Pure Bayesian Nash Equilibria for Bayesian Games with Multidimensional Vector Types and Linear Payoffs

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the attachment. 

Comments for author File: Comments.pdf

Author Response

(1) It would be helpful if the authors argue motivations of their studies more carefully. 
For example, are there specific applications of the setup of the model?
 I added a related section (1.3) at the end of the introduction
(2) “Section 1.3. Related work” can be longer. There may be relevant papers on games 
with multi-dimensions/objects other than Krishna and Perry (1992). Are their 
algorithms proposed in the paper novel and (if so) how? Relatedly, I would like to 
know what is new in this paper compared to Edalat et al. (2018).
 I extended this section (became 1.4 after previous remark)
(3) Why do the authors start the paper with discussing very general background and 
general concepts of game theory (lines 66-119 in pages 103)? (It is like an 
introduction of a textbook of game theory.) If there is no special reason, it may be 
better to begin with what the authors have done.
 I understand that it might seem a bit general. We made this choice to address a 
broad audience.
Minor comments:
(1) Page 2, lines 60-61: It is helpful if the authors explain more about “the example of 
newspaper boxes in New Zealand.” (Is this also a story about cooperation?)
 I made the paragraph clearer, this is indeed a story about cooperation
(2) Page 28, line 773: the information of the book may not be perfectly correct. “William 
Poundstone. Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle 
of the Bomb. New York: Knopf Doubleday Publishing Group, 1993.” (ISB 038541580X, 
9780385415804) may be more appropriate (based on my google search).
 you’re right I made the correction in the reference

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The work under review is distinguished, first of all, by the deep elaboration of mathematical aspects and the focus on solving problems related to the theory of games with incomplete information (Bayesian games).

The main advantage of this paper is determined by the results in the form of algorithms that allow finding specific Nash equilibria in the games under consideration.

Unfortunately, the limited time for reviewing did not allow me to understand how general the authors' proposals are and to what extent they are limited by particular values of the payoff matrices.

Of course, one cannot ignore the fact that the scope of consideration in this work is limited to relatively simple games. The author's promise to address more complex game models in subsequent works with more than two actions available to players still seems like a somewhat comforting declaration.

At the same time, such criticisms should also be taken in a balanced and critical manner. As we can see, even such simple games lead to very complex and difficult to understand mathematical models.

For a better perception of the article by a wider circle of readers, it would be useful to include fragments in it that explain in more detail the applied aspects of the results obtained, at least to some extent highlighting the advantages that they provide.

I believe that if published, this paper will undoubtedly be interesting and useful for researchers working in the field of Bayesian games.

Author Response

For a better perception of the article by a wider circle of readers, it would be useful to 
include fragments in it that explain in more detail the applied aspects of the results 
obtained, at least to some extent highlighting the advantages that they provide.
 I guess that comments in review 1 help address this comment too

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The review is attached.

Comments for author File: Comments.pdf

Author Response

1. In the line 31 it is better "an outcome of a set of actions". In fact, an outcome IS a set 
of actions, like it is said in line 34.
 I replaced the end of the sentence by “[…] and consider an outcome function f
giving an outcome of a set of actions.”
2. Line 90: a concept of strategy is introduced not only in Bayesian games, for example, 
in hierarchical games with perfect information.
 I meant the concept of a “strategy depending on a type”, I rephrased it to make it 
clearer
3. In Definition 4 (p.5) the domain of values for probability should be [0,1] instead of R.
 Indeed, I made the correction
4. In Definition 6 it makes sense to define the utility functions uij. Also, it will be 
interesting to propose an interpretation of the types θij as weight coefficients in the 
linear convolution.
 I added definition of u_ij in point 2 of definition 6 and a small comment on the 
meaning of theta_ij
5. In Tables 1 and 2, if s=z why do you need two variables?
 In the construction of the setup, we start from a Prisoner Dilemma with its own 
variables and a social game with its own variables. We then join both games by 
identifying relations between variables (and set equality on s and z for instance).
6. Why is the paragraph 2.7 called "algorithmic solution"? In fact, an analytical 
investigation is continued there.
 I renamed this part “Equilibrium computation”
7. Line 637: what does mean "less optimal"?
 I meant that the general algorithm is computationally less efficient than the 
DGPD algorithm in the case of DGPD equilibrium search as it explores a larger 
search space without exploiting the DGPD structure (I updated the sentence to 
make it clearer

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I appreciate all accomplishments which the authors have added to the revised draft. I also respect their decision not to consider all comments by the referees in the revised draft. 

I only have a minor comment on their revised draft.  Section 1.3 has better to be supported by some research papers, newspapers or magazines. The current section 1.3 does not include any, which may undermine the credibility of the arguments in this section. 

Author Response

Add references to section 1.3 as suggested by the reviewer.