Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution
There is investigated the noncooperative dyadic three-person game with the parameter, defining the penalty for the reservoir pollution. Given NE-solutions of this game as the environment guard model not only contain simultaneously symmetrical and favorable NE-situations for the pollution subjects, b...
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Інститут телекомунікацій і глобального інформаційного простору НАН України
2010
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| Цитувати: | Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution / V.V. Romanuke // Екологічна безпека та природокористування: Зб. наук. пр. — К., 2010. — Вип. 6. — С. 77-93. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-194142013-02-13T02:55:22Z Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution Romanuke, V.V. Екологічна безпека There is investigated the noncooperative dyadic three-person game with the parameter, defining the penalty for the reservoir pollution. Given NE-solutions of this game as the environment guard model not only contain simultaneously symmetrical and favorable NE-situations for the pollution subjects, but also may be used to force these subjects to apply the cleaning installations completely within the non-equilibrant situation, allowing to lose minimally though. Досліджується безкоаліційна діадична гра трьох осіб з параметром, що визначає стягнення за забруднення водойми. Наведені NE-розв.язки цієї гри як моделі охорони довкілля не тільки містять одночасно симетричні та вигідні NE-ситуації для суб.єктів забруднення, але також можуть бути використані для того, щоб змусити ці суб'єкти застосовувати очисні споруди у рамках нерівноважної ситуації, що, втім, дозволяє досягати мінімальних втрат. Исследуется бескоалиционная диадическая игра трёх лиц с параметром, что определяет взыскание за загрязнение водоёма. Приведённые NE-решения этой игры как модели охраны окружающей среды не только содержат одновременно выгодные NE-ситуации для субъектов загрязнения, но также могут быть использованы для того, что- бы вынудить эти субъекты применять очистительные сооружения в рамках неравновесной ситуации, что, впрочем, позволяет достичь минимальных потерь. 2010 Article Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution / V.V. Romanuke // Екологічна безпека та природокористування: Зб. наук. пр. — К., 2010. — Вип. 6. — С. 77-93. — Бібліогр.: 8 назв. — англ. XXXX-0062 http://dspace.nbuv.gov.ua/handle/123456789/19414 519.833 en Екологічна безпека та природокористування Інститут телекомунікацій і глобального інформаційного простору НАН України |
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Екологічна безпека Екологічна безпека |
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Екологічна безпека Екологічна безпека Romanuke, V.V. Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution Екологічна безпека та природокористування |
| description |
There is investigated the noncooperative dyadic three-person game with the parameter, defining the penalty for the reservoir pollution. Given NE-solutions of this game as the environment guard model not only contain simultaneously symmetrical and favorable NE-situations for the pollution subjects, but also may be used to force these subjects to apply the cleaning installations completely within the non-equilibrant situation, allowing to lose minimally though. |
| format |
Article |
| author |
Romanuke, V.V. |
| author_facet |
Romanuke, V.V. |
| author_sort |
Romanuke, V.V. |
| title |
Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution |
| title_short |
Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution |
| title_full |
Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution |
| title_fullStr |
Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution |
| title_full_unstemmed |
Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution |
| title_sort |
environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution |
| publisher |
Інститут телекомунікацій і глобального інформаційного простору НАН України |
| publishDate |
2010 |
| topic_facet |
Екологічна безпека |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/19414 |
| citation_txt |
Environment guard model as dyadic three-person game with the generalized fine for the reservoir pollution / V.V. Romanuke // Екологічна безпека та природокористування: Зб. наук. пр. — К., 2010. — Вип. 6. — С. 77-93. — Бібліогр.: 8 назв. — англ. |
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Екологічна безпека та природокористування |
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2025-07-02T20:18:14Z |
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| fulltext |
77
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
UDC 519.833
ENVIRONMENT GUARD MODEL AS DYADIC
THREE-PERSON GAME WITH THE GENERALIZED FINE
FOR THE RESERVOIR POLLUTION
V.V. Romanuke, cand. tech. sc.
