Coalition-structured governance improves cooperation to provide public goods

While the benefits of common and public goods are shared, they tend to be scarce when contributions are provided voluntarily. Failure to cooperate in the provision or preservation of these goods is fundamental to sustainability challenges, ranging from local fisheries to global climate change. In the real world, such cooperative dilemmas occur in multiple interactions with complex strategic interests and frequently without full information. We argue that voluntary cooperation enabled across overlapping coalitions (akin to polycentricity) not only facilitates a higher generation of non-excludable public goods, but it may also allow evolution toward a more cooperative, stable, and inclusive approach to governance. Contrary to any previous study, we show that these merits of multi-coalition governance are far more general than the singular examples occurring in the literature, and they are robust under diverse conditions of excludability, congestion of the non-excludable public good, and arbitrary shapes of the return-to-contribution function. We first confirm the intuition that a single coalition without enforcement and with players pursuing their self-interest without knowledge of returns to contribution is prone to cooperative failure. Next, we demonstrate that the same pessimistic model but with a multi-coalition structure of governance experiences relatively higher cooperation by enabling recognition of marginal gains of cooperation in the game at stake. In the absence of enforcement, public-goods regimes that evolve through a proliferation of voluntary cooperative forums can maintain and increase cooperation more successfully than singular, inclusive regimes.


Model
We consider a population of size Z representing the relevant actors of the system and potential members of coalitions. By forming a coalition, players can produce a public good with a specified degree of excludability. We do not assume coalitions to be cooperative. Players chose alternatively to be members of the coalitions, M -who either cooperate, C, or defect, D -or outsiders, O. The lack of punishment mechanisms purposefully creates a difficulty for cooperation. We represent the coalitions as contribution games in which members interact in (sub-) groups of size N to obtain some benefit, B(C), that depends on the total contribution of each coalition, C. In games of loss, the benefit is often thought of as the loss that is not created, which, in reality, can be hard for the players to grasp. Our first strong assumption relies on considering that, at any given time, all existing coalitions have the same size and that the functional form B(C) is the same for all coalitions. Members contribute to the coalition, c c ≥ 0, which conceptually mimics potential running costs and signals shared goals. Cs additionally contribute a given amount, c, to the game considered to be the cost of cooperation, whereas Ds make no other contributions. A fraction, ≤ ≤ e 0 1 , of the benefit produced is shared exclusively among Ms, creating a so-called club good 48 and making Ds the free riders. The remaining fraction, − e 1 , spills over to everyone, including outsiders, making it a public good. This parameter controls the extent to which the GPG can be privatized to the coalition. The model leaves out the institutional mechanisms of forum shopping and advocacy networks, and it is also comprised of homogeneous players, each with equal weight, so that there is no potential for a large player to nucleate or enforce cooperation (i.e., as a hegemon might do in international politics). Where the model could be conceived in terms of relations between cities, nations, and even aggregations such as the EU, we presume each entity makes decisions on its own accord, with no homophily 49 . There is no perception of a collective goal among players 50, and action is based on return on contribution, not on reciprocation or retaliation of others' actions 51,52 ; only the unconditional pursuit of self-interest in each discrete time-step, with no foresight of the future or memory of past interactions. Even where there are gains to full cooperation, there is no capacity for ex-ante coordinated action, including in small groups. There is no potential for collective punishment or enforcement in the model, either among the players or externally imposed. Incorporating those mechanisms would each conduct to further and more stable cooperation in the structured games presented here 16,17,53 . The goal of our model is to show that allowing for coalitions with overlapping membership is on its own a positive mechanism for the sustainability of cooperation under minimal information.
