Does anybody know of any examples in the literature; or otherwise whether it is possible to prove that a certain probabilistic logic satisfies the qualitative constraints given by Luce 1967 (the annals of mathematics 38, 780 -786), which are sufficient for unique strong numerical probabilistic representation, without making some equi-probability assumption or going through some other route.
In particular his condition L. I ask this because whilst Luce's theorem does not appear to make an equi-probability assumption, such as the presumption of finitely many equi-probable atoms (koopman, de finetti and savage in the finite case, or nearly equally probable partitions (which is non-atomicity plus montonone continuity for, villegas and tightness and fineness in the infinite case).
It is sometimes said that Luce's result is stronger because it does not make such presumptions in order to derive a unique strong representation (but i now think what is meant here is that luce's axiomatization allows for a unique representation when the sample space is only finite, which the other do not)
This is because most proofs that aim at deriving Luce's conditions seem to rely on using one of the above methods in any case. This being they entail luce's conditions (but not the converse). So equi-probability is presumed, in some sense, in deriving luce's axioms (despite his theorem which follows theiron, not depending on an equi-probability assumption) even if Luce's theorem, which falls out of said axioms does not itself make such presumptions.
Thus i am wondering if there are examples of Luce's axioms being derived directly and not through one of these other routes (or using some other equi-probability presumption). I also presume i am correct that non-atomicity does implicitly make an equi-probability assumption as an artefact of the mathematics?
I also had some questions as to the some of the assertions of fine 1973 ,'unique to and within a positive multiplicative factors; Is this talking about the uniqueness or homegeneity of the representation (the similarity group of transformations); or the probability measure. I was always under the impression insofar as the probability is concerned, that it just unique (given the probabilistic representing structure).
He says something similar about Savage axioms of tightness, fine-ness in the paragraph above where he mentions Luce L (i thought that Savage's utilities were on an interval scale, and that his 'probabilistic unique-ness, if there is any at all) would not be present without this extra (utility principles, sure thing and the like, constant acts, unless this was his justification for fine, tightness,P, almost uniform partitons) .
By this I mean the homogeneity (or invariance of the representation), the uniqueness of the representation(probabilistic structure, not the values within it) .On the other hand, the values (or function) within the probabilistic representing structure, were unique simpliciter
And even then, his, only up to (but including) positive affine transformations (interval scale) with relata to the uniqueness of the representation, only came through his extended preference model; not simply given his qualitative; if at all. But that nonetheless, the probabilities values themselves . Of course, those may have been mentioned earlier on, and I didnt notice, or were the justification for his partition axioms.
Likewise, otherwise, I would have presumed that the probability values, or the probability function, (granted his constraints and the axiom of probabilities)were just unique, but the representation (ie the axioms structure) would be itself hardly unique at all (ie the permissible transformations, type of talk, the permissible transformation which preserve some structure, ratios of difference etc)
He also says within a multiplicative constant in both case, making it appear that its some halfway house between a ratio scale and strong interval scale (such as a log-interval scale); this may apply to luce, or apply to some of the equi-space or even more complicated PCS, concanted, bi-symmetric structures and other such things he constructed which may use L, and S, but I would not think that satisfying that constraint in and of itself, in the finite case (and it can sometimes be applied) would grant you that from thin fair. Presumably the values themselve would be unique.
On the other hand, one has to be careful, because he does not mention any of this when it comes to villegas or the countably additive uniqueness derivations. He is careful to distinguish P from quantitative P (a numerical function on [0,1], and additive P, which is the traditional numerical finite(or countably additive) probability function.
Athough if i remember correctly, he only uses the words, quantitative P, which sounds very odds (the values might be uniquess, there may be only measure, but I presume that insofar that it is probabilistic unique, or those transformations that would respect the qualitative constraints,whilst preserving some probabilistic structure, would be hardly more then ordinal (or interval at best)- and so possibly 'unique up, to but not including everthing (ie hardly unique at all).
Although Villegas uses those same terms with relata to boolean rings, but I presume he is talking about the values or probability function, given the representing structure and probability axioms (it would depend on presumably, the number of countably infinite or finite disjunctions that are in F, as not all may be included as it would otherwise would be closed-- so he either means that it is essentially unique within tiny difference, or that it varies depending on aforementioned property. He presumably is not claiming THAT MUCH (a ratio scale, one hand, and his convention proof, completely and utterly uniqueness of the function on [0,1] in both sense of the measures,and that addivitity just falls out of thin air, as if he never used it to derive the unique measure to begin with (presumably the same is the case in all the above models)
, that it is unique in the other sense (nor in the original theorem where its atomles monotone sigma algebra)
Unless I am missing something. And that the linear transformations (business, interval scales) just one inform as to which values are needed to be set to pindown the entire (in this case, probability structure) so that one know immediately whether the measure is unique..
The representation in some already or almost always insists on a unique measure, unless it is really case that any [0,1] probability structure, that uses an interval scale (such suppes equi-spacing) will be such that, if unique, all of the [0,1] that agrees on the the first two axioms, will be same as that unique probability function, AND they are additive, as it one had derived the better part of the probabilistic axioms, (most of additivity) out of thin air, and that the unique measure, which presumed employed addivity somehow literally could have been just read off without even assuming addivitiy (but only the first two axioms).
This is where I am sometimes confused as to whether they really mean a constant or an arbitrary constant Ax +b where A is a positive constant within any representation (the values of the outcomes) but not between them, but that B could within (and of course between as well). But that would hardly be a monotonic transformation then. Otherwise the arbirariness must merely be to indicate that it could be either positive or negative, but that within any given representing numeircal structure (probabilistic or not), it will be the same constant. I find that very hard to believe to some degree, at least in some cases. As it would presumbly have aforemetioned consequence that given any strongly representing numerical structure on [0,1] with P(sigma)=1, and P(empty set)=0, all of them (and presumably there is only one) will be a probability function (additive) and the values will be unique (it will just be identical to the unique probability value function). I must clearly be mis-interpreting something, I must be
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