Social Pressure in Networks Induces Public Good Provision

Round 1

Reviewer 1 Report

Overview

The author develops a dynamic model with forward-looking agents explicitly based on social pressure and shows how the social pressure could generate cooperation in the public goods game. Actions are taken by agents once and for all and updating is stochastic with inertia. 

Much emphasis is put on contagion of behaviors and on the role of leaders.  Contagion is the result of best response of fully rational and forward-looking players (under a sufficient condition involving the ratio of contributing cost to benefit of contributing). The intuition is that if this ratio is not so high, there is room for the contributing friends (and also for the so-called predisposed friends) to move the balance towards the rational choice of each agent to contribute in order to get higher payoff. A leader is an agent who starts contributing, so influencing his friends to follow this choice or to predispose to do so.

Assessment

The paper describes a simple model to explain why individuals participate in collective actions (of any nature) in presence of social pressure. The paper is well written and clear, introducing interesting considerations in a well studied strategic setup. I think it is a nice addition to the literature.

Specific comments:

(1) Please, clarify in what sense contagion in this paper is different from what typically happens whenever there is transition from one equilibrium to another one. If it is not different, than plainly acknowledge this.

(2) Please provide a sound explanation of the reasonability of corollary 8 for Spearheaded Equilibria in the case of limited observability, namely how the leader could ever compute equation (4) if knowledge about the network is limited to neighbours.

(3) As the author suggested, heterogeneity in the parameters could be introduced. I would suggest to acknowledge that the heterogeneous problem could be formulated and solved using Scale-Free Networks, i.e., real networks that exhibit fail-tail degree distributions such as Twitter.

Author Response

Thank you very much for your comments. I provide here answers, which I hope properly address them.

(1) Please, clarify in what sense contagion in this paper is different from what typically happens whenever there is transition from one equilibrium to another one. If it is not different, than plainly acknowledge this.

Answer: I have changed the second paragraph in the Introduction, to clarify this difference: “The earlier literature explained contagion over time (Kandori et al., 1993; Young, 1993) focusing on transitions between different equilibria: every individual best-responds to everybody else in society, and behavior changes from generation to generation, due to random shocks that shift behavior between different basins of attraction. In contrast, in this paper behavior change happens as part of the equilibrium: as agents contribute, they increase the incentives for other agents to contribute due to social pressure, generating a snowball effect. Crucially, unlike Kandori et al., 1993; Young, 1993; Morris, 2000, in which behavior change happens as agents change their selected action, in this paper contagion happens instantly, as part of the best response of the individuals in the network. Therefore, rather than analyzing how different conventions evolve over generations, I focus on how behavior can change almost instantly, due to contagion in the network induced by social pressure.”

(2) Please provide a sound explanation of the reasonability of corollary 8 for Spearheaded Equilibria in the case of limited observability, namely how the leader could ever compute equation (4) if knowledge about the network is limited to neighbours.

Answer: This is a point that was unclear, I apologize. After thinking about what condition would be necessary, I have come to the conclusion that the leader would actually need perfect observability, while the rest of the population still has limited observability (this is still a substantial weakening of the assumption of perfect observability by the entire population). Therefore I have changed the relevant paragraph: “On the other hand, for a spearheaded equilibrium under limited observability, we would need one extra condition, and that is that the agent $i$ who is to become leader actually has perfect observability”. I have also changed the proof of Corollary 8 accordingly (the proof does not change substantially).

(3) As the author suggested, heterogeneity in the parameters could be introduced. I would suggest to acknowledge that the heterogeneous problem could be formulated and solved using Scale-Free Networks, i.e., real networks that exhibit fail-tail degree distributions such as Twitter.

Answer: I have changed the Conclusion to reflect this: “Several extensions could be developed for the baseline model. I have assumed homogeneity in the benefits, costs, and punishments. Introducing heterogeneity would generate new interesting predictions; in particular, analyzing how heterogeneity in the parameters interacts with homophily (the tendency of people to have friends like themselves), when both heterogeneity and homophily take place along the same dimension (i.e. agents with lower cost of contribution tend to be friends with each other). This could be modeled using scale-free networks (in which the degree distribution has fat tails, and that better capture real-world networks such as the Internet or Twitter, Barabási and Bonabeau, 2003) and that have already been used to analyze the Prisoner’s Dilemma with heterogeneous investments (Cao et al., 2010; Wang et al., 2018). This would open up new questions, such as whether the network would be robust to a government trying to stifle potential leaders, for example by removing hubs from the network (Perc, 2009).”

