CC BY
3Games with Differently Directed Interests and Their Application to the Environmental Management
Guennady A.Ougolnitsky
Southern Federal University,
8A Milchakov St., 344090 Rostov-on-Don, Russia E-mail: ougoln@mail.ru
Abstract. The resolution of numerous ecological problems on different levels should be realized on the base of sustainable development concept that determines the conditions to the state of environmental-economic systems and impacting control actions. Those conditions can’t be realized by themselves and require special collaborative efforts of different agents using both cooperation and hierarchical control. To formalize the inevitable trade-offs it is natural to use game theoretic models including the games with differently directed interests.
Keywords: differential games, sustainable development, cooperation, dynamic stability, hierarchical control
JEL classification codes: C70, Q56
1. Introduction
A rapid growth of production and human population, urbanization and activity of transnational corporations exert an essential influence on the environment and result in many ecological problems on local, regional, and global level. Those problems are environmental pollution, water and wind erosion, acid rains, global warming, extinction, non-renewable resource exploitation, and so on. A successful solution of the problems is possible only within a sustainable development due to which an economic growth doesn’t violate the ecological equilibrium. Sustainable development requires strong collaborative efforts of states, corporations, social organizations and individuals. Because of many concerned agents it is evident that sustainable development is a conflict controlled process which must be described by game theoretic models. A special attention has to be given to the classes of differential games describing hierarchical relations, cooperation and dynamic stability (time consistency) .
A general survey of non-cooperative differential games is given by Basar and Olsder (1995), hierarchical games are considered by Fudenberg and Tirole (1991). Group optimal cooperative solutions in differential games are proposed in Dockner and Jorgensen (1984), Dockner and Long (1993), Tahvonen (1994), Maler and de Zeeuw (1998), Rubio and Casino (2002). Haurie and Zaccour (1986, 1991), Kaitala and Pohjola (1988, 1990, 1995), Kaitala et al (1995), Jorgensen and Zaccour (2001) presented classes of transferable-payoff cooperative games with solutions which satisfy group optimality and individual rationality. In Leitmann (1974, 1975), Tolwinsky et al (1986), Hamalainen et al (1986), Haurie and Pohjola (1987), Gao et al (1989), Haurie (1991), Haurie et al (1994) are presented solutions satisfying group optimality and individual rationality at the initial time in cooperative games with non-transferable payoffs. In some papers (see Hamalainen et al (1986), Tolwinsky et al
(1986)) threats are used to ensure that no players will deviate from the initial cooperative strategies throughout the game horizon. Environmental applications are considered in Dockner and Long (1993), Hamalainen et al (1986), Jorgensen and Zaccour (2001), Kaitala and Pohjola (1988, 1995), Kaitala et al (1995), Mazalov and Rettieva (2005, 2005, 2006, 2007), Yeung (2009) and others.
The problem of dynamic stability in differential games has been intensively explored in the past three decades (Yeung 2009). Haurie (1976) raised the problem of instability when the Nash bargaining solution is extended to differential games. Petrosyan (1977) formalized the notion of dynamic stability in differential games. Kidland and Prescott (1977) introduced the notion of time consistency related to economic problems. Petrosyan and Danilov (1979, 1982) proposed an ’’imputation distribution procedure” for cooperative solution. Petrosyan (1991, 1993b) studied the dynamic stability of optimality principles in non-zero sum cooperative differential games. Petrosyan (1993a) and Petrosyan and Zenkevich (1996, 2007) have presented a detailed analysis of dynamic stability in cooperative differential games, in which the method of regularization was used to construct time consistent solutions. Yeung and Petrosyan (2001) designed time consistent solutions in differential games and characterized the conditions that the allocation-distribution procedure must satisfy. Petrosyan and Zenkevich (2009) proposed the conditions of sustainable cooperation. Those and other results permit to propose a mathematical formalization of the concept of sustainable development on the base of a game theoretic model (Ougolnitsky 2002).
