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DOI:10.24200/J65.2019.7068.1684

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s1 � 0; s2 � 0 (4)

u}vJty "CU= xOWv |UQQ@ pkDUt CQwY x@ wCU= xOw@ Q=R=@ R= |�RH Q=Pox}=tQUQO "OvyO|t s=Hv= |Q=Pox}=tQU 'Q=R=@ x@ OwQw hOy =@ =yCmQW OQ=wt R= |Q=}U@ QO=yCmQW 'C}r=ai wQW R= TB xm OQ}o|t Q_v QO xvwo u}= Q[=L VywSB xm |r=LQO |Q=Pox}=tQU R= =DU=Q u}= QO w OvQ}o|t OwN C}r=ai K]U |=kDQ= x@ s}tYDQ=R=@ x@ s}kDUt OwQw |=Q@ |k=}DW= Q=Pox}=tQU u}= xm "Ovvm|t xO=iDU= Q=R=@ GQ=N=yCmQW Q=}DN= QO x}=tQU u=wva x@ =Q |er@t =yvD w OQ=Ov O}rwD Ov}=Qi QO Cm =QW wu=wD|t p=Ft |=Q@ "OwW|t s}yU xOt; CUO x@ OwU QO u; |=R= QO w OyO|t Q=Qk|O}OH O}rwD \N |R=Ov=x=Q =} OwN C=R}yHD |v=UQRwQx@ |=Q@ =yxv=NQ=m s}tYD x@R= Q=m u}= |=Q@ R=}v OQwt |x}=tQU xm |}xvwo x@ "OQm xQ=W= OwHwt O}rwD \N Q=vm QO|HQ=N Q=Pox}=tQU O=Qi= =} |YwYN VN@ p}@k R= Q=R=@ R= GQ=N QO |Q=Pox}=tQUpY=L OwU QO Q=Pox}=tQU 'xOW |Q=Pox}=tQU dr@t uOv=OQoQ@ |=Q@ "OwW|t u}t-=D

"OwW|t s}yU O}rwD |xv} Ry Vy=m w CqwYLt |=kDQ= R=wO u}= "CU= xOW xDiQo Q_v QO xOvvmO}rwD wO =@ |v}t-=D |xQ}HvR Qw_vt u}O@w OvyO =kDQ= =Q OwN C=R}yHD '|Qw=:vi CiQW}B =@ u=tRsy OvDUy p}=t xOvvmO}rwDpL |=Q@ "Ovvm xO=iDU= Q=R=@ R= GQ=N QO |}x}=tQU R= OwN |=yxv} Ry u}t-=D |=Q@|Q=Pox}=tQU x@ s=Ok= =}; xm Ovm|t u}}aD |R=@ Q@yQ u=wva x@ Q=Pox}=tQU |R=@ u}=OwN CqwYLt Ct}k xOW O=H}= \}=QW R= |wQ}B x@ =yCmQW TBU "Q}N =} Ovm|t

