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Exploitation of a Mobile Resource with Costly Cooperation

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Abstract

Localized rights to common pool resources such as territorial use rights fisheries are a widely used class of management tools. However, the effectiveness of different designs of localized rights remains relatively unexplored, especially when one considers strategic interactions within and across patches in a metapopulation. Using a conceptual model of a system of localized fishing rights over each patch, we demonstrate how the interplay between the spatial distribution of rights and biological and strategic spillovers map into outcomes. Specifically, we show how accounting for endogenous costs to cooperative exploitation within a patch alters the conclusions derived from models that assume sole ownership within each patch. Moreover, we demonstrate how strategic interactions between patches can cause the costs to cooperative exploitation in any given patch to increase. These results highlight the complex political-economy dimensions that are important to consider in the design and evaluation of localized property rights in fisheries governance and elsewhere.

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Notes

  1. The related literature on the management of straddling and shared stocks (where most of the game-theoretic analysis in fisheries is undertaken), have either addressed interactions between sole owners of patches harvesting a straddling or migratory stock (e.g. Levhari and Mirman 1980; Fischer and Mirman 1996; Hannesson 1997; Bjørndal et al. 2000; Bhat and Huffaker 2007; Kaffine and Costello 2011; Costello and Kaffine 2017) or interactions of multiple agents within the same patch harvesting a common stock (e.g. Clark 1980; Sutinen and Andersen 1985; Richter et al. 2013; Aburto-Oropeza et al. 2017). There is also a related literature on invasive species spread where individual owners face coordination problems in controlling transboundary invasions (Fenichel et al. 2014; Epanchin-Niell and Wilen 2015; Liu and Sims 2016).

  2. Similar assumptions are made in the spatial fisheries ecology literature, e.g. Mumby et al. (2007) and Blackwood et al. (2012).

  3. Physical size, \( m \), is treated as exogenous, which is consistent with TURF size determined via geopolitical forces outside the control of current harvesters.

  4. We treat \( N \) as fixed and exogenous, which is consistent with TURF membership through customary tenure (Cinner 2005). Entry that is endogenous to resource characteristics and coalition formation are interesting areas for further study.

  5. A large body of literature has examined the effects of group size on the likelihood of securing cooperative outcomes, especially in games of public goods provisions. The conclusions are generally ambiguous (see and Chaudhuri (2011) and Nosenzo et al. (2015) for a review). The costs of cooperation are potentially affected in many ways through negotiation, supervision, enforcement, and punishment. If these costs are fixed at the community level, then they help to explain contrasts between cooperative and non-cooperative outcomes at the community level, but do not explain levels of per capita extraction that are between the fully cooperative and non-cooperative cases. See McCarthy et al. (2001) for further discussion.

  6. See Clark (1990) for a thorough exposition of cooperative and competitive harvest of renewable resources.

  7. Barrett finds that if the difference between cooperative and non-cooperative benefits is small, then the agreement can only be sustained by a few members.

  8. For now, we leave the shape parameter general but in subsequent numerical analysis, we relate this parameter to the relative physical size of the patch and to labor costs.

  9. McCarthy et al. (1998) examine pasture stocking rates, Ahuja (1998) and Lopez (1998) examine communal management of agricultural land, Ovando et al. (2013) compile a survey of 67 fishing cooperatives around the world, and Aburto-Oropeza et al. (2017) examine leadership of fishing cooperatives in Mexico.

  10. Kaffine and Costello (2011) examine endogenous cooperation across patches using a Nash reversion strategy to punish defectors. In their strategy, a defector is forever punished by having all others revert to non-cooperative harvest.

  11. For example, the incentives to defect increase with \( N \), implying the minimum investments will increase if \( F_{max} \) is held constant.

  12. Unlike the Japanese and Baja California systems, we do not consider additional coordinating actions that manage the full range of the coastal resource. While cooperation between TURFs is a fascinating topic, it is beyond the scope of this study. For discussion of the impacts of coordination across TURFs, see Kaffine and Costello (2011).

