A Virtuous Circle of Governance Contests with Externalities

Round 1

Reviewer 1 Report

The authors of A Virtuous Circle of Governance Contests with Externalities present a relevant topic, especially in the context in which governments create competitions with financial resources, which also creates a dual impact of both the contestant and the recipient, and the authors develops in the paper the impact of externalities on the efficiency of governance.

The concepts, citations and bibliographic sources are appropriately mentioned by the authors of the research. For example, the authors mention that “the winning-externality competitions are studied by Ref. [5-9] ”.

The research methodology is found in the paper, respectively the authors of the research use models based on probabilities, such as the lottery model of Ref. [16], but it is not highlighted by a separate chapter, which is why we suggest the authors to group the information in the paper.

The results are presented appropriately, with the authors presenting “the impact of externalities on governance efficiency depends on two key factors: (i) the choice of governance contests, the externality of the player and the externality of the winner; and (ii) the relative effectiveness of stakeholder efforts. However, we suggest to the authors of the research to highlight in a separate paragraph the personal scientific contributions to the specialized scientific literature, made in the paper.

The conclusions mentioned by the authors highlight "externalities generated by the efforts of stakeholders invested to win the competition" and present future research, but it should also be mentioned the limitations of the study. Moreover, as I mentioned for the results chapter, it would be appropriate in the conclusions chapter to mention the personal scientific contributions of the authors.

We congratulate the research team for the topic of the paper, and after reviewing according to the above mentioned recommendations we propose for acceptance the paper.

Author Response

#1.

The authors of A Virtuous Circle of Governance Contests with Externalities present a relevant topic, especially in the context in which governments create competitions with financial resources, which also creates a dual impact of both the contestant and the recipient, and the authors develops in the paper the impact of externalities on the efficiency of governance.

The concepts, citations and bibliographic sources are appropriately mentioned by the authors of the research. For example, the authors mention that “the winning-externality competitions are studied by Ref. [5-9] ”.

1. The research methodology is found in the paper, respectively the authors of the research use models based on probabilities, such as the lottery model of Ref. [16], but it is not highlighted by a separate chapter, which is why we suggest the authors to group the information in the paper.

< 1 > We have introduced and classified various contest success functions:

On lines 206-214.

“This logit function has been extensively used in the literature related to contests starting with [17] going through [22]. [23] considers a general logit function as p_i(x_i, x_j) = ax_i^ri/(ax_i^ri + x_j^rj), where a is the ability parameter reflecting the relative strength of players, and ri and rj reflect the marginal productivity of efforts (i ≠ j). [15], [24], and [25] assume that players i and j have access to identical productivities of efforts: ri = rj. Furthermore, [26] assumes that ri = rj = 1. Unlike [15] and [23-26], we focus on the impact of the externalities on Tullock contest outcomes, assuming that a = 1 and ri = rj = 1.”

2. The results are presented appropriately, with the authors presenting “the impact of externalities on governance efficiency depends on two key factors: (i) the choice of governance contests, the externality of the player and the externality of the winner; and (ii) the relative effectiveness of stakeholder efforts. However, we suggest to the authors of the research to highlight in a separate paragraph the personal scientific contributions to the specialized scientific literature, made in the paper. The conclusions mentioned by the authors highlight "externalities generated by the efforts of stakeholders invested to win the competition" and present future research, but it should also be mentioned the limitations of the study. Moreover, as I mentioned for the results chapter, it would be appropriate in the conclusions chapter to mention the personal scientific contributions of the authors.

< 2 > Thank you for your comments. We followed your comments.

On lines 189-193

“Previous studies we consider have focused on efficiency of the contests. In contrast, this study focuses both on efficiency and on fairness of contests with externalities. To this end, we model contests in which there is a difference in the relative efficiency of players’ efforts.”

On lines 488-492.