2. Games with differently directed interests
A general model of game with differently directed interests may be represented as follows (case n = 2 is considered for simplicity):
where terms f describe coincident interests, terms g - antagonistic interests, and terms fi , f2 - independent interests of the players. Specific cases of the model (1) are games with partly coincident interests
U2(xi,X2) = f (xi,X2) - g(xi, X2) + f2(xi,X2) ^ max .
(1)
U2(xi, X2) = f (xi,x2) + f2(xi,X2) ^ max .
(2)
and games with partly antagonistic interests
U2(xi,X2) = -g(xi, X2) + f2(xi, X2) ^ max .
(3)
For example, a well-known Germeyer-Vatel model (1974)
ui(xi, X2) = f (ai — xi, a2 — X2) + fi(xi) ^ max ;
xi€Xi
U2(xi, X2) = f (ai — xi,a2 — X2) + f2(x2) ^ max . (4)
X2GX2
is a specific case of a game with partly coincident interests (2) for which an existence of strong equilibrium is proved.
There are many natural applications of models (1)-(3) to the management problems. Thus, a model of motivation (impulsion) management
ui(xi,x2)= xif (X2) + fi(xi, X2) ^ max ;
0<xi < i
U2(xi,X2) = (1 — Xi)f (x2) + f2(xi,X2) ^ max . (5)
0<x2<i
belongs to the type (3). In our previous paper (Ougolnitsky 2010) a specific case of the model (2) was considered, namely a pricing model of hierarchical control of the sustainable development of the regional construction works complex
ul(p,Po,p) = Spa(p) - Mp(p,p0) ->• max
0 < P0 < p < Pmax
uf(p,Po,p) =pa(p)+p£(p)( 1 ~ol(p)) ->• max 0 < p < Pmax.
Here p is sales price; p0 - normative price of social class residential real estate development; p - limit price of social class residential real estate development; M » 1
- penalty constant; p(p,p0) = { °’ p < pp0, ; 5 - Administration bonus parameter for
social class residential real estate development sales; pmax - ”overlimit” price of social class residential real estate development (there are no sales if p > pmax); a(p) is share of residential real estate development bought by Administration with warranty; £(p) - share of another residential real estate development successfully sold by Developer without help.
There are many other examples of game theoretic models of the described type in management, such as models of resource allocation considering private interests, models of organizational monitoring optimization, models of reward systems, corruption models and so on. The games are considered as hierarchical ones.
3. Cooperative differential game of resource allocation with partly coincident interests on the base of compulsion
A dynamical model of hierarchical control by resource allocation in a tree-like system may be represented as follows:
E / gi(x(t), ui(t))dt ^ max;
0 < qi(t) < 1; ri(t) > 0, i € M;^2 ri(t) = M € [to,T];
i^M
fi(x(t),Ui(t)) = gi(x(t),Ui(t)) + hi(x(t),Ui(t)),i € M,t € [to,T]; qi(t) < Ui(t) < ri(t),i € M,t € [to, T]; dx
— = F(x(t), wi(t),..., un(t)), x(0) = xQ. dt
The tree-like hierarchical structure consists of n +1 elements: one element of the higher level (Leader) designated by index 0 and n elements of the lower level (Followers). Let’s denote the whole set of elements by N = {0,1,..., n}, and the set of elements on lower level by M = {1,...,n}. The Leader controls the Followers separately by control variables of compulsion qi (administrative impacts) and control variables ri of impulsion (resources) (Ougolnitsky 2002). Without lack of generality we may consider the total resource equal to one. After receiving the values of qi and ri each Follower i € M chooses the control value ui (environmental protection efforts). The objective of Leader is to maximize the payoff function f0, and the objective of each Follower is to maximize fi . We suppose that gi represents the ecological interests for the whole system, and hi represents private economic interests of the i-th Follower; functions gi are non-negative, continuous, differentiable, monotonically increase on ui , gi(x(t), 0) = 0; functions hi are non-negative, continuous, differentiable, monotonically decrease on ui , hi(x(t),ri(t)) = 0 for each t € ^ T].
