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The infamous coin toss

ergodicityeconomics.com

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Re: The infamous coin toss

#71

Post by ta1243 »

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

Re: The infamous coin toss

#72

Post by n2d4 »

throwaheyy wrote: post #47
kevinwang wrote: post #39

Earlier quoted context omitted.

Can you clarify? I think you might be mistaken, since a gamble with payoffs of 150% and 60% has positive expected value.

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.

It does! You take insane amounts of risks (that you want to avoid), but the expected value stays positive, even if you repeat it a million times. You might have a 99.9999999999...% chance of going broke, but you'll also have a 1/2^1mil chance of making an absurd amount of money; on average (and not median), you will come out ahead.

Re: The infamous coin toss

#73

Post by [deleted] »

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.

[deleted]

Re: The infamous coin toss

#74

Post by arc-in-space »

HelloNurse wrote: post #70
rerdavies wrote: post #60

Earlier quoted context omitted.

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 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.

Re: The infamous coin toss

#76

Post by drekipus »

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

Re: The infamous coin toss

#77

Post by nhinck »

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.

The claim was: "Each round, the collective wealth goes up 5%, no matter how many rounds you run. "

And this isn't true.

Re: The infamous coin toss

#79

Post by kgwgk »

nhinck wrote: post #77
jstanley wrote: post #66

Earlier quoted context omitted.

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.

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.)

Re: The infamous coin toss

#80

Post by wodenokoto »

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”
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