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2Coalitional Solution in a Game Theoretic Model of Territorial Environmental Production
Nadezhda V. Kozlovskaia
St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes, Bibliotechnaya pl. 2, St.Petersburg, 198504, Russia E-mail: kknn@yandex.ru
Abstract A game-theoretic model of territorial environmental production under Cournot competition is studied. The process is modeled as cooperative differential game with coalitional structure. The Nash equilibrium in the game played by coalitions is computed and then the value of each coalition is allocated according to some given mechanism between its members. The numerical example is given.
Keywords: optimal control, nonlinear system, dynamic programming.
1. Introduction
A game-theoretic model of territorial environmental production is considered. The model is based on the research of Petrosyan and Zaccour (2003). In the paper of Petrosyan and Zaccour (2003) the international environmental agreement is modeled, which provides a time-consistent allocation of total costs for all players under which the pollution is reduced.
The model of territorial environmental production is an extension of above mentioned model (Petrosyan and Zaccour, 2003). The region market is considered, where all firms produce homogeneous product under Cournot competition. The production process damages to the environment. Emission of each player is proportional to its output. Any firm has three types of costs: production costs, abatement costs and damage costs.
We consider the voluntary approach to environmental regulation, which became popular in a series of countries. The cooperation of firms leads to increase their profits and decrease of pollution, but the price of product is increased.
The approach of this paper is different. The more general coalitional setting is considered, when not only the grand coalition, but also a coalitional partition of players can be formed. This kind of approach was considered before in . Coalitional values for static games have been studied in a series of papers (Bloch, 1966; Owen, 1997). In a recent contribution, Owen (1997) proposed a characterization of the Owen value for static games under transferable utility. Owen (1997) defined the coalitional value for static simultaneous games with transferable payoffs by generalizing the Shapley value to a coalitional framework. In particular, the coalitional value was defined by applying the Shapley value first to the coalition partition and then to cooperative games played inside the resulting coalitions. This approach assumed that coalitions in the first level can cooperate (as players) and form the grand coalition. The game played with coalition partitioning becomes cooperative one with specially defined characteristic function: The Shapley value computed for this characteristic function is then the Shapley-Owen value for the game.
The present paper emerges from idea that it is more natural not to assume that coalitions on the first level can form a grand coalition. At first step the Nash equilibrium in the game played by coalitions is computed. Secondly, the value of each coalition is allocated according to the Shapley value in the form of PMS-vector, that was derived in the paper of Petrosyan and Mamkina (2006) . The approach was considered earlier in (Kozlovskaya et al., 2010). The main result of this paper is the calculation of this solution (PMS-vector). The main result of the paper is construction the dynamic PMS-value in the model of territorial environmental production.
2. Problem Statement
Consider a region market with n firms which produce for simplicity the same product. Let I be the set of firms involved in the game: I = {1, 2,..., n}.
Denote by qi = qi(t) the output of firm i at the instant of time t. The price of the product p = p(t) is defined as follows
p(t) = a - bQ(t) , (1)
n
where a > 0, b > 0, Q(t) = ^ qi(t) - the total output. The price function p(t) is
i=1
inverse demand function:
Q = m = ^&.
The production cost of any firm equals
Ci(qi(t)) = cqi(t), c> 0, i e I.
The game r(s0,t0) starts at the instant of time t0 from the initial state so, where
so = s(t0) is the stock of pollution at time t0. Let us denote by ei(t) the emission
of firm i at time t. The emission of firms are linear subject to output:
ei(qi(t)) = aqi(t), a > 0. (2)
Denote by ei maximum permissible emission for firm i:
0 < ei(qi(t)) < ei. (3)
We get from (3) that maximal permissible output of firm i is equal to
ei
nmax ____ _J_
Hi i
a
then maximal permissible total output equals
ee
Qmax ____ __
a
n
where e = ei. Suppose the parameters of model are such that the following
i=1
inequality is true
b
a — c-------e > 0,
a
which guarantees the nonnegativity of price (1).
