Pascal’s Mugging

Nov 10 JDN 2458798

In the Singularitarian community there is a paradox known as “Pascal’s Mugging”. The name is an intentional reference to Pascal’s Wager (and the link is quite apt, for reasons I’ll discuss in a later post.)

There are a few different versions of the argument; Yudkowsky’s original argument in which he came up with the name “Pascal’s Mugging” relies upon the concept of the universe as a simulation and an understanding of esoteric mathematical notation. So here is a more intuitive version:

A strange man in a dark hood comes up to you on the street. “Give me five dollars,” he says, “or I will destroy an entire planet filled with ten billion innocent people. I cannot prove to you that I have this power, but how much is an innocent life worth to you? Even if it is as little as $5,000, are you really willing to bet on ten trillion to one odds that I am lying?”

Do you give him the five dollars? I suspect that you do not. Indeed, I suspect that you’d be less likely to give him the five dollars than if he had merely said he was homeless and asked for five dollars to help pay for food. (Also, you may have objected that you value innocent lives, even faraway strangers you’ll never meet, at more than $5,000 each—but if that’s the case, you should probably be donating more, because the world’s best charities can save a live for about $3,000.)

But therein lies the paradox: Are you really willing to bet on ten trillion to one odds?

This argument gives me much the same feeling as the Ontological Argument; as Russell said of the latter, “it is much easier to be persuaded that ontological arguments are no good than it is to say exactly what is wrong with them.” It wasn’t until I read this post on GiveWell that I could really formulate the answer clearly enough to explain it.

The apparent force of Pascal’s Mugging comes from the idea of expected utility: Even if the probability of an event is very small, if it has a sufficiently great impact, the expected utility can still be large.

The problem with this argument is that extraordinary claims require extraordinary evidence. If a man held a gun to your head and said he’d shoot you if you didn’t give him five dollars, you’d give him five dollars. This is a plausible claim and he has provided ample evidence. If he were instead wearing a bomb vest (or even just really puffy clothing that could conceal a bomb vest), and he threatened to blow up a building unless you gave him five dollars, you’d probably do the same. This is less plausible (what kind of terrorist only demands five dollars?), but it’s not worth taking the chance.

But when he claims to have a Death Star parked in orbit of some distant planet, primed to make another Alderaan, you are right to be extremely skeptical. And if he claims to be a being from beyond our universe, primed to destroy so many lives that we couldn’t even write the number down with all the atoms in our universe (which was actually Yudkowsky’s original argument), to say that you are extremely skeptical seems a grievous understatement.

That GiveWell post provides a way to make this intuition mathematically precise in terms of Bayesian logic. If you have a normal prior with mean 0 and standard deviation 1, and you are presented with a likelihood with mean X and standard deviation X, what should you make your posterior distribution?

Normal priors are quite convenient; they conjugate nicely. The precision (inverse variance) of the posterior distribution is the sum of the two precisions, and the mean is a weighted average of the two means, weighted by their precision.

So the posterior variance is 1/(1 + 1/X^2).

The posterior mean is 1/(1+1/X^2)*(0) + (1/X^2)/(1+1/X^2)*(X) = X/(X^2+1).

That is, the mean of the posterior distribution is just barely higher than zero—and in fact, it is decreasing in X, if X > 1.

For those who don’t speak Bayesian: If someone says he’s going to have an effect of magnitude X, you should be less likely to believe him the larger that X is. And indeed this is precisely what our intuition said before: If he says he’s going to kill one person, believe him. If he says he’s going to destroy a planet, don’t believe him, unless he provides some really extraordinary evidence.

What sort of extraordinary evidence? To his credit, Yudkowsky imagined the sort of evidence that might actually be convincing:

If a poorly-dressed street person offers to save 10(10^100) lives (googolplex lives) for $5 using their Matrix Lord powers, and you claim to assign this scenario less than 10-(10^100) probability, then apparently you should continue to believe absolutely that their offer is bogus even after they snap their fingers and cause a giant silhouette of themselves to appear in the sky.

This post he called “Pascal’s Muggle”, after the term from the Harry Potter series, since some of the solutions that had been proposed for dealing with Pascal’s Mugging had resulted in a situation almost as absurd, in which the mugger could exhibit powers beyond our imagining and yet nevertheless we’d never have sufficient evidence to believe him.

So, let me go on record as saying this: Yes, if someone snaps his fingers and causes the sky to rip open and reveal a silhouette of himself, I’ll do whatever that person says. The odds are still higher that I’m dreaming or hallucinating than that this is really a being from beyond our universe, but if I’m dreaming, it makes no difference, and if someone can make me hallucinate that vividly he can probably cajole the money out of me in other ways. And there might be just enough chance that this could be real that I’m willing to give up that five bucks.

These seem like really strange thought experiments, because they are. But like many good thought experiments, they can provide us with some important insights. In this case, I think they are telling us something about the way human reasoning can fail when faced with impacts beyond our normal experience: We are in danger of both over-estimating and under-estimating their effects, because our brains aren’t equipped to deal with magnitudes and probabilities on that scale. This has made me realize something rather important about both Singularitarianism and religion, but I’ll save that for next week’s post.

