A scoring rule *s* assigns to
a credence *c* on a space *Ω* a score *s*(*c*) measuring the
inaccuracy of *c*. The score is
itself a function from *Ω* to
[−∞,*M*] (for some fixed *M*), so that *s*(*c*)(*ω*) measures
the inaccuracy of *c* if the
truth of the matter is that we are at *ω* ∈ *Ω*. A scoring rule is
proper provided that *E*_{p}*s*(*p*) ≤ *E*_{p}*s*(*c*)
for any probability *p* and any
other credence *c*: i.e.,
provided that the *p*-expected
value of the score of *p* is at
least as good as (no more inaccurate than) as the *p*-expected value of any other score.
It is strictly proper if the inequality is strict. It is quasi-strictly
proper if the inequality is strict whenever *p* is a probability and *c* is not.

Here’s a fun fact about scoring rules.

**Theorem:** Let *s* be any continuous bounded proper
scoring rule that is defined only on the probabilities. Then *s* can be extended to a continuous
bounded quasi-strictly proper scoring rule defined on all credences.

This result works in the finite-dimensional case with standard
Euclidean topologies on the space of credences (considered as elements
of [0,1]^{PΩ}) and on
the scores (considered as values in [−∞,*M*]^{Ω}). But it
also works in countably-infinite-dimensional contexts in the right
topologies (ℓ^{∞}(*P**Ω*) on the
side of the credences and the product topology on the side of the
scores), regardless of whether by “probabilities” we mean finitely or
countably additive ones.

The proof uses three steps.

First, show that the probabilities are a closed subset of the space of credences.

Second, apply the Dugundji
extension theorem to extend the score *s* from the probabilities to all
credences while maintaining continuity and ensuring that the range of
the extension is a subset of the convex hull of the range of the
original score. Let *s*_{0} be the extended score.
The convex hull condition and the propriety of the original score on the
probabilities implies that *s*_{0} is proper, though not
necessarily quasi-strictly so.

Third, let *s*_{1}(*c*) = *d*(*c*,*P*) + *s*_{0}(*c*),
where *Q* is the set of
credences that are probabilities and *d*(*c*,*Q*) is the
distance from *c* to *Q* in the ℓ^{∞}(*P**Ω*) norm.
This equals *s*_{0}(*c*) and hence
*s*(*c*) for a credence
*c*.

I don’t know if there is much of a philosophical upshot of this. Maybe a kind of interesting upshot is that it illustrates that quasi-strict propriety is easy to generate?