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The infamous coin toss

ergodicityeconomics.com

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Re: The infamous coin toss

#21

Post by petermcneeley »

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…

Note a single lucky guy is not enough. P of N heads is (1/2)^N but payout is (1.5)^N for this single lucky guy.

(The question can be reformulated such that this could be the case.)

Re: The infamous coin toss

#22

Post by jakell »

didgeoridoo wrote: post #19
jakell wrote: post #6

The article concludes that a +50%/-40% coin toss on average loses 10% every two tosses because 150% * 60% = 90%, but that ignores the two heads/two tails outcomes. Including those outcomes, ie AVERAGE(225%, 90%, 90%, 36%) = 110.25%, recovering the more intuitive result that the coin toss gains on average. The author seems to be confusing mode and mean; the modal path does approach zero.

You’re taking arithmetic averages of percentages… I don’t think that calculates anything meaningful. Try 225% * 90% * 90% * 36% to get the expected value.

You can substitute $ for % in my comment if it helps. If you start with $100, your expected wealth after two throws is the average of $225, $90, $90 and $36.

Re: The infamous coin toss

#23

Post by [deleted] »

jakell wrote: post #17
bzax wrote: post #15

Earlier quoted context omitted.

If you repeat this game n times (as n goes to infinity), you will have Θ(n) pairs of (heads, tails) and O(sqrt(n)) unpaired wins or losses, except for a vanishingly small fraction of the time when the results fall outside of any fixed number of standard deviations. The point is that you as an individual playing a repeated game don't get to meaningfully sample the expected value of the distribution. You only get to sa…

Absolutely. The individual is long-run guaranteed to be wiped out. But I disagree with the original author’s way of concluding that fact (ie, that it arises from “losing 5% per round”, which is just false).

[deleted]

Re: The infamous coin toss

#24

Post by pclmulqdq »

jstanley wrote: post #11
selcuka wrote: post #10

Earlier quoted context omitted.

> In the given game, the collective loses money just as the individual does. No, they don't. If 100 people starts with $100 (i.e. a total wealth of $10,000), and half of them win as expected, the total wealth at the end of first round will be $10,500. Granted it won't be a 5% increase for the next round, and eventually collective wealth will also approach 0, but that's the point. 100 people playing 1 round is not the…

Yeah, that's after 1 round. Both the individuals and the collective expect to gain if they only play 1 round. The interesting part is that they expect to lose money over time even though they expect to gain money if they only play once. But that holds for both the individuals and the collective. A less confusing game with the same mechanism is "flip a coin, if it's heads I give you 1000x your initial investment, if i…

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)

Gain of $150+$375=$575.

Round 3: 25x$36 ($900) -> half $54 and half $21.60 ($945 total) And so on.

Each round, the collective wealth goes up 5%, no matter how many rounds you run.

Re: The infamous coin toss

#25

Post by D_Alex »

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 disagree, and furthermore I think the entire article is basically wrong - in a way. Specifically, the expected average wealth of both the collective and an individual will increase every round. However the median wealth will fall.

The reason for this is that the very rare outliers will accumulate truly enormous wealth, while the rest of the players will never lose more than they start off with. You can see this in the first graph after say 250 rounds - the richest player has around 10,000x his initial wealth, even as the other 1000(or so) have very little - so on average they have 10x what they started with. And the reason no one is rich after 1000 rounds is simply that there were not enough samples. If there were say a billion samples, you might expect one to come in with say a trillion times its starting wealth, overcompensating for the other 999,999,999 losers - on average.

Re: The infamous coin toss

#26

Post by didgeoridoo »

jakell wrote: post #22
didgeoridoo wrote: post #19

Earlier quoted context omitted.

You’re taking arithmetic averages of percentages… I don’t think that calculates anything meaningful. Try 225% * 90% * 90% * 36% to get the expected value.

You can substitute $ for % in my comment if it helps. If you start with $100, your expected wealth after two throws is the average of $225, $90, $90 and $36.

That average is still greater than $100, because you haven’t yet hit the Kelly point beyond which the downside catastrophe dominates. Play it out a few more rounds and see where the average heads to.

[Edit: delete bad math]

Re: The infamous coin toss

#27

Post by bzax »

jakell wrote: post #17
bzax wrote: post #15

Earlier quoted context omitted.

If you repeat this game n times (as n goes to infinity), you will have Θ(n) pairs of (heads, tails) and O(sqrt(n)) unpaired wins or losses, except for a vanishingly small fraction of the time when the results fall outside of any fixed number of standard deviations. The point is that you as an individual playing a repeated game don't get to meaningfully sample the expected value of the distribution. You only get to sa…

Absolutely. The individual is long-run guaranteed to be wiped out. But I disagree with the original author’s way of concluding that fact (ie, that it arises from “losing 5% per round”, which is just false).

I believe the entire point of the ergodicity question here is "If you apply this process n times, with n approaching infinity, obviously the result may depend on what point in the n-times iterated distribution you sample, but if you choose a volume of vanishingly small measure to exclude, can you make a single concrete statement about what the process is doing without taking an expected value over the different outcomes"

And the answer is yes - with probability approaching 1 as n increases (ie excluding a portion of the distribution whose measure decreases to 0), the random process matches a deterministic process which is described by "you lose 5% each round".

Re: The infamous coin toss

#28

Post by wmorgan »

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 compensate for the unlucky ones. Put bounds on the size of the population, and you see that everyone goes bust in all but a tiny fraction of cases. Put bounds on the bank, and even that tiny fraction can't compensate for all the losers, and EV is negative.

Re: The infamous coin toss

#30

Post by mizzao »

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…

Mmm, seems similar to the game of venture capital!
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