Common pool resources with support

We examine the role of support for coalition stability in common pool resource games such as fisheries games. Some players may not want to join a coalition that jointly manages a resource. Still, because they benefit from spillovers, they may want to support the coalition with a transfer payment to set incentives for others to join. We find that the impact of support on equilibria of this game is limited to games with three or five players.

community forest management, and fisheries policies. This paper explores the impact of support in a CPR game with coalition formation. The type of support we consider is a payment from singleton players to a coalition of players who co-ordinate their extraction efforts to jointly manage the resource. This type of support differs from earlier papers that have considered support in CPR games. The way we model support does not require an outside agency that subsidizes the coalition as in Ansink and Bouma (2013) and Zavalloni, Raggi, and Viaggi (2019) or has agenda-setting power as in Ansink, Weikard, and Withagen (2019). 1 It also does not require commitment as in Long and Flaaten (2011) who consider supporting transfers by the coalition members to compensate singletons for joining the coalition. Rather, we model support in a sequential game following Ansink et al. (2019). They adapt the game setup of Carraro and Siniscalco (1993) to show that the impact of supporting transfers by singletons on coalition formation in public goods games may not require commitment.
While coalitions in public goods games are known to achieve a modest degree of cooperation (Barrett, 1994;Carraro & Siniscalco, 1993;Hoel, 1992), a well-known result in the fisheries literature is that coalitions in CPR games are generally unstable when there are more than two players (Pintassilgo & Lindroos, 2008). Larger coalitions can only be achieved when coalition membership brings about additional advantages through, for example, Stackelberg leadership (Kwon, 2006;Long & Flaaten, 2011) or economies of scale (Johannesen & Skonhoft, 2009). 2 This negative result on cooperation in the commons is driven by strong free-rider incentives with singleton players increasing extraction efforts in response to the coalition's resource conservation measures. Hence, it is worthwhile to assess whether the option of support can improve upon this result. We approach this option without requiring aspects like commitment, an outside agency, or additional advantages of coalition membership.
We model support using a sequential game that extends the standard two-stage coalition formation game. The extension is adding two stages at the start of the game in which the players choose whether to become a supporter in Stage 1, and subsequently the set of supporters chooses the level of a supporting transfer payment to the coalition in Stage 2. In the remaining two stages, a subset of the remaining players forms a coalition in Stage 3 and all players choose their effort level in a standard CPR game in Stage 4. In this last stage only the coalition members co-ordinate their actions while all others, supporters and free-riders, choose their individually optimal effort level. Since the supporting transfer is shared by all coalition members, the purpose of the supporting transfer is to make coalition membership more attractive. For sufficiently high transfers, the incentive for coalition members to leave the coalition is removed completely. The question is, however, whether supporters are willing to pay sufficiently high transfers. We will see that the answer to this question is not always negative.
The main result of this paper is that there exist equilibrium solutions with positive levels of support. There exist only two such solutions, however. One equilibrium occurs when there are three players in the game, one of which is a supporter and the other two are coalition members. The other equilibrium occurs when there are five players in the game with two supporters and three coalition members. In larger games, there are no stable coalitions. At first sight, this does 1 Agenda-setting power can be used to make take-it-or-leave-it offers and can help implement a target solution. Subsidies will usually support the internal stability of coalitions. From a modeling point-of-view, it seems undesirable to have support offered by an outside agency which is why we, in this paper, endogenize support. Which of the two settings-exogenous or endogenous support-is more realistic will depend on the specific case studied. For CPR games, the focus of much of the literature is on the governance of the resource by the resource users themselves (Ostrom, 1990). This focus suggests that the setting with endogenous support is more pertinent. 2 There are many alternative approaches to model cooperation in the commons, including for example, Heintzelman, Salant, and Schott (2009)  not seem to be good news for cooperation. Still, the result may be relevant as it covers the many situations where the number of players is indeed small. For all cases with four, six or a larger number of players we must draw the conclusion that a support option is not effective.
In the remainder of this paper, we introduce the model in Section 2 and present our main result in Section 3. Next, we illustrate this result in Section 4 based on a fisheries example. We conclude in Section 5.

