📓 non-ergodicity

aka. the irrelevance of expected-value in finite world
aka. fat-tails be weird

re: https://ergodicityeconomics.com/2023/07/28/the-infamous-coin-toss/
non-ergodic
- a gamble (or statistical game) where the following aren't equal (or, don't converge to the same)
  - the "ensemble average"
    - many players playing the game once in parallel
    - aka expected value
    - aka parallel worlds
  - the "time-average"
    - individual, playing a repeated game
ergodic
- when the two limits converge to the same
example gamble:
- heads: win 50% (ie. 50% chance of 150% return)
- tails: lose 40% (ie. 50% chance of 60% return)
- "ensemble average"
  - 150 x 0.5 + 60 x 0.5 = 105 (ie. 5% gain)
  - note: additive
- "time average"
  - 50% heads, 50% tails
  - 1.5 x 0.6 = 0.9 (ie. 10% loss)
  - note: multiplicative
- ie. it seems, that each individual is expected to lose, but, as a whole, the population gains
  - how?
    - b/c although it is very rare to win consecutively (all heads), it is made up for a faster growing magnitude of the reward for that player
a second example, for clarity:
- heads: double your wealth + 10%, (ie. 50% chance of 210% return)
- tails: lose everything (ie. 50% chance of 0% return)
- "ensemble average"
  - 210 x 0.5 + 0 x 0.5 = 105 (ie. 5% gain) (as before)
- "time average"
  - 100% chance of eventually hitting tails, thus, 0
- "everyone loses eventually, except that ~one guy who never rolls tails"
  - since there are infinite players in the "ensemble average", there's always someone with all heads

traditional approach to expected value results in "unintuitive" results
- most notably, St. Petersburg paradox
  - https://en.wikipedia.org/wiki/St.Petersburgparadox
- use of logarithmic expected value, results in more "intuitive" results
  - some argue that this more correctly represents human perception of utility
    - ie. going from 0$ -> $10 is different than 1M -> 1M+10
  - Peters argues for log-EV from ergodicity / finiteness
    - and that log-EV is thus a true / correct / rational approach for individuals, not just a whim of human evolution / mental-biases
    - https://pubs.aip.org/aip/cha/article/26/2/023103/134886/Evaluating-gambles-using-dynamics

double integral, long time, and many players
- in the limit -> infinity, no difference in order
- in the limit -> large-but-not-infinite, BIG DIFFERENCE in order
if do the monte-carlo "properly" w/ many millions of iterations both averages are the same
- hence, sometimes very important to get a proper distribution in monte-carlo
- BUT, for "realistic" individual experiences
in discrete statistics, the "time-average" "derivation" is hand-wavy
- but, modeling with stochastic different equations, geometric brownian motion
  - gives us continuous domain
  - also, allows for deriving each of these "different" averages rigorously
    - (allegedly, math is beyond me, see articles linked below)
more fun observations by Peters
- Kelly Criterion, 25% for original the game
  - https://en.wikipedia.org/wiki/Kelly_criterion
  - 0.5/0.4 - 0.5/0.5
- but still not as good as...
- cooperative strategy: individuals play, redistribute, play, etc.
  - tends towards EV as number of cooperators tends -> infinity
- what would the kelly criterion be as a function of number of parallel games?
  - at 1 games, 25% (as above)
  - at infinity, 100% (b/c EV > 0) (or is it infinite? ie. borrow all the money you can)
could we come up with an opposite gamble?
- ie. "most individuals win", "the collective loses"
- would require a rare loser that loses enough dollars to make up for all the others
- ie. the original set up requires an unbounded upside, and finite loss
betting 100$ x 1 =/= betting 10$ x 10 =/= betting 1$ x 100
what percentage of heads needed to net win?
- 1.5^p x 0.6^(1-p) = 1
- ln()
- p ln 1.5 + ln 0.6 - p ln 0.6 = ln 1
- p = ln 0.6 / (ln 0.6 - ln 1.5)
- p = 55.7 %
- but, as N grows, it becomes harder to deviate from 50%
odds of > 55.7% heads, for N tosses:
- sum, x = 0 to 55.7 N, of: N c X 0.5 ^ N
thoughts
- GDP being a terrible measure
  - a GDP that grows, yet ~all individuals feel like they are losing
- VC industry "converts" net-expected-failure to net-expected-success
  - similar to insurance
  - a founder should raise money
    - decreases the losses of the worst-case (~typical case) of venture failure
      - b/c founder is paid a salary, & doesn't pay for everything out of pocket
      - (without necessarily impacting the chances)
    - decreases the gains of the best-case
    - (plus, utility of founders first 1M =/= investor's "last" 1M)
  - good for government to co-invest
    - use investors as basic-threshold-evaluations with skin-in-the-game
    - but, "moral hazard"
  - ideally, probably, government/investors would just blanket invest
    - ie. invest with a minimal threshold, at minimum evaluation cost
    - ~ effectively basic income
- "capitalism" vs "communism"
  - capitalism as favouring the mean over the median ("ensemble-average" over the "time-average")
    - and also believing that this is better for the median anyway
  - communism as the opposite, favouring the median over the mean
    - and also believing that this is better for the mean anyway
- a good foundation for a co-operative board game?
"learnings"
- cooperation good
- insurance good
- redistribution good
- individuals should maximize their rate of return, not expected value
- humans, as individuals are intuitively bad at exponential growth but perhaps decent at non-ergodic risk
  - some people/institutions capitalize on this (for exploitation, or value-creation)
    - insurance, investments, casinos
  - and perhaps the opposite for collectives/institutions
    - collectives are good at exponential growth
    - bad at non-ergodic risk (treat it as ergodic)
      - Taleb's Black Swan, etc.
  - difference in "what's best for me" vs. "what's best for everyone"
related: Taleb, in general (Fooled by Randomness)
- https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3#_edn1
great intro
- https://www.investmentmagazine.com.au/wp-content/uploads/2012/11/Martin-Goss-TOWERS-WATSON-White-Paper.pdf
https://www.advisorperspectives.com/articles/2021/03/01/understanding-fat-tail-returns
https://en.wikipedia.org/wiki/Ergodicity_economics
https://en.wikipedia.org/wiki/Ergodicity

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📓 non-ergodicity

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