Investigating Peer and Sorting Effects within an Adaptive Multiplex Network Model
Round 1
Reviewer 1 Report
The work studies a model where a coordination game is combined with a learning mechanism. To model the different processes at play, the authors use a multiplex representation with two layers. The first one drives the coordination dynamics while social learning takes place in the second layer. For the learning, two different strategies are possible: conform with the majority of the other players in the neighborhood or rewire a connection to o new neighborhood. The results demonstrate that segregation between the two actions is the preferred choice in almost all the cases.
I find the work interesting and scientifically sounding. The manuscript is, in general, clear and well written and the idea behind the work, up to my knowledge, novel. Thus, I feel comfortable in suggesting it.
However, I have some minor comments for the authors:
- The introduction is quite long and verbose. In my opinion, a shorter and more focused intro will encourage potential readers.
- I have some doubt about the informative content of figure 1. I agree that it is important to show the evolution of the three quantities (fraction of rewirers, segregation index and dominant action) as function of R_0 but presenting 18 panels where the effect of the two most important parameters is minimal does not help in demonstrating the importance of the model.
- Even if the text is, in general, clear, the description of the model is quite hard to follow. In particular, the rewiring mechanism and the difference between the dynamics of the first and second layer. Moreover, the descriptive figure (Fig.1) is cited only at the end of the description when the reader is already lost.
- line 159, typo: "Gurthermore"
Author Response
We thank both reviewers for their time and effort in reading and reviewing our manuscript. We believe that incorporating their comments we have been able to palpably improve the quality of the manuscript. Please find below our answers (in blue) to reviewers’ comments.
On another matter, we have realised that the use of the word “Polarization” was improper to describe the outcome in Fig. 3a and we decided to use instead “Full Consensus”, since polarization is more related to the results obtained in Fig. 3b or 3c. To avoid any confusion, we have changed the manuscript accordingly without using the word “Polarization” anymore.
Reviewer 1
The introduction is quite long and verbose. In my opinion, a shorter and more focused intro will encourage potential readers.
Thank you for your comments. We have condensed and focused the Introduction and created a separate section for the literature review.
I have some doubt about the informative content of figure 1. I agree that it is important to show the evolution of the three quantities (fraction of rewirers, segregation index and dominant action) as function of R_0 but presenting 18 panels where the effect of the two most important parameters is minimal does not help in demonstrating the importance of the model.
We believe that the reviewer refers to the first figure of the results, i.e. Fig. 2. We have thus decided to remove two rows (6 panels) for two different beta values, i.e. 0.05 and 0.5, which were perhaps overwhelming. We hope that this (and a larger figure) could help the understanding of the main differences in the panels. It is true that the effect is small for large values of R_0 but beta and rho_L play a significant role (the higher they are, the higher is the SMI) for small R_0 values. We separated these results in two different paragraphs.
Even if the text is, in general, clear, the description of the model is quite hard to follow. In particular, the rewiring mechanism and the difference between the dynamics of the first and second layer. Moreover, the descriptive figure (Fig.1) is cited only at the end of the description when the reader is already lost.
We now cite Fig. 1 right before the strategy definition. We have also shortened the Model section to make it clearer.
line 159, typo: "Gurthermore"
Thank you for spotting it. We have corrected the typos
Reviewer 2 Report
The paper presents a theoretical model with the aim to analyze how cultural transmissions can inform the level of coordination in a network. The authors use a multiplex network model with a hidden layer on which social learning takes place and an evident layer where payoff relevant relationships are observed. They use different topologies to analyze whether these play a role as well in determining the level of coordination in a society. The authors find that the presence of social learning and heterogeneity of actors works in favor of states in which there is segregation between different groups.
Major Comments:
· Unfortunately, the motivation and set up of the paper is extremely unstructured. It feels like the authors mention a lot of different theories and additionally use cultural transmission as a motivating idea but get so bogged down with explaining all different ideas that the main idea behind the model gets lost. On top of that, important concepts such as homophily are defined in a footnote instead of the text. I had to read the introduction three times to get a general idea of what kind of model I might expect. In my opinion, the introduction is way too long, there should be a separate literature section and I would reduce the cultural transmission idea to the motivating example of such a model, that it is. What the authors do, in effect, is to look at a network model with heterogeneous players and different layers, where on the one layer behavior is learned and on the other – payoff relevant - layer it is imitated. The cultural transmission idea could later be discussed in a more stringent way in an applications section.
