“I’ve brought you here to play a game,” Omega says. “The well-being of humanity depends on your choices, so pay very close attention. You start with 100 credits.”

A screen suddenly appears with the number 100 on it. “To play the game,” Omega continues, “just think of some number of credits that you’d like to bet during the next round, and then push the blue button. You can’t bet more credits than you have remaining. Whatever number of credits you are thinking of when you push the button will be your bet for that round.”

A blue button appears.

“With each push of the blue button, there is a 99% chance that you win, in which case the credits you bet that round will increase by 10%. But there is a 1% chance that you’ll lose, in which case you forfeit all the credits you bet.”

“Let me give you an example,” Omega continues. “Suppose that you have 100 credits and bet 10 of them. If you win, which, remember, has a 99% chance of happening, your 10 credits will grow to 11 credits, so you’ll have 101 credits after that round. But if you lose, which, remember, has a 1% chance of happening, you will lose those 10 credits you bet, and so will end the round with only 90 credits. Is that clear?”

The (“maximizing”, “act”) utilitarian nods.

“Good. It’s also important to know that winning or losing in any particular round won’t affect the chance of winning or losing in any other rounds. And you can play as many rounds of the game as you like. Reality is paused while you are in this room, so if you want, you can play for a hundred years or a million years or however many rounds you choose. When you decide to stop playing, just leave the room through that door to return to your normal life. The moment you leave, I will adjust the world by adding an amount of positive utility proportional to the number of credits you have remaining. The amount of utility I will add to the world for each credit you end up with is equivalent to the utility of 1000 of the happiest lives humans have ever lived. That means there is a lot of utility on the line here, so don’t F this up.”

Omega disappears. The utilitarian thinks about the game for a while and performs some calculations.

“If I bet C credits on my first play, then the average number of credits I will get back from my first play is:

Average I get back after one play = 0.99 [chance of winning] * 1.10 [reward if I win] * C [amount I bet]+ 0.01 [chance of losing] * 0 [penalty if I lose] * C [ amount I bet] = 1.089 C ≈ C + 0.09 C

So that means that each time I play, the number of credits I get back is whatever I bet, increased by about 9% on average. So I maximize the average number of credits I end up with by betting all of them in my first play! If I bet 100 credits, then on average I’ll end up with 108.9, whereas if I bet 99 credits, then on average I’ll only end up with 108.811, so it maximizes expected value to bet them all.

However, nothing changes from one round to the next, since each round is statistically independent from the last. That means that every time I play, I should bet ALL my credits. And if I do that, my credits will grow exponentially, by about 9% with each play. That’s amazing!

Wait a moment…that doesn’t feel right. Isn’t it really risky to bet all my credits each round? I risk losing all of them. But that strategy does seem to maximize the expected total utility, so it is the most ethical thing to do.

Let me double-check my reasoning. Suppose that I bet a fraction F of my credits in one round, and I start that round with C credits. That means at the end of that round, on average, I have:

average after one round = 0.99 * 1.10 * F * C + 0.01 * 0 * F * C + (1-F) * C
= (0.99 * 1.10 * F + (1-F) ) C = C + 0.089 F C

This is clearly maximized when F is as large as possible, which is F=1, so I clearly should bet all my credits on the first round to maximize the average utility! That again seems to imply I should always bet all my credits on every round, since each round is identical and independent from the others, so the decision should always be the same as in the first round.

Hmm, still seems odd. To be sure, maybe I need to think about what happens if I play k rounds in a row, rather than just one.

Let’s suppose I play for two rounds in a row, and bet a fraction F1 on the first round, and a fraction F2 on the second round. Let M1 be the multiplier I happen to get on the credits the first round (1.10 with 0.99 probability or 0 with 0.01 probability), and M2 the multiplier on the second round (which is identical to but independent from M1). Then, after two rounds where I bet first F1 then F2, my average payout is:

average after two rounds = average[ M2 * F2 * (M1 * F1 * C + (1-F1) * C) + (1-F2) * (M1 * F1 * C + (1-F1) * C)] = ( 1 + 0.089 F1 + 0.089 F2 + 0.089^2 F1 F2) C

Which, since all the coefficients are positive, will always increase as F1 increases and as F2 increases. So to maximize the expected utility, we want to use the maximum values for F1 and F2, which are just F1=1 and F2=1, implying again that we should bet all our credits on each round even if we’re playing for two rounds!

If I bet all the credits I have at each round, then the expected value after playing k rounds will be my original starting credits multiplied by the product of the expected values of M1, M2, etc., since each is independent from the others (and the expected value of a product of independent variables is the product of the expected values). Therefore:

average after k plays = 100 * 1.089^k

Wow, so the expected utility really will grow exponentially, and the more times I play, the greater that utility will be, on average! And even if there is some other way to produce an equal amount of utility in this game, through some other betting strategy, this approach will clearly do it in fewer plays than any other – and since I myself get slight disutility being stuck here, forced to play this game, all utility (including my own as a tiny part of that) is maximized playing in the more efficient way.”