To define a cooperative differential game with initial state x0 and time horizon [t0,T] on the base of compulsion it is sufficient to build a characteristic function v : 2n ^ R using the definition of compulsion (Ougolnitsky 2002). In this case the values of ri are fixed, and the Leader chooses qi , i € M, as open-loop strategies qi(•) = {qi(t)}, t € [to, T]. We have
v ({0}; xo,T - to) =
max min
o<qi(,)<ri(,),ieMu(t)eRi(qi(»),ri(,)),ieM ^ Jt0
y~] / gi(x(t),ui(t))dt
Jt 0
E / gi(x(t),ri(t))dt; t0
Ri(qi(t),ri(t)) = Arg max fi(x(t),u(t)), i € M,t € [to,T];
qi(*)<ui(t)<ri(*)
the compulsion mechanism has a form qi(t) = ri(t) ^ u*(t) = ri(t), i € M, where u*(t) is an optimal reaction of i-th Follower on qi(t), t € [to,T]. Respectively,
v({i}; xo,T - to) = I gi(x(t),ri(t))dt +f hi(x(t),r (t))dt =
t0 t0
I gi(x(t),ri(t))dt,i € M;
t0
v(K; xo ,T - to) = E/ fi(x(t),ri (t))dt = E/ gi(x(t),r (t))dt,K C M.
t0 t0
So, if Leader has full possibilities of compulsion, he can compel all Followers to follow the ecological interests only. Farther we get
v({0} U K; xo, T - to) =
= max max min [> / Qi(x(t),ui(t))dt
o<qi(^)<ri(.)1ieMqi(.)<ui(t)<ri(.)1ieKqi(.)<ui(t)<ri(.)1ieM\K ^ J ,
ieM t0
r t pT
+ E / gi(x(t),ui (t))dt + E/ hi(x(t),ui(t))dt] =
ieK^to ieK^to
f t
= max max min [>(2 / gi(x(t),ui(t))dt
o<qi(*)<ri(*),ieM qi(*)<ui(t)<ri(*),ieK qi(*)<ui(t)<ri(*),ieM\KfK J^
r T ,-T
+ hi(x(t),ui(t))dt) + ^2 / gi(x(t),ui(t))dt] =
to ieM\K 0
= E/ (2gi(x(t),u*(t))dt + hi(x(t),u*(t)))d,t + ieK^to
f t
+ E / gi(x(t),ri(t))dt, t0
ieM\K '
!■ T
where E / (2gi(u*(t)) + hi (u*(t)))dt =
ieK o
r T
max max > / (2qi(x(t),ui(t)) + hi (x(t),ui(t)))dt,
o<qi(.)<ri(.),ieKqi(.)<^(.)<ri(.),ieKieK./to
u* (•) i G K
the mechanism of compulsion is q* (•) = j , .. . At last,
^ w L ri(^), i € M \ K.
v(N; xo, T — to) = max max / [2gi(x(t),ui(t)) +
o<qi(*)<ri(*),ieM qi(*)<ui(t)<ri(*),ieM ^ Jt0
r T
+hi(x(t),ui(t))]dt = / [2gi(x(t),u^(t))dt + hi(x(t),u*i (t))]dt.
ieM ‘
So, if the Leader forms a coalition with Followers, he begins to consider their private interests, and the point of maximum shifts.
Lemma 1. Function v is superadditive.
