Cooperative and non-cooperative harvesting in a stochastic sequential fishery

Journal of Environmental Economics and Management 45 (2003) 454–473

Cooperative and non-cooperative harvesting in a stochasticsequential fishery$

Marita Laukkanen

Fondazione Eni Enrico Mattei, Palazzo Querini Stampalia, Campo Santa Maria Formosa, Castello 5252, 30122 Venice,

Italy

Received 15 November 2000; revised 26 December 2001

Abstract

We examine cooperative harvesting in a sequential fishery with stochastic shocks in recruitment. Twofleets harvest in a stochastic interception fishery. We analyze cooperative management as a non-cooperativegame, where deviations from cooperative harvesting are deterred by the threat of harvesting at non-cooperative levels for a fixed number of periods whenever the initial stock falls below a trigger level. Weillustrate the sequential harvesting game with an application to the Northern Baltic salmon fishery.Cooperative harvesting yields participants substantial gains in terms of expected payoffs. The greatest gainsaccrue to the fleet harvesting the spawning stock, the actions of which are not observed by the competitor.An explanation for the prevalence of fish wars is provided: even if a cooperative agreement is reached,shocks in recruitment trigger phases of non-cooperative harvesting. Further, the cooperative solution canonly be maintained when stock uncertainty is not too prevalent.r 2003 Elsevier Science (USA). All rights reserved.

Keywords: Fisheries management; Shared resources; Sequential fishery; Non-cooperative games; Stochastic recruit-

ment

1. Introduction

Migratory fish stocks breed in one area, move to another to grow, and return to their breedingarea to spawn. During their migration the stocks are harvested in a sequence of fisheries. Eachfishery affects the stock available to the next fishery in the gauntlet and therefore its economic

$Research for this paper was carried out at the Department of Agricultural and Resource Economics, University of

California at Berkeley. The study has benefited enormously from ongoing discussions with Larry Karp. Further thanks

for helpful comments go to Jeffrey LaFrance and anonymous referees. Financial support from the Yrjo Jahnsson

Foundation and the Finnish Cultural Foundation is gratefully acknowledged.

E-mail address: [email protected].

0095-0696/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0095-0696(02)00020-7

performance. For example, salmon spawn in rivers, feed in the open sea, and return to their homerivers to spawn. Southern bluefin tuna spend most of their juvenile phase along the coast ofAustralia, then migrate to the high seas between South Africa and New Zealand. Pacific Halibuttravel along the coast of Canada and the United States. If a single authority controls each fishery,two basic management options arise: each fishery manager can either maximize the return fromhis own fishery, or cooperate with other fleets in order to maximize the aggregate return, andbargain for a share of that return. In a cooperative solution, agents negotiate over the amount offish they leave behind. When recruitment is deterministic, they can monitor adherence to theagreement through the stock available at the outset of harvest.Hannesson [9] studies cooperative management in a sequential fishery, and examines cooperative

harvesting as a self-enforcing equilibrium supported by the threat of harvesting non-cooperativelyforever if deviations are detected. Transboundary management of sequential fisheries has also beenaddressed by McKelvey [19] who considers side payments in a sequential fishery.1 Otherwise, economicliterature has focused on joint harvest in a single fishery simultaneously exploited by multiple agents.The inefficiency of non-cooperative management of shared resources has been shown by Clark [4],Levhari and Mirman [17], and others. Munro [21] proposed binding agreements to secure the success ofcooperative agreements. Hamalainen et al. [8] and Kaitala and Pohjola [14] consider threat strategies asa means to sustain cooperation. Hannesson [10] studies how critical the number of agents sharing a fishstock is for realizing the cooperative solution supported by threat policies.Stochastic fluctuations in stock recruitment complicate the joint management problem. When

recruitment depends on both the spawning stock level and unobserved shocks, the agent who harveststhe initial stock cannot infer the amount of fish left behind by the other agent. Cooperative harvestingequilibria supported by the threat of non-cooperative harvesting ad infinitum break down. We study anagreement designed to overcome this problem and maintain coordinated harvest as a self-enforcingequilibrium in the case of stochastic recruitment. We incorporate stock uncertainty into Hannesson’s [9]setting using the stochastic population model developed by Reed [24,25]. We then apply the model ofnon-cooperative collusion analyzed by Green and Porter [7] to study joint management of a sequentialfishery. We consider two fleets, each controlled by a sole authority acting independently. Differing fromGreen and Porter [7], one agent’s action follows that of the other.The sustainability of a cooperative solution is an empirical question. We apply the model to

data for the northern Baltic salmon fishery. The numerical results indicate that by cooperatingboth parties could obtain substantial gains in expected payoffs. For most parameter values, theagreement is characterized by an arbitrarily small likelihood of reversion to a punishment phase.For a range of parameter values, the optimal length of the punishment phase is practically infinite.

2. The model2

Assume that a fish stock migrates between two areas. It breeds in Area 2, and migrates to Area1 to feed. Agent 1 harvests in the feeding area. At the outset of the season in year t he has access to

1Side payments as a solution to transboundary resource management conflicts were first suggested by Munro [21].2The bioeconomic model of a seasonal fishery we use has been discussed by Clark [1–3], Spence and Starrett [27],

Levhari, Michener and Mirman [16], Hannesson [9,10], and others. The stochastic population model follows Reed

[24,25].

M. Laukkanen / Journal of Environmental Economics and Management 45 (2003) 454–473 455

the initial stock Xt, which he harvests down to the escapement level St1: The escapement left

behind by Agent 1 migrates to Area 2, where Agent 2 harvests the stock down to the escapementSt2: The escapement left behind by Agent 2 spawns, producing recruitment of young fish that in

turn migrate to Area 1 to feed. The stock available to Agent 1 in year t þ 1 is given by

X tþ1 ¼ ytRðSt2Þ; ð1Þ

where RðS2Þ is the expected or average spawning stock–recruitment relation, and fytg is a

sequence of independent identically distributed random variables with unit mean. Each yt is a

shock to the recruitment that the agents cannot observe directly. The random multipliers yt aredistributed on some finite interval ½a; b�; where 0oao1oboN; with a common cumulativedistribution function F and continuous density f. The recruitment relation RðS2Þ is differentiableand strictly concave, with bR0ð0Þ41 and limS2-N aR0ðS2Þo1: In the absence of harvest, S2 equalsthe stock X, and the process X t would eventually enter and then stay within in the interval ½l; u�;where

u ¼ maxfbRðS2Þ : bRðS2ÞXS2; S2X0g

and

l ¼ maxfaRðS2Þ : aRðS2ÞXS2; S2X0g;

on which it would, in the limit, attain a stationary distribution (Fig. 1). Consider next harvest withconstant Area 1 and Area 2 escapements y and z. When ypaRðzÞ and zpy; the escapements arefeasible in any state of nature. The distribution of X will then be the same as that of yR(z).3 Weshall only be concerned with constant escapement policies z and y that satisfy zpypaRðzÞ:Let x denote the current size of the exploited stock at any moment in time, ci the constant

unit cost of fishing effort for agent i, and pi the constant price of catch. Assuming that theharvest follows the Schaefer production function, the marginal cost of harvest for each agent isgiven by ci=x: The profits in period t to Agent 1 from harvesting the stock from X t down to St

1 are

given by

pt1 ¼

Z X t

St1

p1 c1

x

� �dx ¼ p1ðX t St

1Þ c1ðln X t ln St1Þ: ð2Þ

The period t profits to Agent 2 from harvesting the spawning stock from St1 to St

2 are

pt2 ¼

Z St1

St2

p2 c2

x

� �dx ¼ p2ðSt

1 St2Þ c2ðln St

1 ln St2Þ: ð3Þ

The agents are risk neutral and maximize their expected discounted net revenue EP

N

t¼0 dtpt

i

� �;

where dt denotes the common discount factor d raised to the tth power.The actions available to each agent in period t are setting the escapement levels, St

1 for Agent 1

and St2 for Agent 2. Agent 1’s strategy st

1 : R2tþ1þ -Rþ defines Agent 1’s escapement as a function

3In a deterministic model the steady state recruitment would be a single point. In a stochastic model the steady state

will instead be represented by a stationary probability distribution. If the recruitment were observed over many years,

many different recruitment levels would occur. The distribution of the long-run frequency of these levels will follow

yR(z) (see [24]).

