Strategic behavior and dynamic externalities in commercial fisheries

Ecological Economics 169 (2020) 106503

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Analysis

Strategic behavior and dynamic externalities in commercial fisheries a, ,1

Anna A. Klis * , Richard T. Melstrom a b

b

T

Department of Economics, Northern Illinois University, Zulauf Hall 510, DeKalb, IL 60115, United States Institute of Environmental Sustainability, Loyola University Chicago, United States

ARTICLE INFO

ABSTRACT

Keywords: Bioeconomics Game theory Cooperation Overfishing

This paper uses game theory to analyze commercial fishers’ willingness to practice conservation to recover a depleted fishery. We compare a game in which players choose their conservation effort simultaneously to a game in which there is a leader and a follower. We show that because the players ignore the effect of their conservation effort on the other player's expected benefits, their collective effort is suboptimal. When the players choose their effort sequentially, the leader puts less effort in conservation, reducing the likelihood of recovery and collective welfare.

1. Introduction Externalities are a well-known contributor to overfishing. In an open or limited access system, individual harvesters have little-to-no incentive to consider the effect of their catch on the future returns from the stock, or to account for the effect of their catch on the stock available to other harvesters. These externalities have led to stock depletion and “fishing down the food web,” as seen in a declining average trophic level of worldwide catch (Pauly and Palomares, 2005). Externalities can also arise when behavior outside commercial fisheries affects environmental conditions inside the fishery and hence stocks—for example, when an oil spill or harmful algal bloom limits access to or kills a portion of the stock—potentially creating regime shifts that alter the provision of ecosystem services (Crépin et al., 2012). The current state of global fisheries indicates a critical need to not only address these externalities but also consider strategies for recovery, which will require that commercial fishers adjust and coordinate their behavior. Recovering and protecting fisheries is often challenged by strategic behavior between commercial fishers or between countries. Economists have responded to this challenge by developing game-theoretic models in which individual fishers or countries consider the actions of others when identifying their own preferred action. Typically these models incorporate dynamic aspects in addition to strategic behavior. Economists have used these models to examine outcomes under duopolies (Levhari and Mirman, 1980; Kaitala and Munro, 1993) and limited access systems with several players (Clark, 1980; Lindroos and Kaitala, 2000), risk aversion (Mine Cinar et al., 2013), coalitions

(Kaitala and Lindroos, 1998; Pintassilgo, 2003; Laukkanen, 2003), threat strategies (Cave, 1987), transfer payments (Kaitala and Pohjola, 1988), quota management (Anderson and Uchida, 2014), and resource shocks (Lindahl, 2012). Economists have also used game theory to study coordinating conservation in other natural resource industries with externalities (Reeling and Horan, 2014; Güth et al., 2015; Reeling et al., 2017). Strategic behavior can also occur between individuals and regulators, for example, when regulators attempt to restrict harvest by periodically closing the fishery and fishers respond by fishing more intensively during the season. In this paper, we use game theory to examine commercial fishers’ willingness to invest in recovering depleted fisheries in the presence of a dynamic externality. We develop a new model in which a fish stock faces an exogenous probability of becoming damaged and recovery depends endogenously on the players’ actions. Prior research on strategic behavior in fisheries generally examines dynamic externalities through harvesting, which directly deplete the fishery. In contrast, we examine a situation in which the fishery is depleted due to an external shock. This feature is consistent with real-world fishery catastrophes that arise outside the industry, e.g. due to an invasive species or an oil spill. Conservation effort incurs a private cost but yields benefits that spread across the whole industry. This is because conservation aids recovery, which increases the payoff for every player. The conservation externality and strategic behavior thus create a coordination problem in which the equilibrium amount of conservation effort will tend to be too low compared to the optimal amount. Apart from this new framework, the key contribution of this paper comes from comparing simultaneous and sequential behavior.

Corresponding author. E-mail addresses: [email protected] (A.A. Klis), [email protected] (R.T. Melstrom). 1 Additional Affiliation: NIU Institute for the Study of the Environment, Sustainability, and Energy. ⁎

https://doi.org/10.1016/j.ecolecon.2019.106503 Received 21 May 2019; Received in revised form 23 September 2019; Accepted 2 October 2019 0921-8009/ © 2019 Elsevier B.V. All rights reserved.

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom

Sequential behavior in fisheries can arise when a player is able to fish or implement harvest policies before their peers, which can arise due to exogenous factors such as geography. This is important because the results of prior fisheries research suggests that sequential behavior can lead to more desirable conservation outcomes than the equilibrium reached through simultaneous actions (Benchekroun and Van Long, 2002; Hannesson, 2011). In contrast, we find that in the sequential game total conservation effort is generally less than it would have been if the players chose simultaneously. The results therefore imply that sequential behavior can be bad for conservation of shared fisheries (and other common pool resources). This paper is motivated by sequential regulatory and conservation actions between different political jurisdictions. This includes, for example, the migration of Pacific salmon between the U.S. and Canada, and the migration of North Sea herring between several European counties, as well as conservation actions among U.S. states and sovereign tribes with access to the Laurentian Great Lakes. Sequential actions can also arise between different industries, such as recreational anglers targeting a portion of a stock that may be inaccessible to commercial harvesters. Our results suggest that if countries or user groups move simultaneously with their fisheries management policies following stock collapse, the result would likely be an improvement over the sequential outcome. Section 2 develops a dynamic model of fisheries production and conservation. Section 3 analyzes the implied equilibria under cooperative and non-cooperative management, including sequential, leader–follower decision-making. Section 4 concludes with a discussion of the relevance of these results in the context of fishing rights distribution and invasive species management. All proofs and derivations not present within the text itself are located in Appendix A.

In the damaged state, the transition probability creates a dynamic positive externality because it is a function of both players’ actions. We fix p¯i + p¯j 1 (or 2p¯ 1 in a symmetric game) to keep P0,1 in the range of probability. Our model is a novel form of common resource management and falls under the umbrella of evolutionary dilemma games. Dilemma games arise when the payoff structure between two or more players’ decisions does not yield a dominant cooperative strategy (Tanimoto and Sagara, 2007); “[t]he notion dilemma comes from the fact that acting in one's self-interest is tempting to every individual involved, even though all individuals would benefit from acting in the group interest” (Płatkowski, 2017). Dilemmas are often present in resource coordination problems (Tanimoto, 2018), particularly with regard to how cooperation can evolve over time and with heterogenous populations. In our model, there is no action available in the healthy state, so the dilemma arises from the conservation investment choice in the damaged state. Conservation is costly to the fishery today, but leads to increased expected returns in the future. 3. Effort toward recovery 3.1. Cooperative outcome First, we examine the solution under cooperation, i.e. how society would ideally manage the two fisheries. The cooperative problem has separate value functions for the healthy (h ) and damaged (d ) states, respectively,

