Differential Games in a Stochastic Fishery
Jon M. Conrad †
Dyson School of Applied Economics and Management
Cornell University
Ithaca, New York 14853
Email: jmc16@cornell.edu
Abstract
Let X = X(t) denote the biomass of a fish stock at instant t. Suppose that the
stock is subject to harvesting by i = 1, 2, . . . , n countries, each having adopted a
feedback policy taking the form Yi = Yi (t) = φi (X). The
Pfish stock is an Itô
variable, evolving according to dX = [rX(1 − X/K) − i φi (X)]dt + σXdz. This
stochastic process assumes that the expected change in the fish stock is logistic
net growth
√ less harvest, but with a standard deviation rate of σX, where
dz = (t) dt is the increment of a Wiener Process with (t) ∼ N (0, 1). An
n-country differential game is solved for the subgame perfect harvest policies. The
Kolmogorov forward equation can be solved for the stationary density for X(t).
There are analytic expressions for the expected value of this distribution and for
expected harvest. The parameters r, K, and σ are fitted to the Norwegian
spring-spawning herring (NSSH), a straddling stock that migrates from Norway’s
Exclusive Economic Zone (EEZ) to summer feeding grounds in Iceland’s EEZ.
During migration, the adult population passes through the EEZs of the Faroe
Islands, countries in the European Union (EU), and through international waters.
An Index of Impaired Welfare (IIW) is defined. It measures the expected present
value from a stochastic fishery when it is exploited as an n-country differential
game relative to the expected present value that would be obtained by a sole
owner.
The author gratefully acknowledges the support of the National Science Foundation through
award 0832782.
†
Differential Games in a Stochastic Fishery
1
Introduction and Overview
Differential games have three important features. First, there is at least one
state variable whose evolution is influenced by the actions of two or more agents
(players). Second, the common state variables affect the economic welfare of all
agents. Third, each agent is aware of the other agents and optimizes assuming that
all other agents will optimize as well.
Differential games have played an important role in the sub-fields of industrial
organization and resource economics. They have been used to study the dynamic
behavior of oligopolists when faced with a common but partially-adjusting price;
as in Fershtman and Kamien (1987, 1990). They have been used to analyze the
race to innovate via investment in research and development; as in Reinganum
(1981, 1982). In resource economics, differential games have been used to analyze
oligopolists’ rates of extraction from a nonrenewable resource; as in Dasgupta and
Heal (1979), Lewis and Schmalensee (1980), Loury (1986) and Karp (1992). The
case where agents harvest a common-property, renewable resource has been analyzed by Levhari and Mirman (1980) and Clemhout and Wan (1985). Kamien and
Schwartz (1991, Section 23) provide a nice introduction to differential games. Their
third example deals with two fishermen harvesting the same, deterministicallyevolving, stock and is based on Plourde and Yeung (1989).
1
Reinganum and Stokey (1985) note that deterministic differential games might
be solved for either open loop (optimal time-path) strategies or closed loop (feedback) strategies. Feedback strategies give the ith player’s optimal action as a function of the evolving state variables and do not require the potentially unrealistic
time-commitment implied by open-loop strategies. In stochastic differential games,
where the evolution of the common state variables are subject to stochastic shocks,
only optimal feedback strategies make sense, and they must be derived (if possible)
using stochastic dynamic programming (SDP).
In a now classic paper, Pindyck (1984) considers a continuous-time stochastic
fishery, where the fish stock is an Itô variable. The fish stock evolves according to
the stochastic differential equation dX = [F (X) − Y (X)]dt + σXdz, where F (X)
is a net growth function, Y (X) is an unknown optimal feedback harvest policy,
√
σX is the standard deviation rate, and dz = (t) dt is the increment of a Wiener
process, with (t) ∼ N (0, 1). Pindyck provides three interesting examples where
he identifies the optimal feedback harvest policy for a sole owner. In all three examples the optimal policies are constant proportion policies, where the optimal
harvest at instant t takes the form Y ∗ = φX, where φ > 0 is a constant depending
on the discount rate, parameters in the objective functional, and parameters in
the stochastic differential equation for the evolving fish stock. For a given stock
size, an increase in the standard deviation parameter, σ, reduces optimal harvest
in the first example, leaves optimal harvest unchanged in the second example, and
increases optimal harvest in the third example.