M. Laukkanen / Journal of Environmental Economics and Management 45 (2003) 454–473456

of past and present recruitments and Agent 1’s past escapements by st1 X 0; :::;X t;S0

1; :::;St11

� �¼ St

1:4 The choice of domain reflects the fact that Agent 1 does not observe escapements left

behind by Agent 2 but only perceives the initial stock level X t at the beginning of the fishing

season. Agent 2’s strategy st2 : R2tþ1

þ -Rþ defines Agent 2’s escapement level as a function of past

and present Agent 1 escapements and past Agent 2 escapements by st2ðS0

1; :::;St1;S

02; :::;S

t12 Þ ¼ St

2:Agent 2, unlike Agent 1, observes the competitor’s escapement St

1: A contingent strategy for agent

i is an infinite sequence si ¼ fs0i ; s1i ; :::g: A Nash equilibrium is a strategy profile ðs1�; s2�Þ thatsatisfies

Es1;s2�

XNt¼0

dtp1ðst1;X

tðst12� ÞÞ

" #pEs1�;s2�

XNt¼0

dtp1ðst1�;X

tðst12� ÞÞ

" #;

Es1�;s2

XNt¼0

dtp2ðst2;S

t1�Þ

" #pEs�

1;s�2

XNt¼0

dtp2ðst2�;St

1�Þ" #

; ð4Þ

for each agent i and all feasible strategies si:Reed [24] showed that a constant escapement policy maximizes the expected discounted net

revenue from a stochastic fishery in the case of the Schaefer production function. If it werepossible for Agent i to manage the resource as a sole owner, the first order condition for the soleowner optimal escapement level would be

pi ci=Si ¼ dR0ðSiÞ½pi ci=RðSiÞ�: ð5Þ

(aR

(S2)

S2)

bR

2S

X

l

l

u

u

Fig. 1.

4R+2t+1 denotes the 2t+1 dimensional positive orthant. The expression s1

t : R+2t+1-R+ denotes a mapping from the

2t+1 dimensional positive orthant to the one dimensional positive orthant.


Let S�i denote the solution to (5). Agent i0s sole owner optimal escapement So

i would be Soi ¼ S�

i if

X4S�i ; and So

i ¼ X otherwise. Assuming that S�i is always feasible in that ytRðS�

i ÞXS�i for all

ytA½a; b�; the stochastic and deterministic models then yield the same sole owner optimalescapement (see also [5]).

3. Non-cooperation in a stochastic sequential fishery

We next examine non-cooperative harvesting where each agent makes his harvest decisionwithout considering its effect on the other agent’s expected payoff. There are no negotiations orunderstandings between the agents. We confine our attention to interior solutions where bothagents participate in uncoordinated harvest in every state of nature. Each agent sets hisescapement to maximize the expected present value of his profits, taking as given the other fleet’sescapement which he can only infer from his knowledge of the other fleet’s objective function. Theoptimal escapements are constrained by the non-negativity and feasibility constraints 0pSt

1pX t

and 0pSt2pSt

1:The lowest profitable escapement levels are the zero marginal profit levels c1=p1 and c2=p2:

Agent 1 will harvest in period t only if his marginal net revenue p1 c1=X t at the outset of harvest

is positive. Agent 2 will only harvest if p2 c2=St1 is positive. We assume that ytRðc2=p2Þ4c1=p1

for all ytA½a; b�; and that c1=p14c2=p2: Both agents then harvest at any state of nature. Theexpected discounted payoffs in period t are

EV1ðX t;St1Þ ¼ p1ðX t St

1Þ c1ðln X t ln St1Þ þ dEV1 ytRðSt

2Þ;Stþ11

� ; ð6aÞ

V2ðSt1;S

t2Þ ¼ p2 St

1 St2

� c2 ln St

1 ln St2

� þ dV2ðStþ1

1 ;Stþ12 Þ:5 ð6bÞ

With X t4c1=p1 and St2 given it is optimal for Agent 1 to harvest to the zero marginal profit

level St1 ¼ c1=p1; which maximizes short-term profit. Similarly, with St

14c2=p2 and Stþ11 given,

Agent 2’s optimal escapement is St2 ¼ c2=p2:

6

The competitive harvesting game has a stationary equilibrium in the strategies

SN1 ¼ c1=p1;

SN2 ¼ c2=p2:

(

5Agent 2’s expected non-cooperative payoff and single-period profits depend only on the escapements, which are

deterministic. We therefore leave out the expectation operator in the expressions for Agent 2.6 If c1/p1pc2/p2, Agent 1 will maximize his expected payoff by excluding Agent 2: Agent 1 will be able to harvest

down to a stock level that is unprofitable for Agent 2, and force Agent 2 to leave an escapement S2t=S1

t . Agent 1 will

then maximize (6a) subject to the constraint S1tpc2/p2. Similarly, if bR(c2/p2)pc1/p1, Agent 2 will exclude Agent 1 by

harvesting down to an excapement producing recruitment at which it is not profitable for Agent 1 to harvest, forcing

Agent 1 to leave an escapement ytR(S2t ). Agent 2 will then maximize (6b) subject to the constraint bR(S2

t )pc1/p1. If

ytR(c2/p2)pc1/p1 only for some ytob, Agent 2 will only be able to exclude Agent 1 in bad seasons.


These escapements give rise to the expected non-cooperative equilibrium profits EpN1 and pN2 :The escapements are below optimal levels, since the agents do not account for the effect of

their harvest on the fish stock. Any fish in excess of SN1 left behind by Agent 1 would be harvested

by Agent 2, while a spawning stock larger than SN2 left behind by Agent 2 would only benefit

Agent 1.We study whether preplay communication, without commitment, enables the agents to manage

the resource more successfully. Assume that the agents confer, and agree on a cooperativemanagement scheme that yields higher expected payoffs to each agent. Hannesson [9] examinesthe case where cooperative harvesting is supported by the threat of reverting to non-cooperativeharvesting forever if defection is detected. Stock uncertainty complicates the enforcement ofharvesting agreements since agents are no longer able to directly observe the actions of theircompetitors. We next examine conditions under which cooperative harvesting can be sustained asa self-enforcing equilibrium when stock fluctuations are incorporated into the model.

