Probability Spaces Identifying Ordinal and Cardinal Utilities in Problems of an Economic Nature: New Issues and Perspectives

Round 1

Reviewer 1 Report

The paper describes a new approach to economical problems, using probability spaces identifying ordinal and cardinal utilities. First of all, the author gives a basic introduction to the problem, giving some known facts from the probability theory, and then he formulates the main aims of his work. After that he describes the reasons that lead him to use the probability methods in his work, and shows the connection between risky asset and its probability space. Than he discusses coherence properties of the notion of expected return on risky assets. In particular, he describes the linearity of P-function and pass to the consequences from it. After that the author speaks about expected returns of risky assets, and discusses monetary scale and the one of utility connected with two risky assets studies outside the budget set of the investor. He describes utility function considered inside the budget set for different cases and character of the investor (for more and less risky persons). In the end he comes to a conclusion and tells us about some another applications.

The paper is quite interesting and, in principle, it can be published in the paper. However, the main disadvantage of it is the lack of practical examples. The authors speaks about economics, so it would be better to mention some real cases that lead to numerical results.

Also, there are some less serious remarks:

 - on page 9 the author introduces a theorem and says that it has been proved in a previous paper. However, there is no link to this paper;

 - on page 11 the author uses the covariation Cov (.,.). It would be better to introduce this function;

 - on page 20, it is better to divide (59) to two separate inequalities.

Finally, I would like to describe some minor problems and typos:

page 2, line 42: “a unique” must be changed by “an unique”;

page 2, line 49: “are” must be changed by “do”;

page 3, line 114: “They are then of a…” seems to be a bad wording;

page 4, line 136: “a Euclidean” should be changed by “an Euclidean”;

page 6, line 222: “nothing but a random gain” seems to be a bad wording;

page 10, line 9: “a point” can be changed by “only point” or “one point”;

page 15, line 5: author always emphasizes “he or she”, but here only “she” is used;

page 17, line 463: “A utility” should be changed by “An utility”;

page 22, line 16: “stardard” should be changed by “standard”;

page 24, line 584: “give” should be changed by “gives”.

I strongly recommend to read the text carefully. There are a lot of linguistic mistakes, and all of them should be corrected.

It is hard to read the paper, because there are a lot of mistakes and bad wordings. Small list of them (it does not contain all problems) is given in the review.

Author Response

REVIEWER 1 

Author Response File: Author Response.docx

Reviewer 2 Report

Please, read the attached file review.pdf

Comments for author File: Comments.pdf

The English language needs little editing.

Author Response

REVIEWER 2 

Author Response File: Author Response.docx

Reviewer 3 Report

The article is very interesting and has a lot of technical content. But as a general observation, the article is quite dense. I suggest that it can be broken down into parts and, in each of them, place tables and figures. This could make it easier to understand and read for people who are not immersed in this type of approach.

I cite as a model the article “Normality of demand in a two-goods setting”, authors: LaurensCherchyea, ThomasDemuynckb, BramDeRock (reference [33]) in the text;

Some observations are simple comments and/or suggestions, but there are some small errors that should be corrected.

Is it interesting to always talk about "he or she" or always talk about "investor" or something like that (regardless of gender);

Abstract: What is this? Mathematical classification? e.g., MSC: 60A05? It is not part of this journal;

Line 14, line 573 - Daniel Bernoulli – add reference (if possible);

Line 185: put the reference Bruno de Finetti; Footnote 3: appears before being cited; Line 326 ... in another paper... Which paper?

Cite reference Fréchet (if possible);

Right after equation (41): was placed only “she” and not “he or she”;

Equation 42: explain: Combination of m, two by two will give ½. (m.(m-1)). In fact, this is the combination of (m+1), two by two. Was this clear?

Line 454: error – the word “bettor” appears; Line 560: Bravais-Pearson reference;

Put names “Section 1, 2, 3,...”  with capital letters; Reference 13: put year of publicatio n(1967) in bold.

Author Response

REVIEWER 3 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I think that main problems of the paper have been solved. However, I would like to say that there could be more descriptions of different applications. This can make the paper more useful.

Reviewer 3 Report

The text is dense and very useful for research in the area and will undoubtedly be widely referenced. It is a text with great quality and value. But as a suggestion to reach other areas of applied science, I think that more practical examples could help to spark greater interest in the text.