Very small versus very very small benefits

If the best way to spend $4 billion is to buy everyone a burrito, is the best way to spend $40 million to buy everyone a few beans?

Imagine you have $4 billion to give away, but you face a strange constraint on how you can donate it. You only get to pick a number, call it \(n\), between 1 and 340 million. The $4 billion will be evenly divided among \(n\) randomly selected Americans.¹ No transaction cost to redeem — the money will just show up in each person’s checking account or wallet or whatever. If you pick \(n = \text{340 million}\), then every American gets about $11.75, roughly enough to buy an entree at Chipotle. Anything above that means fewer people get more money each.

Which choice of \(n\) will lead to the greatest improvement in total well-being?

The answer seems easy if you believe — as I do, and as I suspect most social scientists and philosophers do — that people have diminishing marginal utility to money. Bernoulli makes the argument in addressing the St. Petersburg paradox:

There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount.

Redistributive policies, notably progressive income taxation, are built on the idea of diminishing marginal utility to money. To the extent that it’s coherent to make this sort of interpersonal comparison, total happiness is greater when two people each have $500 than when one has $1000 and the other has nothing.

If you accept that there’s diminishing marginal utility to money, then you should also conclude that the most altruistic choice is \(n = \text{340 million}\). The best you can do is to buy every American a burrito. If you’re skeptical, see below for a proof.

Via DALL-E, prompted with “Make a picture of 340 million burritos falling down from the sky like a gift from the gods.”

This doesn’t seem crazy to me, even if I’m skeptical. Once every couple of years, I go to Chipotle and place an order, only to be told at the end that the card reader is broken so my meal is free. It makes me pretty happy when this happens,² even though I’ll spend $12 on a lunch burrito without a second thought. Multiply the free-lunch happiness across 340 million people, and you’ve brought lots of joy to the public. Probably a lot more overall joy than if you’d given 100 free burrito lunches to 1% of people, and nothing to the other 99%.³

The mathematical logic here doesn’t depend on the amount you’re divvying up — yet I have a much harder time accepting the claim at smaller scale. Imagine you had $40 million to give out, one-hundredth the amount we originally thought about. Now if you choose \(n = \text{340 million}\), every American ends up with about 12 cents, roughly equivalent to an extra spoonful of beans on their burrito.

This is where I can’t follow my own logic. I believe in the diminishing marginal utility of money. I’m not a utilitarian, but I can act like one here. And yet I can’t bring myself to pick \(n = \text{340 million}\) here. I can’t help but think total happiness would be greater if 1% of people got a free lunch and 99% got nothing than if everyone got less-than-a-hill of beans.

Maybe the issue is that the spoonful of beans is imperceptible? As with most questions of moral mathematics, Derek Parfit has something to say about imperceptible benefits. He argues from a thought experiment he calls “The Drops of Water”:

A large number of wounded men lie out in the desert, suffering from intense thirst. We are an equally large number of altruists, each of whom has a pint of water. We could pour these pints into a water-cart. This would be driven into the desert, and our water would be shared equally between all these many wounded men. By adding his pint, each of us would enable each wounded man to drink slightly more water — perhaps only an extra drop. Even to a very thirsty man, each of these extra drops would be a very small benefit. The effect on each man might even be imperceptible. (Reasons and Persons §28)

Parfit concludes that the moral thing to do is to pour your pint into the cart. His claim follows from two lines of argument.

1. Imperceptible benefits are still benefits. To believe otherwise, you must admit that “at least as good as” and “no better than” are not transitive, such that some state of affairs \(x\) may be at least as good as the state of affairs \(y\), and \(y\) is at least as good as \(z\), yet \(x\) is not at least as good as \(z\). In other words, you’re trapped in the Sorites paradox.

2. Reasoning from large group actions. If you read Parfit’s parable closely, you’ll notice he specifies “an equally large number of altruists,” so that if each of them pours in a pint, each wounded man ends up with a pint. Parfit relies on logic reminiscent of the categorical imperative. To state his argument loosely: if it’s better for every altruist to pour his pint in than for none to do so, then it’s also better for each individual to pour his pint in than not.

Argument 1 is irrelevant for the random-money-splitting problem I’ve set up here. I’m working with a utility model that meets the transitivity assumptions we would want. Parfit is comparing giving the imperceptible benefit to withholding it. I’m comparing different divisions of benefits, some imperceptible and some not.

Argument 2 also seems irrelevant here. If you apply large-group reasoning to the random-money-splitting problem, you come to the conclusion that the choice of \(n\) doesn’t really matter. If some large number of decision-makers each controls a fraction of the budget to be spent, and each chooses a number of people among whom their fraction will be randomly divided, then the law of large numbers implies that each person will receive very close to \(1/n\) of the budget regardless of the scheme each individual donor uses.

