A cooperative incentive equilibrium for a resource management problem

Journal of Economic Dynamics and Control 17 (1993) 659-678. North-Holland

A cooperative incentive equilibrium for a resource management problem* Harri Ehtamo and Raimo P. H5m5lGnen Helsinki University of Technology, SF-02150

Espoo 15, Finland

Received December 1988, final version received May 1992

A two-country dynamic game model of whaling is considered. It is assumed that the countries have exact information, possibly with some time delay, about each other’s whaling efforts measured by the number of vessels involved in whaling. It is shown how to construct equilibrium strategies which are affine in the available information and which, when jointly carried out, realize a given Pareto-optimal decision. Numerical results suggest that some of these strategies are credible; i.e., the countries do believe that the announced strategies will be followed.

1. Introduction The analysis of cooperative equilibria in dynamic games has recently aroused a great deal of interest among game theorists; see, e.g., Haurie and Pohjola (1987), Haurie and Tolwinski (1984), HBmllHinen, Haurie, and Kaitala (1984, 1985), and Tolwinski, Haurie, and Leitmann (1986). Haurie and Tolwinski (1984) and Tolwinski, Haurie, and Leitmann (1986) have shown how the use of control-dependent memory strategies permits the inclusion of a threat in a cooperative strategy which leads to a class of acceptable equilibria in dynamic games. The equilibrium strategies studied in these papers are discontinuous in the available information. However, there is a class of appealing strategies which are continuous in the information and which are worth studying when considering cooperative equilibria; namely, affine incentive strategies. Such strategies were used by Ho, Luh, and Olsder (1982) and by Zheng, Basar, and Cruz (1982, 1984) to study incentive Stackelberg games. Further results on the existence and construction of affine

Correspondence to: Harri Ehtamo and Raimo P. Htiimlllinen, Helsinki University of Technology, Systems Analysis Laboratory, Otakaari 1, SF-02150 Espoo 15, Finland. *This work has been supported by the Yrjii Jahnsson Foundation.

01651889/93/$05.00

0 1993-Elsevier

Science Publishers B.V. All rights reserved

660

H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

incentive strategies, and equilibria defined by means of these strategies, were obtained by Ehtamo and Hamllainen (1986, 1989). In this paper we shall consider a two-country dynamic game model of whaling. Each country is an independent decision maker (DM). The action sets consist of the DMs’ admissible whaling efforts measured by the number of vessels involved in whaling. It is assumed that at every instant the DMs have exact knowledge, possibly with some time delay, of each other’s actions, and that their strategies make use of this information. Suppose the DMs are in a position to cooperate, and (Ef, E‘j) is an agreed (Pareto-optimal) decision. The problem is to find piecewise linear strategies for both DMs such that when one DM implements or is believed to implement his strategy, no temptation exists for the other to cheat or to break the agreement in the course of the game. It should be noted that the concept of linear equilibrium strategies is not new in the economics literature. In 1976, such strategies were used by D.K. Osborne in a static framework [see Osborne (1976)] to explain the stability of the OPEC cartel of oil-producing countries, which was created in October 1973 and was still very powerful three years later. In Osborne’s work a particular emphasis was put on the linear equilibrium strategies defined by the straight line, tangent to the isoprofit curves at the joint profit-maximizing point, passing through the origin. Along this line market shares for the countries are constant. Since the model for the oil-producing countries as well as the model for the whaling countries describe an oligopolistic market, it is not surprising that the equilibrium strategies studied in this paper share some good properties with those studied by Osborne. We will show how to construct affine equilibrium strategies that can be used to support a given Pareto-optimal decision. In particular we consider such continuous, piecewise linear strategies that the strategy for DMi only calls for a reaction when the number of the active vessels for DMJ’ exceeds the agreed number. Numerical experiments suggest that the constructed strategies have the property that if DMJ’ deviates from his cooperative action Ejd, then it will be more beneficial for DMi to follow his equilibrium strategy than to follow his cooperative action Eid. This property was analytically shown to be true (for a static oligopoly model) in Osborne’s work and it was considered to make the equilibrium strategies credible. For, as was written by Osborne: ‘Member j has every reason to expect the other members to follow their quota rules (equilibrium strategies) and retaliate to his cheating, for in so doing they will lose less than by standing pat at xp (the agreed Paretooptimal point) . . . The quota rule thus incorporates a credible threat.’ Cooperative equilibria in fishery management games have been previously studied by Hamallinen et al. (1984, 1985). However, their strategies are discontinuous and such that even in the case of a slightest deviation from the agreed whaling effort the retaliation of the offended country causes a rapid

H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

661

decrease in the number of whales. After the retaliation period the stock has diminished essentially and the creation of new cooperation will be difficult. Thus the whaling model is a good example of a situation where continuous threat strategies may be preferable to discontinuous ones; a small increase in the whaling effort of one country causes a similar counteract increase. The contents of the paper are as follows. In section 2 we define the whaling model and give its Pareto-optimal solutions. In section 3 the definition of an incentive equilibrium is given and no-memory equilibrium strategies for a class of Pareto-optimal whaling decisions are derived. In section 4 we give a numerical example, and in section 5 the construction of time lag strategies is considered.