2. Dynamic fishery model Consider a stochastic fishing zone model with two commercial fishers. The fish stock exists in one of two states: healthy, with s = 1, or damaged, with s = 0 . When the stock is healthy, the players take no strategic action and receive a deterministic payoff of u1. With probability (1 r ) the fishery remains healthy, and with probability r the fishery becomes damaged. When the fishery is damaged, each player i has a choice variable pi [ pi , p¯i ] [0, 1] that represents their conservation effort to aid recovering the stock. The subscript on the bounds of the action set indicates that players may be capable of different effort levels. In the real world, this choice variable could represent a change in effort or gear to reduce harvest pressure, investing in habitat management, or setting aside some fishing ground as a Marine Protected Area. In our model, this pi is part of the transition probability back to the productive state, so more effort yields a higher chance of getting out of the damaged state. This action is costly, however, and reduces the payoff in the damaged state, which is u 0 (pi ) , a decreasing function in pi , i.e. u 0 ( pi ) > u0 (p¯i ) > 0 and u0 (pi ) < 0 . For our purposes, we will assume the payoff in the healthy-state is more than twice the largest possible payoff in the damaged-state, i.e. u1 > 2u0 ( p ) . This is a stricter condition than is necessary for the functional forms we adopt later, but it sharpens the theoretical analysis. Intuitively, this ensures that the healthy state is worth getting back to; if the minimum effort payoff from the damaged state were too close to the healthy state payoff, then fishers might tolerate the damaged state forever. We ignore static externalities in favor of focusing on the dynamic transition, and so the agent's utility is not a function of the other agent's choice, pj , in either state. The transition probability, on the other hand, is a function of both player’ choices,

P h (pi ,

pj ) = 2u1 + [(1

pj ) + r

P d (pi ,

pj )]

P d (pi ,

pj ) = u0 (pi ) + u 0 (pj ) + [(pi + pj )

P h (pi ,

pj ) + (1

P d (pi ,

pj )

r)

P h (pi ,

(1)

pi

pj )]

(2)

where the P superscript denotes values under cooperation. Eq. (1) is the value of the fishery in the current healthy state, which includes the payoff of the two fishers plus the discounted present value of expected future returns. The higher the probability that the fishery remains healthy, r , the more the healthy-state value counts toward the present value of the fishery. Eq. (2) has a similar combination of current and future payoffs except that both the current payoffs and the future returns depend on the players’ total conservation effort toward recovery, pi + pj . We can solve for hP and dP using Eqs. (1) and (2). The ideal manager will choose conservation effort pi and pj to maximize the value in the damaged state (recall there is no need to choose conservation effort in the healthy state), which we write as:

max

(1

(1

r ))(u0 (pi ) + u 0 (pj )) + 2 (pi + pj ) u1

(1

pi , pj

s. t. pi

pj

)[1

pi

p¯i

pj

p¯j

(1

r ) + (pi + pj )] (3)

Assuming an interior solution, the Markov perfect equilibrium is described by

P0,1 = pi + pj ,

[u 0 (pi ) + u 0 (pj )

2u1]

[1

(1

r ) + (pi + pj )] u 0 (pi ) = 0

[u 0 (pi ) + u 0 (pj )

2u1]

[1

(1

r ) + (pi + pj )] u 0 (pj ) = 0

(4)

based on the first order conditions associated with the maximization problem. If the players are symmetric, the solutions for pi and pj can be

with the transition matrix: 2

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom

stated implicitly as

u 0 (p P ) =

1

We can state the following comparative statics of the preferred actions. First, with respect to the transition from the healthy state to the damaged state, r , both the non-cooperative and cooperative actions are decreasing:

(p P ))

2 (u1 (1

u0 r ) + 2 pP

(5)

where pi = pj p P . The optimal choice equates the marginal cost and the marginal benefit of conservation effort. The left-hand side (LHS) of Eq. (5) is the marginal cost. The right-hand side (RHS) shows the marginal benefit of conservation as the gap between the healthy and damaged-state payoffs when the cooperative action is taken, scaled by the discount rate and recovery probability inherent in the dynamic process, and multiplied by two, meaning that both players are accounted for in their effect on one another. We will return to these conditions after finding the solution to the non-cooperative conservation problem.

pM < 0, r

This is intuitive; putting effort into recovery becomes less worthwhile when a healthy fishery is increasingly likely to be damaged again. Second, given a weakly concave damaged-state reward function u 0 (pi ) and reasonable parameter values, both the non-cooperative and cooperative actions are increasing with respect to the discount factor, :

pM

3.2. Non-cooperative outcome—simultaneous actions

pjM ) = u1 + [(1

i d (pi ,

pjM ) = u0 (pi ) + [(pi + pjM ) + (1

r)

pi

i h (pi ,

pjM )

pjM ) + r

i d (pi ,

i h (pi ,

i d (pi ,

pjM )]

pM > 0, u1

(6)

(7)

Similarly as for the cooperative problem, we can solve for the healthy-state fixed point of Eq. (6) and substitute the resulting expression into Eq. (7). Thus, given player j 's strategy, pjM , player i 's maximization problem is

max pi

(1

(1

(1

)[1

s. t. pi

pi

r )) u 0 (pi ) + (pi + pjM ) u1 (1

r ) + (pi + pjM )]

p¯i

(8)

Assuming an interior solution, the unique Markov perfect equilibrium is described with one condition for each player:

[u 0 (piM )

u1]

[1

(1

r ) + (piM + pjM )] u0 (piM ) = 0

[u 0 (pjM )

u1]

[1

(1

r ) + (piM + pjM )] u0 (pjM ) = 0

(9)

We can note that j 's conservation effort appears in the first order condition for player i : the presence of pjM increases the probability of return to the healthy state and thus perversely, in a sense, lowers the marginal benefit of player i 's own conservation effort. As in the cooperative problem, we consider the case where the players are symmetric, so that the conditions can be written as

u 0 (p M ) =

1

(u1 (1

u0 (p M )) r ) + 2 pM

pP

>0

pP >0 u1

As the base value of fishing in the healthy state increases, players invest more to return to it. Rather than providing the mathematical derivations, which are available upon request, we provide visual evidence of the comparative statics using numerical simulations. We illustrate how the preferred actions change with respect to the parameters by specifying a functional form for the payoff function and simulating the outcome over a range of parameter values. For these simulations we use a squared-loss utility function for the damaged-state payoff: u 0 (pi ) = u 0 2pi2 , which is concave in pi . We use baseline level parameters of = 0.9 for the discount factor, r = 0.1 for the exogenous transition probability from healthy to damaged, u 0 = 1 for the damaged-state reward, and u1 = 2.01 for the healthy-state reward that just satisfies the condition u1 > 2u0 (0) .2 Fig. 1 shows how p M and p P change with respect to the probability r of transitioning from the healthy state to the damaged state. The graph on the left compares symmetric individual effort under simultaneous non-cooperation ( p M ) and cooperation ( p P ), while the graph on the right compares the total effort of both agents under the same situations. For any transition probability r less than about 0.4, the optimal outcome has both players choosing the maximum amount of conservation effort, which is p¯ = 0.5; collectively, this effort level guarantees recovery to the healthy state. Without cooperation, both agents’ conservation effort remains far enough below the efficient level, that the probability the fishery will recover never exceeds 75%, even when the chance of returning to the damaged state is zero. Fig. 2 shows how p M and p P change with respect to the discount factor . The graph on the left compares symmetric individual effort under non-cooperation ( p M ) to cooperation ( p P ), while the graph on the right compares the total effort of both agents under the same conditions. Regardless of cooperation, optimal total conservation effort is zero when = 0 and future outcomes have no present value. Total conservation effort rises more than proportionately with the discount factor under cooperation but less than proportionately when there is no cooperation, so the non-cooperative effort stays persistently below the socially optimal level. Fig. 3 shows how p M and p P change with respect to different levels of the healthy-state payoff u1. As before, the graph on the left compares an individual's level of conservation effort under non-cooperation ( p M ) to cooperation ( p P ), while the graph on the right compares the total

pjM )

pjM )]