Informed players. We start by assessing a game of "informed players, " who have complete information on the game and, hence, by definition, can compute payoffs assuming different behaviors and choose their approach strategically (even if without foresight). More precisely, these players can calculate their payoff and a hypothetical payoff with an alternative strategy. In Fig. 1 we specify six different states representing possible individual perceptions of the game at any given point, resulting from combinations of the shape of the benefit generated by total contribution, B(C), and three effective game values: coalition member share, ε ≡ θ′ e N / 1 ; public-good spillover, ε 2 Figure 1. Possible states within the dynamic system, computed for informed players. Depending on the total contribution of the other players, C′, an informed player faces different scenarios depending on the game parameters. On the top right, we consider the marginal gains from switching between each pair of strategies given by ∆Π ≡ Π − Π XY X Y , where Π X are given in Eqs. (3). Here, we set ; and κ ≡ c c / c . Parameters θ and θ′ control congestion of the public and club goods. The sign of each of these three quantities controls the direction of its respective arrow in the states represented.
; and relative cost of engagement, κ (see Methods for extended parameter description). Figure 1 includes representations of marginal gains from switching from strategy Y to X, ∆Π XY . In the decision-making, the relative cost of coalition engagement κ is judged against the relative benefit, ≡ b B c / , whereas the cost of cooperation is judged against the marginal return on investment, . This means that even if the coalition provides large benefits, with κ ε > + b max (1 )/ 1 , in which case condition ii) is fulfilled, the ratio at which the benefit is produced per unit of investment is crucial for sustaining cooperative coalitions.
For a typical growing sigmoidal shape for b(C), we can reproduce general and well-known results. Starting with a large and highly cooperative coalition, if ( ) max is high enough, outsiders will join. However, for very high levels of cooperation, typically the variation of b is small, and new members join as defectors while cooperators are likely to stop contributing, state B. If contributions decrease, but b is still sufficiently high, marginal returns on cooperation increase, which leads to a stable large coalition, creating a dynamic balance between states A and B. Complementarily, as the coalition grows, member share ε 1 decreases, effectively reducing the balance ε b 1 , which leads to lessened cooperation and, consequently, reduced total benefit produced and appeal of the coalition, leading to one of the remaining scenarios, C to E. We expect state A, even if stable, to reach a dynamic equilibrium with other states. Therefore, as a consequence of the assumption that benefits to contribution present decreasing returns for high contributions, even the most successful coalitions tend to have non-universal engagement and must tolerate free riders. These results are not new, but their application to our model of GPG is essential for comparison with a situation in which players cannot access the value of the return to their contributions.
As we already mentioned, a crucial assumption of our model is that players are entirely consistent in their strategy across different coalitions. Reputational gains and trust-building are examples of mechanisms that create a tendency for decisions being taken by an actor in one set of coalitions to have a bearing for similar actions in other coalitions 54 -e.g., interests shaped in one coalition transfer to another. Overall, considering they use the same strategy in all coalitions, in models with informed players, cooperation increases with ε ε Uninformed players and the mitigating impacts of coalition structure. In contrast to the assumption that players have complete information about their options, individuals in cooperation dilemmas often must make decisions without complete knowledge of the game. The same is true of nations with incomplete knowledge of the interests and strategies of others during complex negotiations 55 . As before, actors cannot act "strategically, " i.e., they do not anticipate the (re)action of others 56 . Contrary to the previous section, we now consider the extreme case in which information is absent, and individuals rely on theirs and others' experiences to make decisions. This assumption is a substantial simplification of the recognition in the theory of world politics that information imperfections and high transaction costs motivate governments to create international regimes 57 . To show that polycentricity offers additional information, we compare two scenarios in which "uninformed individuals, " by definition, use the average payoff of players with a given behavior to evaluate the performance of that behavior. Then, we manipulate the structure of interactions without affecting the source of information. This model of behavioral change is inspired in works in evolutionary game theory applied to social contexts 37,58,59 , where individuals use social learning 60 to adopt the currently best strategy in their neighborhood of influence. In our model, contrary to the standard literature in EGT, the structure of interactions is not limited to a fixed group size. Importantly, the interactions can occur between a fraction of the population that scales with the population size -for instance, when the whole population is engaged in a single interaction. As we will show, this means that the result that Nash equilibria are necessarily equilibria of this evolutionary dynamics no longer hold 61,62 . Accordingly, we develop a model where interaction structure can change over time with changes in behavior. Two factors determine the dynamics: (i) average cooperative behavior within coalitions changes depending on the difference in the average payoff of Cs and Ds; and (ii) the average payoff of those members relative to the average payoff of Outsiders governs the change in the number of coalition members. Considering this interpretation, our results can be compared with the analysis in the previous section.
