In Game Theory and Applications, Volume X WITH MIGRATION: RESERVED TERRITORY APPROACH. Vladimir V. Mazalov. Anna N. Rettieva

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1 In Game Theory and Applicaions, Volume X Edied by L.A. Perosjan and V.V. Mazalov, pp.97{10 ISBN X c 004 Nova Science Publishers, Inc. A FISHERY GAME MODEL WITH MIGRATION: RESERVED TERRITORY APPROACH Vladimir V. Mazalov Anna N. Reieva Insiue of Applied Mahemaical Research Karelian Research Cener of RAS Russia 1. Inroducion A dynamic game model of a bioresource managemen problem (sheries) is considered. The cener (sae) which deermines he reserved porion of he reservoir (where shing is prohibied), and he players (shing farms) which harves sh are he paricipans of he game. Each player is an independen decision maker, guided by he consideraions of maximizing he pro from sh sale. We consider nie and innie planning horizon. Ponryagin's maximal principle and Hamilon-Jacobi-Bellman equaion were applied o deermine Nash and Sakelberg equlibriums.. Game model Le us consider he cener, which deermines he reserved area of he reservoir denoed by s, 0 s 1. We consider he sraegies of he wo players which exploi he sh sock during T ime periods. Le us divide he waer area ino wo pars: S 1 and S, where shing is prohibied and allowed, respecively. Denoe by x 1 and x he size of he populaion per uni area of S 1 and S, respecively. Then s = S 1 =S is he reserved area. There is a migraory exchange beween he wo pars of he reservoir wih he exchange coecien = q=s, where q is he exchange rae. The dynamics of he shery is described by he sysem of equaions: x 0 1 () ="x 1 ()+ 1 (x () ; x 1 ()) x 0 () ="x ()+ (x 1 () ; x ()) ; u() ; v() x i (0) = x 0 i (1) where x 1 () 0 { size of he populaion a ime in he reserved area x () 0 { size of he populaion a ime in he area where shing is allowed " { naural growh rae of he populaion u() 0 { rs farm's shing eors a ime v() 0 { second farm's shing eors a ime s() { reserved porion of he reservoir and i = q=s { coeciens of he migraory exchange, i =1. Then he payos of he wo players over a xed ime period [0T]are J 1 = R T 0 e;r [m 1 ((x 1 () ; x 1 ) +(x () ; x ) )+ u() ; p 1 u()] d J = R T 0 e;r [m ((x 1 () ; x 1 ) +(x () ; x ) )+c v() ; p v()] d () c004 V.V. Mazalov, A.N. Reieva

2 9 V.V. Mazalov, A.N. Reieva where x i () { size of he populaion which is opimal for reproducion, m i { penaly for deviaion from he sae (x 1 x ), c i {caching coss of he i-h player and p i { marke price for each player, i =1. Le's denoe c ir = c i e ;r m ir = m i e ;r p ir = c i e ;r i=1 : We consider dieren opimaliy principles Nash opimal soluion We are ineresed in he opimal soluion of he following problem: The Hamilon funcion for player I is J1 (u v ) J 1 (u v ) u J 1 (u v ) J 1 (u v) v: H 1 = m 1r ((x 1 () ; x 1 ) +(x () ; x ) )+r u() ; p 1r u() ()("x 1 ()+ 1 (x () ; x 1 ()))+ + 1 ()("x ()+ (x 1 () ; x ()) ; u() ; v()) : Le's nd he maximum of H 1 =ru() ; p 1r ; 1 () =0: Then he maximum is achieved a he poin According o he maximum principle [4] () and he ransversabiliy condiions are In a similar manner, for player II I yields u() = 1()+p 1r r : = ;m 1r (x 1 () ; x 1 ) ; 11 ()(" ; 1 ) ; 1 () = ;m 1r (x () ; x ) ; 1 ()(" ; ) ; 11 () 1 1i (T )=0 i=1 : H = m r ((x 1 () ; x 1 ) +(x () ; x ) )+c r v() ; p r v()+ + 1 ()("x 1 ()+ 1 (x () ; x 1 ()))+ + ()("x ()+ (x 1 () ; x ()) ; u() ; v()) : and equaions for addiional variables are () v() = ()+p r r = ;m r (x 1 () ; x 1 ) ; 1 ()(" ; 1 ) ; () = ;m r (x () ; x ) ; ()(" ; ) ; 1 () 1