> 0,

As players grow more patient, they invest more in conservation. Third, with respect to the static healthy-state payoff, u1, both the non-cooperative and cooperative actions are increasing:

Now we consider the players’ non-cooperative problem when they move simultaneously, à la Cournot resource competition. When the fishery is damaged, there exists a Markov perfect equilibrium with a stationary strategy, denoted by piM . Taking the other player's strategy pjM in the damaged state as given, player i faces the following value functions for the healthy and damaged states: i h (pi ,

pP <0 r

(10)

We can compare this equation to the socially optimal equivalent and see that the key missing component is the doubling of the numerator present in Eq. (5), which indicates underprovision of conservation. Proposition 1. Given a weakly concave damaged-state reward function u 0 (pi ) , and assuming u1 > 2u0 ( p ) , the cooperative action p P is larger than the interior non-cooperative action p M for symmetric agents. This is an expected result of public goods provision. Cooperating, both agents would act together as the ideal manager and apply more conservation effort, provided that the healthy-state reward is sufficiently larger than the payoff in the damaged state. Without cooperation the agents reduce conservation effort, preferring to free-ride off the effort of the other.

2 Furthermore, in order to satisfy the condition that u (p¯) > 0 , the maximum amount of effort possible for an individual agent acting alone would be p¯ < 1 0.7071; given there are two identical agents, we will actually set

2

3

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A.A. Klis and R.T. Melstrom

Fig. 1. Comparative statics with respect to the probability of transitioning from the healthy to the damaged state on equilibrium investment.

Fig. 2. Comparative statics with respect to the discount factor.

Fig. 3. Comparative statics with respect to payoff in the healthy state.

effort of both players. The disparity between the optimal actions with cooperation and the preferred actions without cooperation is largest for moderate payoffs. At the extreme ends of the payoff range, the cooperative and non-cooperative actions tend to align. The figure indicates that it is generally optimal to allocate only a small amount of effort toward conservation when the payoff in the healthy state is very close to the payoff in the damaged state, and it is generally optimal to allocate full conservation effort if the healthy-state reward is several times greater than the payoff in the damaged-state. The healthy-state

reward is so crucial, that once the differential is high enough, the noncooperative conservation effort converges to the socially optimal level (due to the upper bound on the action of choosing a probability). These comparative statics also illustrate the shifting nature of the dilemma present in the fishery conservation problem of the model. Both fisheries have a continuum of strategies, and each of the parameters affect the dilemma (Tanimoto and Sagara, 2007). We characterize the game's dilemma strength in Appendix A.

4

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A.A. Klis and R.T. Melstrom

3.3. Non-cooperative outcome—sequential actions

Proposition 2. At an interior Stackelberg Markov perfect equilibrium, the effort of the leader, p L , cannot be equal to that of the follower, p F . Furthermore, the effort of the Stackelberg leader, p L , is smaller than that of the Stackelberg follower, p F , when the following inequality holds: (16)u0 (p)2 < u0 (p)(u0 (p) u1)

We now consider a problem in which the players do not cooperate but choose sequentially, in which one of the players is a leader. Sequential fishing can occur when the stock migrates between different locations, so fishers that leave the port nearest the stock at the start of the season have the first harvest opportunity. Sequential fishing can also arise through the timing of regulated seasonal access, for example when fishers in the jurisdiction that opens the season earliest in the year get to harvest before those in other jurisdictions. We model sequential actions in the manner of Stackelberg competition. In a Stackelberg model, the follower has the same problem as in the simultaneous (Cournot) set-up, but the leader takes into account their effect on the follower. Labeling leader i 's action as piL and follower j 's action as pjF , then the leader understands the follower's best response to be a function of their own action, pjF (pi ) . The leader's maximization problem is then:

max pi

(1

(1

)[1 pi

u 0 (p ) u (p) u1 < 0 u0 (p) u 0 (p )

r ) + (pi + pjF (pi ))]

(1 p¯i

(11)

Under Stackelberg timing, the players are no longer completely symmetric; even if they share parameters and functional forms, the timing is an important source of heterogeneity. This is further reflected in that the upper bound on the follower's effort is now determined by the leader's chosen effort: p¯j = 1 pi . If both agents choose effort levels on the interiors of their action sets, then the Markov perfect equilibrium for Stackelberg conservation is described by:

pjF

1+

piL

[u 0 (piL )

u1]

[1

(1

r ) + (piL + pjF )] u 0 (piL ) = 0 (12)

[u0 (pjF )

u1]

[1

(1

r ) + (piL + pjF )] u 0 (pjF ) = 0

(13)

Proposition 3. If the action of the Stackelberg leader is smaller than that of the follower ( p L < p F ), then the total conservation effort in the leader–follower game is less than total conservation effort in the simultaneous game.

The leader's first order condition can be rearranged to form: pjF piL

1+ L

u 0 (pi ) =

1

(1

(u1

u 0 (piL ))

r ) + (piL + pjF (piL ))

(14)

We prove Proposition 3 in Appendix A. This result is related to whether the fishers’ conservation effort are complements or substitutes and indicates whether the sequential timing awards the first or second mover with an advantage. In the general understanding that agents’ efforts are substitutable, we see that sequential timing gives a firstmover advantage to the leader, pushing them to shirk; this free-riding then decreases total provision in comparison to simultaneous timing, and thus a worse social outcome. On the other hand, we can show that reversing the inequalities in Propositions 2 and 3 implies complementary conservation effort that gives a second-mover advantage, and therefore that total conservation effort would be greater under sequential than simultaneous timing. Using the same functional forms and baseline parameters before, we can demonstrate the comparative statics in the sequential game. For the simulation, we can substitute Eq. (15) into the leader's first order condition (Eq. (12)):

which can be compared to Eq. (10) from the simultaneous timing of the non-cooperative game. The numerator of the RHS is now multiplied by the term 1 +

pjF

piL

, which will be either less than or greater than one

depending on whether the follower interprets the leader's conservation effort as a substitute or complement for their own effort, a relationship captured by the sign of

pjF

piL

. In our functional set-up, both agents treat

the other's conservation effort as a substitute. We can derive the comparative statics of the players’ actions upon each other by considering the follower's first-order condition as an implicit function, which gives the following:

pj pi

=

u0 (pjF ) [1

(1

r ) + (piL + pjF )] u0 (pjF )

