The Evolution of Cooperation in One-Dimensional Mobile Populations with Deterministic Dispersal
Round 1
Reviewer 1 Report
I've found the present paper quite interesting, it is well-written and the results are explained clearly. I would recommend it to accept for publication in Games. However, I would like to see if the author would address some of my remarks listed below. I think the discussion part of the present work could be slightly extended to attract more attention from the audience.
- P2L27: I am not sure that 1D lattice is actually a lattice. I would call it as a 1D chain or something like that. For example, the author could write "a 1-dimensional lattice or a chain".
- P3L65-69: It is a bit unclear to me: can an individual move only to unoccupied nodes within the radius R or, say, to exchange its position with another individuals in chosen occupied node? The same notion relates to P3L89: what are those *potential* locations?
- P3Eq(1): Am I right to say there would be no difference of considering a multiplicative cost of cooperation rather than additive?
- P3Eq(1): Could the author provide some example of a real system, e.g. of interacting bacteria, in the Introduction or Discussion that would fit the proposed framework? I am not sure that a system of siderophore producing bacteria vs cheats would fit to that, because cheats cannot survive without cooperators (see www.pnas.org/cgi/doi/10.1073/pnas.1612522114 or http://rspb.royalsocietypublishing.org/content/284/1859/20171089 for some details). One of the examples from works of Erwin Frey may also work. In any case, it would be nice to draw some analogies with real biological systems and mention this in the present paper.
- P11L257: It looks for me that the results can be regarded as reminiscent to another work of Steward, Parsons, and Plotkin (http://www.pnas.org/content/113/45/E7003.short) that states: "We show that simple strategies, where players do not vary their behavior much at all, can nonetheless be successful, and that access to a broader range of behavioral choices can cause a population to evolve toward lower levels of cooperation." This means that *almost deterministic* approach compared to *deterministic* approach would indeed force a population to evolve toward lower levels of cooperation. Am I right? I think it would be nice to reflect this analogy with Joshua Plotkin paper.
- Dimensionaly: just to be sure, am I right that 2D regular lattice would not change much the results of the present work? I suspect that a small world network would nulify the difference between *almost determenistic* and *deterministic* approaches.
Author Response
See the attached file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Thank you for opportunity to review the manuscript entitled “The evolution of cooperation in 1-Dimensional mobile populations with deterministic dispersal”. In this manuscript, author considers an evolutionary dynamics of cooperation in a structured population with migration and investigates the influence of different factors on the fixation probability of cooperation.
The topic is well presented, the description of the model is clear, and results are solid. This paper will certainly contribute to the field of evolutionary game theory. Nevertheless, at some points the manuscript needs further polishing before this work can be published. Overall I recommend minor revisions. My detailed comments are listed below.
1. L.28: As far as I remember, 1-dimensional lattice with periodic boundaries is a ring and not torus (which is 2-dimensional lattice).
2. L.76: Phrase “The payoff of the individual I_n” reads, as I_n is the notation for the payoff. Later it is revealed that I_n is the notation for an individual, and payoff is denoted as p_n. Overall it creates confusion.
3. L. 89: What does the procedure of “sampling” of the node involves? What value is obtained is unclear. Is it an average payoff in the node, or payoff of the individual of the same type in the node, or what would be the payoff of the migrating individual after it moves to this node?
4. L.108: Reading through all events occurring at one step of simulations, it seems that the order is following: the game is played (L.82) -> the event is chosen (L.84) -> if the event is reproduction, the game is played again (L.108). If my interpretation is correct, at reproduction events, the game is played twice. How two payoffs are handled? Are they summed up, or one is discarded, or something else happens?
5. Figs 1-11: The data was collected for a finite number of values D. Yet, the style of plots (continuous lines) implies that information is available for any value of D. I recommend adding point markers indicating obtained data, or change the figures style to scatter plot.