"CU= xOW |UQQ@ OvQDQ@ - nQ@rmDU= |R=@ l} `k=w QO "Ovvm|t u}}aD =Q

xr�Ut h} QaD "3"CU= xOW xDiQo Q_v QO 'OvR=OQB|t u}Wv=H |q=m O}rwD x@ xm xOvvmO}rwD wO =@ |Q=R=@xOvvmO}rwD wO u}= "Ovvm|t u}}aD =Q OwN Ct}k xv=o =OH CQwY x@ =yu; R= l} Qy|O}rwD C=wO= R= w OvyO =kDQ= =Q OwN C=R}yHD |Qw=:viCiQW}B =@ u=tRsy OvDUy p}=tu}t-=D |=Q@ Ovv=wD|t =yCmQW u}= "Ovvm xO=iDU= 'CU= xOW Q=R=@ OQ=w xm |O}OHx@ s}kDUt OwQw |=Q@|r}=tD xm Ovvm xO=iDU= |Q=Pox}=tQU |x}=tQU R= OwN |=yxv} RyOwU x@ xm OQm Oy=wN |Q=Pox}=tQU x@ s=Ok=|DQwY QO Q=Pox}=tQU|iQ] R= "OQ=Ov Q=R=@OwN |x}=tQU R= A XNWt dr@t '|Qw;OwU CQwY QO |w "O@=} CUO |rw@k p@=kOwUV}=Ri= R= |QUm |Q=Pox}=tQU u}= |=R= QO w Ovm|t |Q=Pox}=tQU =yCmQW QO =QwO Qy |wQ Ov=wD|t w= "Ovm|tCi=} QO OwU u=wva x@ O}OH\}=QW QO =Q =yCmQW u}=CmQW Qy "Ovm |Q=Pox}=tQU O}OH |Qw=:vi O} QN |=Q@ CmQW wO u}= R= |m} \ki =}Ct}kCmQW Qy "CW=O Oy=wN |QDtm O}rwD |xv} Ry 'O@=}CUO Q_vOt |Qw=:vi x@ xm?rH |=Q@ CmQW wO u}= xH}Dv QO "Ovm|t u}}aD pkDUt CQwY x@ =Q OwN pwYLt"OyO|t u=Wv =Q |R=@ Q=DN=U 1 pmW "OvR=OQB|t C@=kQ x@ sy =@ Q=Pox}=tQU xHwD

"OvwW|t |iQat Q=DWwv u}= QO xOW xO=iDU= s�qa =OD@=

"xr�Ut Q=DN=U "1 pmW

13

"""w|Q=

PoCt}k

|=Q@|R=

@|x}Q_

vOQm}wQ"OwW|t xOy=Wt 16 |x]@=Q QO R}v swO |xOvvmO}rwD |Qw;OwU \QW

p1 > ((�c2 + �p2 + (� � 1)(s2� � p02))�2

+(�(c2 � p2)p2 � (� � 1)c02p02 + (� � 1)p022 ):

+(� � 1)p01(s2� + c2 � p02)�1)=�(c2 � p2)�1 (16)

|R=UpOt "2"3"CU= xOW xO=O u=Wv 17 |x]@=Q QO Q=Pox}=tQU x@ \w@ Qt OwU `@=D

�1 =2Xi=1

�(�0fi � �fi) (17)

"O};|t CUO x@ 18 |x]@=Q R= =yxOvvmO}rwD R= l} Qy x@ \w@ Qt OwU `@=D�0fi = (p0i � c0i)D0i � �((p0i � c0i)D0i � (pi � ci)Di) i = 1; 2 (18)

"O};|t CUO x@ 19 |x]@=Q R= O}OH O}rwD |=yxv} Ry 17 |x]@=Q QO

c01 = (c1 � �Ax)

c02 = (c2 � �A(1� x)) (19)

xr�Ut pL "4"OwW|t pL Q=Pox}=tQU OwQw R= TB w Q=Pox}=tQU OwQw R= p@k Cr=L wO |=Q@ |R=@QO w x@U=Lt pt=wa |rO=aD OwU w =[=kD Q}O=kt '|rO=aD |=yCt}k CtUk Qy QO

OwW|t xU}=kt Qo}Om} =@ G}=Dv QN;

Q=Pox}=tQU OwQw R= p@k "1"4OwW|t |UQQ@ 'OQ=Ov OwHw Q=Pox}=tQU xm |]}=QW |va} xr�Ut |x}rw= Cr=L =OD@=OvDUy C}r=ai pweWt u; =@ Q[=L p=LQO =yCmQW xm Ct}k |x}rw= Q}O=kt wu}}aD =Q Ct}k u=tRsy CQwY x@ CmQW wO xm|}=Hv; R= "OwW|t x@U=Lt (p1; p2)