4. Cooperative management

Assume that the agents negotiate and agree on a constrained Pareto efficient joint harvesting

strategy, with escapement levels SC1 and SC

2 such that SC1 4SN

1 and SC2 4SN

2 : Cooperative

escapements give rise to the expected single-period profits EpC1 and pC2 ; where EpC14EpN1 and

pC24pN2 : If side payments are not possible, each agent must harvest to earn a profit. The agents

settle on the threat strategies of reversion to the non-cooperative escapements SN1 and SN

2 ; and onthe length of the punishment phase, T 1 periods, over which the threat strategies will be appliedif violations of the agreement are detected.7 Since Agent 1 only observes recruitment X t; which

depends on both stochastic shocks and Agent 2’s escapement, a trigger level %X is agreed uponwhich recruitment has to exceed for Agent 1 to continue cooperation. Agent 2 observes St

1 and

hence Agent 1’s adherence to the agreement.Suppose the agents then commence harvesting in accordance with their cooperative escapement

levels SC1 and SC

2 in a Nash equilibrium in trigger strategies. They continue to do so until

recruitment X t falls below the trigger level %X; or Agent 1 escapement St1 below the cooperative

level SC1 : Once an St

1 below SC1 has been observed, T 1 periods of punishment follow, during

which the agents harvest to the non-cooperative escapements SN1 and SN

2 regardless of what St1

and X t are. At the conclusion of the T 1 punishment periods, Agent 2 first returns to

cooperation.8 Once resumed, cooperation prevails until the next time that Xo %X or S1oSC1 :

Assuming that SN2 oSN

1 and SN1 oyRðSN

2 Þ for all yA½a; b�; both agents continue to harvest in the

punishment phase.

7Using non-cooperative escapements as threats is plausible because of their Nash equilibrium property.8 In a more symmetric structure, Agent 1 would first resume cooperation had he been the one to start punishing.

However, since only Agent 1 observes the initial stock, Agent 2 does not know whether Agent 1 is cheating, or

punishing after having observed a low recruitment. It will be in Agent 1’s interest to always claim that he is cheating,

and harvest down to S1N once more while expecting Agent 2 to first stop punishing. Hence we have the asymmetry that

Agent 1 in effect punishes for T periods and Agent 2 for T1 periods.


Formally, the agreement is defined as follows. The game has normal and reversionary stages.Agent 1 regards period t as normal if

(a) t=0,

(b) t1 was normal and X t4 %X; or X tTo %X and t T 1 was normal,and reversionary otherwise. Agent 2 regards period t as normal if

(a) St1XSC

1 and t ¼ 0 or t1 was normal,

(b) St T1ð Þ1 oSC

1 and t T was normal,

and reversionary otherwise. The agents act sequentially. Their strategies are defined by

SCi if t is normal;

SNi if t is reversionary:

(

We first determine the conditions under which cooperation is optimal for Agent 1. Table 1summarizes the definitions used in the discussion.9 Agent 1 cooperates in a normal period t if the

expected payoff EVC1 ðSC

1 ;X tÞ from doing so is greater than that from cheating, EVN1 ðSN

1 ;X tÞ:Formally, cooperation in a normal period t is optimal if EVN

1 ðSN1 ;X

tÞpEVC1 ðSC

1 ;XtÞ for X t

X %X:

In order for the threat of reversion to the non-cooperative Nash equilibrium escapement SN1 to be

credible, it must also be optimal for Agent 1 to harvest down to SN1 when X tp %X: We must thus

have EVN1 ðSN

1 ;XtÞXEVC

1 ðSC1 ;X

tÞ for X tp %X: At X t ¼ %X; the two conditions yield

EVN1 ðSN

1 ; %XÞ ¼ EVC1 ðSC

1 ; %XÞ: ð7Þ

For any given SC1 ; SC

2 ; and T condition (7) determines a trigger stock level %X such that Agent 1 will

adhere to the agreement, and harvest cooperatively in normal periods but revert to punishmentphase at low stock levels.

Table 1

Notation in Agent 1’s problem

XC ¼ yRðSC2 Þ; XN ¼ yRðSN

2 Þ Random recruitment with escapements SC2 and SN

2

EVC1 ðSC

1 ;XtÞ Agent 1’s expected payoff from cooperation (leaving the cooperative

escapement SC1 ) in a normal period t, evaluated after observing X t such that

X t4 %X

EVN1 ðSN

1 ;XtÞ Agent 1’s expected payoff from harvesting down to the non-cooperative

escapement SN1 ; evaluated after observing X t

o1 ¼ Eyp1ðSN1 ; XNÞ Agent 1’s expected single period profit in reversionary periods

EVC1 ðSC

1 ; XCÞ Agent 1’s expected payoff from cooperation in period t following a normal

period, evaluated prior to observing X t

EVC1 ðSC

1 ; XCjXC

X %XÞ Agent 1’s expected payoff from cooperation in period t following a normal

period, evaluated prior to observing X t but conditional on XCX %X

g1 ¼ E½p1ðSN1 ; X

CÞjXCo %X� Agent 1’s expected single period profit in the first period of reversionary play,

with Agent 1 the one to start punishing

PrðyRðS2Þo %XÞ ¼ Fð %X=RðS2ÞÞ Probability of reversionary play

9 In order to avoid cumbersome notation, we suppress S2 and T in Agent 1’s expected payoffs here. Agent 1’s payoff

will depend on these variables, but they are not choice variables for him once the agreement has been negotiated.


To write out (7), we first write out Agent 1’s expected payoffs from cheating and from

cooperating. The expected payoff from deviating and harvesting down to SN1 is

EVN1 ðSN

1 ;XtÞ ¼ p1ðSN

1 ;X tÞ þXT1

t¼1dto1 þ dT EVC

1 ðSC1 ; XCÞ: ð8Þ

The current period payoff from deviating is p1ðSN1 ;X tÞ: Agent 2 detects cheating with certainty

and commences retaliation, harvesting to his non-cooperative escapement SN2 . Non-cooperative

harvest then continues through T 1 reversionary periods with the expected profits o1: Inreversionary periods it is optimal for Agent 1 to apply his Nash harvesting strategy. The expected

payoff from resuming cooperation in period t þ T is EVC1 ðSC

1 ; XCÞ; with Agent 2 first returning tocooperation in period t þ T 1:

Agent 1’s expected payoff from cooperating and leaving SC1 in normal periods is

EVC1 ðSC

1 ;X tÞ ¼ p1ðSC1 ;X

tÞ þ ½1 Fð %X=RðSC2 ÞÞ�dEVC

1 ðSC1 ; XCjXC

X %XÞ

þ Fð %X=RðSC2 ÞÞ dg1 þ

XT

t¼2dto1 þ dTþ1EVC

1 ðSC1 ; X

CÞ( )

: ð9Þ

In period t Agent 1 obtains the cooperative profit p1ðSC1 ;X tÞ: If X tþ1 exceeds the trigger stock %X;

cooperative harvest continues with the expected payoff EVC1 ðSC

1 ; XCjXC

X %XÞ: If X tþ1 is below %X; areversionary phase begins. In the first retaliatory period, Agent 1 obtains the expected profit g1;followed by T 1 punishment periods with expected profits o1: Cooperation resumes in period

t þ T þ 1; with expected payoff EVC1 ðSC

1 ; XCÞ:

Using (8) and (9), at X t ¼ %X Eq. (7) can be written as

p1ðSN1 ; %XÞ p1ðSC

1 ; %XÞ ¼ ½1 Fð Þ�dEVC1 ðSC

1 ; XCjXCX %XÞ

þ Fð Þ dg1 þd2 dTþ1

1 do1 þ dTþ1EVC

1 ðSC1 ; X

CÞ� �

d dT

1 do1 þ dT EVC

1 ðSC1 ; XCÞ

� �: ð10Þ

We will henceforth refer to Eq. (10) as the Trigger Stock Equation.We next consider Agent 2’s problem. Table 2 summarizes the notation. Agent 2’s actions are

not observed directly. Unlike Agent 1, Agent 2 then faces a continuous choice of escapements S2

in normal periods. The escapement determines his current payoff and the probability of