Denote by s = s(t) the total stock of accumulated pollution by time t. The dynamics of pollution accumulation is defined by the following differential equation:
n
s(t) = a J2 qi(t) - Ss(t), k=i
s(to) = so, (4)
where S is the rate of pollution absorption, a > 0 is a known parameter. Any firm
has two types of costs, which are not directly connected with the production process:
abatement costs and damage costs. The abatement costs at moment of time t equals
Ei(qi(t)) = — ej(t)(2ej - ei(t)) = -aqi(2ei - aqt),
Y > 0, 0 < ei(t) < ei.
The cost function Ei(qi) increases and reaches the maximum at qi = qmax. The function Ei(qi) is concave. Damage costs depends on the stock of pollution:
Di(s(t)) = nis(t), ni > 0, i e I.
The firm i tries to maximize the profit
CO
ni(so, to; q) = / e-p(t-t0^{pqi - Ci(qi) - Di(s) - Ei(qi)}dt, (5)
to
where q = q(t) = (qi(t), q2(t),..., qn(t)), t > t0 is trajectory of production output,
0 < p < 1 is a discount rate, p is defined by (1).
3. Coalitional Solution
m
Let A = (Si, S2,..., Sm) be the partition of the set I, such that Si n Sj = %, |J Si =
i=l
m
I, \Si| = Hi, J2ni = n.
i=i
Denote by M the set M = {1, 2,...,, m}.
Suppose that each firm i from I is playing in interests of coalition Sk, to which it belongs, trying to maximize the sum of payoffs of its members, i.e.
max V nj(so; q) = qjeSk jesk
(6)
e p(t to) ^2 {pqj- cj(qj) - Dj(s) - Ej(qj)}dt,
to jeSk
where q = q(t) = (qi(t), q2(t),..., qn(t)), t > to - trajectory of production output,
0 < p < 1 - discount rate.
Without loss of generality it can be assumed that coalitions Sk are acting as players. Then at first stage the Nash equilibrium is computed. The total cost of coalition Sk is allocated among the players according to Shapley value of corresponding subgame r(Sk). The game r(Sk) is defined as follows: let Sk be the set of players involved in the game r(Sk) , r(Sk) is a cooperative game.
= max
qj £ Sk ,
Definition 1. The vector
PMS(x,t) = [PMS1(x,t),PMS2(x,t),...,PMSn(x,t)], is a PMS-vector, where PMSi(x,t) = Shi(Sk,x,t), if i e Sk, where
( * Hh '
M Di,M csk
and (S1, S2,..., Sm) is the partition of the set I.
3.1. The Construction of Coalitional Solution
Step 1. Computation of the Nash equilibrium in the game of coalitions Sk, k e M.
Each firm i from I is playing in interests of coalition Sk, to which it belongs, trying to maximize the sum of payoffs of its members (6).
The Nash equilibrium in the game of coalitions is computed by the solution of the following system:
max
CO
^ nj(so; q) = nmx [ e-p(t-to) ^ {pqj-
a ESk J
^£Sk jeSk q £Sk to jESk (7)
-Cj(qj) - Dj(s) - Ej(qj)}dt k e M, subject to equation dynamics (4).
Step 2. Computation of the characteristic function and the Shapley value in the game rS (so), k = 1, 2,...,m. Computation of the characteristic function isn’t standard ( Petrosyan and Zaccour, 2003): when the characteristic function is calculated for K, the left-out players stick to their Nash strategies Step 3. Construction of the PMS-vector.
Payoffs of all players i e I forms a PMS-vector (Petrosyan and Mamkina, 2006). PMS(so) = (PMSi(so),PMS2(so),...,PMSn(so)), PMSi(so) = ShSk (so), where ShSk (so) is the Shapley value in the game rS (so)
The Nash equilibrium is calculated with the help of Hamilton-Jacobi-Bellman equation (Dockner et al., 2000). The total cost of coalition Sk is allocated among the players according to Shapley value of corresponding subgame r(Sk). The game r(Sk) is defined as follows: let Sk be the set of players involved in the game r(Sk) , r(Sk) is a cooperative game.
Computation of the characteristic function of this game isn’t standard. When the characteristic function is computed for the coalition K e Sk, the left-out players stick to their Nash strategies. Payoffs of all players i e I forms a PMS-vector (Petrosyan and Mamkina, 2006).