2
In this paper we consider a differential game in a stochastic fishery. The model
is relevant for highly migratory stocks that pass through the extended economic
zones (EEZs) of two or more countries or through international waters where no
single country has management authority. Previous applications of differential
games to renewable resources have assumed deterministic stock dynamics, with
time typically partitioned into discrete intervals. There appears to be no analysis
of a continuous-time, differential game, where the state variable is a stochastically
evolving renewable resource.
The model in this paper employs a functional form for country-specific utility that allows analytic solutions for value functions and optimal feedback harvest
policies. These analytic solutions permit transparent comparative statics. As in
example one from Pindyck (1984), we observe that an increase in volatility, σ,
will cause each county to reduce its harvest for a given stock size. The model is
calibrated to the Norwegian spring-spawning herring (NSSH), a highly migratory
stock which spawns off the coast of western Norway before migrating to summer
feeding grounds in Iceland’s Extended Economic Zone (EEZ). There are currently
four countries (Norway, Iceland, the Faroe Islands, and Russia) and members of the
European Union that harvest the NSSH. When migrating to Iceland, the herring
stock passes though an area of international waters referred to as the “Ocean Loop,”
where the herring may be subject to harvest by distant-water fleets. This fishery
has undergone intense study, including Touzeau et al. (1998), Patterson (1998),
Bjørndal et al. (1998), Kitti et al. (1999), Bjørndal and Gordon (2000), Toresen
3
and Østvedt (2000), Arnason et al. (2000) and Bjørndal et al. (2004). The preferred biological model is an age-structured model with up to 17 age classes. Such
a detailed biological model will typically preclude optimization and analysts often
rely on simulation to determine the consequences of different harvest policies and
distributions of allowable catch. The simple biomass model in this paper sacrifices
biological reality for mathematical tractability and analytical transparency. However, when our simple biomass model is applied to the NSSH, we obtain numerical
results that are not inconsistent with the differential game being played prior to
the collapse of the fishery in 1969.
In the next section we pose and solve the sole owner (n = 1) problem for the
optimal management of our stochastic fishery. In Section 3 we solve the n-country
differential game. In Section 4 we fit the stochastic biomass model to data for
the NSSH fishery from 1950 - 2011 and compute the solutions for the sole owner
and the differential game between five countries with a common discount rate. An
index of impaired welfare is constructed to measure the reduction in welfare from
the five-country, differential game. Section 5 concludes.
2
The Sole Owner of a Stochastic Fishery
This section stands on the shoulders Pindyck (1984). We assume that the fish
stock is an Itô variable where the expected change in biomass over the time increment dt is determined by logistic growth less harvest, rX(1 − X/K) − Y , where
X = X(t) is the biomass of the fish stock at instant t, r > 0 is the intrinsic growth
4
rate, K > 0 is the environmental carrying capacity, and Y = Y (t) is the biomass
harvested at instant t. The expected change in the fish stock may be greater or
less than the actual change in the fish stock owing to the term σXdz, where σX is
√
a standard deviation rate and dz = (t) dt, with (t) ∼ N (0, 1), is the increment
of a Wiener process. The stochastic proceess describing the evolution of harvested
biomass becomes dX = [rX(1 − X/K) − Y ]dt + σXdz.
The sole owner derives utility only from harvest according to the function
U (Y ) = α − β/Y , where α and β are positive parameters. This utility function is strictly concave and bounded from above with U (Y ) > 0 when Y > β/α,
U (Y ) = 0 when Y = β/α, U (Y ) < 0 when 0 < Y < β/α, U (Y ) → −∞ as Y → 0
and U (Y ) → α as Y → ∞. The sole owner seeks to
Z
Maximize E
Y
∞
(α − β/Y )e−δt dt
0
Subject to dX = [rX(1 − X/K) − Y ]dt + σXdz, X(0) > 0.