<0

(17)

which more clearly indicates a curvature assumption for u 0 (·) . The LHS of Eq. (17) is the instantaneous rate of change of the damaged-state payoff over the “acceleration” of that effect: how quickly does the damaged-state payoff plunge with increased effort? If the payoff is very concave in conservation effort, the denominator will be large and so the LHS will be small. Meanwhile, the RHS is the gap between the damaged-state and healthy-state payoffs over the damaged-state rate of change. If the healthy-state payoff sufficiently dominates any possible damaged-state payoff (an assumption made earlier), then the numerator will have a large absolute value and so the RHS may also be a large number. We find this relationship holds for reasonable functional forms of u 0 (·) , and that the leader will use their timing advantage to free-ride off of the follower, decreasing their own effort and hence compelling the follower to increase their effort. The utility function used in our numerical analysis, u 0 (pi ) = u 0 2pi2 , is an example of such a function. However, we cannot rule out functional forms that facilitate the opposite, i.e. that the leader may choose more conservation effort than the follower. This could occur if the reward gap is not very large and the function is not very concave, but the marginal cost of effort is high. In such a case, the leader may anticipate that the follower is not very responsive and may choose to increase their own effort instead. This leads us to state the following proposition, which posits that, as the leader goes, so the Stackelberg total goes.

r )) u 0 (pi ) + (pi + pjF (pi )) u1

(1

s. t. pi

We prove the first statement and derive this condition in Appendix A. Eq. (16) can be rewritten as:

(15)

This effect is negative, which indicates that if one agent (here, the leader) increases their conservation effort, the other (here, the follower) prefers to decrease their own effort and free-ride. Given this result that players’ conservation efforts are economic substitutes, we ask whether the leader is incentivized to provide conservation or free-ride, and how their action compares to that of the follower. We find that the heterogeneity introduced by Stackelberg timing guarantees that the two fishers’ efforts are not identical, but the difference relative to the simultaneous model depends on the curvature of the damaged-state reward function.

u0 (pjF (piL ))

1

[1 [1

(1 (1

r ) + (piL + pjF (piL ))] u0 (pjF (piL ))

(u 0 (piL )

u1)

r ) + (piL + pjF (piL ))] u0 (piL ) = 0 (18)

In the simulations, we solve for the conservation effort levels in Eq. (18) using numerical approximation. Fig. 4 shows the comparative statics for changes in the probability, r , of transitioning from the healthy state to the damaged state across the 5

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom

Fig. 4. Comparative statics for the probability of transitioning from the healthy to the damaged state.

Fig. 5. Comparative statics for the discount factor.

cooperative, simultaneous and sequential games. As in the previous section, we perform these simulations using a squared-loss utility function for the damaged-state reward and baseline parameters of = 0.9 , r = 0.1, u 0 = 1, and u1 = 2.01. The graph on the left shows that the leader in the sequential game chooses substantially less conservation effort than they would in the simultaneous game, compelling the follower to increase their own conservation effort. Interestingly, the leader's conservation effort is not a monotonic function of r —a result from the higher order terms in Eq. (18)—so that at very small damage probabilities the leader will actually increase their effort when the probability of damage increases. Nevertheless, consistent with Proposition 3, total conservation effort is persistently less in the sequential game than in either the cooperative or simultaneous games, as shown in the graph on the right. The same pattern of outcomes holds for changes in the discount factor and the healthy-state payoff u1. Figs. 5 and 6 show how conservation effort in the sequential game is always less than conservation effort in the non-cooperative simultaneous game. Non-cooperative conservation effort under simultaneous action converges to the socially optimal effort level when the healthy state payoff u1 is between two and three times greater than the damaged state payoff, whereas total conservation effort in the sequential game reaches the socially optimal effort level when the healthy state payoff is between three and four times greater than the damaged-state payoff.3 Thus, in general, total

conservation effort will be less and the stock will take longer to recover in a shared fishery when effort between players is determined sequentially rather than simultaneously. 4. Discussion and conclusion The results indicate that restoring or recovering shared resources will be especially challenging when conservation effort across players is carried out sequentially. We showed that leaders will invest less in conservation than they would have in a simultaneous choice setting. The leader anticipates that the decrease in their own effort will cause the follower to increase their effort beyond that in the simultaneous setting, and in this way, the leader uses the substitutability of their effort levels to free-ride off of the follower. If the players cooperated, they would find their collective and individual payoffs would increase by choosing more conservation effort. We can conclude that sequential management of shared fisheries is likely to be bad for conservation and hence fish production in most reasonably parameterized models. There is still some room to investigate cases when a leader actually chooses to increase their effort in comparison to the follower. Theoretical results imply that this would lead to an improvement in provision compared to simultaneous non-cooperation, but we think such cases are non-generic and limited to possibly unrealistic damaged(footnote continued) by the bound of 1 minus the leader's choice at around u1 = 3.5, causing a kink in their effort curve, which then slowly converges from above.

3

Note, the leader's choice is never constrained in the sequential game because total conservation effort stays well below 1. The follower's choice is constrained 6

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Fig. 6. Comparative statics for the healthy-state reward.

state reward functions. This model of conservation effort and dynamic transition may help explain the conditions in which cooperative management of Great Lakes fisheries developed between state and provincial agencies. These lakes fall under the jurisdictions of eight U.S. states and one Canadian province. By the early 20th century, a lack of collective access limits and harvest restrictions had created a legacy of boom and bust in the lakes’ fisheries. Efforts to establish an international fishery commission had failed repeatedly, which could be attributed to little perceived difference between the healthy and damaged-state payoffs. However, sea lampreys invaded the lakes and by the 1950s had destroyed some of the most valuable commercial fish stocks. This invasion would have substantially lowered the damaged-state payoff, thereby increasing the benefits of investing in recovery and of cooperation. Indeed, the Great Lakes Fishery Commission was created in 1954 to coordinate sea lamprey research and control. Despite the success of the Commission's effort to control sea lampreys (abundance has been reduced by 90% from its peak), cooperative management did not rapidly spread to other aspects of the fishery. After a decade of sea lamprey control, the state of Michigan addressed the lack of predator fish production in the 1960s by unilaterally stocking non-native Pacific salmon for sport fishing, once again altering the ecology of the lakes. Minnesota and Wisconsin followed with their own stocking programs for Pacific salmon in the 1970s. This action illustrates the difficulty of coordinating conservation, even when the players have evidence that cooperation can generate higher payoffs for everyone. How does the result that conservation effort is less when the players behave as in Stackelberg competition compare with prior research? There are several important results that relate to our own. First, prior research shows that in a common pool fishery sequential fishing effort results in a smaller harvest, less economic surplus, and a larger stock than when the players fish simultaneously. In this case, sequential fishing turns out to be bad for welfare (although good for the stock). This is similar to our result that sequentially choosing conservation effort to recover a depleted fishery is harmful to the collective payoff. Second, players can choose at the same time but have unequal access to the fishery, giving one country or industry, as the case may be, more opportunity to harvest fish. The fishery is unambigously worse off when

these players share a common pool and one of the players has greater capitalization or fleet size. However, the fishery is generally better off when the players have unequal shares of the pool; both players then have relatively strong incentives to practice conservation (Hannesson, 2011). This indicates that Stackelberg competition may not reduce conservation effort as much if there is some separation in the pools between players. Third, the distribution of fishing rights has an important effect on the ability of coalitions to achieve cooperative management (Kaitala and Lindroos, 1998; Pintassilgo, 2003; Laukkanen, 2003). Prior research shows that larger coalitions form, or are more likely to form, when there is Stackelberg competition. In particular, sequential behavior is good for social welfare when the coalition acts as the leader, thus providing a strategic advantage over non-members and reducing the incentive to free-ride (Kwon, 2006). This result has important implications for our finding that fisheries are at relatively greater conservation risk under Stackelberg competition, however we do not consider a coalition game. Coalitions are more likely to form when there is a leader, so unconditionally some fisheries could be better off under Stackelberg than Cournot competition. A natural follow-up question is therefore: under what conditions will cooperation form among a number of agents? It is beyond the scope of this paper to analyze the size and stability of coalitions, but we expect the insights would accord with existing research on this topic. There is a large and growing literature on the formation and stability of coalitions in environmental policy, including fisheries policy (Kaitala and Lindroos, 1998; Pintassilgo, 2003). Sequential behavior matters because when an entity considers whether to lead or join an environmental agreement, it must anticipate how the non-signatories will respond to the agreement, i.e. as followers. Our research provides further evidence that the substitutability of public good provision generates a first-mover advantage under sequential timing; we expect that a similar result extends to coalitions attempting to recover shared fisheries, which is a topic for future investigation. Conflict of interest None declared.