3 A Fishery Game Model wih Migraion 99 wih he ransversabiliy condiions i (T )=0 i=1 : Finally, o nd he Nash equilibrium we have o solve he following sysem: x 0 1() ="x 1 ()+ 1 (x () ; x 1 ()) x 0 () ="x ()+ (x 1 () ; x ()) ; 1()+p1r c r1 0 11() =;m 1r (x 1 () ; x 1 ) ; 11 ()(" ; 1 ) ; 1 () ; ()+pr r 0 1() =;m 1r (x () ; x ) ; 1 ()(" ; ) ; 11 () 1 0 1() =;m r (x 1 () ; x 1 ) ; 1 ()(" ; 1 ) ; () 0 () =;m r(x () ; x ) ; ()(" ; ) ; 1 () 1 i1 (T )= i (T )=0 x i (0) = x 0 i : Le's recall our noaions and inroduce new variables: ij = ije r ij=1 : Then for hese variables we ge he sysem: x 0 1() ="x 1 ()+ 1 (x () ; x 1 ()) x 0 () ="x ()+ (x 1 () ; x ()) ; 1()+p 1 ; ()+p 0 11() =;m 1 (x 1 () ; x 1 ) ; 11 ()(" ; 1 ; r) ; 1 () 0 1 () =;m 1(x () ; x ) ; 1 ()(" ; ; r) ; 11 () 1 0 1() =;m (x 1 () ; x 1 ) ; 1 ()(" ; 1 ; r) ; () 0 () =;m (x () ; x ) ; ()(" ; ; r) ; 1 () 1 i1 (T )= i (T )=0 x i (0) = x 0 i : (3) Theorem 1. and u () = 1 ()+p 1 v () = ()+p wih i, i =1 saisng (3), form he Nash opimal soluion of he problem (1)-(). Proof. See []. Example. Modelling was carried ou for he following values: q =0:, 1 = = q=s, " =0:0, m 1 = m =0:09, = c =10,p 1 = p =, T =00,r =0:1. Le he iniial size of he populaion be x 1 (0) = 50, x (0) = 50. And he opimal for reproducion populaion sizes are x 1 =andx =. You can see he opimal values of u () andv () (equal for boh players) in Fig.3. The gure indicaes ha he player's sraegies should equal almos all he ime. The size of he populaion in he reserved area grows from 50 o 1 individuals (Fig.1). The size of he populaion in he area where shing is allowed grows from 50 o 140 individuals (Fig.). The players' pros given ha hey use he opimal sraegies, are J 1 = J =103:

4 V.V. Mazalov, A.N. Reieva Figure 1. Values of x 1() Figure. Values of x () Figure 3. Values of u () 1.. Sackelberg opimal soluion We are ineresed in he opimal soluion of he following problem: max vr(u ) J 1(u v) max vr(u) J 1(u v) where R(u) =fv j J (u v) J (u v 0 )g and v R(u ) : For solving his problem we use Ponryagin's maximum principle modied for a wo level game. We x some sraegy u(), and he Hamilon funcion for player II is Wherefore H = m r ((x 1 () ; x 1 ) +(x () ; x ) )+c r v() ; p r v()+ + 1 ()("x 1 ()+ 1 (x () ; x 1 ()))+ + ()("x ()+ (x 1 () ; x ()) ; u() ; v()) : v() = ()+p r r and he equaions for addiional variables are 0 1 () =; wih he ransversabiliy condiions = ;m r (x 1 () ; x 1 ) ; 1 ()(" ; 1 ) ; () = ;m r (x () ; x ) ; ()(" ; ) ; 1 () 1 i (T )=0 i=1 : We subsiue his sraegy of player II ino he sysem (1) and combine i wih he equaions for addiional variables x 0 1() ="x 1 ()+ 1 (x () ; x 1 ()) x 0 () ="x ()+ (x 1 () ; x ()) ; u ; ()+pr r 0 1 () =;m r(x 1 () ; x 1 ) ; 1 ()(" ; 1 ) ; () 0 () =;m r (x () ; x ) ; ()(" ; ) ; 1 () 1 1 (T )= (T )=0 x i (0) = x 0 i :

