higher payout than any other strategy. The original Kelly paper has more precise maths, but I hope this helps to guide you.

[1] https://en.wikipedia.org/wiki/Kelly_criterion

Re: The infamous coin toss

#175

Post by TheCleric »

robertlagrant wrote: post #157
shove wrote: post #139

Earlier quoted context omitted.

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...…

> Most people are rich ... because they made or did something that's valuable.

This assumption is doing a LOT of heavy lifting here.

Re: The infamous coin toss

#176

Post by jncfhnb »

amelius wrote: post #166
AdamN wrote: post #86

Earlier quoted context omitted.

People do say that. Usually the critique of (perfect) capitalism is that it (structurally and mathematically) skews towards increasing wealth for winners at the expense of the rest of the population and that this is inevitable. Then the response of advocates against this situation is to either commit to more structures to prevent that outcome towards diluted (non-perfect) capitalism or other social forms (socialism,…

> skews towards increasing wealth for winners at the expense of the rest of the population The usual response against that is: what is wrong with being a winner? I did something right, let me have my well-deserved money. The problem with someone getting well-deserved money is that the money starts working against everybody else, so essentially this money gives everybody else negative money.

Many times your money is not well deserved. Many times it’s due to luck, inheritance (see: luck), or dirty tactics.

Regardless, the problem is very rarely that you “earned” money. Almost everyone is fine with you getting wages or perhaps entrepreneurial income.

The problem is that wealth allows people to become more wealthy without really doing anything. Wealth compounds so that the rich get richer. And often the risk of financial loss is so low in a diversified portfolio that it’s basically a guaranteed win. See economic rent.

IMO what people want is neither an income tax, nor a wealth tax, but an income tax rated on wealth. Wealth does get created over time, but despite this, the wealthy also take a continually larger share of the total pie. Society should be the one to get the bulk of passive wealth accumulation gains. Not individuals who happened to be wealthy.

If you have a billion dollars, your income tax rate should be like 99%.

Re: The infamous coin toss

#178

Post by robertlagrant »

vharuck wrote: post #121
robertlagrant wrote: post #110

Earlier quoted context omitted.

Unless you're paid by the word, it'd be better to skip all that and state your third option.

Media could be owned by the staff. That's what the Hell Gate website does. https://hellgatenyc.com/about-us We could also restrict ownership by audience size. So any one person or non-media organization cannot own or control multiple media organizations if their total audience size is above X people. Audience size would be prorated for partial ownership or control (i.e., owning 1% of a newspaper that serves 100K peop…

> Media could be owned by the staff

Agree. That would be private people owning it, though.

Re: The infamous coin toss

#179

Post by dontlaugh »

LeonB wrote: post #107
shove wrote: post #106

Earlier quoted context omitted.

Marx was specifically talking about industrial production and made that pretty clear in his work. “It’s far too low resolution a phrase” when purposefully taken out of context. If “the means of production” applies to the factory owner and the plumber equally, I’d counter its not at all a useful phrase for “teaching people to hate another group of people” because those aren’t at all the same type of people.

Agreed. I have friends that are plumbers and friends that own factories. Both plumbers and factory owners do indeed have opportunities to exploit people. But my plumber friend can only exploit one customer at a time, where as a factory owner multiplies that ability. Multiplication is a powerful operator.

The plumber generally exploits one or two employees. But otherwise your point stands.

Re: The infamous coin toss

#180

Post by eli_gottlieb »

sobriquet9 wrote: post #171
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…

> The only solution: eat the rich. The only? Maybe not bet 100% of your wealth every time? That's what Kelly criterion is about.

You have to have enough in the first place that trying to support yourself isn't betting 100% of your wealth.
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