Are Brier and log-loss proper or strictly proper scoring rules?

Question

(This article nicely explains the difference between proper and proper scoring rules)

According to the Wikipedia entry, and Merkle & Steyvers (2013), these are both strictly proper scoring rules. But in his article Damage Caused by Classification Accuracy and Other Discontinuous Improper Accuracy Scoring Rules Dr Harrel refers to these two as just proper scoring rules:

The two most commonly used proper scoring rules are the quadratic error measure, i.e., mean squared error or Brier score, and the logarithmic scoring rule...

Are these two strictly proper or just proper scoring rules?

Answer 1

Both are strictly proper. See Selten ("Axiomatic Characterization of the Quadratic Scoring Rule", Experimental Economics, 1998), who uses the term "incentive compatible" in place of "strictly proper". His proofs work with distributions with finite support, but apply to the continuous case as well, with the necessary modifications.

See also Gneiting & Raftery ("Strictly Proper Scoring Rules, Prediction, and Estimation", JASA, 2007), who give the Brier and the log scores as examples of strictly proper scoring rules in Examples 1 & 3 in section 3.1 without further comment or explanation.