(Khmelnytskyy National University)
There is investigated the noncooperative dyadic three-person game with
the parameter, defining the penalty for the reservoir pollution. Given NE-
solutions of this game as the environment guard model not only contain
simultaneously symmetrical and favorable NE-situations for the pollution
subjects, but also may be used to force these subjects to apply the cleaning
installations completely within the non-equilibrant situation, allowing to lose
minimally though.
Äîñë³äæóºòüñÿ áåçêîàë³ö³éíà ä³àäè÷íà ãðà òðüîõ îñ³á ç ïàðàìåò-
ðîì, ùî âèçíà÷ຠñòÿãíåííÿ çà çàáðóäíåííÿ âîäîéìè. Íàâåäåí³ NE-
ðîçâ�ÿçêè ö³º¿ ãðè ÿê ìîäåë³ îõîðîíè äîâê³ëëÿ íå ò³ëüêè ì³ñòÿòü îäíî-
÷àñíî ñèìåòðè÷í³ òà âèã³äí³ NE-ñèòóàö³¿ äëÿ ñóá�ºêò³â çàáðóäíåííÿ,
àëå òàêîæ ìîæóòü áóòè âèêîðèñòàí³ äëÿ òîãî, ùîá çìóñèòè ö³ ñóá-
�ºêòè çàñòîñîâóâàòè î÷èñí³ ñïîðóäè ó ðàìêàõ íåð³âíîâàæíî¿ ñèòóàö³¿,
ùî, âò³ì, äîçâîëÿº äîñÿãàòè ì³í³ìàëüíèõ âòðàò.
Èññëåäóåòñÿ áåñêîàëèöèîííàÿ äèàäè÷åñêàÿ èãðà òð¸õ ëèö ñ ïàðà-
ìåòðîì, ÷òî îïðåäåëÿåò âçûñêàíèå çà çàãðÿçíåíèå âîäî¸ìà. Ïðèâå-
ä¸ííûå NE-ðåøåíèÿ ýòîé èãðû êàê ìîäåëè îõðàíû îêðóæàþùåé ñðåäû
íå òîëüêî ñîäåðæàò îäíîâðåìåííî âûãîäíûå NE-ñèòóàöèè äëÿ ñóáúåê-
òîâ çàãðÿçíåíèÿ, íî òàêæå ìîãóò áûòü èñïîëüçîâàíû äëÿ òîãî, ÷òî-
áû âûíóäèòü ýòè ñóáúåêòû ïðèìåíÿòü î÷èñòèòåëüíûå ñîîðóæåíèÿ â
ðàìêàõ íåðàâíîâåñíîé ñèòóàöèè, ÷òî, âïðî÷åì, ïîçâîëÿåò äîñòè÷ü
ìèíèìàëüíûõ ïîòåðü.
Problem statement
Mathematical modeling of eco-defense events and systems is an
actual problem of fundamental and natural sciences. The dyadic
noncooperative game is the most comfortable mathematical model of
V.V. Romanuke, 2010
78
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
conflict events by the interaction amongst several subjects of the
environment pollution. In the game of this class a subject of pollution
has only two pure strategies, one of which is to apply the cleaning
installations, and another is to not apply the cleaning installations. A
classic dyadic game as the environment defense model is the three-
person dyadic game [1, p. 193 � 197], where the j -th subject of
pollution has a pure strategy { }0, 1jx ∈ for 1, 3j = by the zero pure
strategy, meant applying the cleaning installations. There are three
industrial factories, using the water from some reservoir for their
technical needs, and the natural ecological balance is restored only if
not greater than the single factory pours out the wasted water into the
reservoir without cleaning. The violation of this condition drives to the
penalty for three units from each of factories, though the application
of cleaning installations costs only one unit. Meanwhile this game
doesn�t have the simultaneous profitable, symmetric and equilibrant
situation, its formulation is not general, as that fine in three units had
been only locally fixed. And it is unknown, what solutions will be
when that fine is generalized.
