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The infamous coin toss

ergodicityeconomics.com

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Re: The infamous coin toss

#61

Post by jacknews »

kevinwang wrote: post #50
throwaheyy wrote: post #47

Earlier quoted context omitted.

Over time it does not. A related concept in finance/trading is “drawdown”. A single trade can have a positive expected value. But over time, if you take a loss you have to get a bigger win to end up back where you started, because you have less capital to work with.

> Over time it does not. Yes, the article shows that almost certainly, any individual's wealth will approach 0 from repeatedly taking this gamble. However, the comments I replied to say: > But this is purely a result of the distribution of returns from a single toss. which I don't understand.

It will only approach zero because you lose more than you gain.

If the loser got 0.6666c instead of 0.6c, and the winner got $1.50, then over time you'd break even, on average.

And yet the expected return would apparently be 1.08333. If think the conclusion is that 'expected return' is a fallacy, you just can't add proabability-outcomes in this way to get an 'expected outcome'.

Re: The infamous coin toss

#63

Post by DarkNova6 »

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…

Confusing the equivalence of -50% and +50% is very common. In fact, many video games provide wrong bonuses based on this misunderstanding (Age of Empires 2 being a famous example).

Even a simple calculation will show the truth:

-40% = 0.6

+50% = 1.5

0.6 * 1.5 = 0.9

Conclusion: With every coin-toss you lose 10% of wealth on average.

Re: The infamous coin toss

#64

Post by rerdavies »

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 wealth of all investors: $0.00515378

What could have been made more clear in the presentation. Average wealth increases; but average log(wealth) decreases. Investors on the long improbable upper tail make huge amounts of money; investors on the lower tail just don't make much difference to the average, when they lose 40% of $0.01. What causes the wild income distribution: the fact that log(wealth) decreases on average. This doesn't decrease the total money supply, but it does increase the skewed distribution of wealth.

It has nothing whatsoever to do with anything related to flipping consecutive heads and tails. And a great deal to do with the fact that the log(gain) is 0.405465 and the log loss is -0.510826. So the average investors log(wealth) decreases over time even though the average total wealth increases.

Most figures are wildly variable from trial to trial. Most especially, what That Guy makes is highly variable. If he rolls 72 heads and 28 tails instead of 71 heads and 29 tails, then he's worth $2,565,000 more, and single-handedly adds $25.65 to the average net worth! The number of people who lose money is surprisingly, depressingly consistent.

The obvious conclusion: increasing wealth concentration is an unavoidable consequence of capitalism that probably has very little to do with merit.

The only solution: eat the rich.

Re: The infamous coin toss

#65

Post by jstanley »

ActivePattern wrote: post #20
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…

No, your expected value is indeed positive over repeated iterations: (1.05^i)*w, where i is the number of iterations and w is the starting wealth. The intuition for why this happens is that the losses of the majority are made up by the big gains of a minority. You can even see that after two iterations: Case 1: two heads -- 225% wealth Case 2: heads, tails -- 90% wealth Case 3: tails, heads -- 90% wealth Case 4: two…

Wow, great point! I had ran my simulation several times to check that it was reasonably stable and it gave me less than $1 average wealth each time, so I thought that was that. But just now I re-ran it with a much larger population and indeed the average wealth grows now instead of shrinking. Sadly I can't edit my comment above any more, but:

> that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating!

is indeed untrue. Thanks!

Re: The infamous coin toss

#66

Post by jstanley »

selcuka wrote: post #51
pclmulqdq wrote: post #24

Earlier quoted context omitted.

The collective also gains in round 2, and each subsequent round. From the intuition that you have about the first round, treat each group with the same amount of money separately, and you will see that money grows in every round. Example: Round 1: 100x$100 (total $10000) -> 50x$60 + 50x$150 (total $10500) Gain of $500 total Round 2: 50x$60 ($3000) -> 25x$90 + 25x$36 ($3150) 50x$150 ($7500) -> 25x$225 + 25x$90 ($7875)…

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.

Re: The infamous coin toss

#67

Post by n2d4 »

DarkNova6 wrote: post #63
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…

Confusing the equivalence of -50% and +50% is very common. In fact, many video games provide wrong bonuses based on this misunderstanding (Age of Empires 2 being a famous example). Even a simple calculation will show the truth: -40% = 0.6 +50% = 1.5 0.6 * 1.5 = 0.9 Conclusion: With every coin-toss you lose 10% of wealth on average.

While I agree, that conclusion is wrong because you forgot to look at the win-win and loss-loss cases. Even with two coin tosses, as in your example, you'll win on average:

1.5 * 1.5 = 2.25

0.6 * 1.5 = 0.9

1.5 * 0.6 = 0.9

0.6 * 0.6 = 0.36

=> 1.1025x total

It's counter-intuitive because even though you almost always lose, you still win (linear) wealth on average (but not median). The difference is that if you have unlimited tosses available, you don't care about maximizing EV after X tosses. Instead you care about minimizing your risk of losing it all.

Re: The infamous coin toss

#68

Post by jstanley »

dongobread wrote: post #34
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…

I tried this and collective wealth does increase: https://imgur.com/a/Zc2aApj The median person does get poorer over time though, a small percent get extrmeely wealthy: https://imgur.com/a/v2AVyrv https://imgur.com/a/7pPfKk6

Thanks, it turns out to really matter how big the population is, otherwise you aren't likely to have enough lucky members to bring the mean up.

Re: The infamous coin toss

#69

Post by operator-name »

DarkNova6 wrote: post #63
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…

Confusing the equivalence of -50% and +50% is very common. In fact, many video games provide wrong bonuses based on this misunderstanding (Age of Empires 2 being a famous example). Even a simple calculation will show the truth: -40% = 0.6 +50% = 1.5 0.6 * 1.5 = 0.9 Conclusion: With every coin-toss you lose 10% of wealth on average.

For some reason this is not mentioned in the article but is mentioned in the video.

Re: The infamous coin toss

#70

Post by HelloNurse »

rerdavies wrote: post #60
throwaheyy wrote: post #46

Earlier quoted context omitted.

“current bankroll” is the issue here. 1 * 1.5 * 0.6 = 0.9 < 1.

Not getting it. There are FOUR possible outcomes in two consecutive rolls. HH: 2.25! HT: 0.9 TH: 0.9 TT: 0.36. Expectation: 4.41/4 = 1.1025. The losses due to HT/HT get more than offset by the massive gain from HH over TT. I can imagine there are arguments to be made about median expected value, and the effect on concentration of wealth. But whatever they are, they aren't being made.

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 TTxHH sequence is worth the same as HTxHT, losing the usual 10% every two tosses. 2.250.36=0.81=0.90.9.

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