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The infamous coin toss

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Re: The infamous coin toss

#81

Post by kgwgk »

wodenokoto wrote: post #78

Off topic or at least sidetracked, this took me by surprise > For historical reasons, this average is also called the expected value Why is this “for historical” reasons?

Because long ago people started using a word he doesn't like.

https://web.universiteitleiden.nl/fsw/verduin/stathist/1stwo...

EXPECTATION. According to A. W. F. Edwards, expectatio occurs in 1657 in Huygens's De Ratiociniis in Ludo Alae (David 1995).

According to Burton (p. 461), the word expectatio first appears in van Schooten's translation of a tract by Huygens.

The two references above point to the same text as Huygens's De Ratiociniis in Ludo Alae was a translation by van Schooten. NB The word expectatio is used quite frequently throughout the text.

This is the Latin translation by Van Schooten of the first proposition:

Si a vel b expectem, quorum utriusque aeque facile mihi obtingere possit. expectatio mea dicenda est (a+b)/2

This is the Dutch text of Huygens' Van Rekeningh in Spelen van Geluck. This text was published in 1660 but already written in 1656.

Als ick gelijcke kans hebbe om a of b te hebben, dit is my so veel weerdt als (a+b)/2

The literal translation of the Dutch text is: If I have an equal chance to get either a or b, this to me is worth as much as (a+b)/2. There is no explicit mention of expectation only of value, but as the rest of the explanation of the first proposition is concentrated on the possible outcomes of a game of chance, expectation is implicitly around.

Expectation appears in English in Browne's 1714 translation of Huygens's De Ratiociniis in Ludo Alae (David 1995).

This is Browne's 1714 translation of the first proposition:

If I expect a or b, and have an equal chance of gaining either of them, my Expectation is worth (a+b)/2

Re: The infamous coin toss

#82

Post by nhinck »

kgwgk wrote: post #79
nhinck wrote: post #77

Earlier quoted context omitted.

The claim was: "Each round, the collective wealth goes up 5%, no matter how many rounds you run. " And this isn't true.

Do you mean that it's not true that "in one round, the collective wealth goes up 5%"? I could agree with that. (However, if you add "in expectation" it will be true for one round and for each round no matter how many rounds you run.)

For a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds.

Re: The infamous coin toss

#83

Post by sushibowl »

arc-in-space wrote: post #74
HelloNurse wrote: post #70

Earlier quoted context omitted.

In the long time limit, there are about the same number of heads and tails, and since all changes are multiplicative, the coin tosses can be permuted. We can thus divide the game into two sets of coin tosses: excess heads or tails that represent a negligible amount of good or bad luck, and many HT pairs, each of which shrinks EV by 0.9. Put another way, for each HH pair you should expect a TT pair, and a HHxTT or TTx…

Thanks, putting it this way does help me understand a bit better. So a large number of repeated fair tosses can be broken down into a sequence of win-loss pairs, which are always =0.9. The thing that still confuses me is, why the heck is the EV 1.05? It seems to be expressing something true - if you were to split your money into a thousand piles and "play" each individually, you make money overall.

The gamble is set up in such a way that there is a fairly large chance that you will lose money, coupled with a tiny chance of an incredibly large gain.

The article, just like the poster above you, characterizes a series of bets into win/loss pairs that add up to a 0.9 return per pair. There are lots of sequences that can be characterized this way. However there will be a few sequences that contain many, many heads and win a lot of money. There are of course also a few sequences with many tails, but their loss cannot decrease below zero so it is contained.

So it's a little bit like a lottery ticket, where the positive gains are extremely concentrated into a very small lucky group. The more rounds of the gamble you play, the smaller the lucky winning group gets, and the larger their wealth.

Re: The infamous coin toss

#84

Post by kgwgk »

nhinck wrote: post #82
kgwgk wrote: post #79

Earlier quoted context omitted.