Table 2

Notation in Agent 2’s problem

o2 ¼ p2ðSN1 ;S

N2 Þ Agent 2’s single period profit in reversionary periods

p2ðSC1 ;S

C2 Þ Agent 2’s single period profit in cooperative periods.

p2ðSC1 ;S

C2 Þ4o2

EV2ðSC1 ;S

C2 Þ Agent 2’s expected payoff from cooperative harvesting

EVP2 ðSN

1 ;SN2 Þ Agent 2’s expected payoff at the beginning of a punishment phase

PrðyRðS2Þo %XÞ ¼ Fð %X=RðS2ÞÞ Probability of reversionary play


cooperative versus reversionary play in the next period. Given SC1 ; %X and T, Agent 2 chooses a

cooperative escapement S2 that maximizes his expected payoff from cooperative harvesting. Let

SC2 denote Agent 2’s optimal choice. Let EVP

2 ðSN1 ;S

N2 Þ denote Agent 2’s expected payoff at the

beginning of a punishment phase. Agent 2’s expected payoff EVC2 ðSC

1 ;S2Þ satisfies the functionalequation

EVC2 ðSC

1 ;S2Þ ¼ p2ðSC1 ;S2Þ þ ½1 Fð %X=RðS2ÞÞ�dEVC

2 ðSC1 ;S2Þ

þ Fð %X=RðS2ÞÞdEVP2 ðSN

1 ;SN2 Þ

¼ p2ðSC1 ;S2Þ þ ½1 Fð %X=RðS2ÞÞ�dEVC

2 ðSC1 ;S2Þ

þ Fð %X=RðS2ÞÞdXT2

t¼0dto2 þ dT1p2ðSN

1 ;S2Þ"

þdTf½1 Fð %X=RðS2ÞÞ�EVC2 ðSC

1 ;S2ÞþFð %X=RðS2ÞÞEVP2 ðSN

1 ;SN2 Þ

��: ð11Þ

In a normal period t Agent 2 harvests Agent 1’s cooperative escapement SC1 : If X tþ14 %X;

cooperation continues in the next period. If X tþ1o %X; Agent 1 harvests down to SN1 ; and T 1

reversionary periods with the non-cooperative profit o2 follow. In reversionary periods it is

optimal for Agent 2 to harvest to SN2 : In period t þ T, Agent 2 first returns to cooperative

harvesting, with the profit p2ðSN1 ;S2Þ: Depending on X tþ1; cooperative harvesting resumes in

period t þ T þ 1; or another reversionary phase is entered.

Solving (11) for Agent 2’s expected payoff EVC2 ðSC

1 ;S2Þ yields

EVC2 ðSC

1 ;S2Þ ¼p2ðSC

1 ;S2Þ o2 Fð %X=RðS2ÞÞdT ½p2ðSC1 ;S2Þ p2ðSN

1 ;S2Þ�1 dþ ðd dTÞFð %X=RðS2ÞÞ

þ o2

1 d: ð12Þ

Derivation of (12) is presented in Appendix A. Agent 2’s expected payoff consists of the non-cooperative profits, plus the single-period gain from harvesting cooperatively, appropriatelydiscounted. The gain from cooperation is adjusted for Agent 2’s loss from first returning tocooperation after a reversionary phase.

Given SC1 ; %X; and T, Agent 2’s optimal cooperative escapement SC

2 must satisfy

EVC2 ðSC

1 ;S2ÞpEVC2 ðSC

1 ;SC2 Þ for all S2: ð13Þ

Assuming an interior solution, the necessary condition for (13) is @EVC2 ðSC

1 ;SC2 Þ=@S2 ¼ 0;

which from (12) equals

@

@S2p2ðSC

1 ;SC2 Þ½1 dþ ðd dTÞFð %X=RðSC

2 ÞÞ� þ f ð %X=RðSC2 ÞÞ

%XR0ðSC2 Þ

RðSC2 Þ

2

� fðd dTÞ½p2ðSC1 ;S

C2 Þ o2� þ ðdT dTþ1Þ½p2ðSC

1 ;SC2 Þ p2ðSN

1 ;SC2 Þ�g ¼ 0: ð14Þ

Agent 2’s optimal cooperative escapement SC2 balances the marginal benefit of additional harvest

with the marginal increase in the risk of losses resulting from a switch to reversionary play.

Eq. (14) implies that @p2ðSC1 ;S2Þ=@S2jS2¼SC

2o0: Since @2p2ðSC

1 ;S2Þ=ð@S2Þ2o0; comparing (14) to

the condition that determines Agent 2’s non-cooperative escapement level, @p2ðSC1 ;S2Þ=@S2jS2¼SN

2

¼ 0; we see that the cooperative escapement always exceeds the non-cooperative one.


We must now determine how the agents set T and SC1 in an ‘optimal’ manner, given that for any

T and SC1 pair the values of %X and SC

2 are determined by the Trigger Stock Eq. (10) and Agent 2’s

first-order condition (14). Suppose that the agents settle on a cooperative agreement thatmaximizes a weighted sum of their expected payoffs. The agents negotiate in period 0 after the

stock X 0 has been observed. Let a be the weight on Agent 1’s expected payoff and ð1 aÞ that onAgent 2’s expected payoff. The agents then settle on SC

1 and T that maximize the expected joint

payoff

JðS1;S2; %X;T ;X 0Þ ¼ aEVC1 ðS1;S2; %X;T ;X 0Þ þ ð1 aÞEVC

2 ðS1;S2; %X;TÞ; ð15Þ

subject to Eqs. (10) and (14). For any SC1 and T, the constraints determine SC

2 and %X:10 Each agentmust also obtain at least his expected non-cooperative payoff. A cooperative solution

fT ; %X;SC1 ;SC

2 g at which conditions (10) and (14) hold is a self-enforcing equilibrium, and the

strategies are subgame perfect.11

By varying the weight a between 0 and 1, we obtain a constrained Pareto optimal frontier in thespace of expected payoffs ðEV1;EV2Þ: Each point on the frontier corresponds to a different trade-off between the expected payoffs of the two agents. Determining the value of a that is most likelyto arise out of the negotiations requires a theory of bargaining. Nash [22] proposed an intuitivelyappealing solution where the outcome of the bargaining process depends on the relative power ofthe two parties measured in terms of the expected payoffs if no cooperation were forthcoming.

The Nash solution for a is the point on the Pareto frontier for which ðEVC1 o1=1 dÞðEVC

2 o2=1 dÞ is maximized.12 Of course bargaining solutions other than the Nash one may emerge(see for example [23,26,28]). The two agents might also behave in a less sophisticated manner andsimply choose weights equal to their harvest shares in the preceding 2 or 3 years, or assign equalweights to each agent’s expected payoff.We next determine the expected fraction of time spent in cooperation during a cycle and the

expected ratio of cooperative to non-cooperative periods. The number of periods during which theagents continue to cooperate, denoted by i, is a random variable whose distribution depends on

FðSC2 ; %XÞ: Given that period t 1 was normal, the probability of cooperation in period t is

1 FðSC2 ; %XÞ and the probability of reversion FðSC

2 ; %XÞ: The probability of cooperation in i

consecutive periods then is ½1 FðSC2 ; %XÞ�i and the probability of reversion in period i þ 1 is

FðSC2 ; %XÞ: The probability of cooperation lasting exactly i periods thus is ½1 FðSC

2 ; %XÞ�iFðSC2 ; %XÞ:

Given the distribution of i and the length of the punishment phase T 1 that the agents settled

10Mirrlees [20] points out difficulties that may arise in solving a maximization problem of the type discussed here.