Appendix A A.1 Cooperative problem A.1.1 Value function derivations The derivations of the healthy and damaged-state value functions in the cooperative problem are

7

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom P h (pi ,

pj ) = 2u1 + [(1

(1

(1

P h (pi ,

P h (pi ,

r ))

P d (pi ,

(1

(1

P d (pi ,

pj ) + r P d (pi ,

pj ) = 2u1 + r

2u1 + r

pj ) =

P h (pi ,

r)

pj )]

pj )

pj )

r ))

for the healthy-state value function. For the damaged-state value function, P d (pi ,

pj ) = u 0 (pi ) + u0 (pj ) +

[1

(1

pi

P d (pi ,

pj )]

(1

(1 = (1

r ))[1 (1

(1

= (1

pi

pi

P d (pi ,

pj )

pj ) P d (pi ,

2u1 + r (1

(1

pj )

r ))

P d (pi ,

pj )]

pj )

r ))(u0 (pi ) + u 0 (pj )) + (pi + pj ) P d (pi ,

× (2u1 + r [(1

+ (1

r ))

r )) :

(1

(1

(1

pj )

pj ) = u 0 (pi ) + u 0 (pj ) + (pi + pj )

Multiply through by (1

(1

P d (pi ,

2u1 + r

(pi + pj )

r ))[1 (1

pj )) (1

pi

2r (p i

pj )]

P d (pi ,

+ pj )]

pj )

r ))(u0 (pi ) + u 0 (pj )) + 2 (pi + pj ) u1

Using the coefficient simplifications derived for the non-cooperative case (see below), this can be rewritten as:

(1

)[1

(1

P d (pi ,

r ) + (pi + pj )]

pj ) = (1

(1

r ))(u 0 (pi ) + u 0 (pj ))

+ 2 (pi + pj ) u1 P d (pi ,

pj ) =

(1

(1 (1

r ))(u0 (pi ) + u 0 (pj )) + 2 (pi + pj ) u1 )[1

(1

r ) + (pi + pj )]

A.1.2 First order conditions The Lagrangean for the maximization problem described by Eq. (3) is:

(p , µ1i , µ 2i , µ1j , µ 2j ) =

(1

(1

r ))(u 0 (pi ) + u 0 (pj )) + 2 (pi + pj ) u1

(1

)[1

+ µ1i (pi

(1

r ) + (pi + pj )]

p ) + µ 2i (p¯

pi ) + µ1j (pj

p ) + µ 2j (p¯

pj )

where µki represents the multiplier on the respective boundary condition. The Kuhn–Tucker conditions of the cooperative problem are found by taking derivatives with respect to pi and pj , combined with the complementary slackness constraints. There can be corner solutions at the low or high boundaries provided particularly large or small parameters. If we consider the interior case, then the first order conditions are as follows:

[(1 (1

(1 )[1

[(1 (1

(1 )[1

r )) u0 (pi ) + 2 u1] = (1 r ) + (pi + pj )] r )) u0 (pj ) + 2 u1] (1

r ) + (pi + pj )]

=

[(1

(1 (1

[(1

r ))(u0 (pi ) + u0 (pj )) + 2 (pi + pj ) u1] )[1

(1 (1

(1

r ) + (pi + pj )]2

r ))(u0 (pi ) + u0 (pj )) + 2 (pi + pj ) u1] )[1

(1

r ) + (pi + pj )]2

Multiplying both sides by the LHS denominator gives us an intertemporal marginal cost versus marginal benefit formulation:

[(1

(1

r )) u0 (pi ) + 2 u1] =

[(1

(1

r )) u0 (pj ) + 2 u1] =

[(1

(1

r ))(u 0 (pi ) + u0 (pj )) + 2 (pi + pj ) u1 ] [1

[(1

(1

(1

r ) + (pi + pj )]

r ))(u0 (pi ) + u 0 (pj )) + 2 (pi + pj ) u1 ] [1

(1

r ) + (pi + pj )]

Following the simplifications used in the non-cooperative problem (see below), the simplified conditions are

[u 0 (pi ) + u 0 (pj )]

[1

(1

r ) + (pi + pj )] u0 (pi )

[u 0 (pi ) + u 0 (pj )]

[1

(1

r ) + (pi + pj )] u0 (pj )

When agents are symmetric and take action pi = pj =

2 (u 0

(p P )

u1) = [1

(1

r) + 2

pP ] u

0

2 u1 = 0 2 u1 = 0 pP ,

these reduce to the following equation: (A.1)

(p P )

which can be rearranged to form Eq. (5) in the text.

8

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom

A.2 Non-cooperative derivations under simultaneous timing The derivations of the healthy and damaged-state value functions in the non-cooperative problem are i h (pi ,

pjM ) = u1 + [(1

(1

(1

i h (pi ,

i h (pi ,

r ))

pjM ) + r

pjM ) = u1 + r i d (pi ,

u1 + r

pjM ) =

i h (pi ,

r)

(1

(1

i d (pi ,

i d (pi ,

pjM )]

pjM )

pjM ) r ))

for the healthy state, and for the damaged state i d (pi ,

pjM ) = u 0 (pi ) +

(1

(1

pjM ))

pi

i d (pi ,

(1

(1

r )) :

pjM ))(1

(1

r ))

(1

pi

= (1

(1

[(1

(1

= (1

r )) u 0 (pi ) + (pi + pjM ))(1

pi

(1

+ (1

r ))

pjM )

pi

i d (pi ,

u1 + r (1

(1

i d (pi ,

pjM )

pjM ) r ))

pjM ) i d (pi ,

+ r

2r ( p i

pjM ))

+ pjM )]

i d (pi ,

pjM )

r )) u 0 (pi ) + (pi + pjM ) u1

(1

(1

i d (pi ,

pjM )(u1

r ))

Simplifying the coefficient on

(1

(1

pjM )

pjM ) = u0 (pi ) + (pi + pjM )

Multiply through by (1

(1

i d (pi ,

u1 + r

(pi + pjM )

pjM ))(1

pi

i d

(1

on the LHS:

+ pjM )

2r (p i

r ))

=1

(1

r)

(1

pi

pjM ) +

2 (1

r )(1

=1

(1

r)

(1

pi

pjM ) +

2 (1

r )(1

2 (1

=1

(1

r)