We are interested in studying the effect of overlapping coalitions, rather than a single coalition, which requires the typical coalition size to be smaller than the number of members. For this purpose, we use an exogenously determined shape that both constrains the growth of any coalition and leads to coalition proliferation 19 . In Fig. 2A, we show how N y ( ) is an increasing function of the fraction of members in the population, y, which allows for the continuous growth of the typical coalition size as more individuals engage in coalitions (either as Cs or Ds). If α = 1, then the coalition size is simply the number of players engaged in coalitions (i.e., there is only one coalition). However, for any larger value of α, the coalition size is bound to be smaller, until there is universal participation when y approaches 1. Notice that a value of α bigger than 5 already makes the size of the coalition restricted to its small baseline value, g m , until more than half of the population is participating. This also means that its effects on the dynamics are smaller and smaller.
A single coalition (α = 1). The case of a single coalition can be described by setting α = 1. It is trivial to show that for all shapes of B and values of the parameters, the fraction of cooperators always decreases, leading to (2020) 10:9194 | https://doi.org/10.1038/s41598-020-65960-8 www.nature.com/scientificreports www.nature.com/scientificreports/ the collapse of any "uniformed" single coalition. In Fig. 2B, the lowest level of cooperation happens for α = 1, with non-zero values being due to exogenously imposed noise perturbing the system. Accordingly, Fig. 2C shows the dynamics of this system pointing to configurations in which most individuals are Os. Consequently, the probability distribution, represented by the shadow background, is also close to that point.
This proves how a single coalition is very hard to bootstrap with players whose information is bounded to their current interactions, resulting in an uncooperative equilibrium even for very favorable game conditions with a fully cooperative Nash equilibrium -e.g., a marginal return on a unit of contribution much greater than one. This information limitation is a harsh scenario and, in reality, players negotiate on the basis of some information -even if scarce, uncertain or simply created by analogy with previously known/experienced games or dilemmas; for instance, the Kyoto Protocol climate negotiations applied the Montreal Protocol model of regime structure, even though the game was different 10 . However, the extreme case presented here sets a pessimistic baseline, which, as shown below, can be improved without adjusting the capabilities of the players in the game.
