Fisheries Management in Congested Waters: A Game-Theoretic Assessment of the East China Sea

Abstract

Fisheries in the East China Sea (ECS) face multiple concerning trends. Aside from depleted stocks caused by overfishing, illegal encroachments by fishermen from one nation into another’s legal waters are a common occurrence. This behavior presumably could be stopped via strong monitoring, controls, and surveillance (MCS), but MCS is routinely rated as below standards for nations bordering the ECS. This paper generalizes the ECS to a model of a congested maritime environment, defined as an environment where multiple nations can fish in the same waters with equivalent operating costs, and uses game-theoretic analysis to explain why the observed behavior persists in the ECS. The paper finds that nations in congested environments are incentivized to issue excessive quotas, which in turn tacitly encourages illegal fishing and extracts illegal rent from another’s legal waters. This behavior couldn’t persist in the face of strong MCS measures, and states are thus likewise incentivized to use poor MCS. A bargaining problem is analyzed to complement the noncooperative game, and a key finding is the nation with lower nonoperating costs has great leverage during the bargain.

Appendices

Appendix A

Proofs of Theorems 1 Through 4

2.1 Proof of Theorem 1

Theorem 1. Existence of a subgame equilibrium.

Any instantiation of the game’s parameters and choice of the overall game’s decision variables yields a subgame equilibrium.

Proof

Consider the following algorithm which takes the parameters, ${F}_{kl}$, and ${m}_{k}$ as given, and seeks values of the SGVs:

Algorithm A1 Finding a SGE

• 1. Initiate all SGVs to ${F}_{ij,kl}=0$

• 2. Determine the set of SGVs, $F^{{\prime }}$, to increase in the next iteration of the algorithm as follows. Define the maximum achievable rent as:

  $\pi^{{\prime }} = \max \left\{ {0, \max \left\{ {\pi_{ij,kl} | \mathop \sum \limits_{ij} F_{ij,kl} < F_{kl} } \right\}} \right\}$; if no $kl$ exists satisfying $\mathop \sum \limits_{ij} F_{ij,kl} < F_{kl}$, then set $\pi^{\prime} = 0$.

.

If $\pi^{{\prime }} = 0$, stop. Otherwise, include all SGVs in $F^{{\prime }}$ whose corresponding rent equals $\pi^{{\prime }}$, and can be increased without decreasing another SGV; that is, $F_{ij,kl}$ may only be in $F^{{\prime }}$ if $\mathop \sum \limits_{ij} F_{ij,kl} < F_{kl}$.

Before describing step 3, note the following properties are maintained throughout the algorithm:

Algorithm A1, Property 1.

SGVs are never allowed to go below 0, and ${\sum }_{ij}{F}_{ij,kl}\le {F}_{kl}$ remains true for all $kl$.

Algorithm A1, Property 2.

If ${F}_{ij,kl}>0$, then ${\pi }_{ij,kl}\ge 0$ and ${\pi }_{ij,kl}\ge {\pi }_{{ij}{^{\prime}},kl}$ for all ${ij}{^{\prime}}$.

These two properties ensure all subgame constraints are satisfied throughout the algorithm, other than those stating no profitable quotas are left unused. The latter condition is the stopping condition for the algorithm ($\pi^{{\prime }} = 0$).

• 3. Increase all SGVs in ${F}{^{\prime}}$ while obeying the following rules:

• i. Increase SGVs in ${F}{^{\prime}}$ at rates such that their rents remain equal.

• ii. Consider the case where ${\sum }_{ij}{F}_{ij,kl}={F}_{kl}$ for some $kl$ (so that by definition ${F}_{ij,kl}$ are not in ${F}{^{\prime}}$). Modify the SGVs ${F}_{ij,kl}$ at rates such that ${\sum }_{ij}{F}_{ij,kl}={F}_{kl}$ remains true, and ${F}_{ij,kl}>0$ iff ${\pi }_{ij,kl}\ge {\pi }_{i{j}{^{\prime}},kl}$ for all $i{j}{^{\prime}}$.

• iii. Return to step 2 whenever a new SGV enters $F^{{\prime }}$, or when ${\sum }_{ik}{F}_{ij,k{l}{^{\prime}}}={F}_{k{l}{^{\prime}}}$ for any SGVs in $F^{{\prime }}$, or when the rent of a SGV in $F^{{\prime }}$ becomes 0.

The proof this algorithm always produces a SGE is clear, after noting one additional property to accompany Properties 1 and 2.