“The contribution of this study is that it is the first to raise the issue of efficiency and fairness of contests with externalities simultaneously. In addition, this study shows that the comparative-statics results on real rent-dissipation vary by introducing the relative efficiency of efforts in the winner-externality contest.”

We congratulate the research team for the topic of the paper, and after reviewing according to the above mentioned recommendations we propose for acceptance the paper.

Reviewer 2 Report

- removed for peer review -

Author Response

#2.

• removed for peer review -

No comments to respond to

Reviewer 3 Report

A section titled "Discussion" is missing from the article. In this section, the authors should compare their findings with those of other authors.

Only 16 references were used in this paper. This is a rare situation in scientific articles.

The source should be given under each figure.

There are four obligatory elements in the final conclusions: (a) general summary of results, (b) recommendations and implications, (c) research limitations, (d) suggestions for further studies

Author Response

#3.

1. A section titled "Discussion" is missing from the article. In this section, the authors should compare their findings with those of other authors.

< 1> Thank you for your comments. Even if we add a section titled “Discussion”, we have compared the findings of the previous studies with those of this study.

On lines 453-461.

“Many studies have examined the efficiency of contests with externalities. The previous studies that examine the winner-externality contests including [6], [7], and [10] show that nominal rent-dissipation is increased with positive externalities. For example, [9] examines the winner-externality and shows that real rent-dissipation is increased with positive externalities. In comparison, [2] examination of the player-externality shows that rent-dissipation is decreased with positive externalities. If h = 1 in this study, the comparative-statics results of the previous studies still hold.”

2. Only 16 references were used in this paper. This is a rare situation in scientific articles.

The source should be given under each figure.

< 2 > We have added 10 references. Baye et al. (2012) is related to player-externality contests [5] and the others are closely related to the logit functions that we use to illustrate the properties of the equilibrium [18-26].

3. There are four obligatory elements in the final conclusions:(a) general summary of results, (b) recommendations and implications, (c) research limitations, (d) suggestions for further studies

< 3 > In the final conclusions, we have followed four obligatory elements.

Reviewer 4 Report

I believe the topic addressed by the authors, their methodology and results are novel and interesting. The paper is generally well written and structured. 

Given the orientation of the journal toward environmental science, the paper should motivate more clearly the particular institutional setting in which the authors' results could be applied. More detailed examples could be provided in sections 1 and 2. Moreover, I believe the authors should discuss the robustness of their results, especially when more than 2 types of players might participate in every contest or when more than one player of each  type were competing.

More importantly, I miss a clear, formal definition of the notion of "virtuous circle of governance". I believe the authors should clarify explicitly what they understand by this concept. In particular, I suspect it comprises the idea of an aggregate expected payoff that is increasing in the size of the externality; and a difference between the favorite's expected payoff and that of the "underdog" that decreases with the parameter "gamma/delta" as well, while preserving always the highest expected payoff for the favorite player. 

The authors may consider incorporating to Propositions 1 and 2 a comparative static exercise with respect to the asymmetry in the efficiency of players' efforts, measured by parameter "h". It seems to me that the aggregate expected payoff in the contest with player-externality may be increasing in "h", which implies that the least efficient players should be invited to participate in the contest and challenge the favorite player. This might be the case because the favorite will exert much less effort to win.

In any case, I believe the authors should elaborate more on explaining the intuition for the results. They could also organize their findings at the end of section 4, by providing a table with a comparison of every measure (aggregate expected payoff, difference in expected payoffs), both in the player-externality contest and the winner-externality contest.

Finally, in the concluding remarks they acknowledge that the player-externality contest (which is generally preferred to the winner-externality contest, because the differences in expected payoffs decrease with "delta") shows potentially a problem of fairness. That is the case because the (inefficient) underdog's expected payoff may, for some parameter values, be higher than that of the favorite player. However, in my opinion that is not the only potential problem with that contest, since there could be a problem of incentives as well. Especially if every player had to work hard at an earlier stage to increase the efficiency of his/her efforts. Adding a previous stage  in the game could be an interesting extension that might affect their final conclusions.