3.2. The Nash Equilibrium in the Game of Coalitions
The solution of the system (7) is equivalent to the solution if the system of Hamilton-Jacobi-Belman equations
EYa
(qj(a - bQ) - cqj - TYjS + —qj(aqj - 2ej))+
H3 ,j-Sk 2
jESk (8)
dWs
H—~q——{cxQ — ^s)}, k G M,
where Q = qj, WSk is the Bellman function subject to equation od dynamics
jEl
(4). By the first Step to find the Nash equilibrium, consider the system (8).
Differentiating with respect to qi, i e Sk the right hand side of the equation (8) leads to
a — bQ — b qj — c + 7a1 qi — Yaei + a—g Sk =0, i € /, k € M. (9)
jESk
m
Let us denote QSk = qj .Then Q = ^ QSj, the system (9) is obtained in the
jESk j=1
following form
a — b "y ' Qsj — bQsk — c + 7a2® — 7«ej + a—= 0? * € I, k € M. (10)
j=i s
Summing equations (10) with respect to Sk gives
nk(a-c-b^Qs.) -nkbQSk +^a2QSk -7= 0, k G M, (11) j=i s
where eSk = ej. Solving (11) subject to QSk, find
jE Sk k
nk(a-c- bQ) - ^aeSk + ank^^ bnk - a2 y
Summing (12) with respect to the set M leads to
QSk = —------keM. (12)
-S- . dws,
■ ^ I /. | I I. - f - f /1 _ / I - ' V I Vp 1
Q =
x - rij(a - c - bQ) - ^aeSi +
bnj a2Y j=i J
m / . dWS^ s- mi 1
_ n3\a - c+ a~dT~) - Yae‘ 3 Q bnj
bnn — a2Y bnn — a2Y1
j=i j 1 j=i j 1
then one can find:
1 + E
™ nj(a- c + a—of-) - 7ae^ j=i bnn — a2Y
Q = ~-------------m ------------------• (is)
,J3
j=i bnj - a2Y
Substituting (13) in (12) gives the formula for QSk. Then solving (9) leads to:
q% = -------— (a — c — bQ — bQsk + a—-—-), i G Sk (14)
a a2Y ds
It can be shown by the usual way that the Bellman function
Wsk = Ask x + Bsk, k =1, 2,...,m, (15)
satisfies the Hamilton-Jacobi-Bellman equation (8) [13]. One can notice that
dWsk
dx
ASk .
Substituting (15) and (16) in formula (14) gives:
(16)
where
and
ei 1 z\ z\
<li =----------r—(a —c —6Q — bQsk + aAsk), i € Sk,
a a2 Y k k
A _ nk(a - c - bQ) - 7aeSfc + ankASk
Qsk — T 2 ’ ’
bnk - a2Y
Q =
E
j = i
Hj (a - c + a.Asj) - YaeSj bnj - a2Y
m bn
1+ y- ----------------°Jh-----------
j=i b-rij — a2y
it means that
Qi, Qi € [0, f- ]
o, qi < 0
i I.
Substituting (15) and (16) into the formula (8) leads to:
pASk s + pBSk = QSk (a - c - Qn) - njs +
jE Sk
+ “^~“ q™(aq™ — 2ej) + Ask(aQn — 5s), k G M.
2 jE Sk
From (18), we get the coefficients ASk and BSk:
A
Sk
E ^
jE Sk
Bsk = - (QnSk(a - c - bQn) +ASkQn + ^Y E - 2e^'))
jESk
(17)
(18)
(19)
where qin is defined by the formula (17), and
Qsk = E qn, k e m,
jE Sk
Qn = E QnSj,
jE M
(20)
n
q
j
3.3. Computation of the Characteristic function
Computation of the characteristic function of this game is not standard. When the characteristic function is computed for the coalition K c I, we suppose that the left-out players have used their Nash equilibrium strategies. The advantage of this approach is the following: such characteristic function is easier to compute. This approach requires to solve only one equilibrium problem, all others being standard dynamic optimization problems, while standard approach requires to solve 2n — 2 equilibrium problems, which are harder then a dynamic optimization one. But this approach has a limitation, because in general the characteristic function is not superadditive. The superadditivity of the characteristic function was considered in (Kozlovskaya et al., 2010; Zenkevich and Kozlovskaya, 2010).