2. The model and the Pareto-optimal

solutions

We shall consider two countries that exploit the same stock of whales. The dynamics of the fishery is described by the state equation

where x(t) 2 0 denotes the number of whales at time t, F is the natural growth function of the whales, E,(t) 2 0, i = 1, 2, are the countries’ whaling efforts measured by the number of vessels involved in whaling at time t, and qi > 0, i = 1, 2, are the catchability coefficients related to the unit fishing effort of each country. The growth function is assumed to be of the logistic form

(2)

F(x) = rx(1 - x/K),

where r is the intrinsic growth rate, and K is the carrying capacity of the environment. Country i, i = 1, 2, considers the net revenue over a fixed time horizon CO, Tl,

sT

e-P1f(ni[qiEi(t)x(t)]qiEi(t)x(t)

Ji = gi(x(T)) +

0

- c”Ei(t))

dt ,

(3)

where pi is the discount rate, Iii is the inverse demand function of whales defined by ZZi(qiEiX)

=

pi

-

kiqiEiX

y

pi>03

ki>O,

(4)

662

H. Ehtamo and R.P. Hiimiiliiinen. A two-country dynamic game model of whaling

cp > 0 is the unit cost of whaling, and gi(X) describes the salvage value of the stock at time T. We assume that gs are twice differentiable and SixCx)

2

Ov

Sixx(x)

5

O

(5)

on the region of interest. We write

Ji = gi(x(T)) +

sT(-

3 UiEFX’ + biEiX - c,Ei) dt

0

ai =

2kiqf exp ( - Pit),

bi

piqi

=

Ci = Co

9

(6)

exp ( - pit), exp ( - pit),

i=

1, 2.

Clark (1985) gives an extensive description of fishery models. This particular model was previously studied, for instance by Hlmallinen et al. (1984), who give some numerical examples of discontinuous cooperative equilibria, and by Ehtamo et al. (1988), who derive the open-loop Nash bargaining solution for the corresponding N-country model. The following notation will be used. The Banach space C([O, T], R) of real-valued continuous functions with the sup topology is denoted by C and the Hilbert space &([O, T], R) of real-valued, measurable, square integrable functions is denoted by L2. For EEL, x Lz, E = (E,, E2), let xE denote the solution of (1). and set A4 = (M,, M2). It can be shown that for Let Mi, MZ~L2, EE L2 x L2 such that Ei(t) I M,(t) a.e., i = 1, 2, it holds

x& 2 xdt)

3

te[O, T].

(7)

Let the countries’ decision sets D1 and Dz be defined by Di = (Ei E Lz 10 I E,(t) I Mi(t)

a.e. on [0, T]} ,

i= 1,2,

(8)

where M1,M, sL2. In order to make the maximization problems welldefined and easier to handle we shall assume that Mi and M2 [which will be specified later on, in eq. (30)] are such that M,(t) 2 pf/8kiCP 1

i = 1,2,

t E[O, T],

(9)

and XM(t)

>

X,

2 max {ZC~/piqi} , i= 1,2

t E[O, T].

(10)

H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

663

Inequality (9) also has a clear economic interpretation: Namely, for all t and x(t) the maximum whaling effort for each country can be at least the myopic maximum [i.e., the maximum of the integrand in (6)]. Denote D = D1 x D2. We are interested in the Pareto-optimal solutions which solve the problem max(h

Jl(E)

+

subject to (l),

(11)

PzJz(E)),

E ED ,

for some p1,p2 > 0, pi + p2 = 1. The following lemma concerns these solutions. Lemma 2.1. E,d(t)

Let Ed(t), xd(t), and A(t) satisfy eq. (1), and the equations tbi

=

-

11;‘4ii(t))xd(t)

-

ci

i= 1,2,

3

I

(12)

UiXd2(t)

X(t) = -

i

~iEd(t)(bi -

UiEf(t)Xd(t))

-

A(T)

=

i

l-(t)

c

F,(Xd(t))

\

i=l

PiSix(Xd(T))

3

tECO,

Tl,

-

i i=l

qiEf(t)

\

,

/

(13)

i=l

and let xd(t) > xnl 9

(14)

bi - p; ’ qiA(t) 2 3Ci/‘2x, )

i=

1,2,

tE[O, T].

(15)

0 < Ed(t) < pf/‘SkiCP,

i= 1,2,

rEC0, Tl,

(16)

Then

i.e., Ed ED, and Ed solves problem (II).