Multiple coalitions (α > 1). Multiple coalitions allow for experimentation with different actors and under different circumstances, which is essential given the complex interests in commons governance challenges and uncertainties as to the strategic interests and intentions of other players. Lack of information -or a deluge of it shows how coalitions grow and proliferate in the dynamic system; for any fraction of members in the population, y, increasing α constrains the size of the typical coalition. Panel (B) shows the distribution (mean ± standard deviation) of the engagement in any coalition, y, and share of those coalition members who interact cooperatively, x. Increasing α results in both higher levels of coalition engagement and in greater cooperation within coalitions. Panels (C-E) further demonstrate the dynamic benefit for cooperation with increasing values of α, for a specific (sigmoidal) choice of the benefit function. They represent the most likely direction of evolution of the system with warmer colors representing faster rates of evolution whereas the background shadow represents the regions where the system spends more time. These results (panels B-E) indicate that increasing parameter α enables player with limited knowledge of the game to better recognize potential gains of cooperation. Notice that in C, even though the system spends most time near the O vertex, the vertex is unstable due to exogenous factors introduced (see below), creating a cyclic dynamic. In effect, coalition-structured governance reduces the cost of absence of information, K. Parameters: Z = 100, 4 , a sigmoidal function specified here with a sharp threshold at ¾ of the group. In order to guarantee the system has no absorbing states, we introduce the possibility for random changes of strategies -an added factor of noise or exogenous shocks -by resetting 18,63,64 for connection between the arrows indicating the most likely direction of evolution and the prevalence times and for details on their computation). (2020) 10:9194 | https://doi.org/10.1038/s41598-020-65960-8 www.nature.com/scientificreports www.nature.com/scientificreports/ -can hobble recognition of welfare-improving opportunities for cooperation. To examine whether coalition-structured governance can overcome the disadvantages of the single-coalition case, we constrain the size of coalitions as players become Members, thus creating multiple overlapping coalitions for α > 1. We show that this will change the dynamics of coalition engagement and cooperation, all else equal. The dynamics of the fraction of cooperators within the coalitions, x, are still governed by the fitness difference between Cs and Ds. That can be expressed in a way similar to what we showed for informed players (see Supplementary Material for proofs) as The first term corresponds to the marginal gains of cooperating that governed the informed dynamics, ∆Π CD , with R standing as the average return on investment, which compares to a cooperation cost, normalized to 1. The last term, K, corresponds to the difference between informed and uninformed players and entails an effective cost for cooperation that (i) exactly cancels the marginal return on investment -when α = 1 (the single coalition), and ii) vanishes when the coalition size is highly constrained, We can interpret K as the cost of the limitations on information available to Cs, which can be zero in some cases. Looking back at our definition of group size constraint, in its essence, the restriction on information is controlled by α− N Zy y / 1 . For high α, larger values of y, engagement in coalitions, can be attained for which the K term remains small, and the lack of information plays a small role. This effect is not an effect of small group size, as setting α to 1 removes any perception of return of contributions, independently of Z and its consequent N. Instead, the effect is the result of the experimentation with different configurations between updates, which is the only way individuals can access more information about the game and the returns to contributions. Polycentricity -in particular, but not exclusively -can achieve such an outcome.
As for the growth of the coalitions, it can be described as with b standing as the average relative benefit produced. As we saw for the informed case, it is crucial that the benefit being produced is high, which can be directly assessed from a comparison between members and non-members. The creation of multiple coalitions allows players to access the marginal gains of the excludable and non-excludable part of the benefit produced by joining a coalition (see Supplementary Material for details on Κ M and proofs). Going back to Fig. 2B, as α increases, the cost of lack of information, K, decreases, which induces an internal fixed point with high participation and cooperation. Nonetheless, the nature of that fixed point is a complex one. In the bottom panels of Fig. 2, we show how, for the sigmoidal shape of the benefit B with a sharp threshold, the internal fixed point with high participation goes from a (stable) spiral, in panel D, to a sink in panel E as α grows. Different shapes of B allow for different classifications of the fixed point, including for its stability, but also change the basin of attraction. Notice, additionally, that the slow regions, represented with arrows with cooler colors, even if transient in the dynamics, have non-negligible probability, increasing the variance of the distribution around the fixed points.

Discussion
Problems of collective action are more easily overcome in small groups, but effective management of common resources frequently requires broad participation. We allow for cooperation to emerge through polycentric structures through a control variable that determines the rules of coalition size and growth (and, consequently, coalition proliferation). Isolating the effect of this variable, we show that, for a general class of public-goods, higher degrees of coalition-based governance not only facilitate a greater generation of non-excludable public goods, but it may also allow evolution toward a more cooperative, stable, and inclusive singular regime. Contrary to any previous study, our results are applicable to the whole range of excludability of the public good, congestion of the non-excludable public good, and for any shape of the return function, whether it implies a need for behavior coordination, dominance, or an optimal mix.
