If the sequence of coin flips has an equal number of heads and tails
It only does in expectation; the underlying process is a martingale. They’re using an illustrative example to show you that investing everything in that random walk leads to a modal expectation of having the same at the end as you do at the beginning.
But that’s an expected value of 0 in log terms; the expected value in linear terms of course follows 1.25^n, where n is the number of flips. Shannon’s Demon reduces the variance in return at the price of reducing the mean return. If you’re only half in the market, your expected value grows at 1.125^n.
But if you have a log utility function, the decreased variance is helpful because then your expected utility grows each period rather than staying flat. (With 100% exposure, your EV of one period is .5*log(2)+.5*log(.5), which is obviously 0, but with 50% exposure your EV of one period is .5*log(1.5)+.5*log(.75), which is positive.) If you have a log utility function, 50% exposure happens to maximize your growth in expected return.
(I do agree with you that the link saying that the offer is a “wash” without bringing in the log utility function, or the tradeoff between variance and expected return, is bad, but those are somewhat subtle issues that they might not want to introduce along with the game.)
They’re using an illustrative example to show you that investing everything in that random walk leads to a modal expectation of having the same at the end as you do at the beginning.
That illustrative example highly depends on the number of heads being exactly equal. If the number of heads and the number of tails differed even slightly, the result would not be the same amount that you started with, and the fact that the ratio of heads to tails was close to 50% would not affect that. If you had 100 heads and 101 tails, you’d end up with half as much as you started with, and if you had 10000 heads and 10001 tails, you’d still end up with half as much as you started with.
And if the number of heads and the number of tails was exactly equal, I could guarantee doubling my money simply by waiting until the last flip to bet anything.
Everything else you’re saying is correct, but the example is bad. And I still suspect that this just proves it’s impossible for a real life stock to actually have equal chances of doubling and halving.
And I still suspect that this just proves it’s impossible for a real life stock to actually have equal chances of doubling and halving.
Well, real life models generally operate on much smaller timescales, with much smaller step sizes. A model where you increase or decrease by .01 on a log scale (roughly 1% increase and 1% decrease) each step seems much more reasonable, but again the same strategy (of 50% exposure, rebalanced continuously) is optimal for a log utility function.
I have no doubt that a real-life stock can change in a manner similar to a log scale, but if it changed in a manner exactly like a log scale, the company could never fail (sending the value to 0) and it could grow larger than the size of the entire economy.
Given that “it can only grow to a certain size before you exceed the real-life limit” transforms the St. Petersburg paradox from infinite expected value to a small expected value, I would expect to see anyone proposing this model show that such real-life limits don’t destroy this model in the same way.
The optimization that I’ve been linking to- take the derivative with respect to exposure, set it equal to 0- is a 1-step optimization problem. That is, the strategy I’m describing as optimal (Shannon’s Demon) is optimal even if there’s only one coin flip, and because of the nature of the setup and the log utility function what’s optimal for one coin flip is optimal for an arbitrary number of coin flips.
It only does in expectation; the underlying process is a martingale. They’re using an illustrative example to show you that investing everything in that random walk leads to a modal expectation of having the same at the end as you do at the beginning.
But that’s an expected value of 0 in log terms; the expected value in linear terms of course follows 1.25^n, where n is the number of flips. Shannon’s Demon reduces the variance in return at the price of reducing the mean return. If you’re only half in the market, your expected value grows at 1.125^n.
But if you have a log utility function, the decreased variance is helpful because then your expected utility grows each period rather than staying flat. (With 100% exposure, your EV of one period is .5*log(2)+.5*log(.5), which is obviously 0, but with 50% exposure your EV of one period is .5*log(1.5)+.5*log(.75), which is positive.) If you have a log utility function, 50% exposure happens to maximize your growth in expected return.
(I do agree with you that the link saying that the offer is a “wash” without bringing in the log utility function, or the tradeoff between variance and expected return, is bad, but those are somewhat subtle issues that they might not want to introduce along with the game.)
That illustrative example highly depends on the number of heads being exactly equal. If the number of heads and the number of tails differed even slightly, the result would not be the same amount that you started with, and the fact that the ratio of heads to tails was close to 50% would not affect that. If you had 100 heads and 101 tails, you’d end up with half as much as you started with, and if you had 10000 heads and 10001 tails, you’d still end up with half as much as you started with.
And if the number of heads and the number of tails was exactly equal, I could guarantee doubling my money simply by waiting until the last flip to bet anything.
Everything else you’re saying is correct, but the example is bad. And I still suspect that this just proves it’s impossible for a real life stock to actually have equal chances of doubling and halving.
Well, real life models generally operate on much smaller timescales, with much smaller step sizes. A model where you increase or decrease by .01 on a log scale (roughly 1% increase and 1% decrease) each step seems much more reasonable, but again the same strategy (of 50% exposure, rebalanced continuously) is optimal for a log utility function.
I have no doubt that a real-life stock can change in a manner similar to a log scale, but if it changed in a manner exactly like a log scale, the company could never fail (sending the value to 0) and it could grow larger than the size of the entire economy.
Given that “it can only grow to a certain size before you exceed the real-life limit” transforms the St. Petersburg paradox from infinite expected value to a small expected value, I would expect to see anyone proposing this model show that such real-life limits don’t destroy this model in the same way.
The optimization that I’ve been linking to- take the derivative with respect to exposure, set it equal to 0- is a 1-step optimization problem. That is, the strategy I’m describing as optimal (Shannon’s Demon) is optimal even if there’s only one coin flip, and because of the nature of the setup and the log utility function what’s optimal for one coin flip is optimal for an arbitrary number of coin flips.