Proof (of lemma). It is sufficient to consider three cases:
1. VK,L C M(K n L = 0)
v(K; xo, T - to) + v(L; xo,T - to) = E / gi(x(t),r (t))dt +
ieK-*to
+ E/ gi(x(t),ri (t))dt = E / gi(x(t),ri (t))dt = v(K U L; xo,T - to). ieL 0 ieK UL,'to
2. K M
v ({0}uK ; xo,T - to) - v ({0}; xo,T - to) - v (K; xo,T - to) —
,-T ,■ T
= E/ [2gi(x(t),u*(t)) + hi(x(t),u*(t))]dt + E [ gi(x(t),ri(t))dt -
ieK 0 ieM\K t0
, T ,-T
gi(x(t),ri (t))dt gi(x(t),ri(t))dt =
ieM t0 ieK t0
/• T
= E / [2gi(x(t),u*(t)) + hi(x(t),u*(t)) - 2gi(x(t),ri(t))]dt > 0;
ieK t0
3. for VK, L C M(K n L = 0) as far M \ K = M \ (K U L) U L we get
v({0} U K U L; ^ T - to) - v({0} U K; ^ T - to) - v(L; x0, T - to) =
E / [2gi(x(t),u* (t)) + hi(x(t),u* (t))]dt +
ieKUL t0
/• t
+ E / gi(x(t),ri(t))dt -t0
ieM\(KUL)
fT CT
-E / [2gi(x(t),u* (t)) + hi(x(t),u* (t))] - E / gi(x(t),ri (t))dt
ieK t0 ieM\K t0
/■ t
/ gi(x(t),ri(t))dt =
ie L t0
/• T
= E / [2gi(x(t),u* (t)) + hi(x(t),u* (t)) - 2gi(x(t),ri (t))]dt > 0;
ieL t0
Thus, the function v is superadditive and generates a cooperative differential game on the base of compulsion I, = (N; v; x0, T -10) = I,(x0, T -10) Let’s remind that the set of non-dominated imputations in the game 1^,(x0,T - t0) is called C-core and denoted as C, (x0, T -10) , and the Shapley value (x0, T -10) is determined by formulas
(xo, T - to) = E Y(k)[v(K; xo, T - to) - v(K \{i}; xo,T - to)], (6)
K CN (ieK)
i = 1,...,n,
m={n~k)l?~1)\k = \Kln=\N\. (7)
□
Theorem 1. In the game r,(xo,T - to) it is true that (xo,T - to) € C,(xo,T -
to).
Proof (of theorem). Let’s calculate the components of Shapley value subject to (6): $0(xo, T - to) = Y(1)v({0}; xo, T - to) +
+Y(2) [v(K; xo, T - to) - v(K \ {0}; xo, T - to)] + ••• +
{0}eK,|K| = 2
+Y(n) E [v(K; xo, T - to) - v(K \{0}; xo,T - to)] +
{o}eK,|K|=n
+Y(n + 1)[z/(N; xo, T - to) - v(M; xo, T - to)] =
= y(1) E/ gi(x(t),ri(t))dt + y(2) E / [E(2gi(x(t),u**(t)) +
ieM t0 {o}eK,|K|=2 t0 ieK
+ h (x(t),u; (t)))+ £ gi(x(t),ri(t)) - E gi(x(t),ri (t))]dt +-------+
ieM\K ieK
+Y(n) E I [E(2g i(x(t),u* (t)) + hi(x(t),u* (t))) +
{0}eK, |K|=n 0 ieK
+ E gi(x(t),ri(t)) gi(x(t),ri(t))]dt +
ieM\K ieK
+Y(n +1)[E ^ (2gi (x(t), ui (t )) + hi(x(t) ui (t )))dt + E v f gi{.x(~t),'ri (t )) dt]
ieM t0 ieM t0
= y(2) E E/ (2gi(x(t),ui(t)) + hi(x(t),ui(t)))dt +
t0
{o}eK,|K|=2ieK t0
+y(3) E E/ (2gi(x(t),u* (t)) + hi(x(t),u* (t)))dt +-----------+
{0}eK,|K|=3 ieK 0
+Y(n) E E T
(2gi(x(t),u* (t)) + hi(x(t),u* (t)))dt +
{o}eK,|K|=nieK t0
+Y(n +1) E / (2gi(x(t),u*(t)) + hi(x(t),u*(t)))dt] =
ieN 0
= El] y(s) E E / Ai(t)dt,
s=2 {0}eK,|K|=s ieK 0
Ai(t) = 2gi(x(t),u* (t)) + hi(x(t),u*(t)). As each of n players of lower level takes part Cizl times in the set of coalitions of s players than
n “+1 T"1 T"1
^0(xo, T - t0) = E ^(s)gn~j E f Ai(t)dt = 0, 5^ f Ai(t)dt. Thus,
s = 2 ieM^0 ieM^0
(xo,T-to) = E / (di(x(t), ui (t)) + 0, 5hi(x(t),u* (t)))dt = 0, 5v (W; xo,T - to)
A /- li /T ” 10
As far Shapley value is Pareto-optimal we get
'y ' (xo, T - to) = 0, 5v(N; xo, T - to) = (xo, T — to)
ieM
and as far all Followers are completely symmetrical we get
&v(xo, T - to) = I (gi(x(t),u* (t))+0, 5hi(x(t),u* (t)))dt,i G M.
