COVID-19: Data-Driven Mean-Field-Type Game Perspective
Round 1
Reviewer 1 Report
The article under review certainly represents a major work toward modeling the spread of COVID-19. The data-driven approach means that data is used periodically to update model parameters, which naturally results in dynamics that generally fit real-world trends. In this work, the behavior of agents is treated as strategic, and thus the dynamics are the result of a Nash equilibrium. The resulting game is said to be of mean-field type because agents take into account distribution-dependent quantities such as variance; this allows the model to include risk-aversion. Simulations show how the model can track with real-world dynamics.
I have two main concerns about this article. The first is about the data-driven methodology. There is no theoretical discussion of this approach, nor is there literature cited. When I see that the output of a data-driven model closely fits actual data, how am I supposed to interpret this? Does it show that the model is correct, or does it merely show that the data were imposed upon the model frequently enough that the simulation was forced to follow the real world? Now one of the key points of game theory is to determine what agents' strategies will be, which one cannot deduce from the data. However, this point does not seem to be discussed anywhere, and so the game theoretic approach does not seem fully justified. Of course, the sheer number of variables that are accounted for in the optimization problems does indicate an inherent theoretical strength of the model. Still, I am only somewhat convinced by the presentation.
My second concern is about the game theoretical background. When it comes to stating the main mathematical propositions, it does not appear that the key terms are well-defined. What does the equilibrium mean in this context? The author uses phrases like ``best response given $m$" that seem reminiscent of mean field games. But this is a mean field type game, which is by no means a mean field game, and in fact the vast literature on mean field games is not cited. (Subsection 2.4 is entitled ``A basic SIRD using Mean-Field Game," which suggests that mean field games do play a role, but again, the notion of equilibrium is not clearly defined for the reader.) In order for this article to be really useful, the reader needs to be able to cut through all the heavy notation and see clearly what is going on conceptually, to understand the methodology.
In connection with this second concern, I think it is only fair for the author to compare and contrast his approach with that of mean field games. This is true because (1) ``mean field type games" are easily confused with ``mean field games," and especially because (2) mean field games have been used to study COVID-19 transmission. See the work of Elie-Turinici-Hubert, which should certainly be cited, in addition to that of the author himself. It is worth pointing out that in this game, there are quite a number of players--195 countries and countless individuals. The mean field game methodology seems natural in this case, so as to simplify the concept of equilibrium to that between a single representative player (or two or three, perhaps, given that individuals, governments, and businesses have different interests) and an evolving distribution. Naturally, literature on mean field games should be cited, for instance the monographs by Carmona-Delarue and Bensoussan-Frehse-Yam.
Despite these concerns, I think this is an impressive work that contains a thought-provoking discussion about how mathematical models can address both economic and health concerns in a strategic way. I hope my comments here are useful in improving the article. Below I list some particular notes that I made while reading through the manuscript. Because of the sheer number of symbols, I was unable to make a thorough search for typos, so I only noted flaws that immediately stood out.
\begin{enumerate}
\item The article, or at least the introduction, should be thoroughly checked for grammatical mistakes, since they appear regularly throughout the text.
The first instance can already be seen in the abstract.
The last four items in this list do not parse: ``The model integrates
untested cases, age-structure, decision-making, gender, pre-existing health conditions, location, testing capacity, hospital capacity, mobility map on local areas, in-city, inter-cities, and international."
\item In line 186, how are $\hat I(t)$ and $\hat R(t)$ defined for $t$ not in the discrete set $t_0 + j$?
\item In line 226, I'm confused by the phrase ``within two meters radius $\epsilon$."
What is the intended meaning of this phrase?
\item In line 233, the notation $S^*,I^*$, etc.~is not defined, and the game is not formulated.
I presume that there is a fixed point problem: for every $m$ there is a $\mu$, and the fixed point is the Nash equilibrium.
This should be stated clearly.
\item The same comment goes for line 236.
\item In line 431, the table referred to in the words ``(18 in Table ??)" seems to be unlabeled.
\item I am trying to understand Proposition 2.
In terms of setup, it hasn't been made clear enough up to this point that $v$ is also a control for the individual.
Likewise, $\delta$ is at one point mentioned as a control for authorities, but just before Proposition 2 it is not clearly stated.