Last publications analysis
Besides [1, p. 193�197], there are the origins [2, 3], where the
dyadic three-person game [4, 5] as the model of the environment
defense has been investigated. The result of [1, p. 193�197] is that
this game has the nine NE-solutions (Nash equilibrium solutions [6,
7]), four of which are in the pure strategies. Being locally fixed for
the fine in the three units, there [1, p. 193�197] have been found
some non-equilibrant situations [1, 7, 8], that had been appeared to
be simultaneously symmetric and the most advantageous [9, 10].
However, there remains the localization in those three units, which
restricts as the NE-solution, as well as the symmetric advantageous
solution, needed to be generalized.
Paper aim formulation
Assign a pure strategy of the j -th factory as jx by { }0, 1jx ∈
1, 3j∀ = . Further, assign the probability of that the j -th factory
will not use the water cleaning technology for the wasted water before
pouring it out into the reservoir as jα 1, 3j∀ = . In other words,
jα is the probability of using the pure strategy 1jx = 1, 3j∀ = .
The now expanding paper aim is to find the NE-solution of the
described game, if the fine for pouring the wasted unclean water out
79
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
by two or three factories is ( ]1;a d∈ , where d is the maximal fine,
that is possible, as d cannot be infinitely large because of the law.
On the other hand, the great fines may influence negatively upon the
factory functioning [11, 12], and it may want to be stopped in it, and
this yet is not profitable for the state (or its regional economics).
Main section of investigation
Assign the payoff of the j -th factory in the pure strategies situation
{ }1 2 3, ,x x x as ( )1 2 3, ,jK x x x 1, 3j∀ = . Then the payoff of the j -
th factory in the mixed strategies situation { }1 2 3, ,α α α is
( )1 2 3, ,jV α α α = ( )( )( ) ( )1 2 31 1 1 0, 0, 0jK− α − α − α +
( )( ) ( )1 2 31 1 0, 0, 1jK+ − α − α α +( ) ( ) ( )1 2 31 1 0, 1, 0jK− α α − α +
( ) ( )1 2 31 0, 1, 1jK+ − α α α + ( )( ) ( )1 2 31 1 1, 0, 0jK+α − α − α +
( ) ( )1 2 31 1, 0, 1jK+α − α α + ( ) ( )1 2 31 1, 1, 0jK+α α − α +
( )1 2 3 1, 1, 1jK+α α α (1)
for 1, 3j = by
( ) ( ) ( )1 2 30, 0, 0 0, 0, 0 0, 0, 0 1K K K= = = − , (2)
( ) ( )1 20, 0, 1 0, 0, 1 1K K= = − , ( )3 0, 0, 1 0K = , (3)
( ) ( )1 30, 1, 0 0, 1, 0 1K K= = − , ( )2 0, 1, 0 0K = , (4)
( )1 0, 1, 1 1K a= − − , ( ) ( )2 30, 1, 1 0, 1, 1K K a= = − , (5)
( )1 1, 0, 0 0K = , ( ) ( )2 31, 0, 0 1, 0, 0 1K K= = − , (6)
( ) ( )1 31, 0, 1 1, 0, 1K K a= = − , ( )2 1, 0, 1 1K a= − − , (7)
( ) ( )1 21, 1, 0 1, 1, 0K K a= = − , ( )3 1, 1, 0 1K a= − − , (8)
( ) ( ) ( )1 2 31, 1, 1 1, 1, 1 1, 1, 1K K K a= = = − . (9)
Putting (2)�(9) into (1) gives the following payoffs:
80
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
( )1 1 2 3 1 2 1 2 3 2 3 1 3 1, , 1 2V a a a aα α α = − − α α + α α α − α α − α α + α , (10)
( )2 1 2 3 1 2 1 2 3 2 3 1 3 2, , 1 2V a a a aα α α = − − α α + α α α − α α − α α + α , (11)
( )3 1 2 3 1 2 1 2 3 2 3 1 3 3, , 1 2V a a a aα α α = − − α α + α α α − α α − α α + α . (12)
It is interesting to underline, that the payoffs (10) � (12) differ
from each other only for the corresponding probability in the mixed
strategy. This means that the equal payoffs in the being investigated
game are only on the symmetric situations { }, ,α α α by 1 2 3α = α = α .