Algorithm A1, Property 3.

${\pi }{^{\prime}}$ is either constant or decreasing linearly throughout the algorithm. The former case occurs only when SGVs in ${F}{^{\prime}}$ are increasing with a perfectly offsetting decrease in SGVs not in ${F}{^{\prime}}$, a process which eventually stops when one of the decreasing SGV reaches 0, and is followed by a state where the only SGVs being modified are increasing. Thus, every state where ${\pi }{^{\prime}}$ is constant is followed by a state where it’s linearly decreasing, and thus ${\pi }{^{\prime}}$ will eventually reach 0.

${\pi }{^{\prime}}=0$ implies either all quotas are exhausted, or no fisherman type $kl$ can achieve positive rent with their remaining quotas. Thus, properties 1 through 3 imply the algorithm will terminate at a subgame equilibrium.

3.1 Proof of Theorem 2

Theorem 2. Utilities are uniquely defined by decision variables.

Any instantiation of the game’s parameters and choice of the overall game’s decision variables leads to uniquely defined utilities for the players. This is in spite of the fact that multiple subgame equilibria (SGE) may exist.

Proof

This theorem is proved in two parts. First, it’s shown that any SGE leads to the same set of biomasses for the fisheries in the game. Next, it becomes easy to show all players receive the same utility under any SGE, given biomasses are the same.

To show any SGE leads to the same biomasses, assume two distinct SGE exist. These will be referred to as SGE1 and SGE2, and the “prime” and “double prime” notion will be used to distinguish their characteristics. That is, $x_{ij}^{{\prime }}$ denotes the biomass of fishery $ij$ under SGE1, while $x_{ij}^{\prime \prime }$ denotes that for SGE2. Assume $x_{ij}^{\prime } < x_{ij}^{\prime \prime }$ for some fishery $ij$. A consequence of this is that $F_{ij,kl}^{{\prime }} > F_{ij,kl}^{{\prime \prime }}$ for some $kl$; otherwise, $x_{ij}^{\prime } < x_{ij}^{{\prime \prime }}$ would be impossible. A further consequence is that fisherman type $kl$ is using all quotas under SGE2: $\mathop \sum \limits_{ij} F_{ij,kl}^{{\prime \prime }} = F_{kl}^{{\prime \prime }}$. This is the case because fishing yields nonnegative rent for $kl$ at biomass level $x_{ij}^{{\prime }}$ (otherwise $F_{ij,kl}^{{\prime }} > F_{ij,kl}^{{\prime \prime }} \ge 0$ would violate the condition that fishermen only fish if profitable), and thus biomass level $x_{ij}^{\prime \prime } > x_{ij}^{\prime }$ yields strictly positive rent. Were $kl$ to have available quotas ($\mathop \sum \limits_{ij} F_{ij,kl}^{{\prime \prime }} < F_{kl}^{{\prime \prime }}$), they would be used to extract this positive srent in fishery $ij$, eventually exhausting all quotas.

It can also be shown that every fishery where fisherman type $kl$ is operating under SGE2 has less biomass under SGE1. To see this, denote the set of fisheries where $kl$ was using positive quotas under SGE2 as $S_{kl}^{{\prime \prime }} : = \left\{ {mn \left| { F_{mn,kl}^{{\prime \prime }} } \right\rangle 0} \right\}$; the claim is that $x_{mn}^{\prime } < x_{mn}^{{\prime \prime }} \forall mn \in S_{kl}^{{\prime \prime }}$. To prove this claim, note the subgame conditions require each fisherman type, $kl$, to earn equal profits everywhere they fish, and that it’s known $F_{ij,kl}^{\prime } > 0$. Now assume $kl$ is fishing in $ij$ under SGE2: $ij \in S_{kl}^{{\prime \prime }}$. This immediately reveals the profits $kl$ earns under SGE1 are less than under SGE2 (since $x_{ij}^{\prime } < x_{ij}^{{\prime \prime }}$). If profits are less under SGE1, then all fisheries in $S_{kl}^{{\prime \prime }}$ must have less biomass under SGE1; if a fishery existed violating this condition, then fisherman type $kl$ could earn greater profits there and would divert from $ij$ to said fishery. Assume instead $kl$ isn’t fishing in $ij$ under SGE2: $ij \notin S_{kl}^{{\prime \prime }}$. . This simply means that under SGE2, $kl$ had more attractive profits outside of fishery $ij$. These are now gone, since $F_{ij,kl}^{\prime } > 0$ despite $x_{ij,kl}^{\prime } < x_{ij,kl}^{{\prime \prime }}$; these more attractive profits being “gone” is equivalent to stating biasses have declined. In sum, it’s now been shown all fisheries in $S_{kl}^{{\prime \prime }}$ must have experienced a decline in biomass.

