Should You Bring an Umbrella? A Math for Decisions
A Rainy Day Dilemma
Imagine you are about to leave for school. The sky is a mix of clouds and sun. You would rather not carry a heavy umbrella if it stays dry, but getting soaked would be even worse. There are two things you can do: take the umbrella or leave it at home. The weather is out of your control — it might rain, or it might not. Depending on what the weather does and what you choose, you end up in one of three outcomes: dry and unencumbered (the best), dry but carrying an annoying umbrella (okay), or wet (the worst). How should you decide?
This everyday puzzle sits at the heart of a big idea in philosophy and economics: expected utility theory. It gives each possible action a single number — its expected utility — and says you should choose the action with the highest number. That number comes from two things: how much you value each possible outcome (its utility), and how likely each outcome is, given your choice. You can think of it as a mental calculator, weighing the good and bad possibilities. But this simple idea has sparked surprising debates about probability, value, and what it means to be truly rational.
What Are the Chances, Really? A Puzzle with Two Boxes
To calculate expected utility, you need to know how likely each outcome is. But what does “likely” even mean? You might think of it as your personal degree of belief — how confident you are that it will rain. Or you might think of it as an objective chance, like the true probability that a fair coin lands heads. Expected utility theory can work with either idea, but a famous puzzle called the Newcomb problem shows that the difference matters deeply.
Imagine a super‑intelligent predictor who can foresee your choices with 90 % accuracy. She puts a closed box and an open box in front of you. The open box contains $1,000. The closed box contains either $1 million or nothing. If the predictor thought you would take only the closed box (one‑boxing), she placed $1 million inside it. If she thought you would take both boxes (two‑boxing), she put nothing. You know all this. What should you do?
On one way of thinking, one‑boxing gives you a very high chance of getting $1 million, because the predictor would have predicted that. Here, your act is treated as evidence about what is already in the box — so the expected utility of one‑boxing is enormous. This is the approach of evidential decision theory. The other way, causal decision theory, says your choice cannot cause money to have been placed there seconds ago; the contents are already fixed. Two‑boxing always gives you $1,000 more than one‑boxing — no matter what. So causalists say two‑boxing is better. The same problem yields opposite advice depending on how you understand the probability inside the expected utility formula. This puzzle, introduced by philosopher Robert Nozick (1938–2002), has kept thinkers arguing for decades. It shows that the simple idea of “likelihood” is anything but simple.
What Is a “Utile”? Measuring What Matters to You
The other piece of expected utility is utility — the number that captures how good or bad an outcome is. But how do we assign a number to something like “dry and unencumbered” versus “wet”? The mathematician Daniel Bernoulli (1700–1782) argued that money has diminishing marginal utility. An extra dollar means a great deal to someone with little, but barely anything to a millionaire. So you cannot simply say that $1,000 is worth 1,000 “utiles.” Utiles are a unit of personal value, not cash.
Classical utilitarians like Jeremy Bentham (1748–1832) thought utility was a measure of pleasure or happiness. Later decision theorists argued that utility is simply a way to represent your preferences: if you prefer outcome A to outcome B, A gets a higher utility number. This avoids the messy job of comparing completely different kinds of things — friendship, health, ice cream — on a single scale. But there is a catch. If utilities are just numbers from your personal preferences, can we ever compare your utilities to your friend’s? That is the problem of interpersonal utility comparisons. If you would pay $10 for a concert ticket and your friend would pay $100, does that mean she values it more? Not necessarily; she might simply care less about money. This makes it hard to use expected utility to decide how to share resources fairly — a difficulty that still vexes economists and policy‑makers.
Why Should You Trust the Math? Arguments for Expected Utility
Why think that maximizing expected utility is the rational thing to do? One classic reply points to the long run. If you face the same kind of gamble over and over, like betting on a roulette wheel, the average payoff per gamble will, over many trials, almost certainly be very close to the expected utility of a single bet. This is guaranteed by the laws of large numbers. In repeated gambles, following expected utility maximizes your long‑term gain. But many of life’s biggest decisions — whom to marry, what career to pick — happen only once, not hundreds of times. So the long‑run argument may not cover all choices.