Jt 0
To complete the proof it is sufficient to verify directly the inequalities of three types:
1- 'Sy^J (xo, T - to) + @j ^, T - to) > v(K U L; xo,T - to);
ieK jeL
2. (xo, T - to) + (xo, T - to) > V({0} U K; ^ T - to);
ieK
3. (xo,T - to) + E (xo, T - to) + E ^ (xo, T - to) > v({0} U (K U L)),
ieK jeL
K n L = $,K,L C M. □
A cooperative game on the base of impulsion is built similarly. Unfortunately, the optimality principle <PV (xo, T - to) G CV (xo, T - to) is not dynamically stable (time consistent). To provide the dynamic stability a regularization procedure (payoff distribution procedure) has to be used (Petrosyan and Zenkevich 2007). Define
<PV(xo, T - to) = f Bi(s)ds,Bi(t) > 0, E Bi(t) = 1,t G [to, T].
Jto ieN
d^v (x t t )
The quantity B*(t) = —----------------- is the instantaneous payoff to the player i at
the moment t. The vector B(t) = (Bo(t), B1 (t),..., Bn(t)) determines a distribution of the total gain among all players. By the proper choice of B(t) it is possible to ensure that at each instant t G [to, T] there will be no objections against realization of the original agreement <PV(xo, T - to), i.e. the imputation <PV(xo, T - to) is time consistent. It is proved under general conditions that the regularization procedure B(t), t G [to,T], leading to the time consistent cooperative solution, exists and is realizable (Petrosyan and Zenkevich 1996).
4. Conclusion
The resolution of numerous ecological problems on different levels must be implemented on the base of sustainable development concept that determines the conditions to the state of environmental-economic systems and impacting control actions. Those conditions can’t be realized by themselves and require special collaborative
efforts of different agents using both cooperation and hierarchical control. To formalize the inevitable trade-offs it is natural to use game theoretic models including the games with differently directed interests. Unfortunately, the main optimality principles of hierarchical control (compulsion, impulsion) are not dynamically stable and therefore can’t be recommended as the direct base for collective solutions. The most prospective is the conviction method which is formalized as a transition from hierarchy to cooperation and allows a regularization that provides the dynamic stability. However, in current social conditions other methods of hierarchical control also keep their actuality. To provide the dynamic stability of those optimality principles it is necessary to build cooperative differential games on their base. An example of the approach is considered in this paper. The following directions of the game theoretic modeling of the concept of sustainable development are of specific interest: investigation of general game theoretic models of the hierarchical control of sustainable development (Ugolnitsky 2002a, 2005; Ougolnitsky 2009); modeling of the hierarchical control of sustainable development of the ecological-economic systems (Ugolnitsky 1999; Ugolnitsky and Usov 2007a, 2009; Ougolnitsky 2002; Ougolnitsky and Usov 2009); modeling of the hierarchical control of sustainable development of the systems of another types (Ugolnitsky 2002b); modeling of the multilevel hierarchical systems of different structure (Ugolnitsky and Usov 2007b, 2010); modeling of the corruption in hierarchical control systems (Rybasov and Ugolnitsky 2004; Denin and Ugolnitsky 2010); development of the information-analytical systems of decision support in the hierarchically controlled dynamical systems (Ugolnitsky and Usov 2008).