A recursive argument extending the above analysis will lead to a contradicting, thus showing $x_{ij}^{\prime } < x_{ij}^{{\prime \prime }}$ is impossible. First note that there must be some fishery $i_{2} j_{2} \in S_{kl}^{{\prime \prime }}$ such that another fisherman type, say $k_{2} l_{2}$, is fishing more under SGE1 than under SGE2: $F_{{i_{2} j_{2} ,k_{2} l_{2} }}^{\prime } > F_{{i_{2} j_{2} ,k_{2} l_{2} }}^{{\prime \prime }}$ for some $i_{2} j_{2} \in S_{kl}^{{\prime \prime }}$. This is the case because fisherman type $kl$ alone cannot cause a decline in every fishery in $S_{kl}^{{\prime \prime }}$, as if it were possible to do this profitably then SGE2 wouldn’t be a subgame equilibrium. Now repeating the previous logic, because $k_{2} l_{2}$ is fishing (more) in $i_{2} j_{2}$ under SGE1 and $x_{{i_{2} j_{2} }}^{\prime } < x_{{i_{2} j_{2} }}^{{\prime \prime }}$, all fisheries where $k_{2} l_{2}$ is fishing under SGE2 (denote these $S_{{k_{2} l_{2} }}^{^{\prime\prime}}$) have lower biomass under SGE1. Also by the previous logic: (i) $k_{2} l_{2}$ is using all quotas under SGE2; and (ii) fisherman type $k_{2} l_{2}$ can’t cause the lower biomasses in $S_{{k_{2} l_{2} }}^{{\prime \prime }}$ alone. More importantly for the recursive argument, however, is that the lower biomasses for each fishery in the union $S_{kl}^{{\prime \prime }} \cup S_{{k_{2} l_{2} }}^{{\prime \prime }}$ cannot be caused by increased levels of fishing from $kl$ and $k_{2} l_{2}$ alone. This is because each is using all available quotas under SGE2. A third fisherman type, $k_{3} l_{3}$, must be expending more quotas in at least one fishery from $S_{kl}^{{\prime \prime }} \cup S_{{k_{2} l_{2} }}^{{\prime \prime }}$ under SGE1 than under SGE2. Repeating the previous line of reasoning leads to a recursion ending in a contradiction: $k_{3} l_{3}$ is using all quotas under SGE2, all fisheries in $S_{{k_{3} l_{3} }}^{{\prime \prime }}$ have experienced a decline in biomass, and fisherman types $kl$, $k_{2} l_{2}$, and $k_{3} l_{3}$ alone can’t cause the biomass declines in $S_{kl}^{{\prime \prime }} \cup S_{{k_{2} l_{2} }}^{{\prime \prime }} \cup S_{{k_{3} l_{3} }}^{{\prime \prime }}$; an additional fisherman type must be contributing to the declines; repeating, eventually there are no additional fisherman types to explain biomass declines, and thus a contradiction has been found. The conclusion is the original assumption, that $x_{ij}^{\prime } < x_{ij}^{{\prime \prime }}$ for some $ij$, is an impossibility. Thus, all SGE have the same biomasses for all fisheries.

Having established that all SGE lead to identical biomasses, it’s easy to show all SGE lead to the same utilities for all players. Recall that all fisherman types, $kl$, receive equal profits in all fisheries where they’re fishing. Assume type $kl$ isn’t using all available quotas under a particular SGE, which means the best available profit at the biomass levels for this SGE is less than or equal to 0. In this case, fisherman type $kl$ contributes 0 to the utility of Player $k$, regardless of exactly where they are fishing (i.e. regardless of the specific values of ${F}_{ij,kl}$ under this SGE). Assume instead that $kl$ is using all available quotas. A condition of the subgame is that fishermen fish only where profits are highest. Denote the maximal profit achievable by fisherman type $kl$ by the constant $v$, and note this is the same across all SGE since biomasses are the same for all SGE. Thus, under any SGE the contribution of fisherman type $kl$ to Player $k$’s utility is ${F}_{kl}\bullet a$, a constant. It’s thus been shown all SGE yield identical utilities for the players.