References not included in the original manuscript: 

Barabási, A.-L., & Bonabeau, E. (2003). Scale-free networks. Scientific American, 288(5), 60–69.

Cao, X. Bin, Du, W. B., & Rong, Z. H. (2010). The evolutionary public goods game on scale-free networks with heterogeneous investment. Physica A: Statistical Mechanics and Its Applications, 389(6), 1273–1280. https://doi.org/10.1016/j.physa.2009.11.044

Perc, M. (2009). Evolution of cooperation on scale-free networks subject to error and attack. New Journal of Physics, 11. https://doi.org/10.1088/1367-2630/11/3/033027

Wang, H., Sun, Y., Zheng, L., Du, W., & Li, Y. (2018). The public goods game on scale-free networks with heterogeneous investment. Physica A: Statistical Mechanics and Its Applications, 509, 396–404. https://doi.org/10.1016/j.physa.2018.06.033

Reviewer 2 Report

Manuscript Number – 1042193

Title Social Pressure in Networks Induces Public Good Provision

Summary:

The article provides a framework to explain contagion in public good provision. It focuses on how social pressure from ‘friends’ in one’s network create incentives for agents to contribute. The game enfolds across multiple periods. In every period, a randomly chosen agent is provided with an opportunity to contribute to the public good. The game continues with a fixed probability every period and payoffs are realized once the game ends. Payoffs comprise of a common benefit which depends on the number of agents in society who have contributed to the public good, a private cost incurred if the agent contributes, and a private social sanction cost if the agent does not contribute. The social sanction cost, which is the main driver of contagion in the paper, is proportional to the number of the agent’s friends who have contributed. The authors show how this fuels contagion where agents rationally anticipate their friends’ willingness to contribute and best respond by contributing.

The paper provides a simple, clean and coherent framework to understand how social pressure can affect contribution in networks. While simplicity is desirable to lay out the mechanism clearly, my main concern with the paper is that it fails to capture some of the important characteristics of such interactions. While the model is simple, it also lacks some of the relevant complexities of real life. Below I detail some of my concerns about the assumptions used in the model.

1. The main innovation of the paper is the inclusion of social sanctions for non-contribution to the public good in individuals’ utility function (equation 1). The authors assume that the social pressure cost is incurred when an individual does not contribute by the end of the game and at least one of her friends In the model, individuals can only contribute when an opportunity arrives (which is stochastic). Hence, it is possible that an individual is not presented with the opportunity to contribute when the game ends. The current model specification assumes that these individuals who could not contribute face social sanction from the contributing friends. This is unrealistic.
2. One possibility for payoffs can be that individuals’ who do not get to update their decision before the game ends face social pressure only if they were not
3. Another issue is that agents suffer social pressure as a function of the number of friends and not the fraction on friends who For example, assume two agents who each have two contributing friends when the game ends. The first agent only has two friends, and both chose to contribute. The other agent has ten friends but only two did. I would assume, the social pressure faced by both would not be the same.
4. I understand the assumption of irreversibility of contribution, but why are people being drawn with replacement? That does not fit with the motivation of enlisting for war as stated by the author. Especially since those who have already contributed cannot change their decision. The set of agents who will receive an opportunity to contribute should realistically be those who have not yet
5. The notation can be improved in places – for instance, the game could be summarized in every period by state S(t) rather than
6. The authors state “Q is the probability that a given agent i will be able to play before the game ends”. Q should be stated as Q(t) since it would take different values for different t (depending on whether the agent has already been provided with an updating opportunity or not). The authors claim that since the game is symmetric Q is the same for all agents – this is only true for Q(1), at the start of the In subsequent subgames, t=2,3,.., Q(t) =1 for agents who have already received an updating opportunity, and <1 for others. Unless the authors mean that Q(t) is the probability of an agent being able to play from period t onwards to before the game ends. Either way, I think the definition and notation need to be made clearer.
7. In explaining Proposition 1, the authors include agent ‘k’. Its not clear to me whether the ‘k’ is a typo and should be ‘i’, or the authors are referring to a separate agent ‘k’?

Author Response

Thank you very much for your comments. Please my answers below, I hope I have properly addressed them.