  13. Our approach is consistent with the disaggregated population dynamics in Flaaten and Mjølhus (2010). We do not consider cases where the two patch model is aggregated into a single patch. Similarly, we do not start from the thought experiment of going from a single patch and dividing into two patches. Specifically, we consider \( m \) to be exogenous and serves to delineate the metapopulation dynamics (i.e. two patches).

  14. We explore model sensitivity to this assumption in a later section by holding the size of patch 2 constant while varying the size of patch 1. In short, we obtain results that are consistent with the main analysis.

  15. The relative sizes of the cost and price parameters, and the difference between the two, are in line with the spatial bioeconomic analysis in Janmaat (2005).

  16. The assumption of homogeneity yields interior solutions where all agents exert positive effort in the fishery.

  17. Robust bisquare weighting discounts data outliers during fitting. In this way, we can place more weight on patch scale, effort combinations that are most practical in terms of producing positive rents to all members and patches.

  18. Our approach is similar to threats to revert back to non-cooperative behavior and is more likely to be credible than threats of exclusions, especially for small groups controlled by well-defined social units (Cinner 2005).

  19. More detailed discussion of the variable costs to cooperation appears in the “Appendix”.

  20. The short-run case where there is a marginal defection in TURF 2 is bounded below by the case where TURF 2 is internally non-cooperative and above by the case where TURF 2 is internally cooperative.

  21. Rents from defection and rents from cooperation in patch 1 decline if patch 2 is internally non-cooperative, but cooperative rents decline more. Additional intuition is developed in “Appendix 1” using a source-receptor model.

  22. This is to maintain an interior solution consistent with Assumption 1. The smallest size to have profitable harvest with high dispersal is about 0.3.

  23. \( \frac{{\partial^{2} \Delta_{i}^{Ch} }}{{\partial e_{j}^{2} }} = mpq \cdot \left( {\frac{q}{r}} \right) \cdot \left( {N - 1} \right) > 0 \)

  24. \( \frac{{\partial^{2} \tilde{e}_{i}^{*} }}{{\partial \gamma^{2} }} = \frac{{ - 4 \cdot \left( {N - 1} \right)^{3} \cdot \left( {N + 1} \right)r\omega }}{{q \cdot \left( {4N + \left( {N - 1} \right)^{2} 2\gamma } \right)^{3} }} < 0 \)

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Correspondence to Gabriel S. Sampson.

Appendices

Appendix 1

Proof of Proposition 1

The proof is fairly straightforward. Note that when harvest effort by the other members is at the Cournot–Nash equilibrium level, then the incentives to defect are zero. If this was not the case, then it would not be a Cournot–Nash equilibrium. This can be seen by solving expression (5) when \( e_{j} = e^{00} ,\; j = 1, \ldots ,N - 1 \), i.e. \( e_{i}^{0} \left( {\left( {N - 1} \right)e_{j} |e_{j} = e^{00} } \right) = e^{00} \). Taking the derivative of expression (10) with respect to \( e_{j} \) and reducing gives:

$$ \frac{{\partial \Delta_{i}^{Ch} }}{{\partial e_{j} }} = mpq \cdot \left( {\frac{q}{r} \cdot \left( {N - 1} \right)e_{j} - \omega } \right) $$
(21)

To note that expression (21) is negative, evaluate \( e_{j} \) at its highest credible value, which is \( e_{j} = e^{00} \). For \( e_{j} = e^{00} \) expression (21) reduces to \( \frac{{\partial \Delta_{i}^{ch} }}{{\partial e_{j} }} = mpq \cdot \left( {\frac{N - 1}{N + 1}\omega - \omega } \right) \), which is negative for \( N \ge 1 \). Finally, note that the second partial derivative of expression (10) with respect to \( e_{j} \) is strictly positive over the interval of interest.Footnote 23 Hence, for \( N \ge 1 \) and \( 0 < e_{j} < e^{00} \) the condition in (21) holds and the gains to defection are non-negative. Because the gains to defection are clearly positive for \( e_{j} = 0 \) and gains are positive yet decreasing over the interval of interest, proves the proposition.