(1

pi

pjM )

=1

(1

r)

(1

pi

pjM ) +

=1

(1

r)

(1

pi

pjM ) +

=1

2 +

2

= (1

)2

= (1

)[1

2) r

+(

+ (1

+

2 (1 2 )(p i

+(

r +

pjM )

+ pjM )

2r (p i

r + r (pi +

+ pjM )

2 (p i

2r (p i

(pi + pjM ))

(pi +

2

pjM )

pi

+ pjM )

pjM ))

2r (p i

+ pjM )

2r

pjM )

pi

pjM )

)(pi + pjM )

) r + (1

r ) + (pi + pjM )]

(1

Finally, returning to both sides of the equation, we have

(1

)[1

r ) + (pi + pjM )]

(1

i d (pi ,

pjM ) = (1

(1

r )) u0 (pi ) + (pi + pjM ) u1

And dividing both sides by the coefficient on the LHS i d (pi ,

pjM ) =

(1

(1

(1

)[1

r )) u0 (pi ) + (pi + pjM ) u1 (1

r ) + (pi + pjM )]

A.2.1 First order conditions The Lagrangean for the maximization problem described by Eq. (8) is:

(p ,

1i ,

2i )

=

(1

(1

(1

)[1

r )) u0 (pi ) + (pi + pjM ) u1 r ) + (pi + pjM )]

(1

+

1i (pi

p) +

¯ 2i (p

pi )

where ki represents its respective boundary condition. The Kuhn–Tucker conditions include the derivative of the Lagrangean with respect to pi and both slackness conditions. Provided they are non-binding, the interior solution is described by the following:

[(1 (1

(1

)[1

r )) u0 (pi ) + u1] (1

r ) + (pi + pjM )]

=

[(1 (1

(1 )[1

r )) u 0 (pi ) + (pi + pjM ) u1] (1

r ) + (pi + pjM )]2

Multiplying both sides by the LHS denominator, we see an intertemporal weighing of costs and benefits as in the cooperative problem:

[(1

(1

r )) u 0 (pi ) + u1] =

[(1

r )) u0 (pi ) + (pi + pjM ) u1 ]

(1 [1

(1

r ) + (pi + pjM )]

Further rearrangement results in: 9

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom

[1

r ) + (pi + pjM )][(1

(1 = [(1

[1

(1

r )) u 0 (pi ) + (pi +

r ) + (pi + pjM )](1

(1 + [1

(1

= (1 (1

(1

(1

r ) + (pi + (1

= [ [1

(1

r )) u0 (pi )

pjM )] u1 2 (p i

r )) u0 (pi ) +

r ))[ u0 (pi )

r )) u 0 (pi ) + u1]

pjM ) u1]

[1

+ pjM ) u1 r ) + (pi + pjM )] u0 (pi )]

(1

r ) + (pi + pjM )]

(1

2 (p i

+ pjM )] u1

Simplifying the coefficient on u1:

[1

r ) + (pi + pjM )]

(1

=

2

= = (1

2

+

2r

+ (1

2r

+

2 (p i

2 (p i

+ pjM )

+ pjM )

2 (p i

+ pjM )

r ))

Using this simplification:

(1

(1

r ))[ u0 (pi )

= (1 (1

(1

(1 (1

(1

r ) + (pi + pjM )] u0 (pi )]

[1

(1

r ) + (pi + pjM )] u0 (pi )]

r )) u1

r ))[ u0 (pi )

(1

[1

r )) u1 = 0

Finally, dividing through by (1

u0 (pi )

[1

(1

r ) + (pi +

(1

r ))

0 results in the condition provided in the text:

pjM )] u 0 (pi )

u1 = 0

When agents are symmetric and take action pi = pj = p M , these reduce to the following equation:

(u0 (p M )

u1) = [1

(1

(A.2)

r ) + 2 p M ] u 0 (p M )

which can be rearranged to form Eq. (10) in the text. A.2.2 Underprovision (Proof of Proposition 1) Proof. We prove Proposition 1 by contradiction. Subtracting Eq. (A.1) from (A.2): [u 0 (p M ) 2u0 (p P ) + u1] = [1 (1 r ) + 2 p M ] u0 (p M ) (A.3) [1 (1 r ) + 2 p P ] u0 (p P ) The two solutions cannot be equal (i.e. p P = p M ), as the expression would reduce to u1 = u0 (p M ) , a contradiction of the set-up where u1 > u 0 (p) for all possible values of p . Next, suppose p P < p M , which implies u0 (p P ) < u0 (p M ) and that p M u 0 (p M ) p P u0 (p P ) < 0 . This would mean that the RHS of Eq. (A.3) is negative, and therefore the LHS is negative, but this contradicts the assumption u1 > 2u0 ( p ) , which implies the LHS is positive. Thus, it must be that p P > p M . □ A.2.3 Universal dilemma strength If we restrict the fishers to only two strategies—taking the socially optimal action, p P , is considered cooperation, while taking the non-cooperative Cournot-Nash action, p M , is defection—then we can reframe the problem as a 2 × 2 dilemma game4 according to the following:

(A.4) where h and d have been simplified as identical continuation values for all possible actions, and h > d . Bimatrix dilemmas exhibit contrasting incentives that can be captured with the universal dilemma strength measures proposed by Wang et al. T R (2015). The gamble-intending dilemma, defined as Dg = R P , represents the lure of acting selfishly. In our model, this is the increase in damagedstate reward, u 0 (·) , from investing less in conservation. If the gamble-intending incentive is positive, then social defection is made more attractive in P S comparison to cooperation. On the other hand, the risk-averting dilemma, defined as Dr = R P , represents the fear of being taken for a fool; if it is positive, then exploitation is less likely. In our model, this risk is mitigated by the fact that an agent's own action contributes toward the transition

4

The four abbreviations most commonly used in the dilemma game literature stand for Reward, Sucker, Temptation, and Punishment (Tanimoto, 2018). 10

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom

back to the healthy state; one's own conservation is beneficial, even if the other agent does not invest as much effort. These measures can then assist with the classification of dilemma games.5 Using the stage payoffs in (A.4), we can derive the universal measures of dilemma strength for the simplified model as follows:

T R R P [u 0 (p M ) u 0 (p P )] + (p M p P )( h d) = P M P [u0 (p ) u 0 (p )] + 2 (p p M )( h d) [u0 (p M ) u0 (p P )] (p P p M )( h d) = [u0 (p M ) u0 (p P )] + 2 (p P p M )( h d) [u0 (p M ) u0 (p P )] (p P p M )( h d) = [u0 (p M ) u0 (p P )] 2 (p P p M )( h d)

(A.5)

P S R P [u0 (p M ) u0 (p P )] + (p M p P )( h d) = [u0 (p P ) u 0 (p M )] + 2 (p P p M )( h d) [u0 (p M ) u0 (p P )] (p P p M )( h d) = [u 0 (p M ) u 0 (p P )] + 2 (p P p M )( h d) [u0 (p M ) u0 (p P )] (p P p M )( h d) = = Dg [u 0 (p M ) u 0 (p P )] 2 (p P p M )( h d)

(A.6)