| SUPPORT IN A COALITION GAME
Consider a set N n = {1, 2, …, } consisting of n symmetric players. Our game is based on the standard two-stage coalition formation game (cf. Barrett, 1994). We apply this game to a CPR following Pintassilgo and Lindroos (2008) and we extend it with support in line with Ansink et al. (2019). The resulting sequential game has four stages. We first introduce the stages in detail and subsequently solve the game backwards.

| The four-stage game
In Stage 1, all players simultaneously decide whether or not to become a supporter. We denote this decision for each individual player by σ i N {0, 1} i ∈ ∀ ∈ . When σ = 1 i , player i becomes a supporter and otherwise not. The outcome of Stage 1 is a set S i σ = { : = 1} i , which comprises all supporters. Since players are symmetric we will often refer to the set of supporters by its size s S = . In Stage 2, the set of supporters jointly decides on the level of the supporting transfer payment t s ( ) to the coalition, with each supporter paying t s s ( )/ due to (ex ante) symmetry. Since support precedes Stage 3 coalition formation and Stage 4 effort in the CPR game, supporters take into account the effect of their support level on these subsequent actions.
In Stage 3, given t s ( ), the remaining n s − players simultaneously decide whether or not to join the coalition. We denote this decision for each individual player by μ i N S {0, 1} \ i ∈ ∀ ∈ . When μ = 1 i , player i becomes a coalition member and otherwise not. The outcome of Stage 3 is a set M i μ = { : = 1} i , which comprises all members. At the end of Stage 3 the set of players is partitioned into three subsets, S, M, and F , where the set F N S M = \( ) ∪ comprises the free-riders. Similar to our notation s for the number of supporters, we will often refer to the coalition by its size m M = | | and the set of free-riders by its size f F = . In Stage 4, given the partitioning in player types s m f ( , , ), all players simultaneously choose their individual effort level e i in the CPR game. Singletons (i.e., free-riders and supporters) choose their effort level to maximize their individual payoffs while coalition members coordinate their effort levels to maximize joint payoffs.
Payoffs are determined using the canonical version of a CPR game (cf. Mesterton-Gibbons, 1993). In this game, we denote a players' individual harvest levels by h qe X i i ≡ , where q is the socalled catchability coefficient, e i is individual effort and X is the resource stock. The stock grows with intrinsic growth rate r according to a logistic growth function where k is the system's carrying capacity. The analysis is conducted for steady states where harvest equals growth such that ( ) where p is the price for selling one's harvest and c is the marginal effort cost. With support, these payoffs are adjusted by transfers, paid by the supporters to the coalition members. Because the CPR game is standard, its detailed description is relegated to the Appendix 2.2 | Solving the game We apply SPNE and solve the game backwards. The main difference with Ansink et al. (2019) is that we use a CPR game rather than a public goods game, which primarily implies different actions by all players in Stage 4. In this stage, equilibrium effort levels are determined from the best response functions of coalition members and singletons (i.e., supporters and free-riders). In the version of the CPR game that we consider here, the coalition-maximizing the coalition payoff and thereby internalizing within-coalition externalities-chooses the same effort level as the singletons (Pintassilgo & Lindroos, 2008). As a result, if m players form a coalition, we effectively have n m − + 1 players harvesting the resource. Solving the system of best responses, equilibrium singleton payoffs λ m ( ), excluding transfers, are defined as follows (a full derivation of payoffs is provided in the appendix): , such that the numerator consists of bio-economic model parameters only. Clearly, payoffs decrease in the number of players n while they increase in the size of the coalition m. Since the coalition behaves as a single player, payoffs to members κ m ( ), excluding transfers, are a fraction m 1 of a singleton's payoff: Equilibrium payoffs in the CPR game given by (1) and (2) can be used to spell out the Stage 3 equilibrium conditions. Recall that in Stage 3, all players who are not supporters choose whether or not to join the coalition. The equilibrium conditions for this stage are the standard internal and external stability conditions for cartel games due to D'Aspremont, Jacquemin, Gabszewicz, and Weymark (1983), which correspond to a Nash equilibrium in membership choices: Internal stability implies that no coalition member has an incentive to leave the coalition, while external stability implies that no singleton has an incentive to join the coalition. Following Weikard (2009), we use a strict inequality sign in the external stability condition which simplifies the analysis in Section 3. The interpretation of this strict inequality is that a singleton who happens to be indifferent between staying a singleton or joining the coalition, is assumed to join. Condition (3) reveals that the main purpose of a supporting transfer t 0 ≥ is to achieve internal stability for a coalition that would not be internally stable without the transfer. The drawback is that the supporting transfer undermines external stability as can be seen from (4).