3.2 Proof of Theorem 3

Theorem 3. Non-use of MCS.

Assume each player owns one fishery. Consider Player $k$’s decision. For given strategies of all other player, responding with ${m}_{k}>0$ can yield at most equivalent utility as ${m}_{k}=0$. If costs of MCS are nonzero, then ${m}_{k}>0$ yields strictly less utility than ${m}_{k}=0$.

Proof

Because each player owns only one fishery, denote the number of fishermen from country $k$ fishing in country $i$’s waters as ${F}_{i,k}$. Assume player $k$ is using non-zero MCS: ${m}_{k}>0$. Unilaterally changing strategy to ${m}_{k}=0$ only affects the costs imposed on fishermen from country $k$ fishing in foreign waters. For all $i\ne k$, if ${F}_{i,k}$ doesn’t change at subgame equilibrium on account of the change in ${m}_{k}$ (because it was not previously profitable to illegally fish in $i$, and is still not), then all SGVs are unchanged and $k$’s utility is either the same (if MCS is costless) or has increased (if MCS costs money). If ${F}_{i,k}$ does change, then it must have increased due to the reduction in costs. This means either: (i) Player $k$ extracts rent from $i$’s waters, improving her utility; or (ii) Player $k$’s fishermen have diverted from $k$’s waters to $i$’s. In the latter case, $k$ can simply issue more quotas to replace those who diverted to $i$’s waters, establishing a subgame equilibrium equivalent to that when ${m}_{k}>0$ but with a higher value of ${F}_{i,k}$. By the previous logic, Player $k$’s utility has increased. This completes the proof.

3.3 Proof of Theorem 4

Theorem 4. Conditions for the legal optimum to be an equilibrium.

Assume a three-player game where each player owns one fishery. Assume $x_{k}^{OA} < x_{k}^{{m^{*} }} \forall k,m \in \left\{ {J,S,C} \right\}$ (notation is defined within the proof). Each player using the legally optimal level of fishing quotas (i.e. that which would be used if illegal encroachments were not a possibility) represents an equilibrium to the congested environment fishing game if and only if the following conditions 1 and 2 hold (all notation is defined in the accompanying proof):

Condition 1

$p_{k} q_{k} x_{k}^{legal} - c_{k} > p_{m} q_{m} x_{m}^{legal} - c_{k} - \beta_{k} P_{m} \forall k,m \in \left\{ {J,S,C} \right\}$, where $k \ne m$.

Condition 2

For all permutations of $k,l,m$ taking on the values in $\left\{ {J,S,C} \right\}$, $u_{k}^{legal} > \mathop {\max }\limits_{{F_{k} \in \left[ {F_{k}^{l} ,F_{k}^{k,m} } \right]}} u_{k} \left( {F_{k} } \right)$ if $x_{k}^{l} > x_{k}^{m}$. In addition, $u_{k}^{legal} > \mathop {\max }\limits_{{F_{k} > F_{k}^{k,m} }} u_{k} \left( {F_{k} } \right)$ if $x_{k}^{m} > x_{k}^{l,m}$, while $u_{k}^{legal} > \mathop {\max }\limits_{{F_{k} > F_{k}^{l,m} }} u_{k} \left( {F_{k} } \right)$ if $x_{k}^{m} < x_{k}^{l,m}$.

Proof

While the notation used to define the necessary and sufficient conditions is quite cumbersome, the proof straightforward. First note that the legally optimal number of quotas Player $k$ should issue is $F_{k}^{legal} = \frac{{r_{k} }}{{2q_{k} }}\left( {1 - \frac{{c_{k} }}{{p_{k} q_{k} Z_{k} }}} \right)$. This is easily derived using calculus, noting Player $k$’s biomass is $x_{k} = Z_{k} \left( {1 - \frac{{q_{k} }}{{r_{k} }}F_{k} } \right)$ and utility is hence $u_{k} = \left( {p_{k} q_{k} x_{k} - c_{k} } \right)\;F_{k}$. Defining $x_{k}^{legal} = Z_{k} \left( {1 - \frac{{q_{k} }}{{r_{k} }}F_{k}^{legal} } \right)$, condition 1 simply states that when each player issues the legally optimal quotas, no illegal encroachments occur. Were this condition violated, say because $p_{C} q_{C} x_{C}^{legal} - c_{C} < p_{J} q_{J} x_{J}^{legal} - c_{C} - \beta_{C} P_{J}$, then Chinese fishermen would have an incentive to fish illegally in Japanese waters. In turn, China could issue additional quotas to farm Japanese biomass down until $p_{C} q_{C} x_{C}^{legal} - c_{C} = p_{J} q_{J} x_{J}^{legal} - c_{C} - \beta_{C} P_{J}$, at which point China is reaping the legal harvest from its own waters while reaping additional harvests from Japanese waters, thus obviously achieving utility in excess of the legal optimum.