Suppose that for parameters of the model the following conditions hold:
1 f ba(A — je) ^ 1 f b
(2 — C — '
-) < 7----(-ei-aAX
/ b — a2Y Va /
b(n + 1) — a2^\ b — a2Y ' b — a2Y - /00-,
Ei 1 A 2b . ( )
-------b ----------------2— (O' ~ c-b o.A------------------e) > 0, % € 1,
a 2bn — a2Y
where
A =-
jei
E nj
jel p + S'
Conditions (22) are the sufficient conditions of superadditivity of the characteristic function.
Computation of th Nash equilibrium in the game rS (s0) To find the Nash equilibrium the system of Hamilton-Jacobi-Bellman equations must be solved:
maxni(s; q) = max e-p(T-t){pqi — Ci(qi) — Di(s) — Ei(qi)}dr, i G Sk. (23)
ei J t
The solution of the system (23) is equivalent to the solution of the system of Hamilton-Jacobi-Bellman equations.
2
pWi = max{</j(a - bQ) - cqi-TiiS + ——qj - 7aei(jri+
dW- (24)
H—t;—(aQ ~ <^s)}j * € Sk-
ds
Differentiaiting the right hand side (24) with respect to qi and equating to 0 leads to
2 _ dWi
a — bQ — bqi—c + 'ya qi - ^aei + a—— =0, i € Sk
ds
Recall that players from I\Sk stick to the strategies (17), where Qn is defined by the formula (20)
• — b "y ' Q'sj ~ bQsk ~ bqi — c + 7c?qi — Ya&i + a~Q— — ^ (25)
jEM\{k}
Summing (25) by Sk gets nk(a — c — b E Qs,) -nkbQsk ~bQsk +7 a2Qsk -7«eSfc +a^ - 0,
jEM\{k} jESk
We obtain
nk(a — c — b Y, Qs,) -7«eSfc +a J2 ^T
ON _ jeM\{k} jESk
Qs* ~ b(nk +1) - «27 • {2b)
One can find from (25), that
$ — ~ + T~—~(a~ c~b E QrSj ~bQsk+a—^------------------------ei). (27)
a b a y ds a
' jEM\{k}
On account of (22), 0 < qf1 < The Bellman functions have the linear form:
Wi = Ais + Bi, i G Sk. (28)
Substituting (28) into (24), we obtain
pAis + pBi = qN (a — c — b QSj — bQsk) — nis+
jEM\{k}
( E
jEM\{k}
+ ^—qf!(af-2ei)+Ai(a( ^ Qg. + QgJ - Ss)
(29)
from (29) one can find
ni
A i — —
p + 5
1
Bi = — (qf^ (a — c — b ^ Qn _6QgJ + _L_^(a;v_ 2et)+
P jEM\{k} ( )
+aAi( E QSj + QN)),
jEM\{k}
where Qn. is defined by (3.2.), qN is defined by (3.3.) and
nk(a — c — b E Qn) — YaeSk + a.Ask
Ql =----------
N jEM\{k}
b(nk + 1) — a2Y
q^ = — + T — (a — c — b ^ QnSj - bQsk + aAi--------------------------------e*).
a b — a2 Y — j k a
jEM\{k}
Computation of the characteristic function for the intermidiate coalition L in the game rS(so) Let L G Sk, \L\ = l, |Sk\ = nk. Players from L maximize
max n(s; q) = max e-p(t-T){pqi — Ci(qi) — Di(s) — Ei(qi)]dr, (31)
qi,qiEL qi,qiEL J
t
on the assumption of the left-out players stick to their Nash equilibrium strategies qN. The solution of (31) is equivalent to the solution of the following Hamilton-Jacobi-Bellman equation.