He triple and then quadruple checks his math. Satisfied that he’s really found the utilitarian solution, he starts to play the game, betting all of his credits in each round.

After 50 rounds, he’s up to 11,739 credits. After 100 rounds, he’s at 1,378,061 credits, equivalent to more than a billion happy lives. He’s extremely tempted to leave at that point with his utility winnings, but he rechecks his calculation again, and it tells him that the average utility maximizing action (i.e., the most ethical one according to his moral philosophy) is to keep playing. At round 118, he loses, and his credits are all wiped out. With no more credits to bet, he has no choice but to leave. He returns through the door to his normal life, and the world is no better off than when the game began.

His strategy of maximizing total average utility appears to be required of him as the most ethical action according to his utilitarian ethical beliefs, yet it seems to be the worst strategy possible in Omega’s game, guaranteeing that the lowest possible amount of utility will be created with 100% certainty.

Interestingly, I think the same scenario still works (i.e., presents a problem and seeming paradox for the utilitarian) even if Omega says that the game also has a chance of ending on its own (if you decide never to end it on purpose), with the probability of the game ending after n rounds are completed of 1/(n*(n+1)). So if you never end the game on purpose, the game has 1/2 probability of automatically ending after the first round, a 1/6 probability of automatically ending after 2 rounds, a 1/12 probability of automatically ending after 3 rounds, and so on. Other than that change, though, the rules are exactly the same (so you can still choose to end the game any time you want, it’s just that now the game might also end after one of the rounds automatically, even if you want to keep playing, and the game ending automatically has the same result as if you ended it on purpose). Note that this specific probability distribution (of the game ending automatically) is special because it implies that the expected value for the number of rounds played is still infinite, even though the game might automatically end after any round.

This piece was first written on April 7, 2018, and first appeared on my website on September 11, 2025.

What are the possible explanations for (or causes of) this?