"OwW|t x@U=Lt Vv pO=aD ltm x@ x}rw= Ct}k Q}O=kt 'Ovvm|tw pw= |=yxOvvmO}rwD R= s=OmQy |=Q@ Vv pO=aD R= xOW pY=L Ct}k %1 x}[k

"CU= 21 w 20 \@=wQ Q@=Q@ ?}DQD x@ swO

p1 = 2(�1 + c1�1)�2 + (�2 + c2�2)�24�1�2 � �1�2

(20)

p2 = 2�1(�2 + c2�2) + (�1 + c1�1)�14�1�2 � �1�2

(21)

%C=@F=|UQQ@ =@ =OD@= swO w pw= |xOvvmO}rwD |rO=aD |=yCt}k uOQw; CUO x@ |=Q@O}=@ Qw_vt u}= |=Q@ "OvDUy Qakt OwU `@=wD xm CU= xOW xO=O u=Wv 'swO jDWt\w@ Qt u}Uy T} QD=t "OW=@ |ivt u}at xt}v OwU `@=wD x@ \w@ Qt u}Uy T} QD=tT} QD=t xm OwW|t xOy=Wt "CU= xOW x@U=Lt 23 w 22 \@=wQ QO =yxOvvmO}rwD x@

3ZQixO=iDU= x}=tQU R= xm CU= |DmQW OwU Q=Okt QO Cw=iD R= |@} Q[ Q=Pox}=tQU OwUu}= |=Q@ 5 CQ=@a "OwW|t ji=wD pt=wa \UwD ?} Q[ u}= Q=Okt "CU= xOQm

"OwW|t xDiQo Q_v QO ?} Q[0 � � � 1 (5)

4ZQi"CU}v Q=Pox}=tQU xHwDt |}xv} Ry Qo}O 'CmQW QO |Q=Pox}=tQU R= TB

5 ZQiQ=Okt =@ ?U=vDt |QDtm O}rwD |xv} Ry 'O@=} CUO Q_v Ot |Qw=:vi x@ xm CmQW Qy

"CW=O Oy=wN xOW |Q=Pox}=tQU� > 0 (6)

6 ZQi"CU= xOW xDiQo Q_v QO u} Ro}=H CQwY x@ CqwYLt

�1 > 0; �2 > 0 (7)

7 ZQixOW xDiQo Q_v QO Ct}k ?UL Q@ |a@=D w CU= |]N CQwY x@ Q=R=@ |=[=kD"Ovm|t Q}}eD Ct}k x@ xHwD =@ =[=kD Q=R=@ u}= QO xm OwW|t ZQi xvwou}= "CU=D1 = �1 � �1p1 + �2p2 (8)

D2 = �2 � �2p2 + �1p1 (9)

"OvwW|t ^=Lr R}v Q} R \wQW '=[=kD `@=wD x@ xHwD =@�1 > 0; �2 > 0; �1 > 0; �2 > 0 (10)

pwYLt u=ty |=[=kD Q@ pwYLt Qy |Dt}k Q}F-=D uOw@ QDW}@ Qou=}@ 11 CQ=@a"CU= ?}kQ Ct}k Q}F-=D x@ C@Uv

�1 > �1 w �2 > �2 (11)

|xOvvmO}rwD |=Q@ ?}DQD x@ R}v Q} R \wQW 'OQ}PB|t |D@Ft Q=Okt =[=kD xm|}=Hu; R="OwW|t xDiQo Q_v QO swO w pw=

�1 + p02�2 � 0 (12)

�2 + p01�1 � 0 (13)

8 ZQiQDW}@ CqwYLt VwQi Ct}k 'OvDUy |Qw;OwU p=@vO x@ u=oOvvmO}rwD xm|}=Hu; R=

"CU= xOW xDiQo Q_v QO =yu; O}rwD |xv} Ry R=p1 > ci ; p0i > c0i (14)