where E{•} is the expected value of discounted utility and δ > 0 is the sole owner’s
rate of discount. This optimization problem may be solved using dynamic programming. The Hamiltion-Jacobi-Bellman (H-J-B) Equation, post Itô’s Lemma,
requires
5
δV (X) = max{α − β/Y + V 0 (X)[rX(1 − X/K) − Y ] + (σ 2 /2)X 2 V 00 (X)}
Y
(1)
where V 0 (X) and V 00 (X) are the first and second derivatives of the unknown value
function, V (X), which gives the maximized expected present value of utility when
a fish stock of size X = X(t) is optimally harvested for the rest of time. The
solution to Equation (1) entails taking a partial derivative of the right-hand-side
(RHS) with respect to Y , setting it to zero, solving for optimal harvest, Y ∗ , as a
function of V 0 (X), and substituting Y ∗ back into the RHS of Equation (1) to obtain the “optimized” H-J-B Equation. This equation is a second-order differential
equation in the unknown value function, V (X). Performing these steps leads to
p
∗
Y = β/V 0 (X) and
p
δV (X) = α − 2 βV 0 (X) + V 0 (X)rX(1 − X/K) + (σ 2 /2)X 2 V 00 (X)
(2)
It can be shown that the value function V (X) = A − B/X satisfies Equation (2)
provided that A = (αK − rB)/(δK) and B = 4β/[(δ + r − σ 2 )2 ]. The expression for
p
B implies that Y ∗ = β/V 0 (X) = (δ + r − σ 2 )X/2. This is the optimal feedback
harvest policy. One might expect that 1 > (δ + r − σ 2 )/2 > 0 and that it will
be optimal to harvest a fixed proportion (fraction) of the stock as it stochastically
6
evolves through time. Note that for any given X > 0 an increase in δ or r will
increase the fixed proportion, while an increase in σ 2 will reduce the fixed proportion.
Pindyck (1984) and Dixit and Pindyck (1994) show how the Kolmogorov forward equation can be solved to determine the stationary distribution for X(t),
when the fish stock is harvested according to a stationary optimal feedback policy,
Y ∗ = φ(X). In the current model, if r > δ, the nondegenerate stationary distribution when our sole owner faithfully follows Y ∗ = (δ + r − σ 2 )X/2 is a gamma
distribution taking the form
2
2
2
[2r/(σ 2 K)](r−δ)/σ X (r−δ−σ )/σ e−2rX/(σ
ψ1 (X) =
Γ((r − δ)/σ 2 )
2
K)
(3)
This stationary distribution has an expected value E{X} = K(r − δ)/(2r) and the
induced distribution for harvest has an expected value given by E{Y } = K(r + δ −
σ 2 )(r − δ)/(4r). These results establish an analytic benchmark which will prove
useful when analyzing the n-country differential game.
3
The n-Country Differential Game
In the n-country differential game, each participating country has access to
the fish stock, either in their extended economic zone (EEZ), or in international
waters. We will assume that each country presumes that other countries harvesting
7
the fish stock have adopted an unknown optimal constant-proportion policy. Then,
with Yi∗ = φi X for i = 1, 2, . . . , n, the stochastic differential equation describing the
P
evolution of the fish stock takes the form dX = [rX(1−X/K)− i φi X]dt+σXdz.
The H-J-B Equation for the ith country, post Itô’s Lemma, takes the form
0
δi Vi (X) = max{αi −βi /Yi +Vi (X)[rX(1−X/K)−Yi −
n
X
Yi
00
φj X]+(σ 2 /2)X 2 Vi (X)}
j6=i
(4)
Each country follows the same steps as the sole owner did in the previous section
to obtain their optimized H-J-B Equation. Taking a partial derivative of the RHS
p
0
of Equation (4) with respect to Yi and setting it to zero implies Yi∗ = βi /Vi (X).
Substituting this expression back into the RHS of Equation (4) we obtain the optimized H-J-B equation
n
q
X
0
00
0
δi Vi (X) = αi −2 βi Vi (X)+Vi (X)[rX(1−X/K)−
φj X]+(σ 2 /2)X 2 Vi (X) (5)
j6=i
The same form of the value function that worked for the sole owner works for the
ith country; that is, Vi (X) = Ai −Bi /X, provided that Ai = (αi K −rBi )/(δi K) and
p
Pn
0
0
2 2
2
∗
Bi = 4βi /[(δi +r− j6=i φj −σ ) ]. With Vi (X) = Bi /X and with Yi = βi /Vi (X)
some careful algebra will reveal
8
Yi∗
= (δi + r −
n
X
φj − σ 2 )X/2
(6)
j6=i
Equation (6) is actually a system of n simultaneous equations in the unknown
φi , i = 1, 2, . . . , n. If one solves Equations (6) when n = 2 one will obtain
φ1 = (2δ1 − δ2 + r − σ 2 )/3 and φ2 = (2δ2 − δ1 + r − σ 2 )/3.