The proof of this lemma is essentially similar to that of Lemma 3.1 (see appendix B) and is therefore omitted. Remark 2.1. Substituting (12) to (13) and using (5) and (14), it is easy to verify that A(t) > 0 for tE[O, T); cf. the proof of Lemma 3.1. Hence the upper bounds in (16) follow as in (74). The lower bounds in (16) are implied by (14) and (15). On the other hand, inequalities (7), (9), and (10) together with (16) imply that xd(t) 2 xM(t) 2 x, for all t.

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H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

In the following section we will give the conditions under which a given Pareto-optimal decision can be supported by using piecewise linear incentive strategies.

3. Equilibrium

strategies

Consider two decision makers, DMl and DM2, with decision sets Di and objective functionals Ji: Di x D2 -+ R, i = 1, 2. Suppose the DMs have knowledge of each other’s actions, and they employ strategies which make use of this information. To simplify the notation we shall assume that a strategy yi for DMi is a causal mapping yi: Dj + Di. Thus, for given Ej, yi(Ej)EDi. Moreover, depending on the information gained and recalled by DMi at time t, yi(Ej)(t) may depend on any or all of the values E(s), 0 I s I t, where tE [0, T]. Let r1 and r2 denote the strategy sets of DMl and DM2, respectively. Let (E,d, E~)ED, x D2. Dejinition 3.1. A strategy pair (yl, y*)~r, rium at (Et, E,d) if

x Tz is called an incentive equilib-

E,d= ~z(Ei’),

E,d= YI 6%) 9

For example, if (E,d, E,d) is a Nash equilibrium pair, then the constant mappings yi (El) = E,d and y2(EI) = Ei constitute an incentive equilibrium at (E,d, E,d). The meaning of Definition 3.1 is obvious: Assume that the DMs are in a position to cooperate and (Ef, Ei) is a bargaining solution. Assume further that each DM wants to safeguard himself against any attempts by the other DM to break the agreement in the course of the game. One way to accomplish this is to provide the DMs with strategies such that when one DM implements (or is believed to implement) his strategy, the best the other can do, is to act according to what was agreed upon. If such strategies exist, they can help the DMs to come to an agreement, and possibly to comply with it. Let (Et, E,d) be a Pareto-optimal solution satisfying (12)-(15). We shall show that there is I]E C, q(t) > 0 for all t, such that the linear manifold, defined by

El (4 - E;l(t) = E2(0

-

E;(t)

fl(O

3

(17)

H. Ehfamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

665

lies on the joint tangent plane to the isoprofit surfaces of J1 and Jz at (Ei’, E,d). Thus, if DMj takes an action E,(t), t E [0, T], and if DMi follows the quota rule (17), then Jj(Ef, Ej) 2 Jj(El, E2). In other words, the strategies defined by yi(Ej)(t) = Ed(t) + qi(t)(Ej(t) - E;(t)), i #j, where q1 = 9 and q2 = V-l, constitute an incentive equilibrium at (E,d, E,d) . Remark 3.1. Employing strategies of this form does not actually mean that the parties should without time delay know the number of the active vessels of the other party, and move accordingly, which is obviously impossible. Rather, the game described by these kind of strategies approximates the ‘real game, where DMi replaces E,(t) - E;(t) by Ej(t - a) - EJd(t- 0) in his strategy, where 0 is a small number so that its effect can be neglected in theoretical considerations. The effect of cr will be studied in section 5, where time lag strategies are considered. We shall need the following result. Let D2, M1, Mz be as in (@-(lo). Define y: D, -+ L2 by

(18)

E,EDz,

Y(E2) = Eo + rlE2 9 where Eo, q E L2 are such that t?(t)2 0,

M,(t)

2 y(M,)(t)

a.e.,

t E [O, T] .

(19)

Observe that for E = (y(E2), E,), where E2 E D2, it holds that xE(t) 2 xy(t) for all t, as is implied by inequality (7). Next, consider the problem max J2(E1, E2), subject to (l), Lemma

3.1.

(20)

El = y(E2),

Let E;(t),

E2 E D2 .

x*(t), and A,(t) satisfy

the equations

a*(t) = F(x*(Q) - qdE:)(Ox*(Q- q&(4x*(& E:(t)

=

Cb2 - (q2 + qltl(w2@)lX*(~)

a2x*‘(t)

X,(t) = - E;(t)@,

3

(22)

- a,E:(t)x*(t))

- &(W,(x*(0) n2m = s2Ax*(73

- c2

x*(o) = x0 ) (21)

3

- qAEZ*)(t) tEC0, Tl,

- q&W)

3

(23)

666

H. Ehtamo and R.P. Hdmiiliiinen, A two-country dynamic game model of whaling

x*(t) >

(24)

xnl,

b2 - (q2 + 41il(t))w) Then 0 < E;(t) < p$/8k,cg

tE[O, T].

2 3c2/2x,,

for all t, and E;

(25)

solves problem (20).

The proof of this lemma is given in appendix B. Let now Ed, xd, 1 be as in (12)-( 15). Define Xi(t) = - Ed(t)(bi - aiEf(t)Xd(t)) - ii(t)

F,(Xd(L))

-

i j=l

qjEjd(t)

7

> (26)

i=

A(T)=gix(xd(T)),

1,2.