Proof of Proposition 2

The partial derivative of expression (10) when evaluated at \( e_{j} = e_{j}^{*} \) and with respect to \( N \) is \( \frac{{\partial \Delta_{i}^{Ch} }}{\partial N} = \frac{{\left( {N - 1} \right)mpqr\omega^{2} }}{{8N^{3} q}} \). If the other \( j \ne i \) members behave cooperatively by dedicating effort according to expression (3), then the incentives to defect are always increasing in \( N \) for \( N > 1 \).

Proof of Proposition 3

We wish to show that \( e_{i}^{*} = \frac{r}{2qN}\omega < \tilde{e}_{i}^{*} < e^{00} = \frac{r}{{q \cdot \left( {N + 1} \right)}}\omega \) for \( 0 < \gamma < 1 \). When \( \gamma = 0 \) the variable costs to cooperation (i.e. variable in the incentives to defect) are zero and solution to the program in (14) reduces to \( e_{i}^{*} \). Additionally, expression (15) is increasing in \( \gamma \) for \( N > 1 \), such that \( \frac{{\partial \tilde{e}_{i}^{*} }}{\partial \gamma } = \frac{{2 \cdot \left( {N^{2} - 1} \right)\omega r}}{{\left( {\gamma \cdot \left( {N - 1} \right)^{2} + 4N} \right)^{2} q}} > 0 \). When \( \gamma = 1 \), we can show that \( \tilde{e}_{i}^{*} = \frac{r}{2q} \cdot \left( {\frac{{1 + \gamma \cdot \left( {\frac{N - 1}{2}} \right)}}{{N + \gamma \cdot \left( {\frac{N - 1}{2}} \right)^{2} }}} \right) = \frac{r}{2q} \cdot \left( {\frac{{1 + \left( {\frac{N - 1}{2}} \right)}}{{N + \left( {\frac{N - 1}{2}} \right)^{2} }}} \right) = \frac{r}{{q \cdot \left( {N + 1} \right)}}\omega = e^{00} \). Expression (15) is twice continuously differentiable in \( \gamma \) and the second partial derivative is strictly negative for \( N > 1 \) and \( 0 < \gamma < 1 \),Footnote 24 implying that the costly cooperation level is bounded below by \( e_{i}^{*} \). Thus, the level of costly cooperation is bounded above by \( e^{00} \) under Assumption 2.

1.1 Source-Sink Model

We extend the case of spatial independence to a source-sink dispersal model where the focal patch is a source. The biology in patch 1 evolves according to:

$$ \frac{{dx_{1} }}{dt} = rx_{1} \cdot \left( {1 - \frac{{x_{1} }}{m}} \right) - bx_{1} - x_{1} \mathop \sum \limits_{i} qe_{1i} , 0 < m < 1 $$
(22)

where \( e_{1i} \) is the effort of the ith individual in patch 1. The net dispersal in patch 1 is thus \( - b \cdot x_{1} \), where \( b \) is the common dispersal rate. In this representation, biomass flows out of patch 1 into patch 2 at a rate that is proportional to the population of patch 1.

1.1.1 Second Stage

Exploiting the closed form steady state-solution to (19), the per capita profits from harvesting for individual \( i \) is:

$$ \pi_{1i} = e_{1i} \cdot \left( {\frac{pqm}{r} \cdot \left( {r - qe_{1i} - \mathop \sum \limits_{j \ne i} qe_{1j} - b} \right) - c} \right) $$
(23)

Jointly optimal per capital steady-state effort is:

$$ e_{1i}^{*} = \frac{r}{2qN} \cdot \left( {\omega - \frac{b}{r}} \right) $$
(24)

To ensure an interior solution, we make the following assumption:

Assumption 4

Given the size of the patch (\( m \)), growth rate (\( r \)), and dispersal rate (\( b \)), we assume the first unit of effort yields a positive return,\( \omega > \frac{b}{rm} \).