If one solves
Equations (6) when n = 3 one will obtain φ1 = (3δ1 − δ2 − δ3 + r − σ 2 )/4,
φ2 = (3δ2 − δ1 − δ3 + r − σ 2 )/4, and φ3 = (3δ3 − δ1 − δ2 + r − σ 2 )/4. From
these results we infer for n ≥ 1 that
φi = (nδi −
n
X
δj + r − σ 2 )/(n + 1)
(7)
j6=i
To determine its own optimal proportional feedback policy, Country i must know its
discount rate, the discount rates of all other countries harvesting the fish stock, the
P
intrinsic growth rate, r, and the volatility rate, σ. Define Φn = ni=1 φi . Then the
stochastic differential equation defining the evolution of the fish stock may be written as dX = r0 X(1 − X/K 0 )dt + σXdz, where r0 = (r − Φn ) and K 0 = (r − Φn )K/r.
The stationary distribution for the fish stock when harvested by n countries is again
a gamma distribution but with parameters depending on r0 , K 0 , and σ and taking
the form
9
0
2
0
2
0
[2r0 /(σ 2 K 0 )](2r /σ −1) X (2r /σ −2) e−2r X/(σ
ψn (X) =
Γ(2r0 /σ 2 − 1)
2
K 0)
(8)
For the gamma distribution in Equation (8) to be nondegenerative σ 2 < 2r0 =
2(r − Φn ). The special case, when all n countries have the same discount rate
(δi = δ > 0), results in
Φn =
n
X
i=1
n(δ + r − σ 2 )
φi =
(n + 1)
(9)
For the multi-country case, when δi = δ, the expected value of the stationary
distribution for X = X(t) is given by
K[r − σ 2 /2 − n(δ − σ 2 /2)]
E(X) = K (1 − σ /(2r )) =
(n + 1)r
0
2
0
(10)
and the expected value for harvest is
2
E(Y ) = K[(1 − σ /(2r))Φn −
Φ2n /r]
[n(δ + r − σ 2 )K][2(r − nδ) + σ 2 (n − 1)]
=
[2r(n + 1)2 ]
(11)
We now examine the implications of the model when fitted to data for the Norwegian spring-spawning herring, 1950-2011.
10
4
The Norwegian Spring-Spawning Herring
The Norwegian spring-spawning herring (Clupea harengus) has traditionally
been a highly migratory species, with adults spawning off the south-central coast
of western Norway from early February through March. After spawning, the migratory pattern in the 1950s and early1960s saw adult year classes migrate west
by northwest into the Norwegian Sea. During this period, prior to the extension of
territorial waters that took place in the late 1970s, the herring stock passed into a
large area of international waters (high seas) where it was exploited under conditions approximating open access. During this period the technology used in finding
and harvesting herring increased significantly. Herring are a schooling species. As
the total stock declined, remaining adults would re-school and thus facilitate their
location and continued exploitation. Schooling species do not exhibit a significant
“marginal stock effect” (see Clark et al. 2010), and harvest costs do not significantly increase with a decline in total stock abundance. Excessive harvest under
open access conditions resulted a stock collapse in 1969 and the imposition of a
moratorium on fishing in 1970.
An interesting behavioral consequence of stock collapse was the ceasation of
migration. From 1970 until the early 1990s both adults and juveniles remained
in Norwegian waters year round. It was not until the early-1990s when the stock
had been bolstered by several strong year classes that adults resumed migration
after spawning. However, by the mid-1990s bilateral treaties on extended economic
11
zones (EEZs) had significantly altered the ocean-scape through which the NSSH
now travelled. Now after spawning, adult herring would pass through the EEZs
of the Faroe Islands, members of the European Community (EU), and though a
considerably smaller patch of international waters, called the “Ocean Loop,” before
reaching Iceland’s EEZ. Russia has had a stake in the NSSH fishery since the mid1960s when a northern stock developed that spawned south of the Lofoten Islands
in Norway, but which migrated into the Barents Sea and the Russian EEZ. So,
by the late 1990s, there were four countries (Norway, Iceland, the Faroe Islands,
Russia) and the European Union, for a total of five players, who could claim a
stake in the fishery and threaten strategic overfishing if not allowed a “fair” share
of total allowable catch.