In Stage 2, supporters choose the transfer t 0 ≥ with the objective of maximizing their joint payoff. This possibility of payoff-maximizing support was briefly considered in an extension by Ansink et al. (2019, section 5.1) but it turned out not to be very effective in a public goods game. 3 In our CPR game, supporters choose a transfer payment as follows: 3 In Section 5 we discuss a more effective model alternative. With slight abuse of notation, we denote by m t ( ) the set of internally and externally stable coalitions under transfer t, as dictated by conditions (3) and (4). Below we will verify that m t ( ) is unique for each given t. By selecting their transfer t s ( ), the s supporters are implicitly selecting the (internally and externally stable) coalition size m that maximizes their payoffs λ m t s s ( ) − ( )/ . Note that the (implicit) selection of a coalition size which is not internally and externally stable could never constitute an equilibrium outcome of the game.
For Stage 1, again we spell out equilibrium conditions, quite similar to those for Stage 3. Recall that in Stage 1, all players choose whether or not to become a supporter. The equilibrium conditions for this stage are similar to those used in Ansink et al. (2019), although we substitute λ m m ( )/ for κ m ( ) using (1) and (2). In interpreting these conditions, presented below, consider the following. In the subgame following Stage 1, any player who is not a supporter will either become a coalition member or a free-rider in Stage 3. A Nash equilibrium in Stage 1 decisions implies that no supporter deviates from σ = 1 i and no other player deviates from σ = 0 i , taking into account these subsequent Stage 3 decisions. It follows that we have four stability conditions in Stage 1 of the game: Conditions (6a) and (6b) are two internal stability conditions for the set of supporters S. They state that no supporter would deviate in Stage 1 to become a member or free-rider in Stage 3. Likewise, conditions (6c) and (6d) are two external stability conditions for the set of supporters S. They state that no nonsupporter (who would choose to become a free-rider or a member in Stage 3) would deviate to become a supporter in Stage 1. Summarizing, the game is solved by checking the equilibrium conditions for Stage 1 and Stage 3, given an optimal choice of the supporting transfer by the supporters in Stage 2 and optimal effort choices by all players in Stage 4.

| EQUILIBRIA WITH SUPPORT
In Section 2.2 we have derived six conditions for the existence of an equilibrium with supporters. Such equilibria can be identified by the numbers of supporters and members as established in stages 1 and 3 of the game, for some given number of players n. We proceed to check whether there exist such combinations s m f ( , , ) that satisfy all six equilibrium conditions (3), (4), and (6a)-(6d). The answer is yes, there exist two equilibrium solutions with support for, respectively, n = 3 and n = 5: s m f ( *, *, *) = (1, 2, 0) and s m f ( *, *, *) = (2, 3, 0). Deriving these equilibria is rather tedious. This will become clear in the remainder of this section, which serves as a proof of this result, while in Section 4 we provide an example to illustrate. Given Stage 4 optimal harvest levels and associated payoffs, we start with the Stage 3 stability conditions (3) and (4). We first verify the existence of at least one combination of integers m n ( , ) with m n 1 < ≤ such that these conditions hold. 4 Solving for t and substituting λ m m ( )/ for κ m ( ), we combine both conditions in a chained inequality: Substituting λ using (1) in the LHS and RHS of (7) and subsequent rearranging we find n m m n m m n m which holds for all combinations m n ( , ) except for the grand coalition (m n = ) when n 3 ≤ . 5 Note that these exceptions are trivial cases since m n = implies s = 0 and so there is no supporter to pay the transfer. Hence, for all nontrivial combinations m n ( , ) there exist supporting transfers that can internally and externally stabilize coalitions of size m. An important corollary of this finding is that a third party could always stabilize a grand coalition with a sufficiently high transfer payment to implement a cooperative first-best effort level.