Using the Lagrangian method of constrained optimization is only legitimate when for any {S1, %X,T}, there is a unique S2

that maximizes EV2(S1, S2, %X,T). If this is not warranted, an analogue of the Kuhn-Tucker method depicted by Mirrlees

[20] may be used to solve the problem.11The equilibrium is not renegotiation proof, since the agents could negotiate if low stock levels have been observed,

and resume cooperative behavior, avoiding the punishment stage.12The Nash solution is the only solution to the bargaining problem that satisfies axioms of rationality, feasibility,

Pareto optimality, independence of irrelevant alternatives, symmetry, and independence with respect to linear

transformations of the set of payoffs. For further discussion see for example [18]. The Nash bargaining solution was

first applied to a fishery problem by Munro [21]. Hnyilicza and Pindyck [11] is another early natural resource

application.


on, the expected fraction of time spent in cooperation during a cycle is

M ¼XNi¼0

i

T 1þ i½1 FðSC

2 ; %XÞ�iFðSC2 ; %XÞ: ð16Þ

The expected number of cooperative periods in a cycle isP

N

i¼0 i½1 FðSC2 ; %XÞ�iFðSC

2 ; %XÞ: Theexpected ratio of cooperative periods to punishment periods, denoted by N, then is

N ¼ 1

T 1

XNi¼0

i½1 FðS2; %XÞ�iFðS2; %XÞ: ð17Þ

5. Empirical illustration: the Northern Baltic salmon fishery

We illustrate the joint management game with an application to the Northern Baltic salmonfishery. The Northern Baltic salmon stock is harvested sequentially by commercial offshore andinshore fleets. Stock recruitment fluctuates, mostly due to the M-74 syndrome.13 Economicallysound cooperative agreements have not been reached. The computations indicate that thecooperative solution we propose could be sustained for a range of parameter values. Since weconsider a simplified model of the fishery, our application is primarily used to illustrate thesequential harvesting game rather than to prescribe policy in a specific fishery.The offshore fishery corresponding to Agent 1 in our previous discussion harvests the stock

first. The inshore fishery corresponds to Agent 2. The structure of the Northern Baltic salmonfishery is more complicated than the basic case analyzed above.14 Only a fraction m of the offshoreescapement moves inshore to spawn, while a fraction b remains offshore till the next season. Arecreational river fishery harvests the stock left behind by the inshore fishery. River harvest shareis small compared to that of the commercial offshore and inshore fisheries, and was here assumedto remain constant. Only a proportion v of the stock is wild salmon that reproduce.Table 3 displays the parameter values. By assumption, average recruitment follows the Ricker

recruitment relation RðS2Þ ¼ kðvS2 H3Þ exp½lðvS2 H3Þ�; where H3 is harvest of spawningsalmon in the river fishery and k and l parameters, estimated by A. Romakkaniemi, Finnish Gameand Fisheries Research Institute (unpublished results). In addition to natural reproduction,

constant stockings I maintain the stock. The stock available to Agent 1 in year t is X tþ1 ¼I þ bSt

1 þ ytRðSt2Þ: We consider the case of y uniformly distributed:

f ðyÞ ¼1

b afor apypb;

0 elsewhere;

8<: ð18Þ

13M74 is an embryonic mortality syndrome found in salmonids in the Baltic Sea. Symptoms include loss of

equilibrium, swimming in a spiral pattern, lethargy, hyperexcitability, and hemorrhage. Afflicted salmon fry usually die

a few days after symptoms begin. In the early 1990s, mortality rates in some rivers in the Gulf of Bothnia reached 50–

90%. The disease has been linked to a deficiency in vitamin B1. While treating hatchery fry with vitamin B1 has proven

effective, the wild salmon populations in the Baltic Sea remain at risk.14The description of the fishery follows Laukkanen [15].


where a ¼ 1 e and b ¼ 1þ e with 0oep1: The mean of this distribution is 1 and the variance

s2e ¼ e2=3: We let e ¼ 0:5 in the base case.

Adjusted for stockings and a part of offshore escapement remaining offshore till the nextseason, Agent 1’s participation constraint is I þ bc1=p1 þ aRðc2=p2Þ4c1=p1 and Agent 2’smc1=p14c2=p2: Since the population is partially maintained by stockings, the parameters in Table

3 satisfy the participation constraints for all 0oep1: The value of SC2 is no smaller than SN

2 and

no larger than mSC1 ; the part of the cooperative offshore escapement moving inshore. The value of

SC1 is no larger than the maximum offshore escapement with no harvest, ½I þ u�=ð1 bÞ; where u is

the upper bound to recruitment. Since the probability of reversion is 1 for values of %X larger than

u, it suffices to consider trigger stocks %X between 0 and u. The optimal agreement was computedby maximizing J subject to these constraints, with the weight a on Agent 1’s payoff varyingbetween 0 and 1 in steps of 0.1.Figs. 2–4 illustrate the optimal agreement. Table 4 displays the full set of results. Note that each

agent gains from cooperation even when the weight on his payoff is zero. The optimal agreement hasto satisfy the individual optimality conditions (10) and (14). When a ¼ 1; for example, no weight isassigned to EV2 in the joint maximization problem. But EV2 still enters the problem through theconstraint in Eq. (14) which determines Agent 2’s cooperative escapement. The policy that maximizes(15) may then result in an EV2 greater than o2; as is the case for the parameters here.Agent 1’s expected payoff increases monotonically with its weight, whereas Agent 2 is better off

when some weight is given to the competitor’s payoff. The trigger stock %X set by Agent 1 decreasesin a; as Agent 1’s gains from cooperation increase. Agent 2’s gains and thus his optimal

escapement decrease in a: Since the probability of reversion increases in %X and decreases in SC2 ; it

Table 3

Parameters describing the Northern Baltic salmon fishery

Parameter Estimate

Price of salmon offshorea, p1 24.32mk/kg

Price of salmon inshorea, p2 20.56mk/kg

Unit cost of fishing effort offshoreb, c1 9.62� 107mk

Unit cost of fishing effort inshoreb, c2 2.69� 107mk

Discount factor, d 0.95

Fraction of St1 moving inshorec, m 0.44

Fraction of St1 surviving to next yearc, b 0.50

Stockingsc, I 4 170 000 kg

Proportion of wild salmon in stock, v 0.10

Parameters of RðS2Þd, k 20.88

l 2.76� 106

Parameter of the distribution f ðyÞ; e 0.5

aDerived from Finnish Game and Fisheries Research Institute [6].bEvaluated from unpublished data in the 1994 FGFRI study Profiles of Commercial Fishing.cDerived from ICES [12] and ICES [13].dEstimated by A. Romakkaniemi, Finnish Game and Fisheries Research Institute (unpublished results).


050

100150

200250

300350

400

0 100 200 300 400 500 600 700

Age

nt 2

's E

xpec

ted

Payo

ff

Agent 1's Expected Payoff

ω1/(1-δ), ω2/(1-δ)

EV1, EV2

106 mk

106 mk

Fig. 2. Both agents gain from cooperation no matter the value of the weight parameter a: Agent 2 has greater relativegains (Agent 1’s expected discounted non-cooperative payoff is o1=ð1 dÞ and that of Agent 2 o2=ð1 dÞ: In the

modified model XN in o1 ¼ Eyp1ðSN1 ; X

NÞ is given by I þ bSN1 þ yRðSN

2 Þ).