pWL = max{y~^ qj (a — bQ) — c^^qj — nj s+
qj EL jEL jEL jEL
Ya2 s;—' 2 - dWL , „ r. ( )
+— ^ qj - 7« 2^ e3l3 + ~ 5s)>-
jEL jEL
Differentitating the right hand side of (32) with respect to qi and equating to 0 gives:
a — c — bQ — b qj + 7 a1 qi — Y®-Ci + qL = (33)
jEL s
Suppose the players from I\Sk stick to qn (17) and t he players from Sk\L stick to
qN (3.3.), so from (33) it can be obtained
a — c — b ' Qg. — b ' q^ — 2b^^qj-\~Yo^qi — a~Q— = 0. (34)
jEM\{k} jESk\{L} jEL s
By the same way it can be found:
l(a — c — b( Q'nj + E qj)) — iae + oAl
L ___ X-'' ,.L __ jEM\{k} jESk\{L} '
2bl — o2y
L L jEM \{k} jESk\{L}
<1 = 2^ qj =--------------------------jtt;—~-----------------------------. (35)
jEL
and then
qf = — + 7 —(a — c — b( ^ Qsj+ E qf + qL) + aAL---------------------------eL). (36)
a b — a2Y —' j — a
' jEM\{k} jESk\{L}
Because of the condition (22), 0 < qf < The characteristic function is defined by the following formula:
Wl = Al s + Bl, (37)
where
E nj
jEL
p + S
^Sj + qj
AL = —
Bl =-(qL(a-c-b( ^ Qg. + ^ qf + qL)))+
p
1 jEM\{k} jESk\{L}
+~^y^,qf(aqf —2ej)+aAL( Qns. + ^2 qf + qL)-
jEL jEM\{k} jESk\{L}
3.4. Characteristic function
We have proved that characteristic function of the game rSk(s0) is given by the following formula:
V (K,s) =
'О, K = %,
Wi (s), K = {i}, WSk (s), K = Sk, Wl (s), K = L,
where Wi(s), WL(s), WSk(s) is defined by (15), (37), (28).
3.5. The PMS-vector in the game rS (s0)
Let sn(t), t > t0 be the coaltiotnal trajectory, and players from coalition Sk players are agreed to divide the total payoff V(Sk, s0) according to Shapley value:
Sh(s) = (Shi(s),Sh2(s),...,Shn(s)),
where SHi(s) is defined by (38). The structure of the Shapley value is the following
Shi(sn(t))= Aisn(t)+ Bshi, (38)
4. The Numerical Example of the Coalitional Solution
All computations were executed in MAPLE 10.
4.1. Parameters of the Model
Consider the game of territorial environmental production of 7 players:
I = {І, 2, З, 4, Б, 6, 7}. Let the parameters of the model be the following: t0 = О - the initial instant of time , s0 = О - the initial stock of pollution,
7
p(t) = 8ООО — І^ E qi(t) - the price function,
i=1
c = З - specific production costs, p = О.ОТ - discount rate,
а = 4 - coefficient that characterizes the specific emission volume,
S = О.2 - natural rate of pollution absorption,
Y = О.ОББ - abatement costs coefficient
ё = (6ОО, 4БО, БІО, 48О, ББО, 4ІО, 4ЗО) - maximum permissible emissions, n = (4.7, Б.З, Б, Б.І, 4.8, Б.2, Б.ОБ) - damage costs coefficients.
