In my paper on accuracy and non-classical logic/semantics, I adapt Jim Joyce’s accuracy-domination theorem to a non-classical setting. His result shows that (under certain assumptions) an improbabilistic belief state is “accuracy dominated” by a probabilistic one (i.e. the latter is closer to the truth, no matter which world is actual). I generalized this to a case where the “worlds” and “truth values” are non-classical, and proved accuracy domination for a notion of “generalized probability”.
Jeff Paris then gave a talk at Leeds, and chatting to him afterwards, it emerged that he’d been interested in something very similar: his 2001 paper “A note on the dutch book method” shows that belief states that aren’t generalized probabilities are susceptible to a dutch book—again, the results cover non-classical as well as classical settings, and the characterization of generalized probability he uses pretty much coincides with mine. (The paper is great, by the way—well worth checking out).
This looked like more than coincidence: what lies behind it? I’ve written up a quick note on the relationship between dutch books and accuracy. It turns out that Paris’s core result on the way to proving the dutch book theorem (an application of the separating hyperplanes theorem) has both his dutch book theorem, and a version of accuracy-domination, as easy corollaries. (The version of accuracy domination is one that measures accuracy by the Brier Score—the square Euclidean distance).
But that isn’t quite the end of matters—the proof just shows that in one specific case, we can construct a specific dutch-book/accuracy-dominating belief state. In effect, if we’re at an improbabilistic belief state, it shows how to construct a probabilistic one that has a property that turns out to be sufficient for both dutch-booking and accuracy-domination. But the property isn’t necessary for either.
But it’s not hard to figure out the more general connection: every accuracy-dominating point corresponds to a dutch book. And although not every dutch book corresponds to an accuracy-dominating point, there’s always some accuracy-dominating point reachable by manipulation of the dutch book.
So I feel I see why the formal connection between the results is now (and remember that these hold in a very general setting—way beyond the standard classical case). But there remain questions: in particular, what about where we measure accuracy by something other than the Brier score? Is there some kind of liberalization of the assumptions of the dutch book argument that corresponds to loosening of the assumptions about how accuracy is measured (and is it philosophically illuminating?)
Thoughts, criticisms, suggestions, most welcome.
J Robert G Williams 