For searching the set of all the acceptable situations of the
j -th factory it is sufficient to find the solutions of the corresponding
nonstrict inequality [1, p. 192]. In this way, the set is found from
the inequality
( )( ) ( ) ( ) ( )2 3 1 2 3 11 1 0, 0, 0 1 0, 0, 1K K− α − α + − α α +
( ) ( ) ( )2 3 1 2 3 11 1, 1, 0 1, 1, 1K K+α − α + α α . (13)
Then the situations { }2 31, ,α α which satisfy the inequality (13)
with the strict inequality sign, being coalesced into the set , and
the situations { }1 2 3, ,α α α which satisfy the inequality (13) with the
equality sign, being coalesced into the set , and the situations
{ }2 30, ,α α which do not satisfy the inequality (13), being coalesced
into the set , will compound the set
. (14)
As the sets of the pure strategies of each of the three factories are identical,
then their sets of the mixed strategies are identical also, and the sets
(15)
81
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
and
(16)
are found analogously from the inequalities
( )( ) ( ) ( ) ( )1 3 2 1 3 21 1 0, 0, 0 1 0, 0, 1K K− α − α + − α α +
( ) ( ) ( )1 3 2 1 3 21 1, 1, 0 1, 1, 1K K+α − α + α α (17)
and
( )( ) ( ) ( ) ( )1 2 3 1 2 31 1 0, 0, 0 1 0, 1, 0K K− α − α + − α α +
( ) ( ) ( )1 2 3 1 2 31 1, 0, 1 1, 1, 1K K+α − α + α α (18)
respectively.
The set may be depicted graphically, as it was in [1, p. 194],
within the cube of situations (figure 1).
Before making it, there must be considered the inequality (18),
which is
. (19)
The inequality (19), taken as the strict inequality, depicts the subset
lying in the plane 3 1α = . As from (19)
(20)
then
82
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
(21)
for 2
1
; 1
2
α ∈
and
(22)
for 2
1
0;
2
α ∈
. The character of the transformation of the subset
is shown on the frames of the figures 2�8, plotted by
the conditions (21) and (22) in the strict form. It is certain that by
the fine ( )1; 2a∈ this subset becomes the inseparable set, though by
Figure 1. The cube of situations in the dyadic game with the three factories
and their payoffs ( ){ }3
1 2 3 1
, ,j j
K x x x
=
83
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
2a > the set consists of the two separate parts, and by 2a =
the set consists of the two half-open squares, not having the
adjacent points within the unit square.
Figure 2. The set of the two separated hyperbolic parts
on the cube of situations in the plane 3 1α = by { }5, 4, 3a∈
Figure 4. The set of the two separated hyperbolic parts
(continued to be evolved) on the cube of situations
in the plane 3 1α = by { }2.2, 2.1, 2.075a∈
Figure 3. The set of the two separated hyperbolic parts
on the cube of situations in the plane 3 1α = by { }2.5, 2.4, 2.3a∈
84
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
Figure 5. The set of the two half-open squares,
not having the adjacent points within the unit square
on the cube of situations
in the plane 3 1α = by 2a = ,
and the inseparable set on the cube of situations
in the plane 3 1α = by { }1.925, 1.9a∈
Figure 6. The inseparable set
on the cube of situations
(completion of its evolution)
in the plane 3 1α = by { }1.8, 1.7, 1.6a∈
Figure 7. The inseparable set on the cube of situations
in the plane 3 1α = by { }1.5, 1.4, 1.3a∈
85
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
By 2a > the set consists of the curvilinear hexagon, lying
in the plane 3 0α = (it is seen as the white-colored area on the figures
2� 4), where the continuous margin belongs to this set. And the set
consists of the two parts of the parabolic cylinders with
generatrices along the 3α axis, where the projections of these parabolic
cylinders on the plane 3 0α = may be seen as the inner parabolic
borders on the figures 2�4.