Condition 1 ensures the only possible way a player could benefit from a unilateral deviation from the legal solution is to farm its own biomass below the legally optimal level, for the purpose of extracting illegal rent from another’s waters. Condition 2 ensures this is not a viable strategy, as shown next. Without loss of generality, assume $F_{J} = F_{J}^{legal}$ and $F_{S} = F_{S}^{legal}$, and analyze China’s utility, $u_{C} \left( {F_{C} } \right)$, as $F_{C}$ is increased. Denote the level of Chinese biomass such that it becomes economical for Chinese fishermen to divert into the waters of Player $i$ as $x_{C}^{i}$. This is easily calculated by solving: $p_{C} q_{C} x_{C}^{i} - c_{C} = p_{i} q_{i} x_{i}^{legal} - c_{C} - \beta_{C} P_{i}$. The level of Chinese fishing attaining this biomass, $F_{C}^{i}$, can also be easily calculated by solving $Z_{C} \left( {1 - \frac{{q_{C} }}{{r_{C} }}F_{C}^{i} } \right) = x_{C}^{i}$.

Again without loss of generality, assume${F}_{C}^{J}<{F}_{C}^{S}$, so that as ${F}_{C}$ is increased Chinese fishermen encroach on Japanese waters before South Korean. Once${F}_{C}>{F}_{C}^{J}$, each additional unit of Chinese quotas increases actual fishing in Chinese waters,${F}_{CC}$, by less than 1. Denote the increase from a unit increase in $F_{C}$ as$\Delta F_{CC}^{\prime }$, which can be easily computed by solving a linear system of 2 equations and 2 unknowns:$\Delta F_{CC}^{\prime } + \Delta F_{JC}^{\prime } = 1$; $\frac{{p_{C} q_{C}^{2} }}{{r_{C} }}\Delta F_{CC}^{\prime } = \frac{{p_{J} q_{J}^{2} }}{{r_{J} }}\Delta F_{JC}^{\prime }$. The latter equation ensures Chinese fishermen continue to receive equal rent in Chinese and Japanese waters. For any level of Chinese quotas beyond $F_{C}^{J}$ and up to an upper-bound (to be specified momentarily), Chinese biomass is $x_{C}^{\prime } = x_{C}^{J} - \frac{{Z_{C} q_{C} }}{{r_{C} }}\left( {F_{C} - F_{C}^{J} } \right)\Delta F_{CC}^{\prime }$, and by equality of rent all Chinese fishermen receive $p_{C} q_{C} x_{C}^{\prime } - c_{C}$, and hence Chinese utility is simply $u_{C}^{\prime } = \left( {p_{C} q_{C} x_{C}^{\prime } - c_{C} } \right)F_{C}$.