5 A Fishery Game Model wih Migraion 101 Then we repea all operaions using he Ponryagin's maximum principle. The Hamilon funcion for player I is H 1 = m 1r ((x 1 () ; x 1 ) +(x () ; x ) )+r u() ; p 1r u() ()("x 1 ()+ 1 (x () ; x 1 ()))+ + 1 ()("x ()+ (x 1 () ; x ()) ; u() ; ()+pr r )+ + 1 ()(;m r (x 1 () ; x 1 ) ; 1 ()(" ; 1 ) ; () )+ + ()(;m r (x () ; x ) ; ()(" ; ) ; 1 () 1 ) Le's nd he maximum of H 1 =ru() ; p 1r ; 1 () =0: Then he maximum is achieved a he poin u() = 1()+p 1r r : According o he maximal principle [4] 0 11() =;m 1r (x 1 () ; x 1 ) ; 11 ()(" ; 1 ) ; 1 () +m r 1 () 0 1 () =;m 1r(x () ; x ) ; 1 ()(" ; ) ; 11 () 1 +m r () 1 () 1 = 1 ()(" ; 1 )+ () 1 0 = 1() r + ()(" ; )+ 1 () and he ransversabiliy condiions are i (T )=0 i (0) = 0 : Finally, o nd he Sackelberg equilibrium we have osolve he following sysem: x 0 1() ="x 1 ()+ 1 (x () ; x 1 ()) x 0 () ="x ()+ (x 1 () ; x ()) ; 1()+p1r ; ()+pr r r 0 11 () =;m 1r(x 1 () ; x 1 ) ; 11 ()(" ; 1 ) ; 1 () +m r 1 () 0 1() =;m 1r (x () ; x ) ; 1 ()(" ; ) ; 11 () 1 +m r () 0 1() =;m r (x 1 () ; x 1 ) ; 1 ()(" ; 1 ) ; () 0 () =;m r (x () ; x ) ; ()(" ; ) ; 1 () 1 0 1() = 1 ()(" ; 1 )+ () 1 0 () = 1() r + ()(" ; )+ 1 () i1 (T )= i (T )=0 x i (0) = x 0 i i(0)=0: Le's recall our noaions and inroduce new variables: ij = ije r Then for hese variables we ge he sysem: x 0 1 () ="x 1()+ 1 (x () ; x 1 ()) x 0 () ="x ()+ (x 1 () ; x ()) ; 1()+p () =;m 1 (x 1 () ; x 1 ) ; 11 ()(" ; 1 ; r) ; 1 () +m 1 () ; ()+p 0 1() =;m 1 (x () ; x ) ; 1 ()(" ; ; r) ; 11 () 1 +m () 0 1() =;m (x 1 () ; x 1 ) ; 1 ()(" ; 1 ; r) ; () 0 () =;m (x () ; x ) ; ()(" ; ; r) ; 1 () 1 0 1() = 1 ()(" ; 1 )+ () 1 0 () = 1() + ()(" ; )+ 1 () i1 (T )= i (T )=0 x i (0) = x 0 i i(0)=0: (4)

6 10 V.V. Mazalov, A.N. Reieva Theorem. and The sraegies u () = 1 ()+p 1 v () = ()+p wih i, i =1 saisng (4), form he Sackelberg opimal soluion of he problem (1)-(). Example. Modelling was carried ou for he same values as in secion 1.1. You can see he opimal values of u () in Fig.6 and hose of v () in Fig.7. The gures indicae ha player's I saregy should equal 6 almos all he ime, and player's II sraegy { equal 1. The size of he populaion in he reserved area grows from 50 o 00 individuals (Fig.4). The size of he populaion in he area where shing is allowed grows from 50 o 10 individuals (Fig.5), The players' pros given ha hey use he opimal sraegies, are J 1 = ;53:714447, J =943: Figure 4. Values of x 1() Figure 5. Values of x () Figure 6. Values of u () Figure 7. Values of v () Le's compare he players' payos when we use dieren opimaliy principles. Pro of player I, which corresponds o dieren sizes of he reserved area s(), are shown in Table 1, and pro of playerii{intable.

7 A Fishery Game Model wih Migraion 103 Table 1. Player I pro s() Nash Sakelberg Table. Player II pro s() Nash Sakelberg In he case of he Nash equilibrium boh players are in he same condiions, so heir sraegies and pros are equal. In he case of he Sackelberg equilibrium player I is he leader and, as Tables 1, show, his equilibrium is beer for player I, bu worse for player II. This soluion gives he gain o player I, whereas player II carries all he expenses of mainaining a sable populaion developmen.. Model over an innie horizon The dynamics of he shery, as before, is described by he sysem of equaions: x 0 1 () ="x 1 ()+ 1 (x () ; x 1 ()) x 0 () ="x ()+ (x 1 () ; x ()) ; u() ; v() x i (0) = x 0 i (5) where x 1 () 0 { size of he populaion a ime in he reserved area x () 0 { size of he populaion a ime in he area where shing is allowed " { naural growh rae of he populaion u() 0 { rs farm's shing eors a ime v() 0 { second farm's shing eors a ime s() { reserved porion of he reservoir and i = q=s { coeciens of he migraory exchange, i =1. Then he discouned payos of he wo players over an innie horizon a a rae r are J 1 = R 1 0 e;r [m 1 ((x 1 () ; x 1 ) +(x () ; x ) )+ u() ; p 1 u()] d J = R 1 0 e;r [m ((x 1 () ; x 1 ) +(x () ; x ) )+c v() ; p v()] d where x i () { populaion size opimal for reproducion, m i { penaly for deviaion from he sae (x 1 x ), c i {caching coss of he i-h player and p i {marke price for each player, i =1. When we invesigaed he model wih a nie planning horizon, we used Ponryagin's maximum principle. The innie horizon gives us an opporuniy o use Hamilon{Jacobi{ Bellman equaion. Le's consider dieren opimaliy principles for innie horizon model..1. Nash opimal soluion Le's x he second player's sraegy and consider he problem of deermining he opimal sraegy of player I. Dene he value funcion V (x) for our problem Z 1 V (x 1 x )=minf e ;r [m 1 ((x 1 () ; x 1 ) +(x () ; x ) )+ u() ; p 1 u()]g : u 0 (6)