· The research question and main aim of the paper is formulated at multiple points in the introduction (however at a rather late stage in the introduction), in multiple different ways so that in the end I was not sure what your main question, your main contribution and idea was. On page 2, line 55ff. you say your main purpose is to analyze how cultural transmission can inform the level of coordination in large social structures. On page 3, l.109ff. you state, however, that the main question you want to answer is whether the interplay between game theory and social learning can inform us about the level of coordination and whether network topology plays a role. In your contributions, you claim that your main contributions are technical concerning the way how strategies are the outcome of a social learning process and using static and dynamic network configurations at the same time. Overall, this confuses the reader greatly. Please come up with one unified research question and what you are addressing and state it rather earlier in the paper so the reader knows what he is to expect.
· In the modelling section there is no mention of costs. Why should I not simply link to everyone in the evident layer if this does not increase my costs and thereby reduces my benefit. In the payoff table on p. 4, you only state that no benefit is produced by being linked to a neighbor with a different action, while positive benefit is produced by being linked to a neighbor with the same action. If there are no costs of linking, why not link to everyone? Later, on p.5 line 159ff. it is stated that only a small number of links can be supported with respect to average degree distribution but this is not further specified.
· In the introduction it is mentioned that one of the contributions of this paper is that the strategies are not given but determined through social learning. However, in the modelling section we then learn that there are only two strategies and that the ratio of those is predetermined. Agents can later on change their strategy, if they observe this to be a good idea on the hidden layer. What is unclear to me is, what do agents actually learn? Only the payoff of the agents they observe? Which action those agents chose? Both? And what can they change in the evident layer? They can change only one link at a time? Are links determined by both adjacent nodes or just one? Can they change action and strategy at the same time?
Minor Comments:
· The abstract is too long
· Footnote 4: The concept is called “Battle of the Sexes” not battle of sex
· There are a number of smaller typos in the paper
· In the modelling section define the set of players. Is there an upper limit?
Overall, while I think the results you present are interesting, I believe the paper is severely lacking in structure and is not clear in its modelling decisions. I believe you should strictly separate between the model and its concept and the potential application (cultural transmission). This would make the paper clearer and your modelling more easy to follow.
Author Response
We thank both reviewers for their time and effort in reading and reviewing our manuscript. We believe that incorporating their comments we have been able to palpably improve the quality of the manuscript. Please find below our answers (in blue) to reviewers’ comments.
On another matter, we have realised that the use of the word “Polarization” was improper to describe the outcome in Fig. 3a and we decided to use instead “Full Consensus”, since polarization is more related to the results obtained in Fig. 3b or 3c. To avoid any confusion, we have changed the manuscript accordingly without using the word “Polarization” anymore.
Reviewer 2
Unfortunately, the motivation and set up of the paper is extremely unstructured. It feels like the authors mention a lot of different theories and additionally use cultural transmission as a motivating idea but get so bogged down with explaining all different ideas that the main idea behind the model gets lost. On top of that, important concepts such as homophily are defined in a footnote instead of the text. I had to read the introduction three times to get a general idea of what kind of model I might expect. In my opinion, the introduction is way too long, there should be a separate literature section and I would reduce the cultural transmission idea to the motivating example of such a model, that it is. What the authors do, in effect, is to look at a network model with heterogeneous players and different layers, where on the one layer behavior is learned and on the other – payoff relevant - layer it is imitated. The cultural transmission idea could later be discussed in a more stringent way in an applications section.
We thank the reviewer for the detailed comments. We have moved the homophily definition from the footnote to the main text. In order to ease the reading we have created a literature review section, as suggested and moved the results and focus of our work at the end of Introduction. We prefer to leave the cultural transmission example in the main narrative of the paper because it is the main focus of our model and not just an application.
The research question and main aim of the paper is formulated at multiple points in the introduction (however at a rather late stage in the introduction), in multiple different ways so that in the end I was not sure what your main question, your main contribution and idea was. On page 2, line 55ff. you say your main purpose is to analyze how cultural transmission can inform the level of coordination in large social structures. On page 3, l.109ff. you state, however, that the main question you want to answer is whether the interplay between game theory and social learning can inform us about the level of coordination and whether network topology plays a role. In your contributions, you claim that your main contributions are technical concerning the way how strategies are the outcome of a social learning process and using static and dynamic network configurations at the same time. Overall, this confuses the reader greatly. Please come up with one unified research question and what you are addressing and state it rather earlier in the paper so the reader knows what he is to expect.
Thank you for highlighting the lack of consistency between the two pages. Research question is now clearly stated at the end of the Introduction.
In the modelling section there is no mention of costs. Why should I not simply link to everyone in the evident layer if this does not increase my costs and thereby reduces my benefit. In the payoff table on p. 4, you only state that no benefit is produced by being linked to a neighbor with a different action, while positive benefit is produced by being linked to a neighbor with the same action. If there are no costs of linking, why not link to everyone? Later, on p.5 line 159ff. it is stated that only a small number of links can be supported with respect to average degree distribution but this is not further specified.