After having depicted the set (16) within the cube of situations by
2a > , there yet may be pictured the sets (14) and (15), which are
obtained by the corresponding right-angle rotation of the set (16). Thus
the set of all the equilibrium situations
(23)
for the case 2a > turns out to be the same structure as it is in [1,
p. 193�197] with 3a = . That is the set (23) for the case 2a >
consists of the following pure strategies situations:
{ }0, 0, 1 , { }0, 1, 0 , { }1, 0, 0 , { }1, 1, 1 . (24)
The payoffs in the situations (24) are
( ) ( ) ( ){ } { }1 2 30, 0, 1 , 0, 0, 1 , 0, 0, 1 1, 1, 0K K K = − − , (25)
( ) ( ) ( ){ } { }1 2 30, 1, 0 , 0, 1, 0 , 0, 1, 0 1, 0, 1K K K = − − , (26)
Figure 8. The inseparable set on the cube of situations
in the plane 3 1α = by { }1.2, 1.1, 1.05a∈
86
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
( ) ( ) ( ){ } { }1 2 31, 0, 0 , 1, 0, 0 , 1, 0, 0 0, 1, 1K K K = − − , (27)
( ) ( ) ( ){ } { }1 2 31, 1, 1 , 1, 1, 1 , 1, 1, 1 , ,K K K a a a= − − − . (28)
Besides, there are the three mixed strategies situations, lying on the
situations cube faces:
1 1
0, ,
a a
,
1 1
, , 0
a a
,
1 1
, 0,
a a
. (29)
The payoffs in the situations (29) due to (1)�(12) are
1 2 3
1 1 1 1 1 1 1
0, , , 0, , , 0, , , 1, 1
a
V V V
a a a a a a a
+ = − − −
, (30)
1 2 3
1 1 1 1 1 1 1
, , 0 , , , 0 , , , 0 1, 1,
a
V V V
a a a a a a a
+ = − − −
, (31)
1 2 3
1 1 1 1 1 1 1
, 0, , , 0, , , 0, 1, , 1
a
V V V
a a a a a a a
+ = − − −
. (32)
Those six situations in (24) and (29), where the situation { }1, 1, 1
is excluded, are non-symmetrical and, as a corollary, they are not
favorable for the pair of factories, having greater losses in (25) � (27),
and not favorable for the single factory, having greater loss in (30) �
(32). But the symmetrical situation { }1, 1, 1 is not favorable for the
three factories simultaneously, as each of them would have been fined
for a units, where 2a > .
Nevertheless, there are two mixed strategies situations inside the
situations cube, which are constituted due to the nonempty intersection
(23) inside the cube, belonged to the set as the
intersection of the three couples of parabolic cylinders generatrices.
Surely, they should be symmetrical, so then assign them as
{ } { } { }1 1 1
1 2 3, , , , , ,α α α = α α α = α α α , (33)
87
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
{ } { } { }2 2 2
1 2 3, , , , , ,α α α = α α α = α α α . (34)
Letting jα = α 1, 3j∀ = into the inequality (18), will find the
situations (33) and (34), transforming (18) into (19) and stating it as
the equality:
( ) ( )22 2 1 2 1 1 0a aα − α + = α α − + = . (35)
Further, from (35) have the quadratic equation
22 2 1 0a aα − α + = , (36)
which roots are the probabilities 1α and 2α for (33) and (34):
( ) ( )1 2 4 2 2
4 2
a a a a a a
a a
− − − −
α = = , (37)
( ) ( )2 2 4 2 2
4 2
a a a a a a
a a
+ − + −
α = = . (38)
It is easy to check, that ( )1 0; 1α ∈ and ( )2 0; 1α ∈ by 2a > ,
so the symmetrical situations (33) and (34) are
( ) ( ) ( )2 2 2
, ,
2 2 2
a a a a a a a a a
a a a
− − − − − −
, (39)
( ) ( ) ( )2 2 2
, ,
2 2 2
a a a a a a a a a
a a a
+ − + − + −
. (40)
The situation (39) payoffs
( ) ( ) ( ){ }1 1 1 1 1 1 1 1 1
1 2 3, , , , , , , ,V V Vα α α α α α α α α =
88
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
( ) ( ) ( )2 1 2 1 2 1
, ,
2 2 2
a a a a a a a a a − − − − − − − − − =
(41)
are obviously identical. They are less than the situation (40) payoffs
( ) ( ) ( ){ }2 2 2 2 2 2 2 2 2
1 2 3, , , , , , , ,V V Vα α α α α α α α α =
( ) ( ) ( )2 1 2 1 2 1
, ,
2 2 2
a a a a a a a a a − + + − + + − + + = − − −
, (42)
speaking that this situation is not favorable entirely.