8 104 V.V. Mazalov, A.N. Reieva The Hamilon{Jacobi{Bellman equaion is rv (x 1 x ) = minfm 1 ((x 1 ; x 1 ) +(x ; x ) )+ u ; p 1 u ("x (x ; x 1 ))+ ("x + (x 1 ; x ) ; u ; v)g Le's nd he maximum over u: u ; p 1 =0: Then he maximum is achieved a he poin Subsiuing i ino he equaion, we ge u + p 1 )= : rv (x 1 x )=m 1 ((x 1 ; x 1 ) +(x ; x ) ) +p1) ("x (x ; x 1 ))+ ("x + (x 1 ; x ) ; v) I is easy o verify ha he quadraic form provides a soluion o his equaion. Le V (x 1 x )=a 1 x 1 + b 1 x 1 + a x + b x + kx 1 x + l. Then he player's I sraegy is u(x) = a x + b + kx 1 + p 1 where he coeciens saisfy he sysem of equaions ra 1 = m 1 ; k 4 +a 1 (" ; 1 )+k rb 1 = ;m 1 x 1 ; kb ; kp1 ra = m1 ; a +a (" ; )+k 1 rb = ;m 1 x ; ab rk = ; ak + b 1 (" ; 1 )+b ; kv ; ap1 + b (" ; )+b 1 1 ; a v + k(" ; 1 )+a 1 1 +a + k(" ; ) rl = m 1 x 1 + m 1 x ; b 4 ; bp1 ; p 1 4 ; b v (7) Analogously for player II v(x) = x + + k x 1 + p where he coeciens saisfy he sysem of equaions r 1 = m ; k (" ; 1 )+k r 1 = ;m x 1 ; k ; kp1 r = m ; + (" ; )+k 1 r = ;m x ; rk = ; k + 1 (" ; 1 )+ ; k u ; p1 + (" ; )+ 1 1 ; u + k (" ; 1 ) k (" ; ) rl = m x 1 + m x ; 4 ; p1 ; p 1 4 ; u () So, we proved he following heorem.

9 A Fishery Game Model wih Migraion 105 Theorem 3. and u (x) = a x + b + kx 1 + p 1 v (x) = x + + k x 1 + p form he Nash opimal soluion of he problem (5)-(6), where he coeciens are dened from (7) and (). Example. Modelling was carried ou for he following values: q =0:, 1 = = q=s, " =0:0, m 1 = m =0:09, = c =10,p 1 = p =, T =00,r =0:1. Le he iniial size of he populaion be x 1 (0) = 50, x (0) = 50. And he opimal for reproducion populaion sizes are x 1 =andx =. You can see he opimal values of u () and v () (equal for boh players) in Fig.10. The gure indicaes ha players' sraegies should increase from o. The size of he populaion in he reserved area grows from 50 o 10 individuals (Fig.). The size of he populaion in he area where shing is allowed grows from 50 o 90 individuals (Fig.9). The players' payos given ha hey use he opimal sraegies, are J 1 = J =3: Figure. Values of x 1 () Figure 9. Values of x () Figure 10. Values of u ().. Sackelberg opimal soluion Using Hamilon{Jacobi{Bellman equaion for player II we ge v(x) = x + + k x 1 + p + u where he coeciens are dened from (). Dene he value funcion V (x) for player's I problem Z 1 V (x 1 x )=minf e ;r [m 1 ((x 1 () ; x 1 ) +(x () ; x ) )+ u() ; p 1 u()]g : u 0 The Hamilon{Jacobi{Bellman equaion is rv (x 1 x ) = min u fm 1 ((x 1 ; x 1 ) +(x ; x ) )+ u ; p 1 1 ("x (x ; x 1 ))+ ("x + (x 1 ; x ) ; u ; x++kx1+p+u )g