Then

It is shown in appendix A that I]S are well defined [except in the case and vi(t) > 0 for all t. Set

&(T) = 0, ~.j(T) # 0, i # j]

Yi(Ej) = Ed + rli(Ej - Ejd) ,

i#j,

(29)

and choose Ml ,M2 E L2 such that r1W2)

=

MI

(30)

2

and (9), (10) hold. By (28)-(30), y2(M1) = M2. Define D1 and D2 as in (8). The following proposition shows that (rl, y2) is an incentive equilibrium at (E,d, E,d). Proposition 3.1.

Let y1 and y2 be as in (29). Then

Jl(Eld, E,d) 2 JI(EI, J2(Ei’,

E,d)

2

Y~(EIN

J~(YI(E~LE~)~

3

VEIEDI,

(31)

VE2eD2.

(32)

H. Ehtamo and R.P. Hiimiiliiinen, A two-country

Proof.

dynamic game model of whaling

667

From (27), (28), 12(42 +

4191) =

l"242U

+

PlWP2~~2)

(33)

=LG142A.

From (12), (14), (15), (26), and (33) it follows that E,d, xd, and A2 solve eqs. (21)-(25) for y = yr. Thus (32) follows. Changing 1 and 2 in Lemma 3.1 and in (33) we get (31). W In this application

yi, as defined by (29), is not feasible in general, since

yi(Ej)(t) may be negative on a set the measure of which is not zero. We shall

next show that yi and y2 can be modified in such a way that they become feasible while preserving the equilibrium property. Let y be as in (18), (19), and let ET, x*, and A2 be as in Lemma 3.1. Then we have: Lemma

3.2.

Let y’:

D2 + L2 be any mapping such that

r’(E%t) = yU$)(t) a.e., and for

(34)

E2 E D2, M,(t) 2 7’ (Ez)(t) 2 HE,)(t)

a.e

(35)

Then

forall

Jdy’(ET),ET)2 J~(Y’(&), E2)

(36)

&ED,.

The proof of Lemma 3.2 is given in appendix C. For example, consider strategies of the following form: YI (U(t)

= E:(r) + H&(t)

- Ezd(r))~i(N%(t) - E;(r)) 3

(37)

rh (E,)(t) = E:(t) + H@,(r) - Ei’(r))q2(NEr(t) - E:(r)) 1

(38)

where E,d, E,d, yll, and 112are as in (29) and H is the unit step function H(s) = 1 for s > 0, H(s) = 0 for s I 0. Clearly for EjEDj, I: is strictly positive, and Mi(t) 2 y: (E,)(t) 2 yi(Ej)(t) Thus yr: D2+Dl,

y;:

D1 +D2,

a.e.,

2

J,(y;(E2LE2),

1,2,

i#j.

(39)

and by Lemma 3.2,

JI(&‘, E,d)2 J,(E,, Y;(E,)) Jz(Eld,E;)

i,j=

3

VEIEDI,

(40)

VE2~D2.

(41)

668

H. Ehtamo and R.P. Htimtiliiinen, A IWO-countrydynamic game model of whaling

To conclude, let Ed = (E,d, E,d) be as in (12)-(15), and let Di, D2 be defined as in Proposition 3.1. Then (y;, y;), defined by (37) (3X), is an incentive equilibrium at (E,d, E,d). Remark 3.2. following nice the number of end of section

4. Numerical

Note that the strategies defined by (37) and (38) have the property: the strategy for DMi only calls for a reaction when active vessels for DMj exceeds E;(t); cf. the discussion at the 4.

example

In this section a numerical example illustrating the properties of the solution is worked out. The parameter values chosen for the model are comparable to the ones used by Hamallinen et al. (1984) and by Ehtamo et al. (1988): x0

=

150,000

)

T=

10,

K = 300,000,

r = 0.06, p1 = p2 =

0.02,

kl = k2 = 0.8,

p1 = p2 =

6,000,

41 = q2 = 0.002,

co1 = 500 ?000 5

c: = 580,000.

Both countries are identical with respect to the discount rate and demand law. The cost in the country 1 - 500,000 mu (mu = monetary units) per vessel per year - is lower than in country 2 ~ 580,000 mu per vessel per year. The initial stock is 150,000 whales, and the carrying capacity of the fishery is 300,000 whales. The planning period is 10 years. Now pf/8kicQ < 12, i = 1, 2, x, < 96,667, and inequalities (9), (10) hold, if e.g. M,(t),M,(t)~[12, 191 for t~[0, lo]. We set gi(X(T))= 3OOx(T)x exp( - PiT), i = 1, 2. The values of E,d, Ej, and xd were computed by solving the two-point boundary value problem (I), (12), (13) with ~1~= 0.505 and p2 = 0.495; see table 1. Observe that (14) holds. The values of A(t) exp(0.02t) were in the interval [300, 4721. Since for i = 1, 2, piqiLl(piqi

-

~cP/~x,)

>

742 > A(t) exp (0.02~))

Vt ,

(15) holds. In table 1, 59, 59 = Ji(Et, E,d), i = 1, 2, and the values of q1 are also shown. Set, e.g., M2(t) = 16 and M1 = yl(M,). Then (9) and (10) hold.