6. Figs 1-11: The data obtained is the fraction of fixation events, which is close but not equal to the fixation probability (y-axis label). The uncertainty in fixation probability is easy to estimate as the number of fixation events is a random number from binomial distribution B(x| 10000, pf), where pf is the fixation probability. Thus, the variance in the number of fixation events can be easily computed. I suggest adding the error bars indicating some meaningful confidence interval (either 1/2/3 standard deviations or 50/90/95% confidence intervals). With such a large sample size, these confidence intervals might be to small to be visible on figures, then the brief mention of this in the text will be enough.
Author Response
See the attached file.
Author Response File: Author Response.pdf
Reviewer 3 Report
Referee report on the paper „The evolution of cooperation in 1-dimensional mobile population
with deterministic dispersal”
The author addresses the issue how mobility/dispersal of players affects the cooperation level
in populations. He/she considers n-player public-good game on one-dimensional lattice (a circle)
with stochastic Moran process of evolution of population and dispersal of players. The presented model is a modification of the one in the reference [8], where authors considered probabilistic dispersal - players move with a higher probability to places which provided higher payoffs.
In the present model, in deterministic dynamics, players move to the closest location with
the highest payoff. There are two main results of the paper under the review:
1. Deterministic dispersal (as opposed to a probabilistic one) favors cooperation in the case
of high mobility and medium dispersal range.
2. Deterministic dispersal (as opposed to a probabilistic one) does not cause the extinction
of cooperation in the case of high mobility, large dispersal range, small benefit-to-cost ratio,
and small strategic-interaction neighborhoods.
This is an interesting result. But is it a general one? To answer such a question, one has to study other related models. First of all, the payoffs given in eq. (1) are very specific. How two dynamics (with deterministic and probabilistic dispersals) differ if one adopts a usual public-good game payoffs, where each cooperator provides a unit good to a common fund which is then (after rescaling) divided evenly into all players in a given neighborhood? Anyway, eq (1) has not been explained; is the first term in (1) important for obtained results?
What is the situation for canonical games describing social dilemmas, that is for Prisoner’s Dilemma and Snowdrift games? What happens if one uses other updating mechanisms,
for example pairwise comparison?
Fixation probability of A-strategy is usually defined as a probability to arrive at the homogeneous state of A-players starting with only one A-player. The author starts with random initial conditions. We need an an explanation here.
When we allow mutations, to have ergodic stochastic dynamics, one may ask what is the level
of cooperation in stationary states in models with deterministic and probabilistic dispersals.
It would be interesting to know what would happen if we restrict the maximal number of players allowed at any lattice site.
The author has written in the Introduction that „spatial structure may also inhibit cooperation”.
However in reference [10], spatial games were compared to replicator dynamics of well-mixed populations. It is more appropriate to compare spatial games to random-matching models as it was pointed out recently in [ C. Hauert and J. Miękisz, Effects of sampling interaction partners and competitors in evolutionary games, Phys. Rev. E 98: 052301 (2018).]. Such a comparison shows that spatial structure favors cooperation also in the Snowdrift game.
In short, I suggest that the author investigates related models to compare deterministic
and probabilistic dispersals. Only then one can say that deterministic dispersal favors cooperation.
I recommend a major revision of the manuscript.
Author Response
See the attached file.
Author Response File: Author Response.pdf
Reviewer 4 Report
In this paper the author develops an earlier model of evolution in a mobile population, using two alternative methods of movement. This is an interesting and worthwhile paper, as was the original, and the new results definitely add value. The paper is well presented, and the main message clearly expressed. I have a few points for the author to address, none of them particularly serious, although point 6) in particular warrants careful consideration.
1) Page 3, equation 1. In this model a cooperator can give a benefit to all other inidividuals, no matter how many, whilst paying only a single cost. Some real benefits are like this, for instance information, but others are not, for instance food or other physical resources. The author should discuss the nature of costs and benefits here in more detail.
2) Page 3 line 87 To select a dispersal move, I believe that the selected individual compares the payoff it would receive at all given sites within range, assuming that all other individuals stay where they are and it moves. Can the author sligthly reword this text to make this clear?
