Hacker News Reader

Comment by contravariant | Hacker News Reader

Unless I'm very mistaken the example they give is not a dynamic system, so asking whether it's ergodic doesn't make sense. For it to be a dynamic system it would need to have an invariant measure, which I can't even begin to figure out from their description.

The process of simply flipping coins, even a biased coin, does correspond to a dynamic system where the space is all infinite sequences of heads and tails, the invariant measure is the joint probability, and the operator is the left-shift (i.e. if you sample 1 series of coin flips and throw away the first result you still end up with a sample of the same distribution).

But this doesn't translate at all to their proposed scenario of starting everyone at 1 and letting the results diverge from there.

View on HN

Why isn't a dynamic system?

A given point has current wealth as it's level and moves randomly up and down in that level according to a process that's random.

https://en.wikipedia.org/wiki/Dynamical_system https://en.wikipedia.org/wiki/Random_dynamical_system

Well dynamical systems aren't usually random for a start.

I suppose you can make it fit the definition for the random dynamical system, though in that case the underlying dynamical system is ergodic.

I think the article is correct that the system isn't ergodic, because it's not "mixing". A given typical trajectory won't get close to every potential point. Most go to the bottom and few go to the top - basically the point of the article.

The fact that it's biased is irrelevant. The question whether it will get close to any point is quite complicated, though with it being essentially a random walk with drift the chance that will get above its initial point is probably less than 1.

This's got nothing to do with ergodic theory though, but feel free to use that word in whatever weird distorted way you like, I literally can't seem to stop you.

A dynamic system is a system whose evolution doesn’t simply depend on its input at time t, but also to its internal state. In this case the wealth at time t does not simply depend on the outcome of the coin toss, you also need to know the value of the wealth before the coin toss.

A dynamical system is typically 100% internal state. It's just

1. A state space X

2. An evolution map f: X -> X

3. A measure \mu on X, preserved by f (such that f_*\mu = \mu) and where \mu(X) = 1.

To represent a series of i.i.d. random variables you can typically just represent them as a sequence of values with a joint probability distribution as the invariant measure.

If you've got a space X of such sequence you can also add the current value to it to get IR x X, with a map f: (x,t) -> x * 1.5 if t[0] is heads & x * 0.6 otherwise. But you'll have to tell me it's invariant measure because I can't come up with one.

An you example I always liked: baggage fees at airlines. On average a $25 bag fee is the same. For 4 business travelers. For a family of four while the dollar amount is the same it effects the family much more.

I'm confused, what's that got to do with ergodicity?

Reply on news.ycombinator.com