You Should Be a Fanatic

Imagine you are offered the following two options, of which you can only choose one:

Certainty: I will flip a coin. If it lands either heads or tails, 100 people will get an additional year of happy life.

Risky: I will flip a coin 10000000000000000000000000000000000000000 times (10^40). If it lands heads every single time, 10^500 people will get an additional year of happy life.

Which should you choose? To see it from perhaps a more pragmatic perspective, if you had to give a dollar to charity, and one charity will definitely help 100 people, whilst another one has a very slim chance of helping an unbelievably, astronomically high number of people, which should you donate your dollar to (assuming it’s the dollar necessary to keep one of them permanently going)?

Intuitively, many will say Certainty is the better option. 100 additional lives for sure is surely better than what is an effective impossibility of some huge payout.

Fanaticism says that you should take Risky — a very tiny probability of a large enough payout is always better than certainty of a pretty good payout.

Despite its initial implausibility, though, I firmly think that upon reflection fanaticism is clearly true. The arguments in its favor are, to my mind, unbelievably strong. And there are so many, each of which converging on Fanaticism. The strength of the case for Fanaticism is that denying it leads to lots and lots deeply crazy results: bullets that I do not have the gall to bite.

And I hope to convince you of this. This post will go like this: I’ll present a barrage of arguments, showing just how absurd the consequences are of denying Fanaticism.

To those who do not want to read the entire post, this is basically what one will have to accept if they do not accept Fanaticism:

• If you are offered a 60% chance of saving 100,000 lives, or instead a 59.999% chance of saving a Googleplex to the power of a Googleplex number of lives, you should take the 60% chance. (Or: that transitivity is false).

• Your preferences should be such that you can be money-pumped: I can offer you a series of offers such that you’ll have lost money with 0 gain at all.

• In your assessment of two offers, you need to figure out what life was like for the Ancient Egyptians before you can know which you should choose

• Even though you should take Certainty over Risky, if your life is long enough, you should take Risky over Certainty

Most of the arguments for Fanaticism are going to be of the form “Look at the insane results you have to accept if you deny it.” However, first, I want to give an intuition pump for Fanaticism: I’d like to give you an intuitive feeling for why this would be true, which, even if it does not sway you to change your mind, will at least soften you up before you hear about what results from denying it.

Imagine you’re going to live two lives, Life1 and Life2. The only purpose of Life1 is to stack up as much money as possible to use in Life2. Let’s imagine that you live for 1 trillion days in Life1.

Every day, I give you the following offer between two options:

Certain Money: 100% chance of 100 dollars.

Risky Money: 1/trillion chance of 10^100 dollars.

Which do you take? Well, if you’re not a Fanatic, you’ll likely want to pick Certain Money every time. So, by the end of Life1, you’ll have 100 trillion dollars (10^14). If you’re a Fanatic, you’ll pick Risky Money every time. By the end of Life1, you’ll very likely have at least 10^100 dollars — in other words, you’ll very likely have at least trillion trillion trillion trillion trillion trillion trillion TIMES as many dollars (you might object that there is an off chance that you get nothing…but of course there’s also an off chance that you get 10^100 dollars twice, or three times, etc.). And, of course, the more times the bet is offered, the better and better the Fanatic does than the non-Fanatic.

This highlights a really important point: all a probability is is a frequency. It tells you the rate at which something happens. And if your goal is to maximize something, it is simply the case that, if the payout is big enough, you should take the lower rate with a bigger payout. Because, in expectation, it will simply lead to more!

Now, you might object here that if the offer between Certain Money and Risky Money is offered many times, then Risky Money is obviously better. But, in a one off, Certain Money is just better. But this is very odd. If Certain Money is just a better offer than Risky Money, why not take it every time? To illustrate: imagine that you are living through Life1, and you’re on day 20,000. If you really believe that Certain Money is better than Risky Money, shouldn’t you just take Certain Money today? Even if you take Risky Money every other day, and every day to come in the future — isn’t it better to just take Certain Money today? 999,999,999 bets on Risky Money and 1 bet on Certain Money is surely better than a trillion bets on Risky Money, if Certain Money is better than Risky Money. But then this argument can just be applied again and again to conclude that you should take Certain Money every time.

Hopefully this gives an intuition pump for why Fanaticism isn’t so weird: picking risky prospects, if the payout is big enough, is simply picking a rate of return that consistently gives you more money than any other strategy. This becomes obvious when we simply zoom out and think of things in the long run — but it’s hard to see a reason why the number of times I offer you a bet should change which one is better.

Nevertheless, worry not if you are not convinced by this intuition pump. Let’s get to the arguments.

A ‘money pump argument’ is an argument that attempts to show that your preferences should conform to a certain rule R, because if your preferences do not, your preferences will lead to you being exploited for free.

For instance, it is often thought that we can give a money pump argument for the rule that your preferences should be transitive: if you prefer A to B, and B to C, then you should also prefer A to C. Just imagine someone who prefers oranges to apples, apples to kiwi, but, prefers kiwi to oranges. If they have a kiwi, and I offer to swap out their kiwi for an apple (which they would prefer!) for a single penny, they do so (let us say they strongly prefer apples to kiwis). I now offer to swap out their apple with an orange, as long as they pay me another penny. They do so. But because they prefer kiwis to oranges, I now offer to give them back the kiwi for the orange, as long as they pay me a third penny: which they gladly do. They’ve now given me 3 pennies for free. And, of course, I can keep pumping them for all their money, while they sit their smiling  (irrationally!)

This seems like a pretty great argument that your preferences — at least, your strong preferences, the kinds you’d be willing to pay money to satisfy — should conform to transitivity.