In Section 2.2 we claimed that a given transfer stabilizes a unique coalition size. To see that this holds it is sufficient to note that the LHS of (7) for an m + 1 size coalition, m λ m λ m ( + 1) ( ) − ( + 1), is equivalent to the RHS of (7) for a coalition of size m. This equivalence implies that if condition (7) holds for some t and m, then it cannot hold, keeping t fixed, for m + 1 nor m − 1 (notice the strict inequality sign). As a result, only coalitions of one size are both internally and externally stable. By selecting their transfer t s ( ), the set of supporters s is implicitly selecting a unique coalition size m.
Two questions remain. One is, in case of multiple possible transfers t s ( ) and related coalition sizes m t s ( ( )), which of these maximizes supporters' payoffs λ m t s s ( ) − ( )/ . This question corresponds to the Stage 2 decision by the supporters. The other question is whether in equilibrium there exist supporters willing to pay such transfers. This question requires the analysis of the Stage 1 stability conditions.
In Stage 2, we employ (5) to determine the supporters' selection of transfers. Transfers are made to internally stabilize a coalition of size m without wasting money, so we use (3) to write the transfer needed to stabilize a coalition of size m: 6 Note that a similar analysis was done by Ansink and Bouma (2013, proposition 3), but we cannot use their proof to check these conditions since it contains a minor error. In equation (A16) of Ansink and Bouma (2013), the denominator of the first term should have s replaced by s ( + 1), since the coalition payoff is shared by s + 1 members if player j joins. Hence, our analysis is slightly different.

5
Substituting m n x = − with x 0 ≥ and solving for n, the inequality in (8) can be written as n x x x x x x x > ( + 3 + 5 + 17 + 18)/(2 + 9 + 12 + 5) 4 3 2 3 2 . Since we require m 1 ≥ , the minimum number of players given x equals n x + 1 ≥ . We substitute n x = + 1 and solve for x to find that the inequality holds for any integer x 1 ≥ . For the trivial case x m n = 0 = ⇔ we obtain 1 + < n n 9 + 2 4 , which holds for any integer n 4 ≥ , confirming the two exceptions presented in the main text. 6 For the trivial case m = 1 we have mλ m λ m ( − 1) − ( ) < 0, and hence no support will be paid. In this case we set t = 0.
Since we know that each transfer induces a unique equilibrium coalition size, we rewrite the supporters' maximization problem in (5) as a problem of the choice of m. Substituting t using (9) we have m t s s λ m mλ m ( ( )) = arg max (( + 1) ( ′) − ′ ( ′ − 1)).
The domain of m is restricted since we need supporters to provide support, hence m n s − ≤ . We write out the Lagrangian with γ denoting the Lagrange multiplier: The related first-order condition and complementary slackness condition to be solved are: , with a remaining fraction that tends to 0 from below as n increases. We proceed to check limit behavior as n becomes large, while results for small n will be assessed numerically below. For sufficiently large n, this asymptote implies that a 1-player change in the number of supporters induces a 2-player change in the number of coalition members. 7 We proceed to Stage 1 and substitute this relation between s and m in stability conditions (6b) and (6d) (note that it is sufficient to show the violation of just one of the four stability conditions in this stage). We also substitute t s ( ) using (9) and rearrange terms to obtain We will show that there do not exist interior solutions with the support that satisfy both conditions. Combining (15) and (16) Next, we verify the existence of at least one combination of integers m n ( , ) such that (17)  , hence assuming large n, this domain can be written as n m 3 − 2 ≥ . On this domain, we find numerically that condition (17) is violated for any combination m n ( , ) with two exceptions: m n ( = 2 7) ∧ ≥ and m n ( = 3 10) ∧ ≥ . These particular parameter combinations, however, violate either condition (6b) or (6d). Hence, for this interior case with large n, no equilibria with support exist.
We proceed to assess the results of stages 1 and 2 for the interior case and small n. We do so numerically. Since the above-described limit behavior occurs already for n 23 ≥ , we check all cases where n < 23. Doing so, we do not find any equilibria with support.