References
Basar, T. and Olsder, G. J. (1995). Dynamic Noncooperative Game Theory. Academic Press, L.
Denin, K.I. and Ugolnitsky, G. A. (2010). Game theoretic model of corruption in the systems of hierarchical control. Izvestiya RAS. Teoria i systemy upravlenia (Journal of computer and system sciences international), 1, 192-198 (in Russian).
Dockner, E. J. and Jorgensen, S. (1984). Cooperative and Non-Cooperative Differential Game Solutions to an Investment and Pricing Problem. Journal of the Operational Research Society, 35, 731-739.
Dockner, E. J. and Long, N. V. (1993). International Pollution Control: Cooperative versus Non-Cooperative Strategies. Journal of Environmental Economics and Management, 24, 13-29.
Fatkhutdinov, R. and Sivkova, L. (1999). Compulsion, impulsion, conviction: a new approach to control methods. Upravlenie personalom (Human resource management), 2, 32-40 (in Russian).
French, J.P. R.,Jr., and Raven, B. (1960) The bases of social power. In: Group Dynamics (Cartwright, D. and Zander, A., eds), pp. 607 - 623. Harper and Row, N.Y. French, J.P.R.,Jr., and Raven, B.
Fudenberg, D. and Tirole, J. (1991). Game theory. MIT Press, Mass.
Gao, L., Jakubowski, A., Klompstra, M. B. and Olsder, G. J. (1989). Time-Dependent Cooperation in Game. Springer-Verlag, Berlin.
Germeyer, Yu. B., Vatel, I. A. (1974). Games with Hierarchical Vector of Interests. Izvestiya AN USSR. Technical Cybernetics. Vol.3 (in Russian).Springer-Verlag, Berlin.
Haurie, A. (1976). A Note on Nonzero-sum Differential Games with Bargaining Solutions.
Journal of Optimization Theory and Applications, 18, 31-39.
Haurie, A. (1991). Piecewise Deterministic and Piecewise Diffusion Differential Games with Model Uncertainties. Springer-Verlag, Berlin.
Haurie, A., Krawczyk, J. B. and Roche, M. (1994). Monitoring Cooperative Equilibria in a Stochastic Differential Game. Journal of Optimization Theory and Applications, 81, 73-95.
Haurie, A. and Pohjola, M. (1987). Efficient Equilibria in a Differential Game of Capitalism. Journal of Economic Dynamics and Control, 11, 65-78.
Haurie, A. and Zaccour, G. (1986). A Differential Game Model of Power Exchange between Interconnected Utilizers. Proceedings of the 25th IEEE Conference on Decision and Control, Athens, Greece.
Haurie, A. and Zaccour, G. (1991). A Game Programming Approach to Efficient Management of Interconnected Power Networks. Springer-Verlag, Berlin.
Jorgensen, S. and Zaccour, G. (2001). Time Consistent Side Payments in a Dynamic Game of Downstream Pollution. Journal of Economic Dynamics and Control, 25, 1973-1987.
Kaitala, V., Maler, K. G. and Tulkens, H. (1995). The Acid Rain Game as a Resource Allocation Process with an Application to the International Cooperation among Finland, Russia, and Estonia. Scandinavian Journal of Economics, 97, 325-343.
Kaitala, V. and Pohjola, M. (1988). Optimal Recovery of a Shared Resource Stock: A Differential Game with Efficient Memory Equilibria. Natural Resource Modeling, 3, 91-118.
Kaitala, V. and Pohjola, M. (1995). Sustainable International Agreements on Greenhouse Warming: A Game Theory Study. Annals of the International Society of Dynamic Games, 2, 67-87.
Kaitala, V. and Zaccour, G. (1990). Economic Development and Agreeable Redistribution in Capitalism: Efficient Game Equilibria in a Two-Class Neoclassical Growth Model. International Economic Review, 31(2), 421-438.
Kidland, F. E. and Prescott, E. C. (1977). Rules rather than decisions: the inconsistency of optimal plans. J. of Polit. Econ., 85, 473-490.