Minor points:

• The authors could reveal the specific parameter values (for "h" and "theta") they used in Figures 1 and 2.
• On line 238 there is a "differential" sign missing.
• On line 302 (proposition 2), it should read "pi2-pi1".

Author Response

#4.

Comments and Suggestions for Authors

I believe the topic addressed by the authors, their methodology and results are novel and interesting. The paper is generally well written and structured. 

1. Given the orientation of the journal toward environmental science, the paper should motivate more clearly the particular institutional setting in which the authors' results could be applied. More detailed examples could be provided in sections 1 and 2.

< 1.1 > Thank you for your comments. We have added a recent issue as another example.

On lines 40-45.

“Another example is patent races for COVID-19 Vaccines. Recently, it has been argued that cross-vaccination of COVID-19 vaccines made by other pharmaceutical companies can increase the effectiveness of vaccines. The companies that obtain patents for the vaccines do not produce the vaccines for their opponents. A positive assessment of cross-vaccination, however, increases the value of patents.”

< 1.2 > In addition, we specify the differentiation between previous studies and this study as follows:

On lines 189-193.

“Previous studies we consider have focused on efficiency of the contests. In contrast, this study focuses both on efficiency and on fairness of contests with externalities. To this end, we model contests in which there is a difference in the relative efficiency of players’ efforts.”

2. Moreover, I believe the authors should discuss the robustness of their results, especially when more than 2 types of players might participate in every contest or when more than one player of each type were competing.

< 2 > Thank you for the good points. The contests are more realistic with a large number of players participating in the contests. The challenge here thought is to obtain the unique Nash equilibrium in this situation. Considering two different types of groups, we can model the situation in which each group allocates prize according to a predetermined sharing rule. As the number of players increases in this situation, it can be examined how the equilibrium outcomes changes. New research will be needed for this analysis, which we leave for future research.

On lines 499-504.

“This study assumes two-player Tullock-type contests. The contests are more realistic with a large number of players participating in the contests. For example, [2], [4], and [9] analyze the two-symmetric-group contests with externalities. This study can be extended by [2], [4], and [9], considering two-asymmetric-group contests. Additional research will be needed to develop these models.”

3. More importantly, I miss a clear, formal definition of the notion of "virtuous circle of governance". I believe the authors should clarify explicitly what they understand by this concept. In particular, I suspect it comprises the idea of an aggregate expected payoff that is increasing in the size of the externality; and a difference between the favorite's expected payoff and that of the "underdog" that decreases with the parameter "gamma/delta" as well, while preserving always the highest expected payoff for the favorite player. 

< 3 > Thank you for your comments. We have defined the notion of “virtuous circle of governance” as follows:

On lines 47-53.

“…virtuous cycle of governance refers to a situation in which the external effects of governance contests increase the expected payoffs of stakeholders, while lowering rent dissipation, and reducing the inequality of the expected payoffs of them. This study defines the situation in which external effects improve the efficiency and fairness of government contests as a virtuous circle of governance.”

4. The authors may consider incorporating to Propositions 1 and 2 a comparative static exercise with respect to the asymmetry in the efficiency of players' efforts, measured by parameter "h". It seems to me that the aggregate expected payoff in the contest with player-externality may be increasing in "h", which implies that the least efficient players should be invited to participate in the contest and challenge the favorite player. This might be the case because the favorite will exert much less effort to win.

< 4 > We have considered the comparative-statics results.

On lines 259-262.

(c) ∂(p₁^* + p₂^*)/∂h < 0 for 1 < h < (2 – γ –2γ²)/(2 – 3γ) and γ < 2/3, or ‘r ≥ 2/3’

∂(p₁^* + p₂^*)/∂h = 0 for h = (2 – γ –2γ²)/(2 – 3γ) and γ < 2/3; and

∂(p₁^* + p₂^*)/∂h > 0 for h > (2 – γ –2γ²)/(2 – 3γ) and γ < 2/3.”