If all members behave according to (24), then the resulting per capita profits are:

$$ \pi_{1i}^{*} = \frac{{rmp \cdot \left( {\omega - \frac{b}{r}} \right)^{2} }}{4N} $$
(25)

If members do not behave cooperatively, then the best response for individual \( i \) conditional on the effort of individuals \( j \ne i \) is:

$$ e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j} } \right) = argmax \pi_{1i} \left( {e_{1i} , e_{1j} } \right) = \frac{r}{2q} \cdot \left( {\omega - \frac{b}{r}} \right) - \frac{1}{2}\left( {N - 1} \right)e_{1j} $$
(26)

If no member in patch 1 cooperates, then the Cournot–Nash equilibrium from (24) is:

$$ e_{1}^{00} = \frac{r}{q} \cdot \left( {\frac{1}{N + 1}} \right)\left( {\omega - \frac{b}{r}} \right) $$
(27)

Comparing (24) to (3) and (27) to (6) reveals that dispersal that is proportional to local population incentivizes individuals to reduce effort and that the reduction is linear in the dispersal parameter.

Conditional on the behavior of other members, the profits earned by member \( i \) from defecting are:

$$ \begin{aligned} & \pi_{1i} \left( {e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j} , \left( {N - 1} \right)e_{1j} } \right)} \right) \\ & \quad = \left( {\frac{r\omega }{2q} - \frac{N - 1}{2}e_{1j} - \frac{b}{2q}} \right) \cdot \left( {\frac{mpq}{2} \cdot \left( {2 - \omega - \frac{q}{r} \cdot \left( {N - 1} \right)e_{1j} - \frac{b}{r}} \right) - c} \right) \\ \end{aligned} $$
(28)

If all other members behave cooperatively, then the profits from defecting in (28) simplify to:

$$ \pi_{1i} \left( {e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j}^{*} , \left( {N - 1} \right)e_{1j}^{*} } \right)} \right) = \frac{r}{2q} \cdot \left( {\frac{N + 1}{2N}} \right) \cdot mpq\left( {\omega - \frac{b}{r}} \right)^{2} \cdot \left( {\frac{N - 1}{4N}} \right) $$

The gains from defecting are the difference between profits under defection (28) and profits earned from fully cooperative behavior (25):

$$ \begin{aligned} & \Delta_{1i}^{Ch} = \pi_{1i} \left( {e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j} , \left( {N - 1} \right)e_{1j} } \right)} \right) - \pi_{1i} \left( {e_{1i}^{*} , \left( {N - 1} \right)e_{1j}^{*} } \right) \\ & \quad = \left( {\frac{r\omega }{2q} - \frac{N - 1}{2}e_{1j} - \frac{b}{2q}} \right) \cdot \left( {\frac{mpq}{2} \cdot \left( {2 - \omega - \frac{q}{r} \cdot \left( {N - 1} \right)e_{1j}^{*} - \frac{b}{r}} \right) - c} \right) - \frac{{rmp \cdot \left( {\omega - \frac{b}{r}} \right)^{2} }}{4N} \\ \end{aligned} $$
(29)

To examine the influence of the dispersal rate \( b \) on the gains from defection, re-write (29) as

$$ \begin{aligned} & \Delta_{1i}^{Ch} = e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j} } \right) \cdot \left( {pqX^{SS} \left( {e_{1i}^{0} \left( {N - 1} \right)e_{1j} , \left( {N - 1} \right)e_{1j} } \right) - c} \right) \\ & \quad \quad\quad - \pi_{1i} \left( {e_{1i}^{*} , \left( {N - 1} \right)e_{1j}^{*} } \right) \\ \end{aligned} $$
(29’)