10 106 V.V. Mazalov, A.N. Reieva Le's nd he maximum over u: wherefore Subsiuing i ino he equaion, we ge u (1 + ) ; p 1 =0 u ( + )+p 1 c )=4 c : rv (x 1 x ) = m 1 ((x 1 ; x 1 ) +(x ; x ) ) (+) c + p 1 ("x (x ; x 1 ))+ (x 1 ( ; k )+x (" ; ; c )+ c+ ; +p c ) I is easy o verify ha he quadraic form provides a soluion o his equaion. Le V (x 1 x )=a 1 x 1 + b 1 x 1 + a x + b x + gx 1 x + l. Then he player's I sraegy is u(x) =((a x + b + gx 1 )( + )+p 1 c )=4 c where he coeciens saisfy he sysem of equaions ra 1 = m 1 + g (+) +a 1 (" ; 1 )+g( ; k c rb 1 = ;m 1 x 1 +b g (c+) c ra = (c m1+a +) c rb = ;m 1 x + a b (+) rg = a g (c+) c ) + g (c+) c b 1 (" ; 1 )+b ( ; k ) ; g(+p) + g 1 +a (" ; ; c ) c rl = m 1 x 1 + m 1 x + b (c+) c So, we proved he following heorem. + a (+) c + b (" ; ; c )+b 1 1 ; a +p c + g(" ; 1 )+a 1 1 +a ( ; k )+g(" ; ; + b (+) c + p 1 ; b(+p) c ) (9) Theorem 4. The sraegies u (x) = ((a x + b + gx 1 )( + )+p 1 c )=4 c and v (x) =((a x + b + gx 1 )( + )+4 c ( x + + k x 1 )+ (p 1 + p ))= c form he Sackelberg opimal soluion of he problem (5)-(6), where he coeciens are de- ned from () and (9). Example. Modelling was carried ou for he same values as in secion.1. You can see he opimal values of u () in Fig.13 and hose of v () in Fig.14. The gure indicaes ha player's I sraegy should increase from 4:6 o 5:, and player's II sraegy { from 0 o 14. The size of he populaion in he reserved area grows from 50 o 10 individuals (Fig.11). The size of he populaion in he area where shing is allowed grows from 50 o individuals (Fig.1). The players' pros given ha hey use he opimal sraegies, are J 1 =163:4795 and J =1153:6670.

11 A Fishery Game Model wih Migraion Figure 11. Values of x 1() Figure 1. Values of x () Figure 13. Values of u () Figure 14. Values of v () Le's compare he players' pros when we use dieren opimal principls. Pro of player I, which corresponds o dieren sizes of he reserved area s(), are shown in Table 3, and pro of playerii{intable 4. Table 3. Player I pro s() Nash Sakelberg Table 4. Player II pro s() Nash Sakelberg As in he case of he nie planning horizon, he Sackelberg equilibrium is beer for player I, bu worse for player II han he Nash equilibrium. In pracice, he Hamilon{Jacobi{Bellman equaion is more convenien for long{erm planning.

12 10 V.V. Mazalov, A.N. Reieva REFERENCES 1. C.W. Clark, Bioeconomic modelling and sheries managemen, Wiley, NY, H. Ehamo, R.P. Hamalainen, A cooperaive incenive equilibrium for a resource managemen problem, J. of Economic Dynamics and Conrol 17 (1993), 659{ A.A. Perosjan, V.V. Zakharov, Mahemaical models in environmenal policy analysis, Nova Scince Publ., New York, L.S. Ponryagin, V.G. Bolyanski, R.V. Gamrelidze, E.F. Mishenko, Mahemaical heory of opimal processes, Nauka, 1969, (in Russian). 5. V.I. Gurman, G.N. Konsaninov, Planning and predicion of naural{economic sysems, Nauka, Novosibirsk, 194, (in Russian). 6. V.V. Mazalov, A.N. Reieva, Abou one resource managemen problem, Survey in Applied and Indusrial Mahemaics 9, no. (00), TVP, Moscow, 93{306, (in Russian). 7. V.V. Mazalov, A.N. Reieva, A shery game model wih age{disribued populaion: reserved erriory approach, Game Theory and Applicaions 9 (003), Nova Science Publishers, Inc., 15{139.. T. Basar, G.J. Olsder, Dynamic noncooperaive game heory, Academic Press, New York, 19.