H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

669

For comparison, table 1 also contains the open-loop Nash equilibrium solution (Er, EF) associated with (l)-(10). It is worth mentioning that in this example p1 and pZ satisfy

/

k=l

Since (a

+

PzJz(E)

3

VEED,

it follows that

J,(E) - Jr) >

for all EED for which Ji(E) 2 J”, i = 1, 2 [see Ehtamo et al. (1988)]. Hence Ed is a cooperative Nash (bargaining) solution for the bargaining problem associated with (l)-( 10). Further numerical computations show, considering longer time horizons, that the Nash equilibrium and the Pareto-optimal trajectories exhibit the familiar turnpike property [cf. H%m&inen et al. (1984)]. The whale stock

Table Results T

Eid

E2”

0 1 2 3 4 5 6 7 8 9 10

7.08 7.13 7.19 7.24 7.30 7.37 7.43 7.50 7.57 7.65 7.73

6.49 6.54 6.60 6.66 6.73 6.79 6.86 6.93 7.01 7.08 7.17

1

for the numerical Xd

‘11

150000 150408 150772 151092 151367 151597 151781 151918 152009 152051 152045

1.00 1.00 1.00 1.00 1.01 1.01 1.01 1.01 1.01 1.02 1.02

~1 = 0.505

)

example. E:

8.10 8.13 8.16 8.19 8.23 8.27 8.31 8.35 8.39 8.44 8.49

/I2 = 0.495

J: = 8.8417 x lo’,

J: = 8.2036 x 10’

Jr = 8.7965 x lo’,

J: = 8.1575 x 10’

E!

7.51 7.54 7.58 7.61 7.65 7.69 7.73 7.77 7.81 7.86 7.91

XN

150000 149810 149608 149392 149162 148918 148658 148383 148091 147781 147453

670

H. Ehtamo and R.P. Hiimiiliiinen, A iwo-country dynamic game model of whaling

converges to the steady-state value of somewhat over 100,000 whales for the Nash equilibrium solution and to somewhat over 152,000 whales for the Pareto-optimal solution. We finally consider some attractive properties which characterize the equilibrium strategies 7; and y;. For continuous functions h-,J+ EDi, K(t)

3

< @(r)
VtECO,u,

(42)

denote Fi = {EiEDilh-(t) Property 4.1.

I E,(t)
There are sets F,,F2

i=

1,2.

(43)

such that, if El E F1 and E2e FZ, then

JI(Y: (E&E,)

2 J,(Ef,

Ez) 3

Jz@I,Y;@I))

2

E,d).

Jz(E1,

a.e.>,

Suppose Property 4.1 holds and suppose that at some instant DMj deviates from his cooperative action E;. Then it will be more beneficial for DM1’ to follow his equilibrium strategy than to follow his cooperative action Ed. As remarked by Osborne (1976) this property of the equilibrium strategies is apt to make them credible. Although Property 4.1 is difficult to study analytically, it can be studied numerically. We considered three different cases with the FiS defined by (i)

J-(t) = E;(t) - 4.0,

h+(t) = E”(t) - 0.2,

i= 1,2,

tE[O,T],

(ii)

h-(t)

= E;(t) - 2.0,

J+(t) = E:(t)

i = 1,2,

tE[O, T],

(iii)

J-(t)

= E;(t) - 1.0,

J+(t) = E”(t) + 0.2,

i=

tE[O,T],

+ 0.1,

1,2

and made several numerical experiments with piecewise constant functions Ei~ Fi to study the validity of Property 4.1 for the strategies and the game parameters defined above. It held in these experiments. The property mentioned in Remark 3.2 seems natural. Moreover, for the above parameter values the following was found to be true. If E,(t) I Ef(t) a.e., i = 1, 2, then J~(Y;(&),&) Jz(E,,

Y;(EI))

= Ji(E:, =

Jz(E1,

E2)

&!I

2

2

JI(EI,

J2(E,,

E2),

E2).