In 1996, Norway, Russia, Iceland, and the Faroe Islands reached an agreement
to share a total allowable catch of 1,267,000 metric tons of herring (Bjørndal et
al. 2004). The EU did not participate in that agreement and the additional harvest by EU-member countries in that year was estimated at 180,419 metric tons
(International Council for Exploration of the Sea (ICES) Advice 2011, Book 9,
p.60). In 1999, the EU, the Faroe Islands, Iceland, Norway, and Russia agreed to
a management plan with the following elements. (1) Every effort was to be made
to maintain spawning stock biomass (SSB) at a level greater than 2,500,000 metric
tons. (2) For 2001 and years subsequent, the Parties agree to restrict their fishing
on the basis of a total allowable catch (a TAC) consistent with a fishing mortality
rate of less than 0.125 for the appropriate year classes as defined by ICES. (3)
12
Should the estimate of SSB fall below 5,000,000 metric tons the fishing mortality
rate and the TAC will be adjusted downward to ensure a safe and rapid recovery to
a level above 5,000,000 metric tons. (4) The Parties shall, as appropriate, review
and revise the above management measures on the basis of new advice provided
by ICES.
In Table 1 we report the ICES estimates of spawning stock biomass, SSBt ,
for the years 1950-2011, and total harvest, Yt , for the years 1950-2010. We use
these data to determine the best-fit values of r, K, and σ that will approximate the stochastic evolution of spawning stock biomass as reported in Table
1. When dt = 1, the stochastic, first-order, difference equation corresponding to
dX = [rX(1 − X/K) − Y ]dt + σXdz may be written as
Xt+1 − Xt = rXt (1 − Xt /K) − Yt + σXt t+1
(12)
where t+1 ∼ N (0, 1). We generate 61 standard normal variates, corresponding
to the years 1951-2011. Setting X1950 = 13, 984, 000 metric tons, which is the
estimated spawning stock biomass from Table 1, we simulate Xt+1 = Xt + rXt (1 −
Xt /K) − Yt + σXt t+1 forward to the year 2011. This simulation requires an initial
guess for r, K, and σ. We used estimates of r = 0.47 and K = 9, 418, 600 metric
tons from Arnason et al. (2000, Table 1) along with a guess that σ = 0.4. We then
changed the initial guess for r, K, and σ seeking to minimize the sum of absolute
deviations (SAD) of SSBt from our simulated Xt for t = 1951, 1952, . . . , 2011. This
13
fitting exercise corresponds to the nonlinear optimization problem
Minimize SAD =
r, K, σ
2011
X
|SSBt − Xt |
1951
Subject to Xt+1 = Xt + rXt (1 − Xt /K) − Yt + σXt t+1
t = 1950, 1951, . . . , 2010,
X1950 = 13, 984, 000 metric tons, t+1 given,
and SSBt and Yt f rom T able 1.
This minimization problem yielded the best-fit values r = 0.45, K = 9, 930, 000,
and σ = 0.257. The resulting plot of SSBt and Xt for t = 1950, 1951, . . . , 2011 is
shown in Figure 1.
With the best-fit values for r, K, and σ we can plot the stationary gamma
distribution for X and calculate the expected value for stock and harvest for
the sole owner (n = 1) and for five countries, all with the same discount rate
(δi = δ = 0.05, i = 1, 2, . . . , 5). When r = 0.45, K = 9, 930, 000, σ 2 = 0.066049
and δ = 0.05 the stationary proportional harvest policy, expected spawning stock
biomass, and expected harvest for n = 1 and n = 5 are reported in Table 2. The
gamma distributions for n = 1 and for n = 5 are both nondegenerate and are
plotted in Figure 2.
To establish a measure of reduced welfare that may result under an n-country
differential game we define the Index of Impaired Welfare (IIW) as
14
P
n X=b
X=a Vi (X)ψn (X)
IIW = PX=b
X=a V (X)ψ1 (X)
(13)
where Vi (X) is the value function of the ith representative country, δi = δ, i =
1, 2, . . . , n, ψn (X) is the stationary gamma distribution for X(t) under the ncountry differential game, V (X) is the value function of the sole owner, and ψ1 (X)
is the stationary gamma distribution for X(t) when biomass is optimally harvested
by a sole owner.