Ledyaev, V. G. (1998). Power: A Conceptual Analysis. Nova Science Publishers, N.Y.
Leitmann, G. (1974). Cooperative and Non-Cooperative Many Players Differential Games. Springer-Verlag, N.Y.
Leitmann, G. (1975). Cooperative and Non-Cooperative Differential Games. D.Reide, Am-stredam.
Maler, K. G. and de Zeeuw, A. D. (1998). The Acid Rain Differential Game. Environmental and Resource Economics, 12, 167-184.
Mazalov, V. and Rettieva, A. (2004). A fishery game model with migration: reserved territory approach. Game Theory and Applications, 10.
Mazalov, V. and Rettieva, A. (2005). Dynamic game methods for reserved territory optimization. Survey in Applied and Industrial Math, 3(12), (in Russian).
Mazalov, V. and Rettieva, A. (2006). Nash equilibrium in bioresource management problem. Mathematical Modeling, 5(18), (in Russian).
Mazalov, V. and Rettieva, A. (2007). Cooperative Incentive Equilibrium. In: Collected papers presented on the International Conference Game Theory and Management (Petrosjan, L.A., Zenkevich, N.A., eds), pp. 316 - 325. SPb.: Graduate School of Management SBbU.
Ougolnitsky, G. A. (2002). Game theoretic modeling of the hierarchical control of sustainable development. In: Game Theory and Applications (Petrosjan, L.A., Mazalov, V.V., eds), pp. 82 - 91. Nova Science Publishers, N.Y., Vol.8.
Ougolnitsky, G. A. (2009). A Generalized Model of Hierarchically Controlled Dynamical System. In: Contributions to game theory and management. ( Petrosjan, L.A., Zenkevich, N.A., eds), pp. 320 - 333. SPb.: Graduate School of Management SBbU.
Ougolnitsky, G. A. and Usov, A. B. (2009). Problems of the sustainable development of ecological-economic systems. In: Global Climatology and Ecodynamics: Anthropogenic Changes to Planet Earth. ( Cracknell, A.P., Krapivin, V.P., Varotsos, C.A., eds), pp. 427 - 444. Springer-Praxis.
Ougolnitsky, G. A. and Usov, A. B. (2010). Three level fan-like systems of control of the ecological-economic objects. Problemy upravleniya (Control problems), 1, 26-32 (in Russian).
Petrosyan, L. A. (1977). The stability of solutions in differential games with many players. Vestnik Leningradskogo Universiteta, Ser.1, Iss.4, Vol.19, 46-52 (in Russian).
Petrosyan, L. A. (1991). The Time-Consistency of the Optimality Principles in NonZero Sum Differential Games. In: Lecture Notes in Control and Information Sciences (Hamalainen R.P. and Ehtamo H.K., eds), pp. 299 - 311. Springer-Verlag, Berlin, Vol.157.
Petrosyan, L. A. (1993). Strongly Time-Consistent Differential Optimality Principles. Vestnik Saint Petersburg University Mathematics, 26, 40-46 (in Russian).
Petrosyan, L. A. (1993). The Time Consistency in Differential Games with a Discount Factor. Game Theory and Applications, 1.
Petrosyan, L. A. (1997). Agreeable Solutions in Differential Games. International Journal of Mathematics, Game Theory and AlgebraVestnik Saint Petersburg University Mathematics, 7, 165-177.
Petrosyan, L. A. and Danilov, N. N. (1979). Stability of Solutions in Non-Zero Sum Differential Games with Transferable Payoffs. Vestnik of Leningrad University, 1, 52-59 (in Russian).
Petrosyan, L. A. and Danilov, N. N. (1985). Cooperative Differential Games and Their Applications. Tomsk State University, Tomsk (in Russian).
Petrosyan, L. A. and Zenkevich, N. A. (1996). Game Theory. World Scientific Publ.Co., Singapore.
