Another simple way to understand why the coin flip scenario makes you lose money: If you have $100, double it with heads to $200, then lose 60% on tails, you have $80 - less than you started with.
That's what I thought when reading this, the opposite of double (100% gain) is half (50%) loss, so if you've lost by more than 50% on the negative coin toss result you will tend to lose money over time.
If the negative was less than half (say, 40%), you would gain money over time.
Thanks for this. I was confused when I first read it, but I had been skimming and thought it meant "keep 60%" (or in other words, lose 40%).
I was reading this and thought of the Kelly criterion. The choice is not binary, you already know the optimal bet sizing here. The kelly fraction at 0.25 gives you what a 3% growth rate? f=1 is obviously a flawed scenario that should not be happening. ( i dont know enough to know whether SBF made the same point. 3AC did bet the house though). Culprit is all in bet, not linear sizing.
Survivorship bias and worshipping is not a new phenomenon either existed since atleast last 40 years in tech where a median dev is not paid as much as one at Google or Microsoft.
If i were to grant the premise that every individual should use log sizing, how does that ensure that we are devoting enough resources to home run outcomes in biotech, space etc.? That is where the toy model breaks because these spaces demand a lot of money to get started and have any meaningful breakthrough. Beyond the individual level, there is clearly a civilizational utility to be swinging for the fences and maximizing the expected value. At an individual level, if people are making sub optimal choices, perhaps the solution is more financial education.
I think readers using Firefox (or forks thereof) with its "enhanced tracking protection" turned on will not see images here; at any rate, they are blocked for me with the note "Socialtracking" in the network tab of the devtools, which I think means that ETP blocked them.
I don't know whether there's some other place to host images for LW articles that won't provoke such issues.
Apologies if this is a newbie math comment, as I'm not great at math, but is there a way to calculate a kind of geometric expected value? The geometric mean seems to require positive numbers, and expected values can have negative terms. Also, how would you apply probability weights?
Not in general. As you say, GM requires positive numbers, but there's a reason for this: imagine GM as log-scaling everything and then performing AM on the results.
So to get the GM of 10 and 1000:
But now notice that:
and so the GM of 1 million and 0.00000000000000000000000001 is 0.00000000000001, and the GM of 1 billion and 0 is 0. This won't really lend itself to calculating a GM of a list including a negative number.
One thing you can do, though, which makes sense if you are e.g. calculating your utility as log(your net worth) in various situations, is calculate the GM of [your current net worth + this value].
For instance, if you are considering a gamble that has a 50% chance of gaining you $2000 and a 50% chance of losing you $1000:
The geometric expected value for the +100% or -60% gamble in the post is straightforward.
If you gain 100% and then lose 60% (or lose 60% and then gain 100%), overall you'll end up down 20% from where you started since (1 + 100%) * (1 - 60%) = 2 * 0.4 = 80% = 1 - 20%. This loss is what's expected over two iterations, so the loss over one iteration is sqrt(80%) = 89.44% = 1 - 10.56%, so the expected loss is that 10.56% of your starting capital per flip.
Edit: said something very silly about weighting.
Nothing says you can't take the geometric mean of a series that includes negative numbers, just that if you have an even number of elements, but an odd number of negative elements, you'll get a complex answer.
To weight the elements of your series, you should be able to take the geometric mean of p(X)*X, then divide by the geometric mean of your p(X).
The techno-optimist dream of an abundant post-AGI world where humans devote their days to art and leisure will look more like billions of people chasing negative sum capital and status jackpots with UBI stipends.
In a post AGI world, the world mostly looks like whatever the AGI wants. Capitalism has a strong incentive to create sports betting, and wave it in peoples faces. AGI doesn't.
Wanted to get an intuitive feel for it, so here's a quick vibecoded simulation & distribution:
https://github.com/eggsyntax/jackpot-age-simulation
(Seems right but not guaranteed bug-free; I haven't even looked at the code)
Here is a simulation, with the Kelly criterion.
https://claude.ai/public/artifacts/7e88597f-7d41-4c84-a88a-55bc603de88d
This essay is about shifts in risk taking towards the worship of jackpots and its broader societal implications. Imagine you are presented with this coin flip game.
How many times do you flip it?
At first glance the game feels like a money printer. The coin flip has positive expected value of twenty percent of your net worth per flip so you should flip the coin infinitely and eventually accumulate all of the wealth in the world.
However, If we simulate twenty-five thousand people flipping this coin a thousand times, virtually all of them end up with approximately 0 dollars.
The reason almost all outcomes go to zero is because of the multiplicative property of this repeated coin flip. Even though the expected value aka the arithmetic mean of the game is positive at a twenty percent gain per flip, the geometric mean is negative, meaning that the coin flip is actually negatively compounding over the long run.
How can this be? Here's an intuitive way to think about it:
The arithmetic mean measures the average wealth created by all possible outcomes. In our coin flip game the wealth is heavily skewed towards rare jackpot scenarios. The geometric mean measures the wealth you'd expect in the median outcome.
The simulation above illustrates the difference. Almost all paths bleed to zero. You need to flip 570 heads and 430 tails just to break even in this game. After 1,000 flips, all of the expected value is concentrated in just 0.0001% of jackpot outcomes, the extremely rare case where you flip a lot of heads.