Unfortunately, to compute the IIW you need parameter values for α and β
from the utility function. They are needed to get numerical values for the coefficients of the value functions, Vi (X) and V (X). You also need values for a and b
that define the appropriate support for X when computing the expected value of
the n-country differential game (in the numerator of the IIW) and the expected
value of the sole owner (in the denominator of the IIW). Based on the gamma
distributions plotted in Figure 2, we set a = 100, 000 mt and b = 16, 000, 000 mt.
If marginal utility is equal to price (as we might expect in a competitive market
for herring), then U 0 (Y ) = β/Y 2 = p. In the Food and Agricultural Organization
(FAO) Yearbook (2011), which provides fishery statistics up to the year 2009, the
average annual price for Atlantic herring was p = $523/mt in an export market
where Y = 287, 629 mt. This implies that β = pY 2 = 4.33E + 13.
There is no compelling way to calibrate the value for α. We would like
U (Y ) > 0 over the relevant range for recent harvests. If α = 1.15E + 9 then
15
U (Y ) > 0 for Y > β/α = 37, 652 mt. The aggregate harvest of NSSH has exceeded
Y = 37, 652 mt for every year since 1983. See Table 1. These parameter values result in IIW = 0.38, implying that the expected present value under a five-country
differential game is 38% of the expected present value of a sole owner. See Table 2.
5
Conclusions and Caveats
This paper developed a model of a differential game for a continuous-time
stochastic fishery being exploited by n ≥ 2 countries. The structure of the model
allowed analytical expressions for each country’s value function, their optimal (subgame perfect) harvest policies, the stationary distribution of the harvested fish
stock, and the expected value for stock and harvest. The feedback policy of the
sole owner (n = 1) along with the resulting stationary distribution for biomass,
expected stock size, and harvest, provided a benchmark to evaluate the extent of
overfishing when n ≥ 2. An index, the IIW, was developed to measure the reduction in expected present value under an n-country differential game relative to the
expected present value that might be obtained by a sole owner.
The model was calibrated for the best-fit parameters, r, K, and σ, for the Norwegian spring-spawning herring based on harvest and spawning stock estimates for
the years 1950 through 2011. For a common discount rate, δi = δ = 0.05, a sole
owner would adopt an optimal proportional policy where Y ∗ = 0.217X, resulting
in a non-degenerate stationary distribution with expected value E(X) = 4, 413, 333
metric tons and expected harvest of E(Y ) = 957, 585 metric tons. If five countries
16
were harvesting the stock as part of a differential game, the stationary distribution, while non-degenerate, would have an expected value of E(X) = 1, 221, 383
metric tons and an expected harvest of E(Y ) = 441, 684 metric tons. The comP
bined feedback harvest policies resulted in Φ5 = 5i=1 φi = 0.362 as the proportion
of biomass harvested at each instant. While our example is highly stylized, the
stationary distributions for spawning stock biomass and the numerical values for
E(X) and E(Y ) when n = 5 are not inconsistent with estimated biomass just prior
to stock collapse in 1969. In this period, prior to the extension of EEZs, it was
likely that individuals harvesting the NSSH were forced, by open access, to behave
as if they were discounting the future at an infinite rate (δi → ∞) as discussed in
Clark (2010, p.28).
It is also interesting to note that the 1999 agreement between Norway, Iceland,
Russia, the Faroe Islands, and the European Union seeks to maintain spawning
stock biomass above 5,000,000 metric tons. Our estimate of K, similar to estimates
in previous studies, is almost 10,000,000 metric tons. In a deterministic model, this
would imply that the spawning stock which would support maximum sustainable
yield would be K/2 = 5, 000, 000 metric tons. With δi = δ > 0, the optimal stock
for the sole owner in a deterministic world is X ∗ = K(r − δ)/(2r) < K/2. It would
appear that managers are trying to maintain spawning stock biomass in the vicinity of K/2. They know that growth is stochastic (the spawning stock in reality
is an Itô variable) and they are reducing the constant proportion harvest policy
below Φ1 = 0.217 to reflect growth and/or other sources of uncertainty. Thus, our
17
model, when n = 5 provides insight into the forces at work prior to stock collapse
in 1969 and, when n = 1, into the structure of the 1999 agreement that attempts
to manage the NSSH in a more rational, welfare-enhancing, way.