3) Page 4 line 131 ``absence of dispersal (M=0)'' and related text. There is movement in the population because offspring can be placed at a different site to the parent, so that the distribution is constantly changing, as opposed to static models such as the classical evolutionary graph theory ones. This is worth stating explicitly. In fact if M=1 there is an equal chance of the next move being a dispersal move, or a replacement which happens in a similar way to dispersal, so that change in the population distribution over the sites is roughly (exactly?) equally due to dispersal and replacement. For M>1 (M<1) dispersal is the bigger (smaller) contributor.
4) Table 1: there is a large range of all parameters used here, except the value of B when compared to C. Why was this narrow range picked, and what happens outside the range? Note that this does not really keep the overall costs compared to benefits fixed, as if a population size is doubled the potential benefits double but the cost does not, and the population size varies from 10-40.
5) Can the author expand a little on why given results happen in places, in addition to just saying what happens? For instance relating to the paragraphs on page 5 lines 151-156, and page 7 line 189 to page 8 line 194.
6) The author discusses the ``breakdown of cooperation'', and in particular states that this occurred with the probabilistic model but not with deterministic dispersal. There are two issues that need to be clarified here. Firstly, there does not seem to be a definition of what breakdown in cooperation means in the paper. When I look at the figures all I see are different values of the fixation probability, which does not obviously imply breakdown below a certain level, say. Secondly, breakdown of cooperation would imply an existing cooperative population being susceptible to defection; thus the fixation probability of defectors would be more relevant. This needs to be explained/ reworded carefully. I note that the other general conclusions of the paper are clearly valid and worthwhile.
7) The discussion is mosly a (useful) summary of what the author has done in this paper. There should be more relating this work to alternative models of other authors and so on.
8) Page 7 line 185 ``identical'' - surely this is just very similar, but not identical?
9) page 13 line 331 JM Smith should be J Maynard Smith (a small but important point, as John Maynard Smith is the single most prominent name in evolutionary game theory).
Overall this is a very nice paper that deserves to be published, subject to the points above being addressed.
Small points:
Abstract line 9 wth the best
line 11 contrasts with an
line 12 in the breakdown
page 2 line 37 in a generally
line 41 with the best
line 52 of a (or the?) public goods game.
page 3 line 59 different sizes
line 79 providing the benefits
line 83 obtain the associated
page 5 line 135 achieved a fixation
line 139 with a sparse
line 141 in the breakdown
line 144 in a small
page 7 line 175 with the probabilistic
line 177 in a significantly
line 186 with a low
line 191 for a denser
Figure 7 caption when the mobility
page 8 line 196 demonstrated a small
page 9 line 199 in the fixation
line 202 with a high
line 203 demonstrates the significant
line 206 increasing the dispersal
line 209 negating the improved
line 215 because a high
line 215 with a large
Figure 11 caption for a high
Figure 12 caption for a high
of the fixation
page 10 line 232 of a public
Figure 13 caption with a small
Author Response
See the attached file.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
Referee report on the paper „The evolution of cooperation in 1-dimensional mobile population
with deterministic dispersal”
The main result of the paper is that a deterministic dispersal allows to maintain a reasonably high cooperation level as opposed to a probabilistic dispersal. even in sparse populations with small
interaction neighborhoods, high mobility rate, and large dispersal range.
However, the considered model and dynamics are very particular so no general conclusions (as the title suggests) can be attained. The author does not want to do some other simulations to strenghten his/her results. OK, let us stick to this particular public goods game and one-dimensional lattice. But instead of the Moran process let us consider a pairwise comparison process for at least few interesting parameters. Does the conlusion hold? By the way, in the Moran process, deaths should also occur in the strategic neighborhoods, is this true for the author’s model?
I think that this is a minimal improvement which is necessary for the publication of the manuscript.
Methodology question: why if the event chosen is the reproduction, individuals must play again, can they use payoffs already obtained?
Editorial remark: the author of the reference [15] is Miekisz not Mikisz.
Conclusion: some further very limited simulations are needed in order for the manuscript to be published.
Author Response
See attached.
Author Response File: Author Response.pdf