Thus the being investigated game by 2a > has the nine NE-
situations (24), (29), (39), (40), where only three situations are symmetrical
(absolutely fair): situation { }1, 1, 1 and (39), (40). Meanwhile, the
situation (40) is such that the factories bear losses, which are apparently
greater than the losses in the situation (39). Will give evidence of that
the situation (39) payoffs are greater than the situation (24) payoffs:
( ) ( ) ( ) ( )1 1 1 2 1
, , 1, 1, 1
2j j
a a a
V K a
− − −
α α α − = − − =
( )2 1
2
a a a− − +
= 1, 3j∀ = ,
whence by , that for 2a > is
true. So, the situation (39) payoffs (41) are the most favorable for the
three factories simultaneously.
Noticing that
( )1 2
lim lim 0
2a a
a a a
a→∞ →∞
− −
α = =
and 1, 3j∀ =
89
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
( ) ( )0, 0, 0 0, 0, 0 1j jV K= = −
due to (2), will conclude, that the huge fines force the factories,
wanting the symmetrical fairness, to convert totally into the water
cleaning technology for the wasted water before pouring it out into the
reservoir. The maximal fine d may be determined from the reason
of the supportable and negligible volume of the unclean wasted water
sink. For instance, if 1 0.1α = that means a factory will pour out
only one tenth of the unclean wasted water, then completely there will
be poured only 30 percent of the unclean wasted water in relation to
the bearable volume of the unclean wasted water sink (which is
equivalent to the case when only one factory pollutes the reservoir).
By that the rate of the loss for a factory should be closely approximated
to one unit, having been fixed as dl . That is the equation
( )1 2
2 d
a a a
a
− −
α = = α (43)
by the fixed dα will give the value a d= of the maximal fine,
satisfying the condition
( ), ,j d d dV α α α =
. (44)
Exemplifying the above further, take 1.05dl = , and from (43) by
0.1dα = obtain that 5.5556d ≈ . Then
( )0.1, 0.1, 0.1 1.0556 1.05j dV l≈ − < − = −
and the condition (44) is not true, so the 30 percent unclean wasted
water sink is not negligible here. But taking 0.09dα = will give
,
that satisfies (44) with the fine 6.105d ≈ , what restricts the factories
at their 27 percent unclean wasted water sink.
90
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
Now will look at the NE-solution of the being investigated game by
the fine 2a = . In this case, might have been called marginal, there are
still the four NE-situations (24) in the pure strategies, and there appear
the three continuous sets of the NE-situations in the mixed strategies
1
1 1
, ,
2 2
α
[ ]1 0; 1∀ α ∈ , (45)
2
1 1
, ,
2 2
α
[ ]2 0; 1∀ α ∈ , (46)
3
1 1
, ,
2 2
α
[ ]3 0; 1∀ α ∈ . (47)
The payoffs in the situations (45)�(47) are
1 1 2 1 3 1
1 1 1 1 1 1
, , , , , , , ,
2 2 2 2 2 2
V V V
α α α =
1 1
3
, 1 , 1
2
= − − − α − − α
, (48)
1 2 2 2 3 2
1 1 1 1 1 1
, , , , , , , ,
2 2 2 2 2 2
V V V
α α α =
2 2
3
1 , , 1
2
= − − α − − − α
, (49)
1 3 2 3 3 3
1 1 1 1 1 1
, , , , , , , ,
2 2 2 2 2 2
V V V
α α α =
3 3
3
1 , 1 ,
2
= − − α − − α −
. (50)
91
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
Here remains only one symmetrical situation
1 1 1
, ,
2 2 2
in the
mixed strategies, consequential to the symmetrical situations (39) and
(40) junction, and giving the equal payoffs
3 3 3
, ,
2 2 2
− − −
.