9 ZQi15CQ=@a "OW=@ xDW=O|Qw;OwU O}=@ 'OW=@ |R=@ x@ OwQw x@p}=t xOvvmO}rwD xmu}= |=Q@

"Ovm|t |UQQ@ =Q pw= |xOvvmO}rwD |Qw;OwU \QWp2 >

1��c1�2 + �p1�2

(1� �)E1

+(c1�1 � �p1�1 � �c1p1�1 + �p21�1) (15)

14

1 1|xQ

=tW'35

�1|xQwO

'�139

8u=DU@

=D�h}QW

C}Q}Ot

w`}=vY

|UOvyt OwU `@=D xmu}= |=Q@ "OwW|t x@U=Lt u}Uy T} QD=t =OD@= OwU `@=D QakD |UQQ@

u}Uy T} QD=t "OW=@ u}at xt}v |ivt u; R= pY=L u}Uy T} QD=t O}=@ 'OW=@ QaktQO ,q@k \@=wQ u}= "CU= 23 w 22 \@=wQ CQwY x@ OwU `@=D u}= R= xOW pY=L

"Ov=xOW xO=iDU= 1 x}[kd�021d2p01

=

2(�1 + �)�1 00 �2��1

!(30)

d�022d2p02

=

2(�1 + �)�2 00 �2��2

!(31)

u}at '=yxOvvmO}rwD OwU `@=D x@ \w@ Qt u}Uy T} QD=t xm OW C=@F= 1 x}[k QOOwU |xv}W}@ uDi=} |=Q@ u}=Q@=v@ "CU= Qakt OwU `@=wD xH}Dv QO w OvDUy |ivtpw= jDWt uO=O Q=Qk QiY =@ OwU `@=wD |=Q@ |rO=aD \=kv =yxOvvmO}rwD x@ \w@ Qt

"O};|t CUOx@ `@=wDp01 = �1 + c01�1 + p02�2

2�1(32)

p02 = �2c02 + p01�1 + �22�2

(33)

33 w 32 \@=wQ QO p01 w p02 Q}O=kt pw= jDWt x@ \w@ Qt \@=wQ u=tRsy pL =@"OvwW|t pY=L 29 w 28 \@=wQ CqO=at x=oDUO pL R= TB w &OwW|t x@U=Lt

�"OwW|t C@=F 2 x}[k w x@U=Lt |rO=aD Ct}k Q}O=kt u}=Q@=v@35 w 34 \@=wQ QO O}OH \}=QW QO xOvvmO}rwD Qy x@ \w@ Qt |rO=aD |=[=kD

"CU= xOW x@U=Lt

D01 = 2�1��1 + c01�2 +

�2

+(s2� + c2)�2 + c01�1

!�2�

=4�1�2 � �1�2 (34)

D02 = �2��1�1 + �1

2�2 � 2(s2� + c2)�2+(c01)�1

!+ (s2� + c2)�1�2

�=4�1�2 � �1�2 (35)

"CU= xOt;CUO x@ 37w36\@=wQ QO x} wv=F\}=QWQO =yxOvvmO}rwD x@\w@ QtOwU�0F 1 = 1

(�4�1�2 + �1�2)2 = �1((2�1�2 � 2c01�1�2 + (�2 + (s2� + c2)�2 + (c01)�1)�2)2

���(2s1�1�2 � (s2�2 + s1)�2)

(4�1�2 � 2(c01 + c1)�1�2+

(2�2 + (s2� + 2c2)�2 + (c01 + c1)�1)�2)) (36)

�0F 2 = 1(�4�1�2 + �1�22)2 �2

((�1(�1 + (s2� + c2)�2 + �1E2)2

���(�1(2s2�2 + s1�1)� s2�1�2) �4�2�1 + �1(2(A(�1 + x)� + 2c2)�2 � (c01 + c1)�1)��1(2�1 + (s2� + 2c2)�2)

!)