| Boundary solutions
Solutions with γ 0 ≠ give the boundary case where m n s = − , so we have no free-riders. We will see that this boundary case gives rise to two equilibrium solutions with support. To show this we proceed to Stage 1 and assess stability condition (6a) or (6b), depending on which case materializes, as discussed below (again, it is sufficient to show the violation of one of the four stability conditions in this stage). Different cases may arise since at the boundary we are not sure how the number of members changes in Stage 3 in response to a supporter deviating in Stage 1. This change in members is positive if the Lagrange multiplier is sufficiently high, inducing another boundary solution with s − 1 supporters and m + 1 members. This change in members is (weakly) negative, however, if the situation with one supporter less induces an interior solution. From the asymptotic behavior described for the interior case, we know that a reduction in the number of members in response to a deviating supporter can never go beyond −3 members (see Footnote 7). Hence, we have m t s m t s x ( ( − 1)) = ( ( )) + with x {−3, −2, −1, 0, 1} ∈ . For x = 1, condition (6b) does not apply and we focus on condition (6a). Using (9) to simplify this condition and substituting m t s m t s ( ( − 1)) = ( ( )) + 1 we obtain λ m t s t s s λ m t s ( ( ( ))) − ( ) ( ( ( ))). ≥ Since λ m ( ) is increasing in its argument, as can be seen from (1), this condition is violated for any m n s = − . For x {−3, −2, −1, 0} ∈ , we focus on condition (6b), which is stronger than (6a). Starting with x = 0, we find a closed-form solution for the zero of condition (6b), which dictates that condition (6b) is violated for all m 2 ≥ : Repeating this analysis for the other values of x shows that condition (6b) is violated for Combining the above cases, a sufficient condition for violation of condition (6b) is that the number of members is sufficiently large: m 7 ≥ . We proceed to assess the results of stages 1 and 2 numerically for the boundary case and m < 7. Doing so, we find two equilibria with support: No other equilibrium with support exists and this result represents only a marginal improvement on the case without supporters where the only nontrivial equilibrium is m*=2 if n = 2 (Pintassilgo & Lindroos, 2008). Surprisingly, although we find a support equilibrium for both n = 3 and n = 5, we do not find one for n = 4. Going from three to four players, the additional player undermines coalition stability irrespective of his player type. If he were a free-rider such that s m f ( , , ) = (1, 2, 1), he would undermine external stability of the set of supporters; condition (6d) is violated. If he were a member such that s m f ( , , ) = (1, 3, 0), the single supporter is not willing to provide support to the resulting 3-player coalition; applying (10) for s = 1 does not result in m = 3. Finally, if he were a supporter such that s m f ( , , ) = (2, 2, 0), he cannot increase the size of the coalition so he has an incentive to deviate; condition (6b) is violated.

| ILLUSTRATION
In this section, we illustrate our main result with an example. This example highlights the players' choices in Stage 1 and Stage 3 of the game and shows how these choices depend on the number of players and their partitioning in player types s m f ( , , ). Players' choices are guided by comparisons of equilibrium payoffs (1) and (2) as well as transfers (9). These functions depend partly on bio-economic model parameters r,p, k, and β c pqk ≡ . In the simulations, we choose to keep these bio-economic parameters fixed and vary only n and its partitioning s m f ( , , ). In choosing the bio-economic parameters, keep in mind that these only scale equilibrium payoffs and as such, their selection is not crucial for the illustration. We take parameters from an example fishery as reported by Seijo, Defeo, and Salas (1998) ≈ ⋅ . In Stage 3, the question is how many singletons will join the coalition. Figure 1 illustrates, for selected values of n 3 ≥ , which Stage 3 coalition size m t s ( ( )) is implicitly chosen by the set of supporters s through the transfer t s ( ) in Stage 2. Each of the marks depicts the unique internally and externally stable coalition size m that maximizes the supporters' payoffs. Given n, we observe for small s that m t s ( ( )) is weakly increasing, illustrating that larger groups of supporters choose to support larger coalitions, but only if there is a sufficiently large set of supporters. The increasing slope is not linear due to the discrete nature of our game. Once the constraint m n s − ≤ kicks in, boundary solutions imply that the line connecting the m t s ( ( )) marks starts to decrease with slope −1.