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0 0.2 0.4 0.6 0.8 1Weight α on Agent 1's Payoff

Prob

abili

ty o

f R

ever

sion

0

2

4

6

8

10

12

Len

gth

of P

unis

hmen

t Pha

seProbability ofReversion

Length of PunishmentPhase

Fig. 3. The optimal length of the punishment phase decreases in the weight a on Agent 1’s payoff. The probability of

reversion need not be monotonic in a:

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Weight � on Agent 1's Payoff

Exp

ecte

d Fr

actio

n of

Tim

e in

Coo

pera

tion

0

500

1000

1500

2000

2500

3000

Exp

ecte

d R

atio

of

Tim

e in

C

oope

ratio

n to

Tim

e in

Pu

nish

men

t

Expectedfraction oftime incooperation

Expected ratioof time incooperation totime inpunishment

Fig. 4. The expected fraction of time spent in cooperation is close to one. The expected ratio of time in cooperation to

time in punishment increases as the length of the punishment phase decreases along with the weight on Agent 1’s payoff,

ranging from 26 to 2800.


Table 4

Agreement on joint management for different values of a, evaluated at e ¼ 0:5

a S1 (kg) S2 (kg) %X (kg) T1 FðS2; %XÞ EV1 (mk) EV2 (mk) J (mk) M N

0 10,401,000 3,299,000 1,271,000 11 0.00349 467,171,000 338,843,000 338,843,000 0.88975 26

0.1 10,401,000 3,299,000 1,271,000 11 0.00349 467,171,000 338,843,000 351,676,000 0.88975 26

0.2 10,331,000 3,267,000 1,257,000 10 0.00073 476,027,000 345,987,000 371,995,000 0.96764 137

0.3 10,331,000 3,267,000 1,257,000 10 0.00073 476,027,000 345,987,000 384,999,000 0.96764 137

0.4 10,331,000 3,267,000 1,257,000 10 0.00073 476,027,000 345,987,000 398,003,000 0.96764 137

0.5 8,932,000 2,720,000 1,106,000 4 0.00318 553,735,000 295,527,000 424,631,000 0.94941 78

0.6 8,932,000 2,720,000 1,106,000 4 0.00318 553,735,000 295,527,000 450,452,000 0.94941 78

0.7 8,932,000 2,720,000 1,106,000 4 0.00318 553,735,000 295,527,000 476,273,000 0.94941 78

0.8 7,549,000 2,267,000 908,000 2 0.00018 583,881,000 228,083,000 512,722,000 0.99726 2777

0.9 7,549,000 2,267,000 908,000 2 0.00018 583,881,000 228,083,000 548,301,000 0.99726 2777

1 7,549,000 2,267,000 908,000 2 0.00018 583,881,000 228,083,000 583,881,000 0.99726 2777

M is the expected fraction of time spent in cooperation during a cycle and N the expected ratio of time spent in

cooperation to time spent in punishment during a cycle. Expected discounted non-cooperative payoffs: o1=ð1 dÞ ¼311; 560; 000 mk; o2=ð1 dÞ ¼ 24; 147; 800 mk:

0

50

100

150

200

250

300

350

400

0 100 200 300 400 500 600 700

Agent 1's expected payoff

Age

nt 2

's e

xpec

ted

payo

ff EV 1, EV 2 at ε =0.5EV 1 , EV 2 at ε =0.2

ω 1/(1-δ), ω2/(1-δ) at ε =0.5ω1/(1−δ ), ω2/(1−δ ) at ε =0.2

106 mk

106 mk

Fig. 5. With less uncertainty, the agents’ expected payoffs vary more as the weights on the payoffs change. At high

values of a, smaller amount of stock uncertainty yields Agent 1 greater gains from cooperation.

00.0050.01

0.0150.02

0.0250.03

0.0350.04

0.0450.05

0 0.2 0.4 0.6 0.8 1Weight α on Agent 1's Payoff

Prob

abili

ty o

f R

ever

sion

024681012141618

Len

gth

of P

unis

hmen

t Pha

se

Probability ofReversion

Probability ofReversion



ε =0.5

ε =0.2

ε =0.5

ε=0.2

Fig. 6. Similarly to the case of e ¼ 0:5; the probability of reversion and the length of the punishment phase remain low

at e ¼ 0:2:


needs not be monotonic in a: Giving some weight to Agent 1 may then decrease the probability ofreversion and increase Agent 2’s expected payoff, as in our example.For any weights a; Agent 2 has the greatest relative gains from cooperation with expected

payoffs 10–14 times his non-cooperative payoff. Agent 1 is at most able to double his expectedpayoff. Agent 2’s relative advantage is explained by the asymmetric structure driving the game:

Since Agent 2’s actions are not observed, he can choose SC2 optimally given the values of SC

1 ; %X;

and T. Agent 1 only has the choice between leaving SC1 and SN

1 : Since Agent 1’s deviations are

observed directly, the optimal deviation is always to harvest down to SN1 :

The probability FðS2; %XÞ of entering a reversionary period is close to zero for all weights a: Theoptimal length of the punishment phase is finite, ranging from 2 to 11 years. The smaller theweight on Agent 1’s payoff, the longer is the punishment phase. A small value of a results in a

large SC1 : The loss to Agent 2 from entering a punishment phase then increases, producing a large

SC2 : The resulting greater expected recruitment combined with the large SC

1 increase the gain to

Agent 1 from deviating to SN1 : The length of the punishment phase must increase in order for

Agent 1 to prefer cooperation.The expected fraction of time spent in cooperation, M, increases from 0.89 to 0.99 as the weight

on Agent 1’s preferences increases and the length of the punishment phase decreases. The expectedratio of time in cooperation to time in punishment, N, increases rapidly as the probability ofreversion decreases, ranging from 26 to 2,800.

5.1. The effect of stock uncertainty on the optimal agreement

We next examine the sensitivity of the optimal agreement to the level of stock uncertainty. Inaddition to e ¼ 0:5; we computed the optimal agreement at e ¼ 0:2; 0:6 and 0.8. Figs. 5 and 6compare the results at e ¼ 0:2 and 0.5 (see also Table 5). The agreement at e ¼ 0:2 is similar tothat at e ¼ 0:5; but the expected payoffs vary more with the weight on the agents’ payoffs. The

Table 5

Agreement on joint management for different values of a, evaluated at e ¼ 0:2

a S1 (kg) S2 (kg) %X (kg) T1 FðS2; %XÞ EV1 (mk) EV2 (mk) J (mk) M N

0 12,100,000 4,221,000 2,209,000 17 2.43E-03 288,216,000 319,951,000 319,951,000 0.88472 24

0.1 12,100,000 4,221,000 2,209,000 17 2.43E-03 288,216,000 319,951,000 316,778,000 0.88472 24

0.2 12,100,000 4,221,000 2,209,000 17 2.43E-03 288,216,000 319,951,000 313,604,000 0.88472 24

0.3 12,100,000 4,221,000 2,209,000 17 2.43E-03 288,216,000 319,951,000 310,431,000 0.88472 24

0.4 7,561,000 2,787,000 1,837,000 1 4.56E-02 658,277,000 110,461,000 329,587,000 0.85246 21

0.5 7,561,000 2,787,000 1,837,000 1 4.56E-02 658,277,000 110,461,000 384,369,000 0.85246 21

0.6 7,561,000 2,787,000 1,837,000 1 4.56E-02 658,277,000 110,461,000 439,150,000 0.85246 21

0.7 7,561,000 2,787,000 1,837,000 1 4.56E-02 658,277,000 110,461,000 493,932,000 0.85246 21

0.8 7,561,000 2,787,000 1,837,000 1 4.56E-02 658,277,000 110,461,000 548,714,000 0.85246 21

0.9 7,561,000 2,787,000 1,837,000 1 4.56E-02 658,277,000 110,461,000 603,495,000 0.85246 21

1 7,561,000 2,787,000 1,837,000 1 4.56E-02 658,277,000 110,461,000 658,277,000 0.85246 21

Expected discounted non-cooperative payoffs: o1=ð1 dÞ ¼ 124; 470; 000 mk; o2= 1 dð Þ ¼ 24; 147; 800 mk:


lower degree of uncertainty gives Agent 2 less freedom in choosing his optimal escapement,decreasing the relative advantage of Agent 2 over Agent 1. At large weights on Agent 1’s payoff,Agent 1 gains more from cooperation, obtaining expected payoffs over five times hisnon-cooperative payoff. Agent 2 gains less. No cooperative solution agreeable to both agentsexists at e ¼ 0:6 or e ¼ 0:8:15 Large variations in stock recruitment make Agent 2’s optimalescapement small. Agent 1 then is better off cheating and harvesting down to his non-cooperativeescapement. Agreements on cooperation therefore break down at high levels of variation in stockrecruitment.

5.2. Alternative parameter values

The economic characteristics of the offshore and inshore fleets in the Northern Baltic salmonfishery differ markedly. Furthermore, only a fraction of the offshore escapement moves inshore tospawn. To study the extent to which our results were driven by the economic differences of thefisheries, we computed the optimal agreement for an alternative set of parameter values. We usedthe Baltic salmon fishery data but changed the inshore price equal to the offshore price, and setthe inshore fishing costs 20% below the offshore costs.16 The fraction of spawners was set equal toone. Stockings were still included in the stock equation. The alternative parameter values satisfythe participation constraints I þ bc1=p1 þ aRðc2=p2Þ4c1=p1 and mc1=p14c2=p2 for all 0oep1:The results for the alternative parameter values were similar to those for the actual NorthernBaltic salmon fishery data. We next summarize the results. The full results for the alternativeparameters are available from the author upon request.Cooperation was again sustained no matter the weights on the agents’ payoffs, with Agent 1’s

expected payoff practically the same for any weight a: Agent 1’s gains from cooperation were nownegligible. With Agent 2’s prices and costs similar to those of Agent 1, Agent 2’s harvest shareunder non-cooperation is small. Thus Agent 1 who harvests the stock first faces little competition,operating practically as a sole harvester. The cooperative agreement then yields Agent 1 littleadditional profit. The punishment phase was practically infinite for most values of a; at 160periods, with a probability of reversion arbitrarily close to zero. Cooperation could again only besustained at moderate levels of uncertainty. Both agents’ gains from cooperation decreased withstock uncertainty.

5.3. Implicit versus explicit enforcement: joint management with stock monitoring

An interesting policy question is whether an agreement with no escapement uncertainty is morestable than the trigger stock solution. Are there gains from explicit enforcement in the form ofmeasuring the spawning stock? To study joint management with stock monitoring, we appliedHannesson’s [9] cooperative management model to the Northern Baltic salmon fishery data. Wemodified the model to include stochastic shocks in recruitment. Fig. 7 displays the results (see alsoTable 6). Each agent’s escapement is now observed. Cooperation can now only be sustained for a

15Eqs. (10) and (14) were not satisfied simultaneously for any values of 0pap1.16 In the actual data the offshore price is 15% higher than the inshore price, and the offshore unit costs 70% higher

than those inshore.


narrow range of weights on the agents’ payoffs. For small weights a on Agent 1’s payoff, theexpected joint value is maximized when Agent 1 is excluded from harvest, which he will not agreeto without side payments. Similarly, Agent 2 is optimally excluded for high values of a: This isdifferent from the trigger stock equilibrium where the constraints in (10) and (14) ensure that eachagent’s interests are accounted for even with a small value on his payoff. The expected payoffsfrom cooperation differ little from those characterizing the agreement without monitoring. Agent1 benefits from stock measurements, while Agent 2 loses the advantage arising from theasymmetric structure of the game and would therefore be better off with unobserved escapements.The qualitative results for joint management with stock monitoring at the alternative parametervalues were similar to those for the actual Northern Baltic salmon fishery data. Cooperation wasagain only agreeable for a narrow range of weights on the two agents’ payoffs. Agent 1 wouldagain benefit from monitoring the competitor’s escapements, while Agent 2 would be better offwith unobserved spawning stocks.The result that implicit cooperative agreements can only be supported when stock uncertainty is

not too high explains why cooperative harvesting is rare. Explicit enforcement in the form of stockmonitoring is a joint management alternative that can be sustained even when variation in

0

200

400

600

800

1000

0 500 1.000Agent 1's Expected Payoff

Age

nt 2

's E

xpec

ted

Payo

ff

Cooperation:

Non-cooperation:

EV 1 ,EV 2

ω1/(1−δ), ω2/(1−δ )

Fig. 7. Agreement with monitoring. Cooperation can only be sustained for a narrow range of weights on the agents’

payoffs. For small values of a expected joint value is maximized when Agent 1 is excluded from harvest, which Agent 1

will not agree to without side payments. Similarly, Agent 2 is optimally excluded for high values of a:

Table 6

Agreement with monitoring

a S1 (kg) S2 (kg) EV1 (mk) EV2 (mk) J (mk)

0 X 2,605,010 0 800,162,000 800,162,000

0.1 X 2,605,010 0 800,162,000 720,146,000

0.2 X 2,605,010 0 800,162,000 640,130,000

0.3 X 2,605,010 0 800,162,000 560,113,400

0.4 X 2,605,010 0 800,162,000 480,097,200

0.5 8,643,630 2,697,160 585,075,000 269,923,000 427,499,000

0.6 6,913,430 3,023,680 800,494,000 4,262,460 482,001,000

0.7 5,165,130 2,582,565 686,636,000 0 480,645,200

0.8 5,165,130 2,582,565 686,636,000 0 549,308,800

0.9 5,165,130 2,582,565 686,636,000 0 617,972,000

1 5,165,130 2,582,565 686,636,000 0 686,636,000


recruitment is large. An important consideration then is the cost of monitoring. The gains fromcooperation would have to exceed this cost in order for monitoring to be a feasible policyalternative.

6. Conclusion

We examine cooperative and non-cooperative harvesting in a stochastic sequential fishery, anddefine conditions under which cooperative harvesting can be sustained as a self-enforcingequilibrium when the actions of one harvester are not observed. Even when both agentscooperate, reversionary periods may occur with a positive probability. Although the agentharvesting first knows that a low stock level reflects a stochastic shock to recruitment, it is rationalto participate in reversionary periods. Otherwise, the agent harvesting the spawning stock wouldhave no incentive to cooperate. The equilibrium is subgame perfect but not renegotiation proof.Supposedly, the agents could renegotiate and agree to continue cooperation after low stock levelsor low escapements have been observed. However, both parties realize that renegotiating wouldunravel the rational for cooperation.We illustrate the sequential harvesting game with an empirical example based on data for the

Northern Baltic salmon fishery. The cooperative solution was supported at moderate levels ofstock uncertainty. Both agents gained from cooperation, but the agent whose actions are notobserved had the greatest gains. The finding that cooperation can only be sustained when stockuncertainty is not too prevalent sheds light on why we rarely observe cooperation intransboundary fisheries. The policy alternative of measuring the spawning stock provides acooperative solution that both parties can agree to regardless of the extent of stock uncertainty.The agents do not benefit equally from stock monitoring. The agent harvesting the initial stockgains from measuring the spawning stock, while the agent harvesting the spawning stock is betteroff when his escapement remains unobserved.A general view in the resource economics literature is that the introduction of side payments

leads to much superior results, not the least because the cooperative resource managementprograms arising therefrom are relatively simple (see e.g. [21]). Allowing for side payments incooperative management of a stochastic sequential fishery would likely yield greater over-allbenefits than the trigger stock agreement, in particular when there are notable differences in thesequential fisheries prices and costs. Exclusion of one of the fisheries from harvest would precludemonitoring problems—it is more straightforward to control whether a fleet is harvesting than howmuch it is harvesting. However, joint management with side payments is often politicallyinfeasible, especially if one fishery should optimally be excluded from harvest.