While broadly consistent with the history of the NSSH fishery, our model is
overly simple in terms of the biology, game-theoretic structure, and environmental
forces that influence growth and abundance. The preferred biological model for the
NSSH would be an age-structured model, with possibly 17 cohorts. It is not likely
that a stochastic model with 17 cohorts would permit the analytical results and
transparency obtained in the current model. The four countries and the EU that
are currently harvesting the NSSH are not equal participants in the NSSH differential game. Norway may possess the strongest bargaining position, since strategic
overfishing might cause the stock to cease migration, as it did from 1970 until the
early 1990s. Our model also assumed that the fish population was an Itô variable,
evolving according to a random walk with expected drift given by logistic growth
less harvest. Toresen and Østvedt (2000) believe that the long-term fluctuation
in the survival of recruits is positively related to the mean temperature of water
flowing into the north-east Atlantic, where the sub-adult year classes reside before
they begin the migratory pattern exhibited by adults. If the thermal distribution of
inflows is nonstationary, due to climate change, it would also invalidate our simple
model of stochastic growth.
18
6
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Table 1:
Spawning Stock Biomass, SSBt , 1950 - 2011, and Total Harest Yt , 1950 - 2010.
Y ear
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
SSBt
13,984,000
12,440,000
11,482,000
10,613,000
9,445,000
10,223,000
11,740,000
10,129,000
9,280,000
7,350,000
5,817,000
4,230,000
3,465,000
2,635,000
2,795,000
3,067,000
2,595,000
1,145,000
219,000
78,000
31,000
8,000
2,000
74,000
85,000
91,000
146,000
384,000
355,000
386,00
469,000
Yt
723,500
994,200
919,2000
833,700
1,306,400
1,217,000
1,460,600
1,148,300
785,000
883,100
821,100
497,900
551,200
670,800
1,117,900
1,325,600
1,723,600
1,131,600
273,100
24,100
20,900
6,900
13,200
7,000
7,600
13,700
10,400
22,700
19,800
12,900
18,600
Y ear
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
SSBt
503,000
501,000
571,000
594,000
492,000
414,000
1,031,000
2,076,000
3,428,000
4,135,000
4,072,000
4,173,000
4,089,000
4,212,000
4,135,000
4,600,000
5,840,000
6,562,000
6,727,000
5,752,000
4,705,000
4,153.000
5,143,000
6,501,000
6,622,000
7,074,000
8,077,000
8,798,000
9,927,000
9,176,000
7,900,000
Yt
13,700
16,700
23,100
53,500
169,900
225,300
127,300
135,301
103,830
86,411
84,683
104,448
232,457
479,228
905,501
1,220,283
1,426,507
1,223,131
1,235,433
1,207,201
766,136
807,795
789,510
794,066
1,003,243
968,958
1,266,993
1,545,656
1,687,373
1,457,014
- - - -
Sources : ICES (1999, 2011)
Table 2:
Φn , E(X), and E(Y ), for n = 1, and n = 5 and the IIW , a = 100, 000, b = 16, 000, 000
Φn
E(X)
E(Y )
PX=b
n X=a Vi (X)ψn (X)
IIW =
n=1
0.2169755
4,413,333
957,585
219,164
PX=b
n X=a
Vi (X)ψn (X)
PX=b
V
X=a (X)ψ1 (X)
24
=
n=5
0.361625833
1,221,383
441,684
83,066
83,066
219,164
= 0.38
Figure 1:
A Plot of SSBt and Simulated Xt when r = 0.45, K = 9, 930, 000, and σ = 0.257
16,000,000
14,000,000
SSBt
12,000,000
10,000,000
8,000,000
Xt
6,000,000
4,000,000
2,000,000
1940
1950
1960
1970
1980
(2,000,000)
25
1990
2000
2010
2020
The Stationary Gamma Distributions for X when Harvested by the Sole Owner, n = 1,
and by n = 5 Countries with a Common Discount Rate.
Figure 2:
0.0000007 n=5
0.0000006 0.0000005 0.0000004 0.0000003 n=1
0.0000002 0.0000001 0 0 2 4 6 -­‐1E-­‐07 8 10 12 Millions 26
X