Lifting Theorems for Continuous Order-Preserving Functions and Continuous Multi-Utility

Round 1

Reviewer 1 Report

Dear authors,

The paper is nicely written and presents highly valuable results.

We believe that it could be shorten to the essential part : local-compactness and sigma-compactness, other new results would better fit in another paper and their respective proofs.

Some minor notational modifications remain to be corrected. This would facilitate the reading and would avoid extra characters.

Sincerly yours,

Comments for author File: Comments.pdf

Author Response

We are very grateful to Reviewer 1. We have followed all her/his suggestions. All tghe remarks are correct and very useful.

In particular:

 

1) We have presented in a unique theorem 3 the old theorems 3 and 7, providing a full proof;

2) We have deleted theorems 5,6 and 8,9, and we have also deleted the condition cc, to be incorporated in a future paper;

3) We have modified the notations E^C and s^C and i^C, as the reviewer correctly pointed out;

4) We have corrected all the typos;

5) We have cited the old reference [23] in the text.

 

We think we have respected all the suggestions.

 

Best regards and thanks,

 

Gianni

Reviewer 2 Report

The manuscript under review provides several lifting theorems for closed preorders. Beyond the natural mathematical significance, the topic is of interest to utility theory. The main result of the paper, Theorem 3, characterizes the closedness of a preorder in respect to the possibility to lift of continuous order-preserving functions defined on compact subsets of its ground set. This result allows to characterize the existence of continuous multi-utility representations in the last part of the paper.

In my opinion the paper is interesting, non-obvious and essentially well-written: I write essentially because, in my opinion, a reader should be helped a little bit understand the proofs of Theorems 5-9 (whether or not they are perfectly clear in the authors’ minds). As I have already said, beyond the natural mathematical significance, the topic is of interest to utility theory and this fact broadens the range of potentially interested readers. As the authors clarify at the end of their Conclusions, this work has a natural pursuance, which in my opinion is interesting and important.

 

Comments.

1.      Lines 148-150. The inference “Therefore, for every pair ….” is unclear to me. From Theorem 2 of the paper under review (namely, from Theorem 1 in Evren and Ok (2011)) I have been able to deduce the inferred conclusion only for a function f_xy into the real line and not into [0,1]. Maybe I am missing something obvious, but I do not understand such an inference. As far as I saw, the inferred conclusion is true: but I inferred that conclusion from Remark 3 and Theorem 4 in Evren and Ok (2011) and not from Theorem 1 in Evren and Ok (2011).

2.      Line 211. The inference “Lemma 2 implies…” should be “Lemmas 1 and 2 imply…” (or anything that the authors prefer and that mentions Lemma 1). The point here is that the inference in line 211 needs Lemma 1. I add that Lemma 1, in the present version, is never explicitly used in the proofs: Lemma 1 is only mentioned (but not explicitly used) in the proof of Theorem 3.

3.      Theorems 5-9. The proof of these Theorems is left to the reader. However, even a reader full of goodwill (the present reviewer included) might have some difficulty in proving them by herself or himself. I would like to see a proof (or at least a sketch of the proof that provides some help).

4.      Abstract. The Abstract promises some results where pseudocompactness is used, but I do not see any result where this property is explicitly mentioned.

Author Response

We are very grateful to reviewer 2. All her/his sugegstions were very useful and adequate.

In particular, according to the requirements of reviewer 1,

1) We have presented in a unique theorem 3 the old theorems 3 and 7, providing a full proof;

2) We have deleted theorems 5,6 and 8,9, and we have also deleted the condition cc, to be incorporated in a future paper.

As regards the particular suggestions of reviewer 2, 

1) We have presented a Proposition 2 after Lemma 1 and Lemma 2, putting them together, and we have cited and used it in the proof of Theorem 3.;

2) We have referred to Remark 3 in Evren and Ok (2011) in order to Justify the separation of x and y by f_{xy};

3) we have deleted the reference to pseudocompactness in the abstract;

4) Theorems 5, 6, 8 ,9 do no longer appear. They will be presented in a future paper.

 

We thank the reviewer very much.