Taking the partial derivative of (29’) with respect to \( b \) gives

$$ \begin{aligned} & \frac{{\partial \Delta_{1i}^{Ch} }}{\partial b} = \frac{\partial }{\partial b}e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j} } \right) \cdot \left( {pqX^{SS} \left( {e_{1i}^{0} \left( {N - 1} \right)e_{1j} , \left( {N - 1} \right)e_{1j} } \right) - c} \right) \\ & \quad + \,e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j} } \right) \cdot pq \cdot \frac{\partial }{\partial b}X^{SS} \left( {e_{1i}^{0} \left( {N - 1} \right)e_{1j} , \left( {N - 1} \right)e_{1j} } \right) - \frac{\partial }{\partial b}\pi_{1i} \left( {e_{1i}^{*} , \left( {N - 1} \right)e_{1j}^{*} } \right) \\ \end{aligned} $$
(30)

The first and second components of (30) are unambiguously negative because \( \frac{\partial }{\partial b}e_{1i}^{0} \left( {\left( {N - 1} \right)e_{1j} } \right) < 0 \) and \( \frac{\partial }{\partial b}X^{SS} \left( {e_{1i}^{0} \left( {N - 1} \right)e_{1j} , \left( {N - 1} \right)e_{1j} } \right) < 0 \). The third component is unambiguously positive because \( \frac{\partial }{\partial b}\pi_{1i} \left( {e_{1i}^{*} , \left( {N - 1} \right)e_{1j}^{*} } \right) < 0 \). Thus, the sign of \( \frac{{\partial \Delta_{1i}^{Ch} }}{\partial b} \) depends on the relative magnitude of the sum of the first two components of (30) and the third component of (30). The net effect of dispersal in the source patch is a reduction to profits from cooperation and profits from defection. If the variable costs to cooperation are increasing in the incentives to defect, as in (12), then the variable costs to cooperation should be decreasing in the dispersal rate \( b \) for a source patch.

Appendix 2

By construction, there is a coupling between variable cooperation costs and incentives to defect (Fig. 8). The largest costs to cooperation are for the cases of spatial independence and low dispersal (Fig. 7 left). Moreover, cooperation costs for these two cases are increasing in \( m \). With high dispersal or source-sink dispersal, cooperation costs are more variable in relative patch size because there are countervailing factors. For instance, incentives to defect in the sink follow an inverted U-shape (Figs. 7, 8), which is largely due to the relationship between per capita effort and \( m \) (e.g. total settlement in the sink decreases with \( m \), giving a bump in per capita effort, Fig. 3). Additionally, with high dispersal based on differences in relative density, inter-patch competition over the resource is pronounced, so there is little opportunity for any member to benefit significantly from defecting (Table 1; Figs. 9, 10, 11, 12).

Table 1 Coefficient estimates for incentives to defect with spatial connectivity
Fig. 7
figure 7

Variable costs to cooperation (left) and the maximum fixed cost that can be financed by the community (right)

Fig. 8
figure 8

Incentives to defect and variable costs of cooperation at the solution for cooperative harvest effort

Fig. 9
figure 9

Steady state biomass as a proportion of patch size (\( m \)) for Cournot–Nash competition (solid), costless internal cooperation (dash), and costly internal cooperation (dotted) for the five dispersal scenarios

Fig. 10
figure 10

Per capita benefits (solid) and variable costs (dashed) to cooperation for \( {\text{n}}_{1} = 2 \) members and \( {\text{n}}_{1} = 5 \) members (x’s). Note: Per capita value is measured as dollars per community member

Fig. 11
figure 11

Per capita effort levels for patch 1 under Cournot–Nash competition (solid), costless internal cooperation (dash), and costly internal cooperation (dotted) for low and high dispersal. Note: Patch 2 size is fixed at 0.6 (vertical line)

Fig. 12
figure 12

Per capita steady-state rents for patch 1 under Cournot–Nash competition (solid) and costly cooperation (dashed) for low and high dispersal. Note: The y-axis is measured as a proportion of the rents earned under costless internal cooperation

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Sampson, G.S., Sanchirico, J.N. Exploitation of a Mobile Resource with Costly Cooperation. Environ Resource Econ 73, 1135–1163 (2019). https://doi.org/10.1007/s10640-018-0294-0

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