A deeper answer comes from representation theorems. Mathematicians like John von Neumann (1903–1957), Oskar Morgenstern (1902–1977), and Leonard Savage (1917–1971) proved that if your preferences obey a few simple rules (for instance, if you prefer A to B and B to C, then you must prefer A to C), then you can be represented as having a numerical utility for outcomes and a numerical probability for states — and your preferences look exactly as if you are maximizing expected utility. The idea is that rationality partly consists in having consistent preferences, and consistency forces expected utility. But critics point out that being representable as having certain beliefs and desires is not the same as actually having them. Moreover, many people’s preferences seem to violate these rules, yet they do not seem obviously irrational.
When the Calculator Gives Strange Advice
Expected utility can lead to very strange results. Consider the St. Petersburg game, proposed by Bernoulli. A coin is tossed until it lands tails. If it lands tails on the first toss, you win $2; on the second, $4; on the third, $8, doubling each time. The expected utility is an infinite sum: 1 + 1 + 1 + … forever. That means, according to the theory, you should pay any finite amount of money to play this game. But nobody would pay a million dollars for a single flip. Something seems wrong.
Other puzzles, like the Allais paradox (Maurice Allais, 1911–2010) and the Ellsberg paradox (Daniel Ellsberg, born 1931), show that people have preferences that conflict with expected utility but still seem sensible. In the Allais paradox, many prefer a sure $100 million to a gamble with a slightly higher expected value but a tiny chance of getting nothing — yet they prefer a different gamble when the sure thing is removed. These patterns violate the Independence axiom, a core rule of expected utility theory. Defenders of the theory say those preferences are irrational mistakes. Others, like philosopher Lara Buchak, have developed modified versions that allow us to be more cautious about low‑probability risks — a kind of risk‑aversion that expected utility finds hard to explain.
Perhaps the most unsettling challenge is the self‑torturer, a scenario from Warren Quinn (1938–1991). Imagine a dial with 1,000 settings, each slightly more painful than the last. Setting 0 is painless; setting 1,000 is agony. But the difference between any two neighboring settings is too small to notice. At each step, you are offered $10,000 to turn the dial up one notch. For each individual step, it seems rational to accept — the pain increase is imperceptible, and you get $10,000. Yet eventually you end up at setting 1,000, which you would never have chosen from the start. Expected utility theory demands that preferences be transitive (if you prefer A to B and B to C, you must prefer A to C). The self‑torturer’s pattern violates transitivity, but many people find it perfectly understandable — a direct challenge to the theory.
How Expected Utility Shapes Your World (and Your Thinking)
You might not realize it, but expected utility thinking surrounds you. In law, juries must decide whether evidence meets the standard of “beyond a reasonable doubt.” Decision theorists like John Kaplan (1968) argued that this standard can be understood as a calculation: how bad is it to convict an innocent person compared to acquitting a guilty one? The right threshold for guilt depends on those values. In medicine, doctors use quality‑adjusted life years (QALYs) — a measure that blends length of life and quality — to decide which treatments to fund. This is essentially expected utility applied to health choices, with utilities representing well‑being rather than personal preferences.
In your own life, every time you check the weather before deciding what to wear, you are informally doing an expected utility calculation. When you pick a video‑game character with a certain attack power and defense, you are weighing probabilities of winning against the utility of different play styles. Even deciding whether to study for a test based on the chance it will be a pop quiz is a tiny decision under uncertainty. Expected utility theory offers a powerful language for talking about such choices — and the debates it sparks remind us that what seems like simple math is actually a deep mirror of what we value, how we think about the future, and what it means to be rational.
Think about it
- If a friend offers you a 50/50 chance to win $20 or lose $10, would you take the bet? What does your answer say about how you weigh gains against losses?
- Imagine a machine that can perfectly predict any choice you will ever make. If such a machine existed, would it make sense to try to surprise it? What does that tell you about choice and prediction?
- A doctor has a treatment that gives a 10 % chance of curing a deadly disease but an 80 % chance of causing severe pain. How would you decide whether to use it? Can a formula capture everything that matters?