H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

671

5. Time lag strategies In practice the DMs need some time to detect and infer the available information. In this case the admissible strategies for DMi, at time t, can only depend on the values of Ej(e) for 0 I 0 I t - 0, where CJ> 0. In this section we shall consider the construction of cooperative equilibria by using such strategies. Let D1,D2,MlrM2 be as in (8)-(lo), let CJ> 0, and set tl = T - 0. Suppose Ed and xd solve problem (11) subject to the additional constraints E,(t) L

=

ki(t)

=

(bi

-

4iKi(t))x(t)

-

ci

UiX’(t)

-

Ei(t)(bi

-

(44

’

aiEi(t)X(t))

-

Kt(t)

F,(X(t))

(45)

Kt(T)

= stx(X(T)),

i= 1,2,

tE[tl,T],

and suppose xd(t) > x, for all t. Note that E’(t), t 2 tl, satisfies the open-loop Nash equilibrium conditions for the dynamic game defined by (l)-(6), the time interval [tl, T], and the initial state x(tl) = xd(tl). Remark 5.1. In order to carry out the analysis of this section it suffices that Ed, xd is any pair satisfying (44) and (45) on [tl, T], together with the condition xd(t) > x, for all t. The choice above is a Pareto-optimal choice in

this class of efforts. For i = 1, 2, define li as in (26) and ?i by rliqi~jXd= - ajxd2Ejd + bjxd - Cj - qjAjxd, Vi =

t >

O,

i #j, tECO,hl, (46)

t1.

Suppose for i = 1, 2, Vi(t)

2

O 9

tECO,

Tl,

bj - (qj + qiVi(t))Aj(t) 2 3cj/2x,,

(47) i #j,

tE[O, T].

(48)

Define y1 and yz as in (29), where Ef and vi are as above. Then Ef, xd, and Aj satisfy equations similar to (21)-(25) for y = yi, i #j. Moreover, it follows

672

H. Ehtamo and R.P. Hiimiiliiinen, A Iwo-country dynamic game model of whaling

that A,(t) > 0 for t E [O, T), i = 1, 2; cf. the proof of Lemma Let MisDi be such that MI(t) 2

pf/SkicP

B.

(49)

3

i= 1,2, i#j,

Mitt) 2 Yi(MJ)(t) 3

and

3.1 in appendix

tE[O,T],

(50)

set 0: = {Ei~L2 IO

s Ei(t)

I

Mf(t)

a.e. on [O, r]},

i=

1,2.

Define strategies yr: 0; + D1 and y;: 0; + D2 as in (37) and Ej’ and vi are as above. Then, according to Lemma 3.2,

We shall

next construct

time lag strategies

(51)

(38), where

of the form

yY(Ej)(t) = Et(t) + H(t - a)H(Gj(t - a))qp(t)Gj(t - CT), Gj _ Ej - Eg ,

(54) i#j,

such that inequalities (52) and (53) hold for these strategies. Let y”: 0; -+ D1 be any mapping such that y”(E$(t)

= -y: (E;)(t)

a.e. ,

(55)

and

where yr is as in (53), E = (y”(E,), E2), and the notation xE is the same as in (7). Then, since (55) and (56) also hold for 7: replaced by yl, it follows from Lemma 3.2 and Remark C.l in appendix C that J2(Ef,

E,d)

2

52W(E2),

E2)

,

VE,eD;.

(57)

H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

673

Now (56) holds for y” = y: defined by (54), if rf is such that

s sT-“~2(t)XE(t)H(G2(t))~l(t)G2(t) s T

~,Wx,W(G2(t

-

4h’iW2(t

-

4

dt

0

dt

2

0

T

=

Ll

i2(t - C)X& - a)H(G,(t - a))rjI(t - 0)G2(t - cr)dt

,

V E2e D;

,

(58)

where in the second integral we took into account the fact that q,(t) = 0 for t > t1.

Let R be such that

(59) and set 6 = 1 + (Ro/x,,,) ,

?Y@)= 6

226

4

A2(t)

(60)

v1(t

-

4

3

o
Suppose yT(M;)(t) I M,(t) for all t, where $ is defined as above. Let E2eD; and set E = (y7(E2), E,). Then, by the mean value theorem, (10) and (59), xE(t - a)/~&)

I

1 + (Ra/x,)

= 6,

V~ECO,Tl,

(62)

so that

Since (63) holds for all E2~ D;, (58) holds for ~7 defined by (61). Hence inequalities similar to (52) and (53) hold for the time lag strategies y: and ye defined by (54), where

674

H. Ehtamo and R.P. HiiMtilbinen. A two-country dynamic game model of whaling

provided

there are M; and M; such that, in addition to (49) and (50),

Mitt) 2 X’(Mj)(t)

9

i= 1,2,

izj,

tE[O,T].

(65)

To test the last hypothesis we estimate the magnitude of 6. If, e.g., r, K, qi, and x0 are as in section 4 and Ml(t) = M2(t) = 19 for all t, (59) holds if R = 22,800, which is obtained by inserting E,(t) = E*(t) = 19 and x(t) = K to the right-hand side of (1). If, e.g., (T= 0.1, then 6 < 1.024. In order for time lag equilibrium strategies of the form (54) to exist, it is necessary that inequalities such as (58) hold for these strategies. This obviously implies that vi(t) must be zero for t > tl, i = 1, 2, as is required by (46). Eq. (22) thus implies that the conditions (44) and (45), required from Ed and xd, are necessary; cf. Remark 5.1. A pair which satisfies these conditions, and which is a good approximation to the solution of problem (11) (measured by the payoffs) when c is small, is the following: Ed(t) = Ed(t), Z”(t) = xd(t) for t < tl, where Ed,xd solve problem (11) and Ed,Zd satisfy eq. (1) with .?‘(ti) = xd(tl), and eqs. (44) and (45) for t 2 tl. If CJ= 0.1 and the other parameters are as in section 4, then Ji’ and 52” are the same as in table 1, to the accuracy of four digits, and q,(t) < q:(t) < ql(t) + 0.01 for te[c7, T].