The discrepancy between arithmetic and geometric means forms what I call the Jackpot Paradox. Economists call it the ergodicity problem and traders call it volatility drag. You can’t always eat the expected value when it’s squirreled away in rare jackpots. Risk too much hunting jackpots and the volatility will turn positive expected value into a straight line to zero. In the world of compounded returns, the dose makes the poison.
Crypto culture in the early 20s was a living example of the Jackpot Paradox. SBF started the conversation in a tweet about wealth preferences.
SBF proudly proclaimed that he had a linear wealth preference. Since he aimed to donate everything, he argued that doubling from $10 billion to $20 billion mattered just as much as going from $0 to $10 billion, so swinging for huge, risky bets were logically worth it from the perspective of civilization.
Su Zhu of Three Arrows Capital echoed this preference for linear wealth and even took it a step further in introducing the exponential wealth preference.
Here’s how the three wealth preferences map to our coin flip game from above.
Given our understanding of the Jackpot Paradox, it’s obvious that SBF and 3AC were figuratively flipping this coin infinitely. That mindset was how they built their fortunes in the first place. Equally unsurprising and obvious in hindsight was that SBF and 3AC both ended up vaporizing ten billion dollars. Perhaps they are quintillionaires in a distant parallel universe which justifies the risks they took.
These blowups aren’t just cautionary parables about the mathematics of risk management, but rather a reflection of a deeper macrocultural shift toward linear and even exponential wealth preferences.
Founders are expected to adopt a linear wealth mindset and take big risks that maximize expected value as cogs in the venture capital machine dependent on power law home runs. The tales of Elon Musk, Jeff Bezos, and Mark Zuckerberg risking everything they had and emerging with the largest personal fortunes on planet Earth reinforce the mythos that drives the entire risk taking sector, while survivorship bias conveniently forgets the millions of founders who go to zero. Salvation comes only to the select few who clear an ever steepening power law threshold.
That taste for outsized risk has seeped into everyday culture. Wage growth has severely lagged compounded capital, causing ordinary people increasingly see their best shot at real upward mobility in negative EV jackpots. Online gambling, 0DTE options, retail meme stocks, sports betting, and crypto memecoins all testify to the phenomena of exponential wealth preference. Technology makes speculating effortless, while social media spreads the story each new overnight millionaire, luring the broader population into one giant losing bet like moths to a light.
We’re becoming a culture that worships the jackpot and increasingly prices survival at zero.
And AI exacerbates the trend by further devaluing labor and intensifying winner take all outcomes. The techno-optimist dream of an abundant post-AGI world where humans devote their days to art and leisure will look more like billions of people chasing negative sum capital and status jackpots with UBI stipends. When society removes left tail outcomes, it shifts risk appetite far right, where only jackpots feel meaningful. Perhaps the up only e/acc logo should be redrawn to reflect the blizzard of paths that bleed to zero along the way, the true silhouette of the Jackpot Age.
In its most extreme form, capitalism behaves like a collectivist hive. The Jackpot Paradox math says it’s rational for civilization to treat humanity as interchangeable labor, sacrificing millions of worker bees to maximize the linear expected value for the colony. That might be the most efficient for aggregate growth, but it distributes purpose and meaning miserably.
Marc Andreessen’s techno-optimist manifesto warns that “man was not meant to be farmed; man was meant to be useful, to be productive, to be proud.”
But the rapid acceleration of technology and the shifts towards higher ever more aggressive risk taking incentives have pushed us precisely towards the outcome he warns against. In the Jackpot Age, growth is fueled by farming fellow man. Usefulness, productivity, and pride are increasingly reserved for the privileged few who win the competition. We have boosted the mean at the cost of the median, leaving a widening gap in mobility, status, and dignity that have breeds entire economies of negative sum cultural phenomena. The resulting externality shows up as social unrest, starting with the election of demagogues and ending with violent revolution, which can be quite costly to civilizational compounded growth.
Having made a living trading crypto markets, I’ve witnessed the degeneracy and desperation that has arisen from the culture shift firsthand. Much like the jackpot simulation, my victory stands on a pile of a thousand corpses of other traders. A monument to wasted human potential.
When people in the industry come to me for trading advice I almost always spot the same pattern. They all risk too much and drawdown too deep. Underneath there is typically a scarcity mindset driving it, a gnawing sense of feeling “behind” where they should be and a compulsion to make it fast.
My answer is always the same. build more edge rather than risk more size. Don’t kill yourself chasing the jackpot. Log wealth is what matters. Maximize the 50th percentile outcome. Make your own luck. Avoid drawdowns. Eventually you will get there.
But most people can never generate consistent edge. “Just win more” isn’t scalable advice. In the technocapitalist rat race, meaning and purpose is ever more winner takes all. It brings us back to meaning. Perhaps what we need is some sort of second coming of religion that reconciles old spiritual teachings with the realities of modern technology.
Christianity once scaled because it promised salvation to anyone. Buddhism spread on the claim that anyone could reach enlightenment.
A modern analog would have to do the same in offering dignity, purpose, and an alternative path forward for all people so they don’t destroy themselves chasing jackpots.