The model caricaturizes dynamic cooperation dilemmas in the provision of GPGs, with non-hierarchical actors making decisions that are both normative and selfish in an information-poor environment. The advantages of increasing multi-coalition regime structure for generation of GPGs, relative to fixed coalition size involving the total population, are a result of minimizing the apparent advantage of free-riders (ubiquitous in the provision of GPGs). By creating multiple references for cooperative action, the multi-coalition structure enables the recognition of marginal gains of cooperation in the game at stake. Indeed, the value of regime creation for overcoming the costs of information has long been known in world politics; governments create regimes to correct for "market failures" in international relations, enabling them to develop agreements in their mutual interest. We argue that coalition-structured governance strengthens the mechanism of information transmission relative to inclusive governance approaches, allowing actors to recognize opportunities for cooperative gains better. This finding builds on benefits already well understood of cooperation in smaller groups, including the development of trust, reciprocity, and ease of enforcement via punishment and ostracism [65][66][67] . These other mechanisms are important, since they may help to sustain cooperation once domestic spillovers are exploited, for instance. We notice that coalition structure governance is not a sufficient guarantee of success. Whenever the incentives are not present, marginal gains cannot be accessed by this mechanism. As an example, Fig. 2E evidences rare bursts, with different sizes, where cooperation collapses before coming back up, a phenomenon observed in other systems, including recently in spatially explicit dynamics 68 . Additionally, we note that the added information created by the coalition-structured governance is (2020) 10:9194 | https://doi.org/10.1038/s41598-020-65960-8 www.nature.com/scientificreports www.nature.com/scientificreports/ reliant on a couple of key assumptions we identified in the text. The first is the tendency for players to have consistent strategies in different coalitions. We argued that non-modeled reputational gains could induce this, but it requires recognition that all coalitions have a common goal even if the lines of action are quite distinct. A balance between redundancy and efficiency would need to be studied. To some extent, this effect can be rephrased on the ability of players to associate actions to an outcome, and that might not be the case if (i) the stance of the individual players is not public or if (ii) the players are involved in coalitions across which they have very heterogeneous responses. The second assumption regards the existence of an identical game being played by all coalitions. Naturally, the different coalitions could be structured to derive co-benefits conditional on overall climate mitigation performance -and that would be a policy recommendation for the coalition co-benefit distribution. However, we expect heterogeneity in terms of efficiency and that the game being played by each coalition to be different. Nonetheless, in an environment where information about the returns of investment is scarce, decisions might be made according to an overall perceived benefit, which can homogenize the effective games different coalitions play.
Defining the conditions under which coalition-based governance may be more effective will require further study. This work has suggested additional merits for multi-level governance and plurilateral coalitionsfor regimes lacking outside external authority, with incomplete information, and strong free-riding dynamics. These merits need to be weighed against inefficiency costs arising from having multiple coalitions. The interplay between the scale of the public good and the scale of the decisions for participation in the governing institutions may also play a paramount role that needs further exploration and in-depth analysis, both in theoretical and practical settings. Our work is also not intended to suggest that fragmentation would benefit cooperation in otherwise functional inclusive regimes 69 . However, the multi-coalition structure -mainly the encouragement of initiatives with domestic co-benefits -may be the most productive way to "boot-strap" a coalition with broadening participation, including for instance the potential for sub-national, national, and regional carbon markets to eventually harmonize into a global market, gradually removing inefficiencies once behavioral coordination barriers are overcome 70 . Our results suggest that reducing the scale at which the coordination game occurs -as with overlapping coalitions -can facilitate larger-scale cooperative outcomes.
Community-based natural resource management is an example of decentralized management of public goods 71 . There, the identified need of power transfer to the local institutions and of accountable representation could be achieved through this overlapping structure, not only by contributing to the management of multiple parcels in diverse groups but also by involving the local institutions in a panoply of related issues, rather than isolating their presumably increased control. Our model suggests that decentralization without overlap may not provide enough incentives, especially when information about the resource dynamics -and the corresponding response to different extracting behaviors -is limited.