What if the charitable deduction were larger?

Nov 3 JDN 2458791

Right now, the charitable tax deduction is really not all that significant. It makes donating to charity cheaper, but you still always end up with less money after donating than you had before. It might cause you to donate more than you otherwise would have, but you’ll still only give to a charity you already care about.

This is because the tax deduction applies to your income, rather than your taxes directly. So if you make $100,000 and donate $10,000, you pay taxes as if your income were $90,000. Say your tax rate is 25%; then you go from paying $25,000 and keeping $75,000 to paying $22,500 and keeping $67,500. The more you donate, the less money you will have to keep.

Many people don’t seem to understand this; they seem to think that rich people can actually get richer by donating to charity. That can’t be done in our current tax system, or at least not legally. (There are fraudulent ways to do so; but there are fraudulent ways to do lots of things.) Part of the confusion may be related to the fact that people don’t seem to understand how tax brackets work; they worry about being “pushed into a higher tax bracket” as though this could somehow reduce their after-tax income, but that doesn’t happen. That isn’t how tax brackets work.

Some welfare programs work that way—for instance, seeing your income rise high enough to lose Medicaid eligibility can be bad enough that you would prefer to have less income—but taxes themselves do not.

The graph below shows the actual average tax rate (red) and marginal tax rate (purple) of the current US federal income tax:

Average_tax_rate
From that graph alone, you might think that going to a higher tax bracket could result in lower after-tax income. But the next graph, of before-tax (blue) and after-tax (green) income shows otherwise:

After_tax_income

All that tax deductions can do is shift you left on the green line. Without the tax deduction, you would instead shift left on the blue line, and then read off your position on the green line. Thus the tax deduction benefits you if you were already donating, but never leaves you richer than you would have been without donating at all.

For example, if you have an income of $700,000, you would pay $223,000 in taxes and keep $477,000 in after-tax income. If you instead donate $100,000, your adjusted gross income will be reduced to $600,000, you will only pay $186,000 in taxes, and you will keep $414,000 in after-tax income. If there were no tax deduction, you would still have to pay $223,000 in taxes, and your after-tax income would be only $377,000. So you do benefit from the tax deduction; but there is no amount of donation which will actually increase your after-tax income to above $477,000.

But we wouldn’t have to do it this way. We could instead apply the deduction as a tax credit, which would make the effect of the deduction far larger.

Several years back, Miles Kimball (an economist who formerly worked at Michigan, now at UC Boulder) proposed a quite clever change to the tax system:

My proposal is to raise marginal tax rates above about $75,000 per person–or $150,000 per couple–by 10% (a dime on every extra dollar), but offer a 100% tax credit for public contributions up to the entire amount of the tax surcharge.

Kimball’s argument for the policy is mainly that this would make a tax increase more palatable, by giving people more control over where their money goes. This is surely true, and a worthwhile endeavor.

But the even larger benefit might come from the increased charitable donations. If we limited the tax credit to particularly high-impact charities, we would increase the donations to those charities. Whereas in the current system you get the same deduction regardless of where you give your money, even though we know that some charities are literally hundreds of times as cost-effective as others.

In fact, we might not even want to limit the tax credit to that 10% surcharge. If people want to donate more than 10% of their income to high-impact charities, perhaps we should let them. This would mean that the federal deficit could actually increase under this policy, but if so, there would have to be so much money donated that we’d most likely end world hunger. That’s a tradeoff I’m quite willing to make.

In principle, we could even introduce a tax credit that is greater than 100%—say for instance you get a 120% donation for the top-rated charities. This is not mathematically inconsistent, though it is surely a very bad idea. In that case, it absolutely would be possible to end up with more money than you started with, and the richer you are, the more you could get. There would effectively be a positive return on charitable donations, with the money paid for from the government budget. Bill Gates for instance could pay $10 billion a year to charity and the government would not only pay for it, but also have to give him an extra $2 billion. So even for the best charities—which probably are actually a good deal more cost-effective than the US government—we should cap the tax credit at 100%.

Obvious choices for high-impact charities include UNICEF, the Red Cross, GiveDirectly, and the Malaria Consortium. We would need some sort of criteria to decide which charities should get the benefits; I’m thinking we could have some sort of panel of experts who rate charities based on their cost-effectiveness.

It wouldn’t have to be all-or-nothing, either; charities with good but not top ratings could get an increased deduction but not a 100% deduction. The expert panel could rate charities on a scale from 0 to 10, and then anything above 5 gets an (X-5)*10% tax credit.

In effect, the current policy says, “If you give to charity, you don’t have to pay taxes on the money you gave; but all of your other taxes still apply.” The new policy would say, “You can give to a top-impact charity instead of paying taxes.”

Americans hate taxes and already give a lot to charity, but most of those donations are to relatively ineffective charities. This policy could incentivize people to give more or at least give to better places, probably without hurting the government budget—and if it does hurt the government budget, the benefits will be well worth the cost.