The upper-bound on this utility function’s applicability is the value of $F_{C}$ such that Chinese and Japanese biomasses have been so degraded that either Chinese or Japanese fishermen find it profitable to start fishing in South Korean waters. There’s no guarantee who will start encroaching on South Korean waters first, so each possibility will be enumerated. It’s already been stated that Chinese fishermen will encroach on South Korean waters once $x_{C} = x_{C}^{S}$, and Japanese fishermen will once $x_{J} = x_{J}^{S}$. To frame everything in terms of Chinese biomass, $x_{J}^{S}$ can be converted to a corresponding biomass in Chinese waters using the fact Chinese fishermen are receiving equal rents. Denoting $x_{C}^{J,S}$ as the level of Chinese biomass where Japanese fishermen begin encroaching on South Korean waters, equality of Chinese rents implies: $p_{C} q_{C} x_{C}^{J,S} = p_{J} q_{J} x_{J}^{S} - \beta_{C} P_{J} \to x_{C}^{J,S} = \frac{{p_{J} q_{J} x_{J}^{S} - \beta_{C} P_{J} }}{{p_{C} q_{C} }}$. Also denote the corresponding level of Chinese fishing $F_{C}^{J,S}$, which is found by solving: $F_{C}^{J,S} = F_{C}^{J} + a$, where $a = \frac{{\left( {x_{C}^{J} - x_{C}^{J,S} } \right)}}{{\frac{{Z_{C} q_{C} }}{{r_{C} }}\Delta F_{CC} }} = \frac{{r_{C} \left( {x_{C}^{J} - x_{C}^{J,S} } \right)}}{{Z_{C} q_{C} \Delta F_{CC} }}$, the additional level of fishing to deplete Chinese biomass from $x_{C}^{J}$ to $x_{C}^{J,S}$, provided that $x_{C}^{J,S} > x_{C}^{S}$. Likewise, the level of fishing needed to deplete Chinese biomass to $x_{C}^{S}$ is $F_{C}^{C,S} = F_{C}^{J} + \frac{{r_{C} \left( {x_{C}^{J} - x_{C}^{S} } \right)}}{{Z_{C} q_{C} \Delta F_{CC} }}$. The upper-bound on the applicability of utility function $u_{C}^{\prime }$ is the lesser of $F_{C}^{C,S}$ and $F_{C}^{J,S}$; optimizing $u_{C}^{\prime }$ within the bounds $\left[ {F_{C}^{J} ,\min \left\{ {F_{C}^{C,S} ,F_{C}^{J,S} } \right\}} \right]$ is a straightforward, one-dimensional quadratic maximization ($u_{C}^{\prime }$ is quadratic because $x_{C}^{\prime }$ is linear in $F_{C}$).

Now assume ${F}_{C}^{C,S}<{F}_{C}^{J,S}$, so that Chinese fishermen begin encroaching on South Korean waters first, and assess China’s utility function for ${F}_{C}>{F}_{C}^{C,S}$. Beyond ${F}_{C}^{C,S}$, a unit increase in $F_{C}$ changes $F_{CC}$ by $\Delta F_{CC}^{{\prime \prime }}$, which is computed by solving a linear system of 3 equations and 3 unknowns: $\Delta F_{CC}^{{\prime \prime }} + \Delta F_{JC}^{{\prime \prime }} + \Delta F_{SC}^{{\prime \prime }} = 1$; $\frac{{p_{C} q_{C}^{2} Z_{C} }}{{r_{C} }}\Delta F_{CC}^{{\prime \prime }} = \frac{{p_{J} q_{J}^{2} Z_{J} }}{{r_{J} }}\Delta F_{JC}^{{\prime \prime }}$; and $\frac{{p_{C} q_{C}^{2} Z_{C} }}{{r_{C} }}\Delta F_{CC}^{{\prime \prime }} = \frac{{p_{S} q_{S}^{2} Z_{S} }}{{r_{S} }}\Delta F_{SC}^{{\prime \prime }}$. The latter two constraints ensure Chinese fishermen continue to receive equal rent in all fisheries. Chinese biomass for $F_{C} > F_{C}^{C,S}$ is therefore $x_{C}^{{\prime \prime }} = x_{C}^{S} - \frac{{Z_{C} q_{C} }}{{r_{C} }}\left( {F_{C} - F_{C}^{C,S} } \right)\Delta F_{CC}^{{\prime \prime }}$, and utility is $u_{C}^{{\prime \prime }} = \left( {p_{C} q_{C} x_{C}^{{\prime \prime }} - c_{C} } \right)F_{C}$, which is again quadratic in $F_{C}$. Unlike $u_{C}^{\prime }$, the domain of $u_{C}^{{\prime \prime }}$ has no upper-bound on $F_{C}$, provided China does not find it optimal to use so many quotas that either Japan or South Korea can no longer earn any profits in its own waters. This is because as Chinese quotas increases, by necessity Chinese fishermen’s rents decrease by equal amount, hence revenues ($p_{i} q_{i} x_{i}$), which don’t depend on who is doing the fishing, decrease by equal amounts, and hence neither Japanese nor South Korean fishermen will change behavior for some critical value of $F_{C}$. To formalize the assumption that Japan and South Korea continue to earn domestic profits as China seeks its optimal response to $F_{J}^{legal}$ and $F_{S}^{legal}$, it’s assumed $x_{k}^{OA} < x_{k}^{{C^{*} }}$ for $k \in \left\{ {J,S} \right\}$, where $x_{k}^{{C^{*} }}$ is the level of biomass corresponding to China’s optimal response given $F_{C} > F_{C}^{C,S}$ and $x_{k}^{OA}$ is the “open access” level of Player $k$’s biomass, where their rents are fully depleted: $p_{k} q_{k} x_{k}^{OA} = c_{k} \to x_{k}^{OA} = \frac{{c_{k} }}{{p_{k} q_{k} }}$. Finding China’s optimal utility subject to $F_{C} > F_{C}^{C,S}$ is again a straightforward quadratic maximization.

