Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space

Round 1

Reviewer 1 Report

I found the manuscript rather enjoyable. It is written very clearly, even for readers that are not familiar with quantum games theory. Obviously the model analysed is quite limited, but the results are clear and interesting. I detected some typos and minor grammatical issues. 

Author Response

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Reviewer 2 Report

The authors, in the paper entitled “Nash equilibria of quantum games in the special two-parameter strategy space” examine pure Nash equilibria in a quantum game that extends the classical bimatrix game of dimension 2,formulate general statements for the existence and form of Nash equilibria and discuss their Pareto efficiency and prove that depending on the payoffs of a classical game the corresponding quantum game may or may not have Nash equilibria.

The authors do not cite some important and recent papers.

As an example:

 THE DYNAMICS OF NASH EQUILIBRIUM UNDER NON-MARKOVIAN CLASSICAL NOISE IN QUANTUM PRISONERS’ DILEMMA

“Reports on Mathematical Physics Volume 81, Issue 3, June 2018, Pages 399-413;

Quantum Prisoner’s Dilemma and High Frequency Trading on the Quantum Cloud

Faisal Shah Khan and Ning Bao  Front Artif Intell. 2021; 4: 769392. doi: 10.3389/frai.2021.769392

The authors do not introduce some particular upgrade in the previous models and do not add significative improvement in the theory. The paper is a good exercise but it is not ready for publication.

Author Response

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Reviewer 3 Report

I discovered a recent paper by Landsberg (doi: 10.48550/arXiv.1110.1351 ) defining a notion of equivalent mixed strategies. I should appreciate if the authors could comment on the use of quaternions to analyze transformations of mixed strategies (for instance, starting from their Example 1).

 

1. What is the main question addressed by the research?  
The paper is aimed at providing new insights on Nash equilibria of quantum games in the particular case of a two-parameter strategy space.
  
2. Do you consider the topic original or relevant in the field? Does it  
address a specific gap in the field?  
  The paper is well written, however, it is a sort of "toy model".
3. What does it add to the subject area compared with other published  
material?  
  Not much. The authors themselves have contributed heavily to a long list of references along the same research lines. And the current paper contains a small addition to previous contributions.
  
4. What specific improvements should the authors consider regarding the  
methodology? What further controls should be considered?  
For instance, to consider more closely the physical realization of the quantum game, where the strategy becomes non unitary because of dissipation.
  
5. Are the conclusions consistent with the evidence and arguments  
presented and do they address the main question posed? 
The conclusions are consistent with the hypotheses assumed. For that reason, I approve of its publication. The importance of the queries addressed here is not clear to me.
  
6. Are the references appropriate?  
  There is room for improvement. I added a recent reference in my review asking the authors to compare their results with the classification of strategies proposed by other authors. Here below I provide other refrences:
  
-Saitoh, Akira & Rahimi, Robabeh & Nakahara, Mikio. (2009). Yet another Framework for Quantum Simultaneous Noncooperative Bimatrix Games. 10.1142/9789812838681_0008. 

-Saitoh, Akira & Rahimi, Robabeh & Nakahara, Mikio. (2009). Quantum Metagame Extensions of Noncooperative Bimatrix Games. 1110. 10.1063/1.3131357. 

- Iqbal et al.,Stability of Mixed Nash Equilibria in Symmetric Quantum Games
doi: 10.1088/0253-6102/42/3/335.

7. Please include any additional comments on the tables and figures.
No comments. As I point out before, the queries considered by the authors in this paper are not surprising.

Author Response

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Author Response File: Author Response.docx