Furthermore, the definition of Nash equilibrium should be stated just before Proposition 2 (or at least \emph{somewhere} in the text).
In terms of the proposition's statement, I cannot see where $(\eta^*,\delta^*)$ are defined in terms of the Hamilton-Jacobi equation.
So the conclusion of the statement is a bit mysterious.
\end{enumerate}
Author Response
Please find attached the file
Author Response File: 
Author Response.pdf
Reviewer 2 Report
This paper proposes a version of the well-known SIR model　based on modeling of mean-field games (more specifically, mean-field-type games) in the context of the current pandemic of COVID-19.
This topic itself is obviously important just now, in this truly evolving situation. I hoped that the paper would show how game theorists, especially those on theory of game dynamics, can contribute to the world to resolve the pandemic anyhow.
Yet, the current presentation of the paper might rather deter a reader like me, who are neither experts in epidemiology nor experts in mean-field games. While I believe that the topic itself is worthy of timely publication to Games, there must be overhaul of writing so I can better assess the actual value of this paper and also the paper can be read by a wide range of readers. For this, I will make a few general suggestions below.
1. What is eventually the merit of the approach that the author is proposing? Yes, it looks like that the author's data-driven MFTG includes more variables. So, it may be obvious that it might have more explanatory power, especially about fitness with data, if the estimated model is completely identified. But, it will need more data and actually, in the last section where the model is matched with data, the author finds most of needed data are not available.
I first thought that the novelty might be in math. But, the proofs of the two propositions only refer to some existing proofs for other mean-field game models. About fitting with data, no estimation method is proposed; indeed, no rigorous criterion about validness of estimation is stated. (Also, if it was the catch of this paper, shouldn't it be sent to a statistic journal?) So, I am not sure what is the goodness of this complicated model. (Yes, there are too many variables, so the goodness is not about tractability or clear illustration of a mechanism behind the pandemic and its mitigation policy.)
In particular, you should give some concrete, quantitative measure about advantage of your model, compared to those incumbent models.
2. This paper is too long for a regular article in Games. (The only paper that exceeds 20 pages in Games is a survey paper as far as I remember.) Actually, readers need to go through 15 pages to reach the author's own model in Section III. Section II is indeed too painful to read, since readers have no clue (only with a hope that it goes forward as a model increases the number of variables and complexity) about how the author's own model looks like and is related with those basic models. Then, in Section IV, readers would be very disappointed to find the data about most of those variables are not available though the catch of this paper about fitting the data.
I'd suggest to minimize the model only to what is eventually estimated and used when the model is matched with the real data. Then, you'd never say that a proper time unit is a millisecond (p.19), which only made me skeptical about reality behind this model. You would not show each of those basic models. Perhaps, you can just present your own model and then explain what components of your model is missing in each of the incumbent basic models.
In the next version, I'd hope to see this paper drastically shorter, say in 15 pages.
3. While the paper gives too much, unnecessary, redundant explanation about minor issues or unused factors of the model (e.g. "spectrum of gender" though you don't have the data about it and thus it would not be considered in the last step to match with the data), there are too many places where the author does not give any information about key components of the model.
The author does not explain how those (quite general or vague) data are put into computation of the model. For example, flight data in Flighrader 24 is used for \eta_1 (p.33). But, isn't eta a real number, while Flighrader 24 shows traces and current locations of specific flights over the world? (And, what was the time range when retrieving the data? What flights did you look at?)
While the author gives casual explanation about the objective of each decision maker, there is no explicit formulation of the objective function (despite those huge notation about variables!). Many notations or even equations lack explanation. For example, what is the list of equations for "D_intra" on p.21? At many places, I doubt if the writing makes any sense...
4. I suppose that the author did not pay much time to polish writing. This is evidenced by a very odd placement of "Acknowledgement" section, which is put before Conclusion. I am not sure if the author is seriously sending this paper for a peer review and publication... Definitely, this paper is not ready for submission.
Author Response
Please find attached the file
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
I see that the author tried to respond to our comments as much as possible given this quite short period for revision. Although I wished that the paper could be wholly written for more concise, clearer exposition, I suppose that it may be fair not to demand too much.
Author Response
Dear Editor and Reviewer
Thank you very for your time and effort in handling our manuscript.
Please find attached the revised version.
Author Response File: Author Response.pdf