(37)

u}at |ivt '0 � � < 1 xmu}= x@ Q_v w �i Q=Okt uOw@ C@Ft x@ xHwD =@ u}Uy"OvQakt OwU `@=wD 'CU=

H(�0f1) =

2(� � 1)�1 00 �2��1

!(22)

H(�0f2) =

2(� � 1)�2 00 �2��2

!(23)

|=Q@ x}rw= Ct}k |rO=aD Q}O=kt 'Vv pO=aD w =yxOvvmO}rwD OwU `@=wD R= xO=iDU= =@CUO x@ 21 w 20 \@=wQ CQwY x@ x]w@ Qt x=oDUO pL R= TB xOvvmO}rwD wO Qy

�"OwW|t C=@F= 1 x}[k w O};|t|x}rw= Cr=L QO =[=kD Q=Okt 'xOW x@U=Lt |rO=aD Ct}k Q}O=kt x@ xHwD =@

"CU= xOy=Wt p@=k 25 w 24 |=yx]@=Q QO |R=@D1 = �1(2(�1 � c1�1)�2 + (�2c2�2 + c1�1)�2)

4�1�2 � �1�2(24)

D2 = �2(�1�1 + �1(2�2 � 2c2�2 + c1�1) + c2�1�2)4�1�2 � �1�2

(25)

"CU= xOt; CUO x@ 27 w 26 |=yx]@=Q QO |rO=aD OwU Q}O=kt u}vJty

�F 1 = �1

2(�1 � c1�1)�2+(�2 + c2�2 + c1�1)�2)

!2�(�4�1�2 + �1�2)2

(26)

�F 2 = �2(�1�1 + �1(2�2 � 2c2�2 + c1�1)

+c2�1�2)2=(�4�1�2 + �1�2)2 (27)

�OvQDQ@ - nQ@rmDU= |R=@� Q=Pox}=tQU OwQw "2"4u}}aD OwN OwU `@=D x@ xHwD =@ =OD@= '|R=@ Q@yQ u=wva x@ '|R=@ x@ OwQw =@ Q=Pox}=tQU=yCmQW TBU w Ovm |Q=Pox}=tQU CmQW Qy QO Q=Okt xJ CU= Q[=L xm Ovm|t"OvR=OQB|t OwN pwYLt Ct}k u}}aD x@ OvQw;|t CUO x@ xm |er@t x@ xHwD =@pL |=Q@ "Ovvm|t xv}W}@ =Q OwN OwU 'Q=Pox}=tQU R= |wQ}B x@ =yxOvvmO}rwD `k=w QO|xr�Ut� s}vm|t xv}W}@ =Q =yxOvvmO}rwD OwU `@=D =OD@= 'nQ@rmDU= pO=aD j@] '|R=@|xv}y@ Q=Okt =D s}vm|t |Q=Po}=H �Q@yQ� Q=Pox}=tQU `@=D QO =Q Q}O=kt TBU w �wQ}Bx@U=Lt x} wv=F |=yCt}k O}=@ =OD@= wQ}B |xr�Ut pL |=Q@ "O}=}@ CUO x@ Q=Pox}=tQUxOvvmO}rwD wO Qy xm ?r]t u}= uDiQo Q_v QO =@ |rO=aD Q}O=kt |x@U=Lt |=Q@ "OwWVv pO=aD ltm x@ =yCt}k |rO=aD Q}O=kt 'Ovvm|t |Q}os}tYD u=tRsy CQwY x@

"OwW|t x@U=LtR= TB =yxOvvmO}rwD R= s=OmQy |=Q@ Vv pO=aD R= xOW pY=L Ct}k %2 x}[k

"CU= 29 w 28 \@=wQ Q@=Q@ ?}DQD x@ |Q=Pox}=tQUp01 = 2(�1 + c01�1)�2 + (�2 + c02�2)�2

4�1�2 � �1�2(28)

p02 = 2�1(�2 + c02�2) + (�1 + c01�1)�14�1�2 � �1�2

(29)

