The infamous coin toss
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The infamous coin toss
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Re: The infamous coin toss
#2[dead]
Re: The infamous coin toss
#3But this is purely a result of the distribution of returns from a single toss.
If I win I get 1.5 times my money, if I lose I'm left with less than 1/1.5 times. When I lose, I lose more than I win.
I agree it's counterintuitive how any individual (on average) will lose over time, yet the entire system grows.
Re: The infamous coin toss
#4But this is purely a result of the distribution of returns from a single toss. If I win I get 1.5 times my money, if I lose I'm left with less than 1/1.5 times. When I lose, I lose more than I win. I agree it's counterintuitive how any individual (on average) will lose over time, yet the entire system grows.
Yeah, as soon as I realized that, I rolled my eyes at not immediately seeing that. When expressed as 150% and 60%, it became obvious to me.
Re: The infamous coin toss
#5The +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 terms than decreases (i.e. not just a 10% difference) in order to balance out.
Re: The infamous coin toss
#6The 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.
Re: The infamous coin toss
#7The +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…
Yes, this is called the Kelly Criterion[1] which tells us we should optimize the expected value of the log of wealth, not the expected value of wealth directly. On a log scale, it's clear that doubling and halving have the same magnitude but opposite signs so are perfectly balanced.
Re: The infamous coin toss
#8I 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 wealth could grow while the individual wealths drop, and the answer is that it does not.
The only way it looks like the collective makes a gain is if you try to say that x_i(t) = x_i(t-1)*1.05, but that's a simplification that doesn't hold in either the individual or the collective case.
But if you forget the part about the supposed misalignment between individual and collective, then the idea that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating!
Re: The infamous coin toss
#9I 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…
Take a simplified system: two players, both with $10. Flip a coin. Let’s say one comes up heads, the other tails. Total is now $15 + $6 = $21. A $1 increase for the whole system.
I agree the distinction is not at all between individual and collective, but rather between one-off and repeated games.
Re: The infamous coin toss
#10I 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 same as 1 person playing 100 rounds.