The NE-solution of the being investigated game by the fines
( )1; 2a∈ may be deduced right from the discourse on the NE-
situations (24), (29), (39), (40), using the case 2a = border conception.
So, by ( )1; 2a∈ there are still the four NE-situations (24) in the
pure strategies, and there the three mixed strategies situations, lying
on the situations cube faces:
1 1
1, ,
a a
a a
− −
,
1 1
, , 1
a a
a a
− −
,
1 1
, 1,
a a
a a
− −
. (51)
The payoffs in the situations (51) due to (1)�(12) are
1 2 3
1 1 1 1 1 1
1, , , 1, , , 1, ,
a a a a a a
V V V
a a a a a a
− − − − − − =
21
, ,
a
a a
a
−= − −
, (52)
1 2 3
1 1 1 1 1 1
, , 1 , , , 1 , , , 1
a a a a a a
V V V
a a a a a a
− − − − − − =
21
, ,
a
a a
a
−= − −
, (53)
92
Åêîëîã³÷íà áåçïåêà òà ïðèðîäîêîðèñòóâàííÿ
1 2 3
1 1 1 1 1 1
, 1, , , 1, , , 1,
a a a a a a
V V V
a a a a a a
− − − − − − =
21
, ,
a
a a
a
−= − −
. (54)
As for the NE-situations within the situations cube, then their set
is empty because of the quadratic equation (36) by ( )1; 2a∈ does not
have real roots.
Conclusions and perspective of further investigations of the
noncooperative models of the environment guard
The investigated dyadic game for 2a > has the single simultaneously
symmetrical and favorable NE-situation (39). This situation must be favorable
also for the local government, wanting the factories to be in functioning state,
as well as the reservoir to be clean. It draws the fine 2a > to be increased
up to the maximal fine d , being fixed with the conditions (43) and (44).
In the marginal case 2a = there is the single symmetrical NE-situation
1 1 1
, ,
2 2 2
with the corresponding payoffs
3 3 3
, ,
2 2 2
− − −
. Obviously,
this situation is pretty unfavorable for each factory [13�15], and the
reservoir would be polluted much irreversibly. Furthermore, the
investigated dyadic game does not have the simultaneously symmetrical
and favorable NE-situations by any ( )1; 2a∈ . But its NE-solution as
the aggregate of (24) and (51) may occur useful anyway. Hence for
both the environment guard and the factories the selection of some
fine 2a > will be the most profitable, as then the factories most likely
will use the strategies from the symmetrical favorable NE-situation
(39), giving the losses, which are greater for the one unit loss:
, what is checked elementarily. But for the
situation (39) equilibrium, all the stated conditions may influence on
93
ÏðåðîìàíòèçìÐîçä³ë 1. Åêîëîã³÷íà áåçïåêà
the factories to claim the treaty about the joint application of the
cleaning installations, that is the non-equilibrant situation { }0, 0, 0
with the minimal payoffs { }1, 1, 1− − − (or losses just in the one unit).
And the greater fine a the more probable that the factories will make
their pact on the joint holding the reservoir clean.
Further investigations of the noncooperative models of the environment
guard should apparently be directed to increasing the number of the
factories (or, generally, potential subjects of the environment pollution),
taking there general fines and finding the most acceptable fine for both
the factories and the environment guard or local government.
* * *
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