• A Causes B – Increasing A is a cause of increasing B but not the reverse. [e.g., more happiness causes more willpower]
• (2) B Causes A – Increasing B is a cause of increasing A but not the reverse. [e.g., more willpower causes more happiness]
• (3) A Causes B Causes A – Increasing A and B are both causes of increases of the other, leading to a positive feedback loop between them. [e.g., more willpower causes more happiness which causes more willpower which causes more happiness, etc.]
• (4) X Causes A and B – There exists at least one other variable X (potentially many more than one) such that increases (or decreases) in X lead simultaneously to increases in both A and B. [e.g., living in a stable environment and having a supportive romantic partner are both situations that increase happiness and are also both situations that increase willpower]
• (5) Non-linearity – A and B actually have a much stronger relationship than it appears (in fact they could be fully deterministically related) because correlation only captures the average extent to which A exceeding its mean coincides with B exceeding its mean, and can understate the strength of relationships that go both up and down or that go up in an inconsistent fashion (that is, it captures linear relationships, but can sometimes mask non-linear ones). [e.g., as willpower goes up happiness tends to go up as people become increasingly good at making good longer-term effective choices, but as willpower gets to extremely high levels, people start experiencing less happiness again because extreme levels of willpower are linked to reduced emotional responses of all kinds]
• (6a) Noise – The correlation is actually much larger or smaller than it seems as a result of sampling error or noise (e.g., the sample size is too low to measure it with reliability, and so it is really more like 0.32 plus or minus 0.3 and could realistically be near zero) [e.g., with a sample size of n=30, a measured correlation of 0.32 has an insanely wide 95th percentile confidence interval of -0.045 to 0.609]
• (6b) Too Many Tries – You found this correlation by looking at a long list of correlations that you calculated (and it stood out to you because it was one of the highest correlations), but there is actually not a significant correlation at all, it just looks that way because you tested so many hypotheses and each one had some chance of looking significant due to random fluctuations (so it’s unsurprising that one ended up being moderately large, and that’s the one you honed in on) [e.g., if you look at all pairs of correlations for 15 variables that leads to 105 distinct correlations, and with a sample size of 50 data points used to calculate each correlation, it is quite likely that you’d get a correlation of 0.32 or higher even if the true correlation between all pairs of variables was 0]
• (7) Outliers – There is an extreme outlier in the data (e.g., a transcription error or data corruption or disgruntled survey respondent), and without it, the calculated correlation would decrease or increase dramatically. [e.g., with n=100 data points, a crazy outlier pair of 30 for A and 1.35 for B when the rest of your values are between 0 and 1 can make the calculated correlation come out to 0.32 when the correlation calculated without that one outlier is actually -0.12]
• (8a) Selection Bias – Your data is a biased sample of the population that you actually care about due to unintentional selection effects in your sampling procedure, and so the 0.32 correlation, which you assumed applies to the broader population from which your data was drawn, is not actually representative of the population you are attempting to draw conclusions about. [e.g., you inadvertently surveyed only younger people, and in younger people, the correlation between willpower and happiness is positive, whereas in older people it is a negligible correlation, yet you’re trying to draw conclusions about the entire population which includes both young and old]
• (8b) Correlation From Conditioning – The variables A and B are actually uncorrelated if you had looked at them in the full population of interest, but you only sampled them in some subpopulation out of convenience, and A and B are both correlated with how likely you are to end up in that subpopulation, creating an observed correlation in the subpopulation that doesn’t exist in the full population [e.g., happiness and willpower are actually uncorrelated, but you only collected data on people at camp, and whether or not a person decides to go to camp is strongly positively predicted by happiness and strongly negatively predicted by willpower; therefore campers are likely to have high happiness or low willpower (though not both) creating a correlation between happiness and willpower among campers that doesn’t exist among non-campers]
• (9) The World Is Too Complex For Mere Mortals – Some mixture of (1), (2), (3), (4), (5) and (6a), (6b), (7), (8a), and (8b) are true.
• (10) Wrong Problem – Of the literally infinite variety of formulas that capture the extent to which A and B vary together, the correlation (i.e., the average of the product of standard deviations from the mean) is not the formula you are truly interested in, what you actually care about is, say, the median of the product of absolute deviations from the median, but you calculated the ordinary correlation because that’s all that you could think of to do, or that’s the only relevant function the software you used had available.
• (11) Undefined – A and B were generated from Cauchy distributions, and the true correlation between these two variables is mathematically undefined (much like how the Cauchy distribution has an undefined expected value), so your calculated value of 0.32 is a meaningless estimate of an undefined quantity.
• (12) Bug – there was a bug in the correlation calculation or data loading code, or data cleaning code and your software just happened to output 0.32, but that’s not actually the correlation value or a value of interest at all.
• (13) Wrong Calculation – You only thought you had run your correlation coefficient code, but you accidentally ran the wrong program (the one you wrote as a teen), and the number 0.32 was the fraction of baseball cards you had at that time that were in mint condition.
• (14) Memory Mistake – Your faulty memory only makes you think the correlation between A and B is 0.32, when in fact, the correlation number you saw on your computer screen (a mere 60 seconds ago) was 0.23 (thanks be to the unreliable three pounds of jello between your ears).
• (15) Dreaming – You are actually asleep right now (with your head slumped down on your keyword, and your knocked-over cup of tea dripping Earl Grey on the carpet), and you only dreamed that you calculated a 0.32 correlation, but when you wake up the correlation will turn out to be, say, -0.134 (or maybe you never collected the data in the first place and will get an F on your assignment).
• (16) God – God, it turns out, predetermined everything, and is, therefore, the one and only cause, so is the true (and only) explanation for this correlation of 0.32.
• (17) Solipsism – Metaphysical Solipsism turns out to be true, and the world has no independent existence outside of your mind, so in effect, you are the only cause, and therefore you are the cause of this experience of observation of 0.32 correlation (but what does it say about you that you would create this experience?)
• (18) Many Worlds – The Many World’s hypothesis is true, and the laws of physics don’t forbid any possible values of this particular correlation, hence for any number r within the range -1 to 1, there is a you who witnessed that correlation r, but most you’s most likely witnessed a correlation close to 0.32 (as evidenced by the fact that you witnessed it and that you are unlikely to have witnessed an unlikely event).
• (19) Simulation – The Simulation Hypothesis is true, you are part of a simulation, and the (simulated) world you live in was designed by conscious beings to have a 0.32 correlation between variables A and B (the true reason for which you will surely never understand, perhaps it has to do with getting published in a 5th dimension outer-reality academic journal on the niche topic of civilizational development in 4d space-time universes).
• (20) Relativism – Truth and meaning are relative and culturally determined, and the definition and usage of correlation, as well as the ritualistic “scientific” process of calculating it, and the belief that it should be calculated at all, are merely an artifact of your time and place, which props up the power of the current ruling elite, yet is so deeply rooted in your culture that you fail to see any other possibility.
• (21) Evil Demon – Descartes’ Evil Demon, as “clever and deceitful as he is powerful,” has been tricking you for the entirety of what you call your life, and all that you think you know of the external world is merely an illusion designed by this demon, including this godforsaken correlation.
• (22) Boltzmann Brains – The universe is infinite (in space, or time) and hence contains an infinite number of brains produced by the brief chance alignment of particles into structures capable of consciousness, and given that only a finite number of brains could evolve on planets, you have a 100% chance of being one of these randomly coalescing “Boltzmann brains,” and you just happen to be one that momentarily believes there is such a thing as correlation and that you measured a 0.32 correlation between A and B (in a moment you will pop back out of existence – bye!)
• (23) Contradiction – There is an as of yet undiscovered (but as the Incompleteness Theorem shows, impossible to rule out) contradiction in the axioms of mathematics as we know it, and a valid mathematical proof exists to show that A and B have any correlation including -1, Pi or 0.32.
• Or perhaps I’m overthinking this, and just: “as A goes up, B tends to go up a bit, on average.”