Subjective Probabilities should be Sharp

By Vasco Grilo🔸 @ 2026-07-10T16:47 (+9)

This is a linkpost to https://www.princeton.edu/~adame/papers/sharp/elga-subjective-probabilities-should-be-sharp.pdf

This is a linkpost for Subjective Probabilities should be Sharp by Adam Elga, which was originally published in Philosophers' Imprint in May 2010. Here is an errata for it. Below is a summary from Claude Opus 4.8 High. Adam said "I took a quick look and at first glance I saw nothing wrong with the summary". I also think the summary is accurate based on my read of the article. I used the following prompt. "Hi. Make an in-depth summary of the paper "Subjective Probabilities should be Sharp", which I send attached".

I very much agree subjective probabilities should be sharp. So I am not concerned about the unawareness argument for "no impartial altruistic justification for preferring any action over another", which relies on unsharp probabilities.

The target debate

Elga takes aim at a popular view in formal epistemology about how to respond to weak or unspecific evidence. He opens with a contrast between three kinds of evidential situations. Sometimes evidence is sharp (you've watched a biased coin land heads 83% of thousands of tosses, so your credence should be ~83%). Sometimes it's sparse but with a clear upshot (you have almost no evidence about whether the number of humans born in 1984 was even, but symmetry still pushes you to ~50%). And sometimes it's sparse and unspecific—his memorable example is a stranger pulling objects from a bag (a regular tube of toothpaste, a live jellyfish, a travel-sized tube of toothpaste), where there's no obvious basis for any particular credence that the next object is toothpaste.

In that third kind of case, many philosophers find it natural to say your degree of belief shouldn't be any single precise number but rather should be indeterminate, vague, or interval-valued—represented by a range like [10%, 80%] (or, more powerfully, by a set of probability functions rather than one). Elga cites Joyce, Levi, Sturgeon, and Walley as defenders of this idea. Their shared claim is not that some computational or representational limitation stops you from forming a precise credence; it's that the evidence itself makes a precise credence unwarranted, no matter how idealized the agent is.

Elga distills the view into a deliberately cautious thesis:

• UNSHARP: It is consistent with perfect rationality to have unsharp degrees of belief.

His own position is the negation:

• SHARP: Perfect rationality requires sharp degrees of belief.

An important clarification

Before arguing, Elga separates SHARP from a stronger doctrine called Uniqueness (the idea that each body of evidence permits exactly one rational credence function). SHARP does not say there's only one permissible function per evidential situation; it allows that several functions might be permissible. It just insists that whichever one(s) you adopt must each be perfectly precise. So this is not a defense of a uniquely rational credence—only of precision.

The strategy: demand a decision rule

Elga's whole case rests on a challenge. On the standard story (expected utility theory), a rational agent's beliefs are a probability function, and she acts to maximize expected utility—and the paper simplifies by assuming utility is linear in dollars. Anyone who says rational agents can have unsharp credences owes a corresponding account of how unsharp probabilities constrain rational action. Elga argues that no acceptable such account exists. If there's no good answer to "how do unsharp credences guide choice?", then the view collapses.

The "great series of bets"

The engine of the argument is a sequential betting setup on some proposition H (say, that it rains tomorrow):

• Bet A: If H is true you lose $10; otherwise you win $15.
• Bet B: If H is true you win $15; otherwise you lose $10.

You're told the full setup in advance, Bet A is offered first and Bet B immediately after, and crucially your opinion about H won't change during the process (no new evidence, no reinterpretation, just the passage of time). The bets are mirror images, so accepting both guarantees a net $5 gain no matter how H turns out.

Elga grants you're not required to accept both (a very confident or very doubtful agent might prefer just one). But he insists on the key premise: a rational agent must accept at least one of the two bets, because rejecting both is dominated—it's worse than accepting both in every outcome, and you can see this in advance. This premise is easy for a sharp-credence theorist to honor. The rest of the paper argues that no version of the unsharp view can.

To set up the problem: suppose your credence is the wide interval P(H) = [10%, 80%]. A sharp agent evaluates Bet A by a clean threshold—accept if P(H) < 60%, reject if P(H) > 60%, optional at exactly 60%. (Bet A's expected value is positive precisely when P(H) is below 60%.) Your interval straddles 60%, and that's where the trouble starts.

