Alright, this doesn't answer the question directly (I'll wait for someone who knows more about the process of ordeals to answer that to the standards of this sub), but I must admit that as a medieval historian-in-training this paper raises all sorts of red flags for me: there are enough of them for me to be able to say with some confidence that I would not be taking this paper seriously in any writing that I would do on ordeals. Here's a breakdown of my concerns with the paper:

• The paper was not written by a medieval historian, or indeed, any sort of historian. This is pretty apparent not only from your OP, which identifies him as an economist, but also from his choice of publication venue (I have no idea how reputable The Journal of Law & Economics is, mainly because I don't think many medievalists read or cite it) and, most damningly of all, his bibliography. While he does seem to cite all the major works about ordeal and trials that I know of (Bartlett, Davies and Fouracre, White, etc.), there's also a large amount of very outdated works cited in a paper which has no really big discussion of previous historiography (e.g. Gibson, Maitland) and, critically, no primary sources at all. Now don't get me wrong, I'm not saying that unless you have a PhD in Medieval History from an Ivy League college then you can't write insightfully about history (this sub is living proof of that), but there's a reason that historians work in certain ways, there's a reason that historians always back their work up with primary sources and there's a reason that historians spend a lot of time gaining a broad, contexualised understanding of the historiography. Unfortunately, economists have a reputation for trampling into other disciplines without contexualising them far more than other social sciences, and this seems to be one of those cases, borne out by the other two major red flags.

• Leeson is basing his theory of ordeals on rational-choice theory, which he claims that no-one has used to study ordeals, or even the convergence of belief and law more generally (p. 693). Rational-choice theory evolved in economics and holds that people are inherently selfish and will always pursue their individual material self-interests within the confines of the information available to them. It's a legitimate theory within economics and is particularly associated with the 'Neoliberal' economics, dominant from the 1970s (Leeson's citing of Hayek, the high-priest of Neoliberal economics is telling). It's also a heavily contested theory within economics, and often struggles outside the theoretical vacuum to explain real-life instances when people act manifestly against their own material self-interest (the existence of PhD students in History is, I believe, the perfect counter-example to rational choice theory). Frankly, I, and more or less every other historian I've met, believe that it's rubbish, and has no value in explaining historical events and trends. I'm aware that there are historians who use it and find it valuable (especially in the US), but they're very much in the minority. It's the second major reason I have serious doubts about the paper.

• The final reason I have doubts is apparent in the very condescending word which appears thoughout the paper: 'superstition'. This sort of attitude to medieval belief, with its subtext of ignorance, is rarely found in medieval history writing these days (unfortunately there are exceptions) because these kinds of 'superstitions', like belief in ordeals, miracles, saints and demons, were wide-spread, common and sincerely held. The crux of Leeson's paper is an attempt to 'rationalise' what is (to our eyes) an irrational belief. It presupposes a 'rational' cleric who, going through the logic and working out the probabilities as Leeson does as if Ordeals weren't really judge by God, and working out that they were a good way to trick the credulous masses into revealing their guilt or innocence. Except, of course, the vast majority of clerics sincerely believed that God (or the presiding saint) would protect the innocent, and there's, as a result, no motive for them not to apply the ordeal to the letter (by, say, replacing hot iron with warm iron). The argument is only logical if we accept the 'logic' that God really didn't intervene in an ordeal. In order to really understand how ordeals worked in the medieval mind, we need to accept the fact that the vast majority of those involved believed that God or another supernatural being would be actively involved in the ordeal.

Like I said at the start, I'm not an expert on ordeal and I'll await someone with a better knowledge of that to comment on the substance of the question and to back it up with primary and secondary sources (which Leeson did poorly). However, given that he seems to have little background knowledge, uses no primary sources, a questionable theory and a faulty understanding of medieval belief, I would take all of his conclusions with a very large pinch of salt.