In Stage 1, the question is how many players will choose to become a supporter. These choices are driven by payoff comparisons, taking into account subsequent decisions in the remaining stages of the sequential game. Figure 2 illustrates payoffs to each of the three player  Figure 1: n = 5 and n = 10. In both panels, for each s and related m t s ( ( )) (as shown in Figure 1), we observe free-rider payoffs λ m t s ( ( ( ))), supporter payoffs λ m t s t s s ( ( ( ))) − ( )/ and member payoffs κ m t s t s m t s ( ( ( ))) + ( )/ ( ( )). Several features of this figure are noteworthy. First, payoffs to each player type at s = 1 and s n = − 1 are identical: these solutions correspond to the trivial case where each player behaves as a singleton. Second, for each player type, payoffs are hump-shaped in s, peaking at the point where the constraint m n s − ≤ kicks in, implying that all players are better off with few freeriders and a large coalition. Third, sharp increases in member and free-rider payoffs occur when a change in s induces a large jump in the number of members: m t s m t s ( ( )) − ( ( − 1)) > 1. One such example is in the right panel when s increases from s = 4 to s = 5, which induces a jump of m t m t ( (5)) − ( (4)) = 2 as is shown in Figure 1. Fourth, member payoffs stay close to freerider payoffs due to internally stabilizing transfers; since these transfers are paid by the supporters, their payoff is generally low.
Our main result is illustrated in the left panel of Figure 2. Recall that one of the two equilibrium solutions that we found in Section 3 equals s m f ( *, *, *) = (2, 3, 0) which occurs for n = 5. The gray arrows in the left panel of Figure 2 depict changes in payoffs upon deviation for this solution. The arrows correspond to stability conditions (6a)-(6d). Since each of the arrows points downward (note that the arrows for (6a) and (6b) overlap), the figure confirms that all stability conditions hold and s m f ( *, *, *) = (2, 3, 0) is an equilibrium solution. Visual inspection reveals that none of the other solutions m s f ( , , ) depicted in both panels of Figure 2 shares this feature, which confirms the result found in Section 3.

| CONCLUSION
Given the bleak results on coalition formation in CPR games, in this paper we explored the prospect of support in such games. We find that support yields an additional equilibrium when there are three or five players. No equilibria with support exist for games with more than five players which confirms that cooperation is particularly hard in larger CPR games, when one does not consider aspects like communication, sanctioning, reputation or norms (cf. Ostrom, 1990). We conclude with descriptions of two model alternatives and their impacts on results.
The first alternative is restricting the supporters' choice of transfers in Stage 2 of the game. Our paper is closely related to Ansink et al. (2019), who studied a public goods game with support. For two common specifications of this game, they did not find any equilibrium with support and resorted to introducing an agency that designs a transfer scheme which the supporters can either reject or accept in Stage 2 of the game. Their existence result for equilibria with support in this restricted version of the game depends on condition (6b) as well as the assumption that λ m ( ) is increasing in its argument. Since our λ m ( ) satisfies this property, their existence result can be applied directly to our setting. Introducing an agency that sets the transfer scheme would result in multiple equilibria with support s m f ( *, *, *) for every n, including some with large m*. An important reservation about this positive result is of course the pertinence of introducing an agency for the specific CPR problem at hand.
The second alternative is to consider the Stackelberg version of the CPR game assessed here. In this game, the coalition chooses its Stage 4 effort level before the singletons do. The Stackelberg game leads to the alternative payoff functions λ m r β n m ( ) = (1 − ) /4( − + 1) 2 2 and κ m r β m n m ( ) = (1 − ) /4 ( − + 1) 2 (Long & Flaaten, 2011). Without support, this game already leads to large coalitions with about half the number of players. Using a similar analysis as for the Cournot case assessed above, we find that adding support does not yield any additional equilibria. We conclude that support has little impact on the outcomes in symmetric CPR games. More research is needed to explore the potential impacts of support when players are heterogeneous and differ, for example, in effort costs, in particular if these are nonlinear.