Dg =

Dr =

> u0 < We know that and that u 0 We also anticipate that h > d , because otherwise there is no purpose to returning the healthy state. The first interesting point to notice is that the two dilemmas are of equal strength in this game, when d is taken to be the same across all futures. The second is that both numerator and denominator consist of a positive term subtracted from a positive term, and so it is clear that the denominator is smaller than the numerator. When u 0 (p M ) u 0 (p P ) > 2 (p P p M )( h d ) , both numerator and denominator are positive, and so both measures are negative. (Similarly when u 0 (p M ) u 0 (p P ) < (p P p M )( h d ) .) This pattern would describe a “Trivial” game, in which no dilemma exists; if fishers only chose between these two strategies (as opposed to the whole continuum), they would choose to cooperate. However, when the single-stage temptation payoff falls between these two discounted expected future payoffs, i.e. M (p P p M )( h u0 (p P ) < 2 (p P p M )( h d ) < u 0 (p ) d) , then both dilemma measures are positive, which transforms the game into a Prisoner's Dilemma. Furthermore, we can see how each of the exogenous parameters affect the dilemma. A decrease in the healthy-state transition probability r , an increase in the discount factor , or an increase in the healthy-state reward u1 increases the future value of the healthy state, H . This, in turn, decreases both Dg and Dr , which strengthen or weaken the dilemma, depending on the parameter space and functional forms. This is, however, a major simplification of the original model which has more than two strategies for each player and continuation payoff values that are functions of those strategies. pM

(p P ) .

(p M )

pP

A.3 Non-cooperative derivations under leader–follower timing The Lagrangean for the leader's maximization problem described by Eq. (11) is:

(p ,

1i ,

2i )

=

(1

(1

(1

)[1

r )) u0 (pi ) + (pi + pjF (pi )) u1 r ) + (pi + pjF (pi ))]

(1

+

1i (pi

p) +

¯ 2i (p

pi )

where, like for the Cournot timing, ki represents the boundary conditions, and the full Kuhn–Tucker conditions include the derivative of the Lagrangean with respect to pi and the slackness conditions. If they are non-binding, the interior solution is described by:

) 2 [1

(1 × (1

(1 )[1

× (1

(1

(1

r )) u 0 (pi ) +

r ) + (pi + pjF (pi ))]

[(1

(1

r )) u 0 (pi ) + (pi + pjF (pi )) u1]

(1

) 1+

pjF

1 × r ) + (pi + pjF (pi ))]2

pjF pi

1+

pi

u1

=0

When distributed and rearranged, this becomes:

(1 (1

(1 )[1

r )) u0 (pi ) + (1

1+

r ) + (pi +

pjF pi

pjF

u1

(pi ))]

[(1 =

(1 (1

r )) u 0 (pi ) + (pi + pjF (pi )) u1 ] 1 + )[1

(1

r ) + (pi +

pjF

(pi

pjF pi

))]2

Multiplying both sides by the LHS denominator, we obtain the same intertemporal weighing of costs and benefits as for the earlier derivations:

5 Ito and Tanimoto (2018) write that “[i]f both Dg and Dr are positive, the game is PD, whereby (D) dominates (C). If Dg is positive and Dr is negative, we face the so-called Chicken game, which has an internal (polymorphic) equilibrium. If Dg is negative and Dr is positive, the game, characterized by bi-stability, is called the SH game. Finally, if both Dg and Dr are negative, we deal with the Trivial game, whereby (C) dominates (D) (i.e. no dilemma exists).”

11

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom

(1

(1

r )) u0 (pi ) +

pjF

1+

pi

[(1

r )) u 0 (pi ) + (pi + pjF (pi )) u1 ] 1 +

(1

u1 =

[1

(1

r ) + (pi +

pjF pi

pjF (pi ))]

Further rearrangement results in:

(1

(1

r )) u0 (pi ) +

= [(1 (1

pjF

1+

pi

+ (1

(1

u1 +

pi

pjF

r )) 1 +

2 (p i

pi

pjF

+ pjF (pi )) 1 + 2 (p i

u 0 (pi ) +

pi

pjF

r ))(pi + pjF (pi )) u0 (pi )

(1 pjF

r )) 1 +

(1

= (1

r )) + (pi + pjF (pi ))]

(1

r )) u 0 (pi ) + (pi + pjF (pi )) u1 ] 1 +

(1

r )) 2u0 (pi ) + (1

(1

u1 [(1

pi

u1

+ pjF (pi )) 1 +

pjF pi

u1

Canceling like terms:

(1

r )) 2u0 (pi ) + (1

(1 + (1

(1

Divide through by (1

(1

pi

u1 = (1

(1

pjF

r )) 1 +

pi

u0 (pi )

r )) :

(1

r )) u0 (pi ) + (pi + pjF (pi )) u0 (pi )

(1 +

pjF

r )) 1 +

r ))(pi + pjF (pi )) u0 (pi )

(1

pjF

1+

pi

u1 =

1+

pjF

u 0 (pi )

pi

A.3.1 Derivation of i pj We use the follower's first-order condition to define an implicit function as follows: p

F (pjF , piL )

[u0 (pjF )

u1]

[1

r ) + (piL + pjF )] u 0 (pjF ) = 0

(1

(A.7)

Taking the total derivative of the Implicit Function described by (A.7):

F (pjF , piL ) pi

= u0 (pjF ) [1

pj

pj

pi

pi

r ) + (piL + pjF )] u0 (pjF )

(1

u0 (pjF ) [1

+ 1 u0 (pjF )

pj

pj

pi

pi (1

pj pi

=0

u0 (pjF ) pj

r ) + (piL + pjF )] u0 (pjF )

pi

= u0 (pjF )

Proposition A.1. An interior Stackelberg solution must have 0 < 1 +

pF pL

< 1.

Proof. By rearranging Eq. (12), we can examine the signs of both sides as follows:

1+ >0

pjF

piL

[ u0 (piL ) <0

u1 ] = [1

(1

r) +

(piL + pjF ) ] u0 (piL )

>0

0

<0

We know that < 1 and r < 1, so (1 r ) < 1 and the bracketed term on the RHS must be positive. This, in turn, is multiplied by u0 (·), so the RHS must be negative. The LHS has a positive discount factor multiplying the gap between damaged-state and healthy-state rewards, which is always negative. This means the remaining multiplier must be positive for LHS to also be negative. We already know that remaining term to be positive we must have that

pjF piL

< 1. □

A.4 Substitutability (Proof of Proposition 2) Proof. The first order conditions of the leader (Eq. (12)) and the follower (Eq. (13)) together imply 12

pjF

piL

< 0 , so in order for the

Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom pjF

(A.8) 1 +

[u0 (p L )

piL

u 0 (p L )

u1 ]

[u 0 (p F )

=

u 0 (p F )

u1 ]

Since u1 > u 0 (·) for all possible values of p and u0 (pL ) < 0 , we know that both sides are well-defined and that neither side is equal to zero. Suppose that p F = p L ; then we could divide both sides by

[u0 (p L )

u 0 (p L )

u1 ]

and be left with the statement 1 +

pjF

piL

= 1. Under substitutability, this is a

contradiction of Proposition A.1 in Appendix A, which states that an interior Stackelberg solution must have 0 < 1 +

pF pL

. Therefore, the interior

actions taken by the leader and follower cannot be equal and so the first part of the proposition is shown.It not possible to determine whether this statement implies that p L < p F or p L > p F without a functional form or restrictions on the curvature of utility. Define the function

g (p )

u 0 (p) u1 u 0 (p )