It follows from (2) and (3) that maximum permissible outputs of players are equal to
qmax = (ІБО, ІІ2.Б, І2Т.Б, І2О, ІЗТ.Б, ІО2.Б, ІОТ.Б)
4.2. Results
Consider the following cases:
1. the Nash equilibrium
2. full cooperation
3. coalitional partition Л1 = ({І, 2, З}, {4, Б}, {6,7})
4. coalitional partition A2 = ({1, 2}, {3,4}, {5, 6,7})
5. coalitional partition A3 = ({1, 2,3,4}, {5, 6,7})
6. coalitional partition A4 = ({1, 2}, {3,4}, {5}, {6}, {7})
7. coalitional partition A5 = ({1, 2,3,4}, {5}, {6}, {7})
Table 1: Results
max NE COO A! Ai A 3 a4 A 5
p -575 1085.99 4292.47 2155.02 2156.65 2869.6 2098.54 3173.57
qi 150 96.64 80.46 82.34 117.6 85.3 79.7 56.29
<12 112.5 99.28 42.96 44.84 80.1 47.8 42.2 18.79
<13 127.5 98.32 57.96 59.84 102.9 62.8 65 33.79
<14 120 98.89 50.46 90.21 95.4 55.3 57.5 26.29
<15 137.5 97.68 67.96 107.7 84.5 109 135.7 137.5*
102.5 100.41 32.96 97.28 49.5 74 102.5* 102.5*
<?7 107.5 100.17 37.96 102.3 54.5 79 107.5* 107.5*
The first string of the table contains the prices of product in all 7 cases. The price
of product is the highest in the case off full cooperation, the price is the lowest, when
the players compete. The dynamics of pollution in any of 7 cases are the following:
sN(t) = 13828.02 - 13828.02e-02t, s1 (t) = 7415.07 - 7415.07e-02t,
sAl (t) = 11689.95 - 11689.95e-02t,
sA2 (t) = 11686.7 - 11686.7e-02t,
sAa (t) = 10260.8 - 10260.8e-02t,
sA (t) = 11802.9 - 11802.9e-02t,
sA (t) = 9652.9 - 9652.9e-02t.
Functions sAl (t), sA2 (t) sA (t) are almost coincides, so let us denote it by sAl (t) (Pic. 1). The emissions are maximin in the case of competition at any t and minimum in the case of cooperation. On Fig. 2-8 profits of any player are represented. The profit is lowest in the Nash equilibrium (competitive case) for any player. On Fig. 9 and 10 the profit functions of players in the case of cooperation and competition are represented. On Fig. 11-15 the the profit functions of players in the case of coalitional partitions are represented.
Appendix
V({1}, sN(t)) = 443161.9+240710e-02t V({2}, sN(t)) = 410614.1 + 271439e-02t V({3}, sN(t)) = 436648.3 + 256074.5e-02t V({4}, sN(t)) = 434696.7 + 261196e-02t V({5}, sN(t)) = 454212 + 245831.5e-02t V({6}, sN(t)) = 460253.9 + 266317.5e-02t V({7}, sN(t)) = 479901 + 258635.2e-0'2t Sh1(sI (t)) = 2596830.2+ 129041.3e-02t Sh2(sI (t)) = 2534222.5+ 145514.7e-02 Sh3(sI (t)) = 2633017.4+ 137278e-02t Sh4(sI (t)) = 2643934.6+ 140023.6e-02t Sh5(sI (t)) = 2630935.5+ 131786.9e-02t Sh6(sI (t)) = 2693927.9 + 142769.1e-02t Sh7(sI (t)) = 2704317.1+ 138650.8e-02t PMSis (t)) = 1004458.7 +203491.8e-02 PMSis (t)) = 9836653.3 + 229469.5e-02t PMS^s1 (t)) = 1019834.5+ 216480.6e-02t PMSis (t)) = 2116292.1 + 220810.2e-02t PMS^(s1(t)) = 2116722+ 207821.4e-02t PMS\(s1 (t)) = 2135539.1 + 225139.9e-02t PMS^s1 (t)) = 2149420.1 + 218645.4e-02t PMS2(s2 (t)) = 2098449.6+ 203435.1e-02t PMS?,(s2 (t)) = 2106725.7 + 229405.6e-02t PMS32(s2 (t)) = 2110817.6+ 216420.4e-02t PMSl(s2 (t)) = 2116851.4 + 220748.8e-02t PMS^(s2 (t)) = 1004927.8+207763.6e-02t PMS2&(s2 (t)) = 1021950.9+ 225077.2e-02t PMS%(s2 (t)) = 1052447.7+ 218584.6e-02t PMS3(s3(t)) = 1618274.1 + 1786139e-02t PMS3(s3(t)) = 1620465.4 + 2014156.7e-02t PMSi(s3(t)) = 1634972.8+ 1900147.8e-02t PMSl(s3(t)) = 1638652.2+ 1938150.8e-02 PMS52(s3(t)) = 2698271.6+ 1824141.9e-02t PMS3(s3(t)) = 2740960.6+ 1976153.7e-02t PMS7(s3(t)) = 2289477 + 1919149.3e-0'2t