On lines 353.

(c) ∂(p₁^** + p₂^**)/∂h > 0.

5. In any case, I believe the authors should elaborate more on explaining the intuition for the results. They could also organize their findings at the end of section 4, by providing a table with a comparison of every measure (aggregate expected payoff, difference in expected payoffs), both in the player-externality contest and the winner-externality contest.

< 5.1 > We have explained the intuition for the results.

On lines 269-288.

“Our findings are interpreted as follows. If the efficiency of the underdog’s efforts is close to that of the favorite, the two players increase their equilibrium effort levels in γ, increasing the external benefits. The external benefits generated by the two players’ effort are evenly distributed to both the favorite and to the underdog. In contrast, when the increase in the equilibrium effort level of the underdog is small compared to that of the favorite, the equilibrium expected payoff of the underdog is greater than that of the favorite.

Proposition 1(c) indicates that equilibrium total expected payoff decreases and increases with the relative efficiency of players’ efforts: ∂(p₁^* + p₂^*)/∂h= {2h(2 – 3γ) + 2γ² + γ– 2}/(1 + h – 2γ)³. When h increases, there may be a situation in which equilibrium total expected payoff decreases: ∂(p₁^* + p₂^*)/∂h < 0. The reason relates to the size of external effects and free riding behavior. Free riding is a more critical issue in the player-externality contest than in the winner-externality contest. If the difference in the relative efficiency of efforts increase in the case of large external effects, the underdog reduces more effort than the favorite (see Appendix 1). This is a major factor in lowering the equilibrium total expected payoff.”

< 5.2 > Even if we do not provide a table with a comparison of every measure, we explained why the comparative-statics results between two contests are different.

On lines 366-377.

“The outcomes of the winner-externality contest in Proposition 2 is different from those of the player-eternality contest in Proposition 1. The key reason is related to the problem of fairness. The underdog cannot obtain external effects in the winner-externality contest, while he can in the player-externality contest. Herein, in the player-externality contest, fairness problem can arise when external effects are large and/or the relative efficiency of efforts are not significant (see Proposition 1(a). However, it is not yet clear that which contests are more efficient. This is because efficiency must be compared through the results of both contests, but fairness can be analyzed as a result of each game. In next section, we analyze efficiency of the two contests by comparing rent dissipation.”

6. Finally, in the concluding remarks they acknowledge that the player-externality contest (which is generally preferred to the winner-externality contest, because the differences in expected payoffs decrease with "delta") shows potentially a problem of fairness. That is the case because the (inefficient) underdog's expected payoff may, for some parameter values, be higher than that of the favorite player. However, in my opinion that is not the only potential problem with that contest, since there could be a problem of incentives as well. Especially if every player had to work hard at an earlier stage to increase the efficiency of his/her efforts. Adding a previous stage in the game could be an interesting extension that might affect their final conclusions.

< 6 > Thank you for your comments. We have done.

On lines 493-504.

“In addition, this study has yet to answer various parts to consider in governance contests with externalities. For example, if both players worked hard at an earlier to win the endogenous prize including external effects, the conclusions this study has obtained may be changed. [15] and [16] explore the repeated contests without externalities. We could extend this study by using the models in [15] and [16], and adding in externalities. This study assumes two-player Tullock-type contests. The contests are more realistic with a large number of players participating in the contests. For example, [2], [4], and [9] analyze the two-symmetric-group contests with externalities. This study can be extended by [2], [4], and [9], considering two-asymmetric-group contests. Additional research will be needed to develop these models.”

Minor points:

• The authors could reveal the specific parameter values (for "h" and "theta") they used in Figures 1 and 2.
• On line 238 there is a "differential" sign missing. 
• On line 302 (proposition 2), it should read "pi2-pi1".

• Thanks – we have done this now.