Lastly, assume $F_{C}^{C,S} > F_{C}^{J,S}$ so Japanese fishermen begin encroaching on South Korean waters first. For this case, when $F_{C} > F_{C}^{J,S}$ the change in Chinese fishing in Chinese waters, $\Delta F_{CC}^{\prime \prime \prime }$, caused by a unit increase in $F_{C}$ can be found by solving a system of four linear equations and four unknowns: $\Delta F_{CC}^{\prime \prime \prime } + \Delta F_{JC}^{\prime \prime \prime } = 1$; $\frac{{p_{C} q_{C}^{2} Z_{C} }}{{r_{C} }}\Delta F_{CC}^{\prime \prime \prime } = \frac{{p_{J} q_{J}^{2} Z_{J} }}{{r_{J} }}\left( {\Delta F_{JC}^{\prime \prime \prime } + \Delta F_{JJ}^{\prime \prime \prime } } \right)$; $\frac{{p_{S} q_{S}^{2} Z_{S} }}{{r_{S} }}\Delta F_{SJ}^{\prime \prime \prime } = \frac{{p_{J} q_{J}^{2} Z_{J} }}{{r_{J} }}\left( {\Delta F_{JC}^{\prime \prime \prime } + \Delta F_{JJ}^{\prime \prime \prime } } \right)$; $\Delta F_{SJ}^{\prime \prime \prime } = - \Delta F_{JJ}^{\prime \prime \prime }$. The second and third constraints ensure Chinese and Japanese fishermen, respectively, maintain equality of rent in the waters they fish in, and the last constraint reflects the fact the total quotas issued by Japan has not changed. Similarly to previous cases, Chinese biomass can be written as $x_{C}^{\prime \prime \prime } = x_{C}^{J,S} - \frac{{Z_{C} q_{C} }}{{r_{C} }}\left( {F_{C} - F_{C}^{J,S} } \right)\Delta F_{CC}^{\prime \prime \prime }$, and China’s utility function for $F_{C} > F_{C}^{J,S}$ is $u_{C}^{\prime \prime \prime } = \left( {p_{C} q_{C} x_{C}^{\prime \prime \prime } - c_{C} } \right)F_{C}$, which is quadratic and easily maximized.

To summarize the proof, when $F_{J} = F_{J}^{legal}$ and $F_{S} = F_{S}^{legal}$, China cannot improve upon $F_{C} = F_{C}^{legal}$ from a unilateral change, assuming $F_{C}^{J} < F_{C}^{S}$, if:

Condition 1

$p_{C} q_{C} x_{C}^{legal} - c_{C} > p_{m} q_{m} x_{m}^{legal} - c_{C} - \beta_{C} P_{m}$ for $m \in \left\{ {J,S} \right\}$.

Condition 2

$$ \mathop {\max }\limits_{{F_{C} \in \left[ {F_{C}^{J} ,\min \left\{ {F_{C}^{C,S} ,F_{C}^{J,S} } \right\}} \right]}} u_{C}^{\prime } < u_{C}^{legal} , $$

$$ \mathop {\max }\limits_{{F_{C} > F_{C}^{C,S} }} u_{C}^{{\prime \prime }} < u_{C}^{legal} \,if\,F_{C}^{C,S} < F_{C}^{J,S} ,\,and $$

$$ \mathop {\max }\limits_{{F_{C} > F_{C}^{J,S} }} u_{C}^{\prime \prime \prime } < u_{C}^{legal} \,\,if\,\,F_{C}^{C,S} > F_{C}^{J,S} . $$

Analogous statements can be made for the case where $F_{C}^{J} > F_{C}^{S}$, as well as for Japan and South Korea seeking improvements on their legally optimal levels of quotas. This completes the proof.

Appendix B

Comments on Parameter Selection

The primarily purpose of this paper was to present an explanation for observed behavior in the ECS, backed by analytical results that apply to any congested maritime environment. While detailed parameter estimation of specific fisheries was not conducted, parameters were chosen to be sensible, as described in this appendix.