Permissive rules are too permissive

The first and most natural family of rules: since your interval spans the 60% threshold, neither accepting nor rejecting Bet A is mandatory—the bet is optional. By the same reasoning Bet B is optional too. But then it's permissible to reject both. Elga says this is plainly absurd: a money-loving agent who knowingly walks away from a guaranteed $5 has departed from perfect rationality. He notes that a wide swath of decision rules in the literature (Levi, Walley, Good, Seidenfeld, Gärdenfors–Sahlin, Gilboa–Schmeidler) deliver exactly this verdict, and so are unacceptable as accounts of ideal rationality. (He's careful to flag that some of these authors—e.g. Gilboa–Schmeidler—may only be theorizing about non-ideal agents, in which case his critique doesn't touch them; and that Levi explicitly embraces the reject-both consequence.)

Strict rules are too strict

At the opposite extreme is the midpoint rule: evaluate bets using the midpoint of your interval, so [10%, 80%] behaves like a precise 45%. This does yield the correct verdict (you'd never reject both bets, since no sharp agent does). But Elga argues it's self-defeating for the unsharp camp. The original motivation for unsharp credences was that the evidence fails to "nail down" any exact probability. Yet the midpoint rule lets the evidence nail down a completely precise pattern of betting odds: an H-ticket worth $100 if true gets valued at exactly $45.000…. If it's fishy for rationality to require an exact credence of 45%, it's equally fishy to require valuing the ticket at exactly $45. So strict rules buy the right behavior only by smuggling precise constraints back in—which removes any reason to have rejected precise credences in the first place. The midpoint rule (and its strict relatives) thus robs unsharpness of its point.

What's needed, and the three "global" attempts

So the unsharp theorist needs a rule that is strict enough to forbid rejecting both bets, yet permissive enough to leave a range of options open when, say, Bet B is offered alone—and that motivates both verdicts naturally. Elga sees only three candidate strategies, all "global" in that they assess choices in light of other (past or future) choices.

1. NARROW — Acting sharpens your interval. Reject Bet A and your P(H) narrows (e.g. to [60%, 80%]), which then disposes you to accept Bet B; in general your intervals contract so as to block predictably inferior sequences. This delivers the right strictness/permissiveness mix. Elga's objection: it forces a rational agent to change her opinion without any change in relevant evidence. His illustration: if your credence about rain is unsharp enough that wearing either a rain-poncho or a non-water-resistant suede jacket is permissible, NARROW says choosing the poncho makes you confident it'll rain and choosing the jacket makes you confident it won't—even though your choice of jacket carries no evidence about the weather (you have no rain-sensing powers, and your clothing doesn't control the sky). Rationality never requires revising an opinion when relevant evidence is unchanged, so NARROW fails.

2. PLAN — When you act, you simultaneously form a plan binding your later choices to cohere with it (reject Bet A → plan to accept Bet B → follow through), but without changing any beliefs. Elga refutes this with the case of Sally, who cares only about money and has a highly unsharp credence about rain. Compare two scenarios: in the first she rejected Bet A and is now offered Bet B; in the second she's offered Bet B alone. PLAN permits rejecting Bet B in the second but not the first. Yet the monetary consequences of accepting and of rejecting Bet B are identical across the two scenarios, and her beliefs are identical, and money is all she cares about—so the situations are alike in every respect she cares about. Rationality can't impose different requirements on choices that are identical in all relevant respects. To the rejoinder "but rejecting Bet B would break her plan," Elga replies that either plan-breaking is something Sally finds costly (contradicting the stipulation that it's costless for her), or it isn't—in which case "Don't break plans!" is as groundless a constraint as "Don't break mirrors!" He flags but sets aside the resolute-choice tradition (Gauthier, McClennen) that would defend plan-following.

3. SEQUENCE — Sequences of actions can be assessed for rationality independently of their parts: each of "reject A" and "reject B" can be individually permissible while the sequence "reject-A-then-reject-B" is impermissible. Elga turns the same Sally argument on it. SEQUENCE makes rejecting Bet B fine when no Bet A preceded it but irrational when it would complete the bad sequence—so it, too, imposes different requirements across two situations Sally can see are identical in everything she cares about. Hence SEQUENCE fails.

Conclusion

Every candidate account of how unsharp probabilities guide action falls into one of the traps: permissive rules wrongly license rejecting both bets; strict rules pin down precise betting odds and thereby destroy the motivation for unsharpness; and the global rules (NARROW, PLAN, SEQUENCE) either demand belief change without evidence change or treat choice-identical situations differently. Since there's no good answer to how unsharp credences constrain rational action, UNSHARP is false—and so perfect rationality requires perfectly sharp probabilities.