As we have reinforced throughout the text, climate will be a crucial test case. Following decades of diplomatic effort, the Paris Agreement has achieved (nearly) full participation by being "catalytic and facilitative" of overlapping cooperative arrangements, including carbon markets and sub-national actions in synergy with country contributions 6 . This structure much more closely resembles a multi-coalition regime than a "comprehensive regime" structure with uniform rules for all countries that had long been sought as a replacement to the Kyoto Protocol. Nonetheless, key questions remain. Does coalition structured governance continue to excel when excludable benefits for cooperators start to run out? Does polycentricity still perform better than an inclusive coalition when the largest player in the system free-rides across coalitions? We leave these important questions for further study.

Methods
We consider a population of finite size Z. Players interact in groups/coalitions of size N to obtain some benefit, B(C), that depends on the total contribution of each coalition, C, and this dependence is the same for all existing coalitions. Players can adopt one of three strategies: cooperate, C, defect, D, or remain outside, O. Cs and Ds contribute to the coalition, c c , but Cs contribute an additional amount, c. A fraction, e, of the benefit produced is shared exclusively among Ms, whereas the remaining fraction, − e 1 , spills over to everyone, including outsiders. This parameter allows us to interpolate between games with and without co-benefits. With this, we can write the payoff of a player with a given strategy when the amount contributed by the remaining players in the coalition is ′ C : Notice that θ′ and θ, defined between 0 and 1, control the "congestibility" of the good 72 . For θ = 1, the good is fully congestible, meaning the participation of one player reduces the spillover produced by the next, a characteristic of common-pool resources; for θ = 0, there is no congestion, which is a common property of air quality and other public goods that spill to other players outside of the game. A variation of this model, which studies a particular functional form of B and type of public good, is discussed in Hannam  , where g m is the minimum group size for a single coalition before multiple coalitions may form.
For the evolutionary game, we use a well-documented dynamic of imitating the better neighbor 73,74 . In each time step, we randomly select a player, X, to potentially change strategy. Player X randomly selects another player, Y, and compares her own average payoff, f X , obtained in all the coalitions that she is part of, to that of player Y, f Y .
player X changes her strategy to that of player Y, where β controls the intensity of selection or, equivalently, the level of errors/certainty in the imitation process. This is often called the logit rule, or Fermi update, also known as pairwise comparison rule 59 , which is the noise version of the "best response" dynamics we discuss, and which entails very interesting and convenient statistical properties 75 With this, one can write the probability that a player with strategy X changes into strategy Y as . Given that we are interested in a scenario in which information is scarce, certainty on the payoff difference is likely small, so β is small in comparison. In this case, one can show that the average fraction of each of the three strategies, , }, can be described by the so-called replicator equation 64,76 . for all shapes of B and values of the parameters. Thus, <  x 0 for any fraction of cooperators and, therefore, no cooperation, = x 0, is the only stable state. In turn, <  y 0, and, necessarily, = x y ( , ) (0, 0) is the final stable state, in which there is no coalition or contributions.
Multiple coalitions (α > 1). We obtain Eq. (1) by plugging Eq. (4.1) and Eq. (4.2) in Eq. (5.1). We can rearrange the latter as . In models of infinite populations that do not consider outsider and consider groups of finite and fixed size which experience all possible configurations, the Κ term is zero, reflecting no cost of lack of information (see an illustrative example of such a case in 77 ). In this paper, however, our control structure allows for coalitions that are of a size comparable with that of the population.

Data availability
Source data for Fig. 2 were generated at the authors' computers and are available from the corresponding author on request. The authors declare that the data supporting the findings of this study are available and reproducible with the information provided within the paper.

Code availability
To generate data for Fig. 2, the authors used Mathematica's implementation of Arnoldi's method based on the "ARPACK" library 78 . The code is available from the corresponding author on request.