5.1 Operating and Opportunity Costs, ${c}_{k}$

Operating costs will vary wildly by vessel type, but this paper assumed a value of 50,000 USD, informed by Cabral et al. (2018) which includes cost estimates of vessels of approximately 1000 gross tons (GT). Assuming 10 fishermen per vessel, an assumption on annual salaries provides an estimate of total operating plus opportunity costs. Salaries for fishermen by country were taken from https://www.salaryexpert.com/salary/browse/countries/fisherman-deep-sea. Chinese annual salaries were 14,700 USD, South Korean 34,752 USD, and Japanese 57,624 USD. Final figures were rounded off while maintaining orders of magnitudes.

5.2 Price of Fish, ${p}_{ij}$

As with fishing costs, prices vary significantly (this time by species). Tai et al. (2017) estimated the average price of fish used for direct human consumption to be 1750 USD per ton in 2010. Considering price increases from inflation and the general depletion of fish stocks, and the variation in high-quality and low-quality species, the parameter values of 3000 USD and 1500 USD per ton used in this paper are reasonable.

5.3 Biological Parameters ${Z}_{ij}$, ${q}_{ij}$, and ${r}_{ij}$

These parameters again vary significantly, based on the specific fishery being modeled. This paper didn’t model specific fisheries (e.g. tuna, herring, etc.), so did not have much to base parameter values on. They were chosen to be relatively aligned to those found in Chen and Andrew (1998). To generalize results, multiple examples were presented with differing values of ${Z}_{ij}$. Examples for alternative ${r}_{ij}$ values weren’t presented because the results would have been similar to shocking ${Z}_{ij}$; a slower natural growth rate has a similar affect to a smaller carrying capacity in the Gordon-Schaefer model. Alternative values of ${q}_{ij}$ weren’t presented because the fundamental insights would continue to hold. A higher ${q}_{ij}$ implies fewer fishermen are needed to reap the same harvest; however, China would still possess the same asymmetric cost advantages over South Korea and Japan, and would use this to issue excessive quotas which encroach on foreign fisheries. In turn, South Korea and Japan also issue excessive quotas rather than allowing China to extract profits unimpeded, as before.

5.4 Maritime Law Enforcement Parameters ${P}_{k}$, ${\beta }_{k}$, ${\beta }_{m}$, ${a}_{1}$, and ${a}_{2}$

The choices of patrols per 100,000 square kilometers of EEZ, ${P}_{k}$, were informed by (Petrossian 2015) and Petrossian (2019). Only Japan is listed explicitly with a value of ${P}_{J}=20$, which is close to the average value of 20.28. It was assumed South Korea would be slightly above average (hence ${P}_{S}=30$ was used in Example 1), and that China would be significantly above (${P}_{C}=50$) based on its well-document patrol investments in recent years (Erickson 2018). When patrols were shocked to higher values in Fig. 2.b and Table 5, the maximum value found in Petrossian of 100 (Taiwan) was kept in mind.

${\beta }_{k}$ and ${\beta }_{m}$ were also informed by Petrossian (2015), where a linear regression model was used to show each patrols and MCS as significantly explainers of illegal fishing. While linear regression coefficients don’t translate directly to the model used in this paper, ${\beta }_{k}$ values were chosen so that costs imposed bprols amounted to roughly 10–20% of operating and opportunity costs. MCS was shown to be a more decisive deterrent in Petrossian (2015), and thus ${\beta }_{m}$ was chosen such that no Chinese fishermen would enter Sth Korean waters illegally in Example 1 when $=100$ and $x_{S} = 1.1667$ (the legal optimum in Example 1). That is, the following was solved: $p_{S} q_{S} x_{S} = c_{C} + \beta_{C} P_{S} + 100\beta_{m} \to \beta_{m} = 6 \times 10^{ - 6}$. (after rounding to the nearest significant digit).

Lastly, the shape parameter for the cost of MCS, $a_{2}$., was arbitrarily set to 0.5. The parameter $a_{1}$. was then selected so that the cost of $m_{S} = 100$. was similar to the cost spent by Norway (Mangin et al. 2018), whom the FAO has rated highly for quality of MCS and which has similar sized fisheries to those used in this paper (Pitcher, Kalikoski, and Pramod 2006). This led to $a_{1} 100^{{a^{2} }} = .034 \to a_{1} = 3.5 \times 10^{ - 3}$.