The infamous coin toss • Hacker News Board

Re: The infamous coin toss

#11

Post by jstanley » Thu, Aug 03, 2023, 2:03 AM UTC

selcuka wrote: post #10

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…

> 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 it's tails I wipe you out completely" - you expect to make a massive gain on any given turn, but if you play enough rounds you'll eventually lose everything.

Re: The infamous coin toss

#12

Post by selcuka » Thu, Aug 03, 2023, 2:05 AM UTC

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…

Ah, I get your point now. True, there is no need to consider multiple people, the expectation is the same for a single person, too.

The collective wealth calculation just makes it easier to visualise.

Re: The infamous coin toss

#13

Post by bogtog » Thu, Aug 03, 2023, 2:08 AM UTC

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.

This was my reaction too, I feel like there is something I'm not getting. I get that the bet is virtually guaranteed to go negative if the number of rounds is high enough (100+ rounds), but the opportunity for huge payouts still pulls the average up.

Re: The infamous coin toss

#14

Post by jakell » Thu, Aug 03, 2023, 2:12 AM UTC

bogtog wrote: post #13

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.

This was my reaction too, I feel like there is something I'm not getting. I get that the bet is virtually guaranteed to go negative if the number of rounds is high enough (100+ rounds), but the opportunity for huge payouts still pulls the average up.

Exactly

Re: The infamous coin toss

#15

Post by bzax » Thu, Aug 03, 2023, 2:13 AM UTC

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.

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 sample once, and you will almost surely (i.e. with probability approaching 1 as n goes to infinity) sample a point in the distribution where you lose nearly all of your money.

Re: The infamous coin toss

#16

Post by petermcneeley » Thu, Aug 03, 2023, 2:13 AM UTC

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…

This simulation is the collective wealth over N players for M times. It is increasing. https://coin-n.streamlit.app/?utm_medium=oembed

Re: The infamous coin toss

#17

Post by jakell » Thu, Aug 03, 2023, 2:15 AM UTC

bzax wrote: post #15

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.

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).

Re: The infamous coin toss

#18

Post by bzax » Thu, Aug 03, 2023, 2:15 AM UTC

bogtog wrote: post #13

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.

This was my reaction too, I feel like there is something I'm not getting. I get that the bet is virtually guaranteed to go negative if the number of rounds is high enough (100+ rounds), but the opportunity for huge payouts still pulls the average up.

The "average" of the distribution goes up as you increase the number of rounds, but the probability that you get an average or above value when you sample that distribution once goes to zero as the number of rounds increases.

Re: The infamous coin toss

#19

Post by didgeoridoo » Thu, Aug 03, 2023, 2:15 AM UTC

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.

Re: The infamous coin toss

#20

Post by ActivePattern » Thu, Aug 03, 2023, 2:17 AM UTC

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 tails -- 36% wealth

So, most people are losing money but a single lucky guy makes enough money that the expected value is still positive.

If you run a simulation with a low sample size, you might not see this effect because the "lucky guy" case is rare (and gets more rare with more iterations).