If this function is decreasing in p , then the inequality in Eq. (A.9) would indicate that the leader's action is smaller than the follower's action ( p L < p F ). Taking the derivative of g (·) , we have

g u (p ) 2 = 0 p

u0 (p)(u0 (p) u 0 (p ) 2

u1 )

(A.10)

The denominator is positive, so the expression is negative if u0 (p)2 < u0 (p)(u0 (p) follower's effort. □

u1) , and in this case the leader's effort is smaller than the

A.4.1 Further underprovision (Proof of Proposition 3) Proof. Suppose 2p M = p F + p L . Rearranging Eqs. (A.2), (12) and (13), we have (A.11)

[u0 (p M )

(A.12)

1+

(A.13)

[u0 (p F )

u 0 (p M )

u1 ]

pjF

piL

u 0 (p F )

u1 ]

= [1 [u0 (p L )

u 0 (p L )

(1 u1 ]

= [1

r ) + 2 pM ]

= [1

(1

r ) + (p L + p F )]

r ) + (p L + p F )]

(1

The RHS of each equation is the same if 2p M = p L + p F . This would imply that the LHS of Eqs. (A.11) and (A.13) are equal, and that Eqs. (A.11) and (A.12) are also equal. But these two expressions imply that p M = p F and therefore p L = p F , which contradicts Proposition 2. Next, suppose 2p M < p F + p L . From Eqs. (A.11) and (A.12), we have:

[u 0 (p M ) u1 ] u 0 (p M )

2 pM = 1

(1

r) =

1+

pjF [u 0 (p L ) u1 ] u 0 (p L ) piL

(p L + p F )

which can be rearranged to form:

[u 0 (p M ) u1 ] u 0 (p M )

1+

pjF [u 0 (p L ) u1 ] = 2p M u 0 (p L ) piL

(p L + p F )

If 2p M < p F + p L , then the RHS is negative. This implies the LHS is negative, and therefore the first inequality in the following statement (the second inequality stems from Proposition A.1):

[u 0 (p M ) u1 ] < 1+ u 0 (p M )

pjF [u0 (p L ) u1 ] [u (p L ) u1 ] < 0 L L u 0 (p ) u0 (p L ) pi

which, if g (p) is decreasing, indicates p M > p L . However, Eqs. (A.11) and (A.13) imply:

[u 0 (p M ) u1 ] u 0 (p M )

[u 0 (p F ) u1 ] = 2p M u0 (p F )

(p L + p F )

If 2p M < p F + p L , then the RHS is negative, implying the LHS is negative, and therefore:

[u 0 (p M ) u1 ] [u (p L ) u1 ] < 0 M u 0 (p ) u0 (p L ) which, if g (p) is decreasing, indicates p M > p F , but this contradicts 2p M < p L + p F . Thus, it must be that 2p M > p L + p F . □

Güth, W., Vittoria Levati, M., Soraperra, I., 2015. Common and private signals in public goods games with a point of no return. Resour. Energy Econ. 41, 164–184. Hannesson, R., 2011. Game theory and fisheries. Annu. Rev. Resour. Econ. 3 (1), 181–202. Ito, H., Tanimoto, J., 2018. Scaling the phase-planes of social dilemma strengths shows game-class changes in the five rules governing the evolution of cooperation. R. Soc. Open Sci. 5, 1–9. Kaitala, V., Lindroos, M., 1998. Sharing the benefits of cooperation in high seas fisheries: a characteristic function game approach. Nat. Resour. Model. 11 (4), 275–299. Kaitala, V., Munro, G.R., 1993. The management of high seas fisheries. Mar. Resour. Econ. 8 (4), 313–329. Kaitala, V., Pohjola, M., 1988. Optimal recovery of a shared resource stock: a differential game model with efficient memory equilibria. Nat. Resour. Model. 3 (1), 91–119.

References Anderson, C.M., Uchida, H., 2014. An experimental examination of fisheries with concurrent common pool and individual quota management. Econ. Inq. 52 (2), 900–913. Benchekroun, H., Van Long, N., 2002. Transboundary fishery: a differential game model. Economica 69 (274), 207–221. Cave, J., 1987. Long-term competition in a dynamic game: the cold fish war. RAND J. Econ. 18 (4), 596–610. Clark, C.W., 1980. Restricted access to common-property fishery resources: a game-theoretic analysis. Dynamic Optimization and Mathematical Economics 117–132. Crépin, A.-S., Biggs, R., Polasky, S., Troell, M., De Zeeuw, A., 2012. Regime shifts and management. Ecol. Econ. 84, 15–22.

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Ecological Economics 169 (2020) 106503

A.A. Klis and R.T. Melstrom Kwon, O.S., 2006. Partial international coordination in the great fish war. Environ. Resour. Econ. 33 (4), 463–483. Laukkanen, M., 2003. Cooperative and non-cooperative harvesting in a stochastic sequential fishery. J. Environ. Econ. Manag. 45 (2), 454–473. Levhari, D., Mirman, L.J., 1980. The great fish war: an example using a dynamic Cournot–Nash solution. Bell J. Econ. 11 (1), 322. Lindahl, T., 2012. Coordination problems and resource collapse in the commons—exploring the role of knowledge heterogeneity. Ecol. Econ. 79, 52–59. Lindroos, M., Kaitala, V., 2000. Nash equilibria in a coalition game of the Norwegian spring-spawning herring fishery. Mar. Resour. Econ. 15 (4), 321–339. Mine Cinar, E., Johnson, J., Palmer, A., 2013. Decision making: fishing production and fishers in the Black Sea. Fish. Res. 147, 296–303. Płatkowski, T., 2017. On derivation and evolutionary classification of social dilemma games. Dyn. Games Appl. 7 (1), 67–75. Pauly, D., Palomares, M.-L., 2005. Fishing down marine food web: it is far more pervasive

than we thought. Bull. Mar. Sci. 76 (2), 197–212. Pintassilgo, P., 2003. A coalition approach to the management of high seas fisheries in the presence of externalities. Nat. Resour. Model. 16 (2), 175–197. Reeling, C.J., Horan, R.D., 2014. Self-protection, strategic interactions, and the relative endogeneity of disease risks. Am. J. Agric. Econ. 97 (2), 452–468. Reeling, C., Palm-Forster, L.H., Melstrom, R.T., 2017. Policy instruments and incentives for coordinated habitat conservation. Environ. Resour. Econ. 1–23. Tanimoto, J., Sagara, H., 2007. Relationship between dilemma occurrence and the existence of a weakly dominant strategy in a two-player symmetric game. Biosystems 90 (1), 105–114. Tanimoto, J., 2018. Evolutionary games with sociophysics: analysis of traffic flow and epidemics. Evolutionary Economics and Social Complexity Science, vol. 17. Springer Nature, Singapore, pp. 1–221. Wang, Z., Kokubo, S., Jusup, M., Tanimoto, J., 2015. Universal scaling for the dilemma strength in evolutionary games. Phys. Life Rev. 14, 1–30.

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