Petrosyan, L. A. and Zenkevich, N. A. (2007). Time Consistency of Cooperative Solutions. In: Contributions to game theory and management. Collected papers presented on the International Conference Game Theory and Management (Petrosjan, L.A., Zenkevich, N.A., eds), pp. 413 - 440. SPb.: Graduate School of Management SBbU.
Petrosyan, L. A. and Zenkevich, N. A. (2009). Conditions for Sustainable Cooperation. In: Contributions to game theory and management. Vol. II. Collected papers presented on the International Conference Game Theory and Management (Petrosjan, L.A., Zenkevich, N.A., eds), pp. 344 - 354. SPb.: Graduate School of Management SBbU.
Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F. (1962). The Mathematical Theory of Optimal Processes. Wiley, N.Y.
Rubio, S. J. and Casino, B. (2002). A Note on Cooperative versus Non-Cooperative Strategies in International Pollution Control. Resource and Energy Economics, 24, 251-261.
Rybasov, E. A. and Ugolnitsky, G. A. (2004). Mathematical modeling of the hierarchical control of ecological-economic systems considering corruption. In: Computer modeling. Ecology. Iss.2. (Ugolnitsky, G.A., ed), pp. 44 - 65. Vuzovskaya kniga, Moscow (in Russian).
Sorokin, P. (1957). Social and Cultural Dynamics.A Study of Change in Major Systems of Art, Truth, Ethics, Law and Social Relationships. Porter Sargent Publisher, Boston.
Tahvonen, O. (1994). Carbon Dioxide Abatement as a Differential Game. European Journal of Political Economy, 10, 685-705.
Tolwinsky, B., Haurie, A. and Leitmann, G. Tahvonen, O. (1970). Cooperative Equilibria, in Differential Games. J. of Math. Analysis and Applications, 119, 182-202.
Ugolnitsky, G. A. (1999). Control of Ecological-Economic Systems. Vuzovskaya kniga, Moscow (in Russian).
Ugolnitsky, G. A. (2002). Game theoretic investigation of some methods of hierarchical control. Izvestiya RAS. Teoria i systemy upravlenia (Journal of computer and system sciences international), 1, 97-101.
Ugolnitsky, G. A. (2002). Hierarchical control of the sustainable development of social organizations. Obshchestvennye nauki i sovremennost (Social sciences and modernity), 3, 133-140 (in Russian).
Ugolnitsky, G. A. (2005). Game theoretic principles of optimality of the hierarchical control of sustainable development. Izvestiya RAS. Teoria i systemy upravlenia (Journal of computer and system sciences international), 4, 72-78 (in Russian).
Ugolnitsky, G. A. and Usov, A. B. (2007). GMathematical formalization of the methods of hierarchical control of ecological-economic systems. Problemy upravleniya (Control problems), 4, 64-69 (in Russian).
Ugolnitsky, G. A. and Usov, A. B. (2007). Hierarchical systems of diamond-shaped structure for river water quality control. Upravlenie bolshimi systemami (Control of big systems), 19, 187-203 (in Russian).
Ugolnitsky, G. A. and Usov, A. B. (2008). An information-analytical system of control of the ecological-economic objects. Izvestiya RAS. Teoria i systemy upravlenia (Journal of computer and system sciences international), 2, 168-176 (in Russian).
Ugolnitsky, G. A. and Usov, A. B. (2009). Control of the complex ecological-economic systems. Avtomatika i telemechanika (Automation and remote control), 5, 169-179 (in Russian).
United Nations (1987). ’’Report of the World Commission on Environment and Development.” General Assembly Resolution 42/187, 11 December.
Yeung, D. W. K. (2009). Cooperative Game-Theoretic Mechanism Design for Optimal Resource Use. In: Contributions to game theory and management. Vol. II. Collected papers presented on the International Conference Game Theory and Management (Petrosjan, L.A., Zenkevich, N.A., eds), pp. 483 - 513. SPb.: Graduate School of Management SBbU.
Yeung, D. W. K. and Petrosyan, L. A. (2001). Proportional Time-Consistent Solution in Differential Games Saint Petersburg State University.
