I stumbled into Ergodicity early in my career without even knowing it. I was working at Susquehanna Investment Group and struggling to fully internalize what they were teaching me. Feeling spurned after I was not invited into their trading class I turned into an apostate of all that Susquehanna teaches. I set out to prove that none of us can actually realize our expected value (EV) in a consistent manner because we are mortal and cannot physically play enough games to increase to realize our EV. This idea is best captured by this chart: ![[Pasted image 20230724100952.png]] This chart shows the cumulative P&L per roll of a favorable (ie. +EV) die rolling game and your cumulative P&L over time. As you can see, despite the game being +EV you experience many losses early in the game and its not until you've played the game around 400 times that you begin to consistently earn your EV. The green line shows your cumulative P&L and while it is a great, upward sloping equity curve, you can see many drawdowns over the number of trials. I was playing a lot of poker at the casinos during this time and playing with +EV (I recall my hourly rate was ~$50/hour) but I would inevitably have bad sessions where I would walk away from the table having lost a thousand dollars or so (which was a lot of money to a kid in his 20s back then). I remember one trip to Vegas with some friends where I set out to play poker for the whole weekend. We arrived on Thursday evening and after dinner my friends went off to play roulette and craps and eventually hit the clubs while I set off for the poker lounge for a long night of grinding. Unfortunately after some bad play and a few bad beats I had exhausted my bank roll for the trip and being dead sober I wanted nothing to do with whatever debauchery my friends were into so I headed back to the room and went to bed. I was rudely awoken at 4am by my drunk friends who were chugging champagne straight from the bottle and boasting of their conquests at the tables and all the comps they had received. The next day I was smarting from the beating I had taken at the poker table and my friends were nursing their hangover. I was staring at another two days in Vegas with no bankroll and feeling miserable while my friends had not only made a ton of money but also had the night of their lives. I was doing something wrong. I knew better than to just go play the -EV games and hope for the best, but my strategy of grinding it out at the poker table was not working for me. After I got home from Vegas I set out to figure out how to beat craps. I built a Monte-Carlo simulation in Excel and ran through thousands of games of craps. I analyzed each roll for duration and optimal betting strategies. My hypothesis was that while the game was a -EV game and there was no changing that, could I find a strategy that would minimize the -EV or even give me +EV. Years later I stumbled across [[Parrondo's Paradox]] which shows that this is in fact possible, but at the time I bootstrapped a strategy that ended up working out quite well for me. Years later after my Susquehanna rebellion had run its course, I read Nassim Nicholas Taleb's *The Black Swan* which reinvigorated my interest in the art and science of [[Decision-Making]] under [[Uncertainty]]. Since that book I have followed his work closely and saw him speak about Fragility at Wharton, but it wasn’t until a few years ago (this being 2023) that I stumbled across this article of his on Medium that explained the concept of Ergodicity with this thought experiment: >Consider the following thought experiment. > >First case, one hundred persons go to a Casino, to gamble a certain set amount each and have complimentary gin and tonic –as shown in the cartoon in Figure x. Some may lose, some may win, and we can infer at the end of the day what the “edge” is, that is, calculate the returns simply by counting the money left with the people who return. We can thus figure out if the casino is properly pricing the odds. Now assume that gambler number 28 goes bust. Will gambler number 29 be affected? No. > >You can safely calculate, from your sample, that about 1% of the gamblers will go bust. And if you keep playing and playing, you will be expected have about the same ratio, 1% of gamblers over that time window. > >Now compare to the second case in the thought experiment. One person, your cousin Theodorus Ibn Warqa, goes to the Casino a hundred days in a row, starting with a set amount. On day 28 cousin Theodorus Ibn Warqa is bust. Will there be day 29? No. He has hit an uncle point; there is _no game no more_. > >No matter how good he is or how alert your cousin Theodorus Ibn Warqa can be, you can safely calculate that he has a 100% probability of eventually going bust. > >The probabilities of success from the collection of people does not apply to cousin Theodorus Ibn Warqa. Let us call the first set _ensemble_ probability, and the second one _time_ probability (since one is concerned with a collection of people and the other with a single person through time). Now, when you read material by finance professors, finance gurus or your local bank making investment recommendations _based on the long term returns of the market_, beware. Even if their forecast were true (it isn’t), no person can get the returns of the market unless he has infinite pockets and no uncle points. The are conflating ensemble probability and time probability. If the investor has to eventually _reduce_ his exposure because of losses, or because of retirement, or because he remarried his neighbor’s wife, or because he changed his mind about life, his returns will be divorced from those of the market, period. >-Nassim Nicholas Taleb, *[The Logic of Risk Taking](https://medium.com/incerto/the-logic-of-risk-taking-107bf41029d3)*, And there it hit me. The difference between the *ensemble* probability, or EV, and the *time* probability is what had frustrated me at the poker table and had blocked my ability to learn what Susquehanna was trying to teach me. Now this revelation opens the door to some really complex and high-minded statistical and philosophical issues that are not in the scope of this article, but what we can take away in layman's terms is that ergodicity is another way to say that the path to the outcome matters just as much as the outcome. **Case Studies** - [CHG Issue #141: Reframing Markets](https://open.substack.com/pub/cedarshillgroup/p/chg-issue-141-reframing-markets) - [CHG Issue #124: Path Dependency](https://cedarshillgroup.substack.com/p/chg-issue-124-path-dependency) - [CHG Issue #117: The Game of Life](https://cedarshillgroup.substack.com/p/chg-issue-117-the-game-of-life) - [CHG Issue #101: Russian Roulette](https://cedarshillgroup.substack.com/p/chg-issue-101-russian-roulette) - [CHG Issue #87: Fragility](https://cedarshillgroup.substack.com/p/chg-issue-87-fragility) Explore Further: [[Shannon's Demon]] | [[Portfolio Construction]] Tags: #Concepts Your support for Cedars Hill Group is greatly appreciated <form action="https://www.paypal.com/donate" method="post" target="_top"> <input type="hidden" name="hosted_button_id" value="74PGN8ZXHQVHS" /> <input type="image" src="https://www.paypalobjects.com/en_US/i/btn/btn_donate_LG.gif" border="0" name="submit" title="PayPal - The safer, easier way to pay online!" alt="Donate with PayPal button" /> <img alt="" border="0" src="https://www.paypal.com/en_US/i/scr/pixel.gif" width="1" height="1" /> </form>