User:IssaRice/Ergodicity

Here's my suggested path to learning about this, assuming you just know basic probability and statistics and calculus but you know nothing about ergodicity. It took me around two days to go through all the things here, but i think you can do it much quicker or much longer depending on your interests. (also i was fumbling around because i didn't know what i was doing, so you will save some time probably.)

read http://squidarth.com/math/2018/11/27/ergodicity.html and http://squidarth.com/math/2019/04/13/ergodicity-animated.html
- Really understand the 1.5x/0.6x game
- Try to formalize the game using random variables
- Come up with a formal expression for the time-average winnings vs ensemble winnings. I think you come up with something like $\sum _{k=0}^{n}{\binom {n}{k}}1.5^{k}0.6^{n-k}{\frac {1}{2^{n}}}$ . Can you prove this is always greater than 1?
- figure out what percentage of flips need to be heads in order for you to have positive winnings. you want $1.5^{k}0.6^{n-k}>1$ so the break-even value of k is $k=-n\log 0.6/\log(1.5/0.6)$ so the fraction of heads is $k/n\approx 0.557$ . Understand that by law of large numbers, your fraction of heads will be super close to 0.5 in the long run, so you will almost never get anywhere close to having 56% of your flips be heads, so you almost certainly lose.
read https://ergodicityeconomics.files.wordpress.com/2018/06/ergodicity_economics.pdf sections 1.1, 1.2, 2.1, 2.2, 2.3, 2.4, 2.5. something like that.
read https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0065
- code up a model of the iterated st petersburg game and play around with it. My own result was https://gist.github.com/riceissa/f0679ff2f032ca16d46ba539e40b6623
read https://www.greaterwrong.com/posts/gptXmhJxFiEwuPN98/meetup-notes-ole-peters-on-ergodicity
- don't miss comments by abram demski and me
  - the point of my example is that there is an additive dynamic with positive growth rate, but where log-wealth utility doesn't take the bet. another question to ask is, is there an additive dynamic with negative growth rate where log-wealth utility does take the bet -- this seems to be impossible.
- code up models of both multiplicative and additive environments, where different organisms exist and can take or refuse bets. Here's my result https://gist.github.com/riceissa/08cdf34f999ff3cc84be8e84373e47e7 (there isn't any evolution going on here; it's just a single generation that consists of a large number of timesteps)
understand the difference between $\lim _{n\to \infty }a_{n}$ and $\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}a_{i}$ . What happens if $a_{n}\to a$ ? if $a_{n}\to \infty$ ? if $a_{n}$ doesn't converge to any number but doesn't go off to infinity either (e.g. it oscillates between 1 and -1)? what if $a_{n}=(-1)^{n}n$ ?
is maximizing time-average the same thing as maximizing growth rate? does one imply the other?
is maximizing time-average the same as maximizing your wealth in the limit as time goes to infinity?

See also