6. Conclusions The paper considers the construction of cooperative equilibrium strategies for a two-country dynamic game model of whaling. It has been assumed that the countries have information about each other’s whaling efforts measured by the number of the active vessels involved in whaling. We have shown how affine incentive strategies can be used to construct credible incentive equilibria which, when jointly carried out, realize a given Pareto-optimal decision. It is evident that similar results can be obtained for many other fishery and resource management games, e.g., for those studied by Hamalainen et al. (1985) as well as for more general dynamic game problems describing oligopolistic markets. Instead of considering decision-dependent equilibrium strategies, one may also consider state-dependent equilibrium strategies. However, the conditions for such strategies will be much more stringent than those for the decisiondependent equilibrium strategies. Moreover, monitoring the whale stock might, in practice, be even more difficult than monitoring the number of the active whaling boats of the opponent. The latter can be done, e.g., by observing the entering and leaving whaling boats in the opponent’s whaling harbours.

H. Ehtamo and R.P. Hiimiiliiinen. A two-country dynamic game model of whaling

675

Appendix A

In this appendix we show that ql, q2, eqs. (28), are well-defined and strictly positive. Since A(t) > 0 for TV[0, T), recall Remark 2.1, it follows from (12) that E/(t) < bi/CJiXd(t) for all t. Set J(t)

=

EP(t)(bi

-

Then J(t) is continuous

UiEf(t)Xd(t))

3

i=

1,2.

and fi(t) > 0 for all f. Ai, i = 1, 2, can be written as

C’4(r, t)f;:(z)dz

A(t) = 4(T, t)gtx(Xd(T)) + 4(z, t) = exp

(66)

r -

s

Jt

Xd(S) -

i

qiEf(S)

i=l

>12t,

3

ds

t E [0, T] .

(67)

We consider two separate cases. (i) glX(xd(7’)) = gzX(xd(T)) > 0: In this case l,(t),A2(t) > 0 for all t, and ql, q2 are well-defined. (ii) g&x”(r)) = g2,(xd(T)) = 0: In this case &(t)J2(t) > 0 for te[O, T). Moreover X(T) = --f,(T) < 0, so that for TV[O, T), vi can be defined by (28), and for t = T, by

Vi(T)= !i; Vi(t)= qjPth(T)/qiPjfj(T).

Appendix B: Proof of Lemma

(68)

3.1

Let Ez, x*, and A2 satisfy (21)-(25). Using (22), (23) can be written as ;Z2(t)= -

~)(b,-~)-i,(t)(f,(x*(t))-~-q,E,O), (69)

22(T)

= gzx(X*(T))

9

where

4 = q2 + 41'1.

(70)

By (24), the first parenthesis in (69) is strictly positive for all t, and we conclude [recall also (5)] that A,(t) > 0 for t E [0, T).

616

H. Ehtamo and R.P. Hiimiiliiinen. A two-country

dynamic game model of whaling

Define H: R3 x [0, T] + R, H(x, E, /I, t) = - ;u2E2x2 + W(x)

+ b2Ex - c2E - 4(t) Ex - 41 E,(t)x)

(71)

,

and set G={(x,ri,t)~R~~x~
012,

OstsT}.

(72)

For (x, 1, t)~ G the maximum of H is attained at the point E(x, 1, t)~ R, E(x A t) = (bz - q(tV)x 7 3 a2x2

-

~2

(73)

and E(x, ,I, t) <

b2x - c2 a x2 = p2q2x2-2c’ 2

< d8k2c;

5

M,(t)

(74)

9

2k2q2x

where in the second inequality we used the fact that the function y + (cry - a)/~‘, y > 0 and cr,p > 0, has the global maximum at y = 2/I/u. Observe that an argument similar to that used in Remark 2.1 shows that 0 < E;(t) < p#k,ci for all t. Define Ho: G-R and E”: G+R by H’(x,

1, t) = max H(x, E, 1, t) = H(x, E’(x, A, t), I, t).

(75)

EER

Then E” is given by (73), and c: H’(x, 1, t) = +--2

-

cz(b2 - q(t)4

+

+ + (bz - dt)Q2

*2x

a2x

W’(x) -

a2

q1EoW).

(76)

Let &(t) be as in (23) and (25). To show that H’(x, l,(t), t) is concave on X = {XER 1x > x,} for each t E [0, T], differentiate (76) twice to obtain

H%, 22(0,t) =

-

- (b2 - q(t)&(t))

- $1,(t)

.