OK, so maybe Parfit’s arguments for the Drops of Water problem don’t overcome my preference to hand out 3.4 million burritos instead of 340 million spoonfuls of beans. But nor does Parfitian logic provide an affirmative case for my approach to the problem. If I had to place bets on what Parfit himself would do here, I’d bet on the beans.

I think the answer is that I’ve talked myself into an apples-and-oranges comparison — a common pitfall of Parfit-style reasoning from moral word problems. The key, I suspect, is that knowing you’ve gotten a benefit is, itself, beneficial. Moreover, the add-on benefit from knowing you’ve lucked out doesn’t necessarily have the same kind of diminishing marginal utility. My subjective well-being is higher in the “oops, your meal is free because the card reader is broken” situation than if Chipotle secretly refunded my credit card transaction without my knowledge. Honestly, in terms of subjective well-being, I’d probably be happier if I knew I was getting a free soda than if they secretly refunded my whole meal. If I could choose which of those would happen to my future self, knowing I was about to get hit with the MiB memory eraser (so that my future self wouldn’t know my past choice was responsible), I think I’d pick the free soda.

It gets easier to make the “right” choice — to give everyone the spoonful of beans — if I assume away any kind of noticeability, so that the direct benefits are realized but not the “I got this free thing!” warm glow. Given a choice between (a) everyone’s burrito has 1% more filling, unbeknownst to them and (b) 1% of people’s burritos are secretly refunded later that night, I’m choosing (a). Or at least I think I am.

Appendix: Proof

Assume there is a finite collection of utility functions \(\{u_1, \ldots, u_m\}\). A benevolent, risk-neutral social planner has a budget \(B > 0\) and chooses a number \(n \in \{1, \ldots, m\}\) as the number of agents across whom to randomly and evenly allocate the budget.

For each agent \(i \in \{1, \ldots, m\}\) and each potential choice \(n \in \{1, \ldots, m\}\), let \(X_i^n\) be a random variable denoting the allocation to \(i\). Note that \(X_i^n = 0\) with probability \(\frac{m - n}{m}\) and \(X_i^n = \frac{B}{n}\) with probability \(\frac{n}{m}\). Given the choice of \(n\), expected social welfare is \[\begin{align} \mathbb{E}\left[\sum_{i=1}^m u_i(X_i^n)\right] &= \sum_{i=1}^m \mathbb{E}[u_i(X_i^n)] \\ &= \sum_{i=1}^m \left[\frac{m - n}{m} u_i(0) + \frac{n}{m} u_i\left(\frac{B}{n}\right)\right] \\ &< \sum_{i=1}^m u_i\left(\frac{m - n}{m} \cdot 0 + \frac{n}{m} \cdot \frac{B}{n}\right) \\ &= \sum_{i=1}^m u_i\left(\frac{B}{m}\right), \end{align}\] where the strict inequality follows from the strict concavity of each \(u_i\). Therefore, \(n = m\) is the unique social welfare maximizing choice.

AI addendum: At first I tried to go at this using Jensen’s inequality, but I couldn’t quite see how to apply it with heterogeneous utility functions. So I used ChatGPT’s o3-mini-high model — the one with the most sophisticated reasoning abilities I have access to — as an intuition pump. At first in our chat, it actually came the wrong conclusion, claiming that \(n < m\) might be optimal if some \(u_i\)’s are much flatter than others. This didn’t seem crazy to me, but I regained my confidence in my original hypothesis after it failed to come up with a counterexample (and in fact proved my claim) for \(m = 2\). With a bit more prompting I was able to get it to the proof strategy here.

Generative AI models have come a long way with their ability to do mathematical reasoning in the time since I first started playing with them three years ago. But even with a pretty simple problem like this one, the model still came to the wrong answer at first. It was nevertheless helpful — after additional probing and prompting.

This work is licensed under CC BY-NC-SA 4.0

Footnotes

1. I’m focusing on distribution among Americans to sideskirt issues of purchasing power parity.

2. Note the difference from “free burrito day” promotions, where you get a smaller-than-usual burrito for no money after waiting in a godawful long line. Between the waiting and the quality differential, the surplus consumer value there is less than the $12 ordinary cost — and in fact may be negative depending on your opportunity cost of waiting. In the card-reader-unexpectedly-down scenario, the cost I would have ordinarily paid is exactly the bonus I’m getting (at least as long as word hasn’t gotten out about the card reader, thereby lengthening the line).

3. Keep in mind that we’re assuming no ability to target. The analysis would be different if you could make sure the 100 lunches went to the 1% who most needed food.