Do you mean that it's not true that "in one round, the collective wealth goes up 5%"? I could agree with that. (However, if you add "in expectation" it will be true for one round and for each round no matter how many rounds you run.)

For a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds.

So you say that it's false that "in one round, the collective wealth goes up 5%", right?

However you agree that it's true that "in one round, the collective wealth goes up 5% in expectation", right?

(I agree that "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds ALMOST SURELY". The last bit is important though. Otherwise it's like saying that if you flip a coin enough times you will get heads - which is not exactly true.)

Re: The infamous coin toss

#85

Post by [deleted] »

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…

[deleted]

Re: The infamous coin toss

#86

Post by AdamN »

drekipus wrote: post #76
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…

> increasing wealth concentration is an unavoidable consequence of capitalism My bad I thought you made a coin flip simulation, not a capitalism simulation. At what point do we stop wagging the finger at the ghost of capitalism and just say it's mathematics? You don't hear people crying for the upheaval of mathematics

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, etc...)

Re: The infamous coin toss

#87

Post by geminger »

wodenokoto wrote: post #80
wmorgan wrote: post #28

This is the St. Petersburg paradox with an extra variable. In SPP, EV approaches infinity as the bank's resources approach infinity. Put bounds on the bank's resources, and you find that even with trillions of dollars your EV is less than $50. Here, not only are we assuming that the bank's resources are infinite, we're also assuming that the population is large enough that there are always enough lucky players to com…

It also made me think of the St. Petersburg paradox and I think the authors did too, as they call their toss “Peter”

He called it "Peters", after himself, which is off-putting.

Re: The infamous coin toss

#88

Post by nhinck »

kgwgk wrote: post #84
nhinck wrote: post #82

Earlier quoted context omitted.

For a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds.

So you say that it's false that "in one round, the collective wealth goes up 5%", right? However you agree that it's true that "in one round, the collective wealth goes up 5% in expectation", right? (I agree that "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds ALMOST SURELY". The last bit is important though. Otherwise it's like saying that if you flip a coin eno…

Yeah I would say "in one round, the collective wealth goes up 5% in expectation" but "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds" not almost, but absolutely.

Re: The infamous coin toss

#89

Post by kgwgk »

nhinck wrote: post #88
kgwgk wrote: post #84

Earlier quoted context omitted.

So you say that it's false that "in one round, the collective wealth goes up 5%", right? However you agree that it's true that "in one round, the collective wealth goes up 5% in expectation", right? (I agree that "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds ALMOST SURELY". The last bit is important though. Otherwise it's like saying that if you flip a coin eno…

Yeah I would say "in one round, the collective wealth goes up 5% in expectation" but "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds" not almost, but absolutely.

Maybe you're not aware but almost surely is a technical term - which makes the statement correct.

Just like you cannot be absolutely certain to get heads if you flip a coin enough times even though it will happen almost surely.

Re: The infamous coin toss

#90

Post by kybernetikos »

jstanley wrote: post #66
selcuka wrote: post #51

Earlier quoted context omitted.

No, it will approach zero like the individual one. Think of it this way: If everyone's individual wealth approaches zero, why would the total go up? Just run the following simulation in a python REPL: import random POPULATION = 100 INITIAL_MONEY = 1000 ROUNDS = 10000 wallets = [INITIAL_MONEY for _ in range(POPULATION)] for iteration in range(ROUNDS): for person in range(POPULATION): if random.random() > 0.5: wallets[…

Try it with a larger population and a smaller number of rounds. It turns out the mean gain actually is positive, but the median gain is negative.

I think that the effect is from the fact that you're sampling from all possible outcome history paths. To realise the positive expected value, your sample needs to include some of the increasingly rare possibilities where you gain enormously. This means that if your population is large compared to the number of branches (2^rounds) then you probably do sample that and the overall gain is positive. If your number of branches is large compared to your population, you probably don't sample the branches where you make lots of money, and your overall gain is negative.
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