Appendix A. Closed-form solution for Agent 2’s expected cooperative payoff

The functional equation for Agent 2’s expected cooperative payoff is

EVC2 ðSC


2 ðSC1 ;S2Þ

þ Fð %X=RðS2ÞÞdEVP2 ðSN

1 ;SN2 Þ: ðA:1Þ


Writing out the expected payoff at the beginning of a punishment phase, EVP2 ðSN

1 ;SN2 Þ; (A.1)

becomes

EVC2 ðSC


2 ðSC1 ;S2Þ

þ Fð %X=RðS2ÞÞdXT2

t¼0dto2 þ dT1p2ðSN

1 ;S2Þ"

þdTf½1 Fð %X=RðS2ÞÞ�EVC2 ðSC

1 ;S2Þ þ Fð %X=RðS2ÞÞEVP2 ðSN

1 ;SN2 Þg

#: ðA:2Þ

We next solve Eq. (A.2) for EVC2 ðSC

1 ;S2Þ: Note that the subgame starting in period t þ ðT þ 1Þand the subgame starting in t þ 1 both follow a period where Agent 2 left the cooperativeescapement S2: In any such period cooperation continues with probability ð1 FÞ and apunishment phase begins with probability F. Since the time horizon of the game is infinite, theexpected payoffs from the two subgames are equal, and we can replace the period t þ ðT þ 1Þexpected payoff in Eq. (A.2), ð1 FÞEVC

2 þ FEVP2 ; by the expected payoff ð1 FÞEVC

2 þ FEVP2

in period t þ 1; which from (A.1) equals 1d½EVC

2 ðSC1 ;S2Þ p2ðSC

1 ;S2Þ�:We then use the formula for

a geometric sum, to obtain

EVC2 ðSC


2 ðSC1 ;S2Þ

þ Fð %X=RðS2ÞÞd1 dT1

1 do2 þ dT1p2ðSN

1 ;S2Þ�

þdT1fEVC2 ðSC

1 ;S2Þ p2ðSC1 ;S2Þg

�: ðA:3Þ

We solve (A.3) for EVC2 ðSC

1 ;S2Þ and simplify the result by adding and subtracting o2 in the

numerator and rearranging, which yields

EVC2 ðSC

1 ;S2Þ ¼p2ðSC

1 ;S2Þ o2 Fð %X=RðS2ÞÞdT ½p2ðSC1 ;S2Þ p2ðSN

1 ;S2Þ�1 dþ ðd dTÞFð %X=RðS2ÞÞ

þ o2

1 d: ðA:4Þ

References

[1] C.W. Clark, Economically optimal policies for the utilization of biologically renewable resources, Math. Biosci. 12

(1971) 245–260.

[2] C.W. Clark, The dynamics of commercially exploited animal populations, Math. Biosci. 13 (1972) 149–164.

[3] C.W. Clark, Profit maximization and the extinction of animal species, J. Polit. Economy 81 (1973) 950–961.

[4] C.W. Clark, Restricted access to common property fishery resources: a game theoretic analysis, in: P.T. Liu (Ed.),

Dynamic Optimization and Mathematical Economics, Plenum, New York, NY, 1980, pp. 117–132.

[5] C.W. Clark, Mathematical Bioeconomics, Wiley, New York, NY, 1990.

[6] Finnish Game and Fisheries Research Institute (FGFRI), Prices Paid to Fishermen in 1993, Official Statistics of

Finland, 1994:5, Helsinki, Finland, 1994.

[7] E. Green, R. Porter, Noncooperative collusion under imperfect price information, Econometrica 52 (1984) 87–100.

[8] R.P. Hamalainen, V. Kaitala, A. Haurie, Bargaining on whales: a differential game model with pareto optimal

equilibria, Oper. Res. Lett. 3 (1984) 5–11.


[9] R. Hannesson, Sequential fishing: cooperative and non-cooperative equilibria, Nat. Resour. Modeling 9 (1995)

51–59.

[10] R. Hannesson, Fishing as a supergame, J. Environ. Econom. Management 32 (1997) 309–322.

[11] E. Hnyilicza, R. Pindyck, Pricing policies for a two-part exhaustible resource cartel: the case of OPEC, Eur.

Econom. Rev. 8 (1976) 139–154.

[12] ICES, Report of the Baltic Salmon and Trout Assessment Group. ICES CM 1994/Assess: 15. Ref. M, 1994.

[13] ICES, Report of the Baltic Salmon and Trout Assessment Group. ICES CM 1995/Assess: 16. Ref. M, 1995.

[14] V. Kaitala, M. Pohjola, Optimal recovery of a shared resource stock: a differential game model with efficient

memory equilibria, Nat. Resour. Modeling 3 (1988) 91–119.

[15] M. Laukkanen, A bioeconomic analysis of the northern Baltic salmon fishery: coexistence versus exclusion of

competing sequential fisheries, Environ. Resour. Econom. 18 (2001) 293–315.

[16] D. Levhari, R. Michener, L.J. Mirman, Dynamic programming models of fishing: competition, Am. Econom. Rev.

71 (1981) 649–661.

[17] D. Levhari, L.J. Mirman, The great fish war: an example using a dynamic Nash–Cournot solution, Bell J. Econom.

11 (1980) 322–334.

[18] R. Luce, H. Raiffa, Games and Decisions, Dover Publications, New York, NY, 1989.

[19] R. McKelvey, Game-theoretic insights into the international management of fisheries, Nat. Resour. Modeling 10

(1997) 129–171.

[20] J. Mirrlees, The theory of moral hazard and unobservable behavior: Part I, Rev. Econom. Stud. 66 (1999) 3–21.

[21] G.R. Munro, The optimal management of transboundary renewable resources, Canad. J. Econom. 12 (1979)

355–376.

[22] J. Nash, Two-person cooperative games, Econometrica 21 (1953) 128–140.

[23] H.J. Peters, Axiomatic Bargaining Game Theory, Kluwer, The Netherlands, 1992.

[24] W.J. Reed, The steady state of a stochastic harvesting model, Math. Biosci. 41 (1978) 273–307.

[25] W.J. Reed, Optimal escapement levels in stochastic and deterministic harvesting models, J. Environ. Econom.

Management 6 (1979) 350–363.

[26] E. Roth, Axiomatic Models of Bargaining, Springer, Berlin, Germany, 1979.

[27] M. Spence, D. Starrett, Most rapid approach paths in accumulation problems, Internat. Econom. Rev. 16 (1975)

388–403.

[28] W. Thomson, Cooperative models of bargaining, in: R.J. Aumann, S. Hart (Eds.), Handbook of Game Theory,

Elsevier, Amsterdam, 1994.


Documents

Cooperative and non-cooperative harvesting in a stochastic sequential fishery