(77)

H. Ehtamo and R.P. Hiimiiliiinen, A two-country dynamic game model of whaling

611

By (25), the term in the big parentheses in (77) is negative for x > x,. Hence for each t E [0, T], H$Jx, A,(t), t) I 0 for all XE X, and the concavity of Ho follows. To conclude, let E:(t), x*(t), and A,(t) be as in Lemma 3.1. By definition, H(x(r), &(r), A,(t), r) 5 HO(x(t), J,(t), r) 2

(78)

and from concavity of Ho, HO(x(r), n,(r), t) I HO(x*@), Mt), r) +

H,O(x*(t)~ b(t), W(t) - x*(t))

(79)

7

for each Ez E D2, XE C such that x(t) > x, for all t, and TV[0, r]. Since A, = - &‘(x*, &, t), (78), (79), and the concavity of g2(x) imply [integrate (78) and (79) from 0 to T and use the state equation] that E: solves problem (20). n

Appendix C: Proof of Lemma 3.2 Let E:, x*, and A2 be as in Lemma 3.1, and let yz be as in (34) and (35). Define H’: CxD2+L2 by H’(x,

E*)(t) = - t a2 E;(t)x’(t) +

&WV’W)

+ b,E,(t)x(t)

- cz E,(t)

- q~y’@,Wb(t) - q&W@)) . (80)

Let E2~ D2, and let x(t) be the solution of

W = FW) - qlf(ENb(t)

- q2&(t)x(t)

a.e., DECO, Tl ,

(81) x(0) =

x0 .

Since A,(t) 2 0 for all t and y(Ez)(t) I y’(E2)(t)

H’ (x, b)(t)

I

a.e., we get

HMO, E,(t), b(t), t) a.e. , t E [0, T] ,

(82)

where H is as in (71). Further, using (78), (79), and (82), we get

H’(x, -b)(t) I H’(x*(O,h(t), t) + Hz(x*(t),

A,(t), t)(x(t) - x*(t))

a.e. ,

TV[0, T] .

(83)

678

H. Ehtamo and R.P. Hiimiiliinen,

A two-country

Eqs. (21), (81), (83), and the concavity J2WZ)~

m

Since y(Ef)(t) = I’

2

J,(f@,),

dynamic game model of whaling

of g2(x) now imply that (84)

E2).

a.e., (36) follows.

n

Remark C.1. Note that the right-hand side of (35) may be replaced by the following condition without affecting the validity of the result: If E2 ED~ and x is the solution of (81), then

s

~2WW’(E2Nt)

0

s T

T

dt

2

~2@bW@2)(t)

dt.

(85)

0

Clark, C.W., 1985, Bioeconomic modelling and fisheries management (Wiley, New York, NY). Ehtamo, H. and R.P. Hlmlllinen, 1986, On affine incentives for dynamic decision problems, in: T. Basar, ed., Dynamic games and applications in economics (Springer-Verlag, Berlin) 46-63. Ehtamo, H. and R.P. Hamlliiinen, 1989, Incentive strategies and equilibria for dynamic games with delayed information, Journal of Optimization Theory and‘ Applications- 63, 3551370. Ehtamo. H.. J. Ruusunen. V. Kaitala. and R.P. Hamiillinen. 1988. Solution for a dynamic bargaining problem with an application to resource management; Journal of Optimization Theory and Applications 59, 391-405. Hamiillinen, R.P., A. Haurie, and V. Kaitala, 1985, Equilibria and threats in a fishery management game, Optimal Control Applications and Methods 6, 315-333. Hiimlllinen, R.P., V. Kaitala, and A. Haurie, 1984, Bargaining on whales: A differential game with Pareto-optimal equilibria, Operations Research Letters 3, 5-l 1. Haurie, A. and M. Pohjola, 1987, Efficient equilibria in a differential game of capitalism, Journal of Economic Dynamics and Control 11, 65-78. Haurie, A. and B. Tolwinski, 1984, Acceptable equilibria in dynamic games, Large Scale Systems 6, 73-89. Ho, Y.C., P.B. Luh, and G.J. Olsder, 1982, A control theoretic view on incentives, Automatica 18, 167-179. Osborne, D.K., 1976, Cartel problems, American Economic Review 66, 835-844. Tolwinski, B., A. Haurie, and G. Leitmann, 1986, Cooperative equilibria in differential games, Journal of Mathematical Analysis and Applications 119, 182-202. Zheng, Y.P. and T. Basar, 1982, Existence and derivation of optimal affine incentive schemes for Stackelberg games with partial information: A geometric approach, Internal Journal of Control 15, 997-1011. Zheng, Y.P., T. Basar, and J.B. Cruz Jr., 1984, Stackelberg strategies and incentives in multiperson deterministic decision problems, IEEE Transactions on Systems Man and Cybernetics 14, 10-24.