PMSf(s4(t)) = 912508.3 + 205458.4e-0'2t PMS|(s4(t)) = 893504 + 231687e-0'2t PMS|(s4(t)) = 941322.5 + 218572.7e-0 2 PMS4(s4(t)) = 879177.1 + 222944.2e-02t PMS44(s4(t)) = 3133855.8 + 209829.8e-0'2t PMS4(s4(t)) = 2125635.4 + 2273156.4e-02t PMS74(s4(t)) = 2294008.6 + 2207584.7e-02t PMSf(s5(t)) = 718845.5+ 168031.3e-02t PMS25(s5(t)) = 702952+ 189482.1e-02t PMSf(s5(t)) = 724518+ 178756.7e-0'2t PMS|(s5(t)) = 724522.1 + 182331.8e-02t PMS77(s5(t)) = 5447153.4 + 171606.4e-02t PMS55(s5(t)) = 3859509.6+ 185907e-02t PMS7(s5(t)) =4100063.7+ 180544.2e-02t
8,000
6,000
4,000
2,000-
0 1 2 3 4 5
t
Fig. 1: Dynamics of pollution
Fig. 2: Profit functions of 1st player Fig. 3: Profit functions of 2nd player
10*
10*
25- f $
20-
15-
""" J —
5- K(4),.'(()) TTTTTTTTTTTTTTTTTTTl
Fig. 4: Profit functions of 3st player Fig. 5: Profit functions of 4nd player
10®
10*
§
Shi
13
VIIS},***'»
1 1 1 1 1 1 1 I it rn t'H i i i i |
00 25 50 7.5 100
t
Fig.6: Profit functions of 5st player
Fig.7: Profit functions of 6nd player
Fig. 9: Profit functions of player in the Fig. 10: Profit functions of player in the Nash equilibrium cooperation
104 104
Fig. 11: Profit functions of player in the Fig. 12: Profit functions of player in the
case Ai case A2
Fig. 13: Profit functions of player in the Fig. 14: Profit functions of player in the case _i.: case Zi4
10*
I I I I I I I I I I I III I I F I I |
00 25 50 7.5 100
Fig. 15: Profit functions of player in the case A5
References
Albizur, M. and J. Zarzuelo (2004). On coalitional semivalues . Games and Economic Behaviour, 2, 221-243.
Bloch, F.(1966). Sequantal formation of coalitions with fixed payoff division. Games and Economic Behaviour, 14, 90-123.
Demsetz, H. (1967). Toward a theory of property rights. The American Economic Review, 57(2), 347-359.
Dockner, E. J., Jorgensen, S., van Long, N. and G. Sorger (2000). Differential Games in Economics and Management Science. Cambridge: Cambridge University Press, p.485.
Kozlovskaya, N., Petrosyan, L. and N. Zenkevich (2010). Coalitional Solution of a Game-Theoretic Emission Reduction Model. International Game Theory Review, 12(3), 275286.
Nash, J. F. (1950). Equilibrium points in n-person games. Proc. Nat. Acad. Sci. USA. Vol. 36, pp. 48-49.
Owen, G. (1997). Values of games with a priory unions. Mathematical Economy and Game Theory / R. Henn and O. Moeschlin (eds.). (Berlin), 78-88.
Petrosyan, L. and S. Mamkina (2006). Dynamic games with coalitional structures. Intern. Game Theory Review. 8(2), 295-307.
Petrosyan, L. and G. Zaccour (2003). Time-consistent Shapley value allocation of pollution cost reduction. J. of Economic Dynamics and Control, 27, 381-398.
Shapley, L. S. (1953). A value for n-person games Contributions to the Theory of Games II. Prinneton: Princeton University Press, 57-69.
Zenkevich, N. and N. Kozlovskaya (2010). Stable Cooperation under Environmental Constrains. International Game Theory Review, 12(4), 453-470.