Thank you for the comment. We do not mention the cost because the consequences of being linked to an agent with different action cause a “missed benefit” (benefit=0) that is lower than the benefit coming from the interaction with an agent sharing the same action. We have also considered other models in Antonioni et al. (2014) using a negative payoff for opposite action interactions but preferred to keep the model in the simplest case (=0).
Regarding the number of links, the agents cannot link to everyone else because we assume they have a physical constraint, that is reasonable in many socio-economic context (it is almost always impossible to be connected to the entire population, or a big fraction of it). To model such constraint we impose that agents cannot increase the number of links, i.e. the average degree, of the initial network.
At line 159, we realised that the sentence was confusing and we removed it.
In the introduction it is mentioned that one of the contributions of this paper is that the strategies are not given but determined through social learning. However, in the modelling section we then learn that there are only two strategies and that the ratio of those is predetermined. Agents can later on change their strategy, if they observe this to be a good idea on the hidden layer. What is unclear to me is,
What do agents actually learn? Only the payoff of the agents they observe?
Which action those agents chose? Both?
And what can they change in the evident layer?
They can change only one link at a time?
Are links determined by both adjacent nodes or just one?
Can they change action and strategy at the same time?
Thank you for the observations, we reply point by point.
The agents start with a specific strategy that is not fixed through the simulation. Hence, they can change it if, by social learning, they actually learn that there is another strategy, more profitable, that can be implemented. Thus, they learn from their neighbours both the payoff, obtained in the evident, and the strategy, in the hidden layer.
The action are A/B and they are chosen according to the strategy (R-agents never change, S-agents conform), both are equivalent.
In the evident layer they can change the action (A or B), if the players are S-agents and their links if the players are R-agents.
Yes, they can only change one link at the time.
The link creation and cut are determined unilaterally.
No, only one of the two dynamics is implemented according to rho_L, which is the probability of the learning dynamics.
Please let us know if there is still anything unclear.
The abstract is too long.
Thank you, we have reduced it of about 5 lines.
Footnote 4: The concept is called “Battle of the Sexes” not battle of sex.
Thank you, we have corrected it.
There are a number of smaller typos in the paper
Thank you. We have checked again the text for correcting the typos.
In the modelling section define the set of players. Is there an upper limit?
Thank you. Even though there is not upper limit (for the number of players as for the number of strategy/action), the set of player is fixed at 1000, which we noticed was enough to avoid any model finite-size effects.
Round 2
Reviewer 2 Report
I find the revised version of the paper to be much improved. However, there are some (minor) comments remaining.
The introduction is more concise and mostly clear now. I have, however, still a problem with the second paragraph (lines 27-50). I still do not think that this motivation of the cultural transmission phenomenom is described here in a way that makes it very clear what the subsequent model has to do with this. It might help, if you give some examples of cultural traits you are referring to. I think giving more intuition here, would lead to a better understanding of what you are modelling later one and how this refers to the cultural transmission idea.
In the introduction (line 24) you are suddenly talking about an opponent, while otherwise talking about coordination. I don't think this wording is a good choice. It is not explained what you mean by opponent and as you are otherwise not using technical terms here, I would suggest using a less "framed" wording, such as simply "easier to predict the behavior of others". At the same time, I think an example of what kind of coordination you are talking about would improve this paragraph, as so far it is used without much context to guide the reader.
In the modelling section, unless I am missing something, you are now no longer explaining that an agent has an upper limit of links he can have. I would re-introduce this, as it is a crucial aspect of your model.
While the overall language and stile is good, I found a number of instances where articles and propositions are not used correctly. If possible please have a native speaker check this. I will point out several instances of this, and other typos I found, below.
l.8 --> on such multiplex networks
l.12 --> endurance of a segregated society
l. 48 --> homophily can be the expression of
l.57 --> heterogeneous behavior of agents
l.73-74 --> Coordination games are characterised by the presence of multiple equilibria that are often equivalent in terms of payoff.
l.77 --> that states that in some games
l.115 --> In recent years
There are a number of other instances of these types of errors that should be corrected.
Author Response
We thank the reviewer for the comments.
We have now added several examples in the second paragraph of the introduction in order to provide some general frameworks of our study, lines 37-38.
We have changed the word "opponent" in the first paragraph of the introduction implementing the suggestion. However, we prefer to not give a more specific example of coordination here because our framework remains quite general and it can be implemented in many social contexts.
We re-introduced the sentence related to the constrained number of links in the network at lines 154-155.
We have also corrected the typos and checked again the whole manuscript.