If you agree, I’ve got bad news for you. It turns out that we can show that, if your preferences do not conform to 3 other conditions (what usually get called Continuity, Independence, and Completeness), then you can also get similarly money pumped. So your preferences should conform to Transitivity, Continuity, Independence, and Completeness. But it turns out that these 4 conditions are the 4 axioms of Expected Utility Theory! If you follow these 4 conditions, it strictly follows that you should be a Fanatic and prefer Risky to Certainty.

For purposes of space, you can find the proofs of these money pumps here. But if you take seriously the money pump argument against transitivity … you should likewise take seriously the problems the money pump arguments against denying Fanaticism!

The next argument for Fanaticism is a continuum argument. Imagine that you’ve chosen Certainty: I’m about to flip a coin, and no matter how it lands, 100 lives will be saved. However, before I flip, I offer you the option to switch out Certainty with the following prospect:

MoreLives₁: 99.9999% chance of saving 100,000 lives.

Would you take an almost identical probability of success (just 0.0001 less) to save a thousand times as many lives? Well … that sure seems way better! So, you take MoreLives₁. But, before we proceed, I offer you:

MoreLives₂: 99.9998% chance of saving 100 million lives.

Again, you ecstatically take MoreLives₂ over MoreLives₁: to save a thousand times as many lives for an almost identical probability is a no brainer.

But you likely see the problem now: I can keep doing this, reducing the probability of success by ever-so-slightly whilst also increasing the payout by 1000x. Eventually, we’ll reach Risky! By Transitivity, then, Risky must be better than Certainty.

If you want to avoid this conclusion, here’s what you’ll have to say (other than denying transitivity): if you have a (say) 60% chance of saving 10 million lives, that is always better than a 59.9999% chance of saving any number of lives! Imagine that in front of you is a button that, if pressed, has a 60% chance of saving 10 million lives. I then tell you that I can increase the number of lives that the button will save to be every human who has and who ever will be in danger: the button will prevent climate change, every existential risk, cure every disease, prevent every war, etc. (and there will be no negative effects from this). And the button still works with a 60% chance! The only catch is this: if the payout of the button is improved in this way, before pressing it, you and I have to each think of a 100-digit number. As long as we don’t think of the same number, you can press the button. But if we do, then the button becomes defective.

Would you play the 100-digit-number game, in order to save all of humanity with the button? Surely you should!

But, if you’re going to deny Fanaticism, you’re going to have to say that we should be timid and not play this game despite the prospect of saving all of humanity … or else, deny transitivity.

There’s another clever continuum argument in support of Fanaticism that I recently read in this paper which I found quite convincing. It’s short enough to paste in full, so I will do so:

Consider, for the sake of concreteness, the prospect 𝑃 of saving 1,000 lives with probability 0.1. A theory that is timid with respect to lives saved might say that 𝑃 is no worse than the prospect of saving even a vast number of lives with probability 0.099, only one percent smaller. We can imagine that the outcome of 𝑃 depends on the outcome of a raffle: if the golden ticket is drawn, then a mechanism will be activated that will save 1,000 lives. Now consider a prospect 𝑃1 that differs from 𝑃 in two ways: first, the mechanism is designed to save many more lives—let’s say 10,000; but second, there is a one-percent failure rate, and in the fail-state, 𝑃1 saves only 999 lives. So 𝑃1 almost certainly leads to ten times as many lives saved as in 𝑃, and at worst, in the highly unlikely fail-state, it will save only one life fewer. Intuitively, 𝑃1 is much better than 𝑃. But now consider a prospect 𝑃2. In 𝑃2, the mechanism usually saves 100,000, but it only saves 998 in the failstate. So, compared to 𝑃1, there are almost certainly 10 times as many lives saved, and at worst one fewer. 𝑃2 is clearly better than 𝑃1, so also better than 𝑃. But if we continue this sequence along, we get to 𝑃1000, in which no lives are saved in the fail-state. Thus 𝑃1000 leads to a vast number of lives saved (namely, 101003) with probability 0.099. By transitivity, 𝑃1000 is better than 𝑃. And so, the argument concludes, our theory of value must not be timid in this particular way. Nor is there anything special about this case; mutatis mutandis, we have an argument against timidity in general.

The last argument for Fanaticism I’ll give you is that, if you deny it, you’ll have to say that what is going on in distant galaxies should affect which choices you should make, even if your choices have zero influence on what goes on in those galaxies. [I adapted this argument from here and here].

To see why, imagine you are offered two prospects, where 3 different results are possible:

Seems pretty clear a 95% chance of saving 1 million lives is better than a 0.1% chance of saving 1 life. But now imagine I inform you of the following: on Krypton, there’s someone who is about to press a button, which has a 95% chance of saving 1 life.

So, really, your decision now looks like this:

Now, your decision is between a 95% chance of 1 million and 1 lives, or a 95.1% chance of saving 1 life. Of course, if we are not timid — if we are willing to take a 0.1% hit in probability to save a million more lives — MoreLives is still the better choice, and what is going on in Krypton doesn’t change anything. But, as we learned from the previous continuum argument, those who reject Fanaticism must be timid (or else reject transitivity)! They must say that there are cases where a 0.1% hit in probability is never worth it, even if it leads to much larger payouts. But that means that MoreLives, because of what is about to happen on Krypton, is now worse than LessLives.

If you reject Fanaticism, you have to either reject transitivity, or else you have to say that, before we decide what is the better decision, we must first figure out what is going on in causally isolated regions of space that are completely unaffected by what decisions we make!