1. The main innovation of the paper is the inclusion of social sanctions for non-contribution to the public good in individuals’ utility function (equation 1). The authors assume that the social pressure cost is incurred when an individual does not contribute by the end of the game and at least one of her friends In the model, individuals can only contribute when an opportunity arrives (which is stochastic). Hence, it is possible that an individual is not presented with the opportunity to contribute when the game ends. The current model specification assumes that these individuals who could not contribute face social sanction from the contributing friends. This is unrealistic.
   • Answer: I agree with this suggestion and it has been changed in the agents’ utility. While this does not change the results of the paper (as the utility is ultimately irrelevant for those who do not get the chance to play), it is important if the model is used for welfare calculations. 
2. One possibility for payoffs can be that individuals’ who do not get to update their decision before the game ends face social pressure only if they were not
   • Answer: It seems this comment was somehow cut short, but I believe it would suggest to not punish individuals who do not get to play, which is what I have changed the assumption to, in this new version. 
3. Another issue is that agents suffer social pressure as a function of the number of friends and not the fraction on friends who For example, assume two agents who each have two contributing friends when the game ends. The first agent only has two friends, and both chose to contribute. The other agent has ten friends but only two did. I would assume, the social pressure faced by both would not be the same.
   • Answer: This is a fair criticism, for which I have a two-fold response. First, the  assumption on the fraction of friends (rather than the total number) is exactly what is assumed by the influential paper Morris (2000), and therefore some of the questions about contagion for the model with the fraction of friends have been answered there. Using that framework would require re-writing this paper entirely (as all of the calculations would need to be re-worked), and given the previous work by Morris (2000), that would reduce the contribution from doing so in this paper. Second, the models are not so dissimilar when it comes to the economic intuition. In my model, for a person with 10 friends, the punishment ex-ante is larger than for a person with only 2 friends. While it is true that ex-post, the punishment would be the same for both people, if they have 2 contributing friends, that is only the ex-post realization of randomness, and an unlikely scenario (as the most likely scenario is that the person with 10 friends will have more than 2 contributing friends, conditional on a person with 2 friends having 2 contributing friends). Hence, while the math for both models would be different, the economic intuition remains quite similar. 
4. I understand the assumption of irreversibility of contribution, but why are people being drawn with replacement? That does not fit with the motivation of enlisting for war as stated by the author. Especially since those who have already contributed cannot change their decision. The set of agents who will receive an opportunity to contribute should realistically be those who have not yet
   • Answer: In this case, I think the assumption should be understood slightly differently. In this model time is measured in discrete periods, and the fact that people are drawn with replacement corresponds to the idea that the rate at which the game progresses is constant. If people were not drawn with replacement, then as more people contributed, there would be less people left, and hence a higher likelihood of being chosen. This would be akin to "time slowing down" as more people contribute, as now it would be more likely that those who had not contributed would be chosen to play before the game ends. To correct this, and keep the time rate constant, it would be necessary to adjust the probability that the game ends, so that it also keeps track of how many people have contributed. The assumption that people are drawn with replacement avoids these complications, and does not affect the strategic behavior in the game, as those who have contributed do not affect the game anymore.
5. The notation can be improved in places – for instance, the game could be summarized in every period by state S(t) rather than
   • Answer: I have changed the notation as suggested; the set is now $S^t$, and the action of player $i$ is now $s^t_i$, to indicate the dependence on either on the time $t$. 
6. The authors state “Q is the probability that a given agent i will be able to play before the game ends”. Q should be stated as Q(t) since it would take different values for different t (depending on whether the agent has already been provided with an updating opportunity or not). The authors claim that since the game is symmetric Q is the same for all agents – this is only true for Q(1), at the start of the In subsequent subgames, t=2,3,.., Q(t) =1 for agents who have already received an updating opportunity, and <1 for others. Unless the authors mean that Q(t) is the probability of an agent being able to play from period t onwards to before the game ends. Either way, I think the definition and notation need to be made clearer.
   • Answer: I have revised the last part of Section 2, to make the definition of Q more clear. My answer to this comment builds on my answer to Comment #4. If the answer to Comment #4 is accepted, it follows that Q should be constant, in order that the time rate at which the game proceeds also remains constant. In that case, there would be no need to index Q with time. 
7. In explaining Proposition 1, the authors include agent ‘k’. Its not clear to me whether the ‘k’ is a typo and should be ‘i’, or the authors are referring to a separate agent ‘k’?
   • Answer: I apologize for the typo, it has been corrected.

Round 2

Reviewer 2 Report

I have no idea why the system decided to cut off my comments mid-sentence. 

My second comment was: 

One possibility for payoffs can be that individuals’ who do not get to update their decision before the game ends face social pressure only if they were not predisposed.

But I am happy with the updated utility function. 

Author Response

Thanks for your comments