%C=@F==OD@= 'p01 w p02 O}OH \}=QW QO =yxOvvmO}rwD |xv}y@ Ct}k uOQw; CUO x@ |=Q@|=Q@ "OvDUy Qakt x} wv=F Ct}k ?UL Q@ =yv; OwU `@=wD xm CU= xOW xO=O u=Wv

15

"""w|Q=

PoCt}k

|=Q@|R=

@|x}Q_

vOQm}wQ'|Q=Pox}=tQU R= TB xH}Dv QO w CU= x}rw= \}=QW QO Ct}k R= QDtm O}OH \}=QW

"Ow@ Oy=wN x}rw= Cr=L R= V}@ =[=kDp1 � p01 = A��2(2x�1 � (x� 1)�2)

4�1�2 � �1�2(44)

(�1 >�22 ; x >

�2�2�1 + �2

)) (45)

%=[=kD x@ \w@ Qt C=Q}}eDCUO x@ 25 w 24 \@=wQ QO |Q=Pox}=tQU R= p@k =yxOvvmO}rwD x@ \w@ Qt |=[=kD35 w 34 \@=wQ QO R}v |Q=Pox}=tQU s=Hv= R= TB =[=kD |rO=aD Q}O=kt u}vJty "Ot;46 |x]@=Q CQwY x@ Cr=L wO QO =[=kD Q}O=kt x@ \w@ Qt hqDN= "OW x@U=Lt

"CU=D1 �D01 = 2(�1 � c01�1)�1�2+

(�2 + (s2� + c2)�2 � c01�1)�2�1

=4�1�2 � �1�2 (46)

QO =[=kD R= QDW}@ O}OH Cr=L QO =[=kD 'OW=@ xDW=O =Q 47 \QW �2 Qo ="CU= x}rw= sDU}U

�2 < �2�1�2 + 2�1�2(c01)+

�2�2(A� � s1� � c2)� �1�2c01=�2 (47)

QO =[=kD hqDN= 'x@=Wt CQwY x@ R}v swO |xOvvmO}rwD |=[=kD OQwt QO"CU= xOW x@U=Lt 48 |x]@=Q

D2 �D02 =��2(�1(2s2�2 + s1�1)� s2�1�2)

4�1�2 � �1�2(48)

QO `k=w QO "CU= |ivt xQ=wty '49 |x]@=Q QO xOW u=}@ \QW =@ 48 |x]@=QCr=L R= QDW}@ |Q=Pox}=tQU R= TB swO |xOvvmO}rwD |=[=kD xQ=wty \}=QW u}=

"Ow@ Oy=wN x}rw=x < �1�2 � 2�1�2

�1�2 � 2�1�2 � �1�1(49)

CQwY x@ 'Ow@ xOW x@U=Lt |rO=aD OwU \@=wQ QO xm OwU Q}O=kt u}@ hqDN="CU= xOt; CUO x@ swO w pw= |xOvvmO}rwD |=Q@ ?}DQD x@ 51 w 50 \@=wQ

�0F 1 � �F 1 = �(A�(� � 1)�1

(2x�1�2 + ((x� 1)�2 � x�1)�2)

(2(2�1 + (s1� � 2c1)�1)�2+

(2�2 + (s2� + 2c2)�2 + (�s1� + 2c1)�1)�2))

=(�4�1�2 + �1�2)2 (50)

"CU= O}OH Cr=L QO OwU V}=Ri= Qou=}@ xm CU= C@Ft xQ=wty jwi CQ=@a�0F 2 � �F 2 = A�(� � 1)�2

(�1(2(x� 1)�2 + x�1)� (x� 1)�1�2)

(2�1�1 + �1(4�2 � 2(s2�2c2)�2+

(�s1� + 2c1)�1) + (s2� + 2c2)�1�2)

=(�4�1�2 + �1�2)2 (51)

'CU= nQ@rmDU= wv R= Q=Pox}=tQU =@ =yxOvvmO}rwD u}@ C@=kQ xmu}= x@ xHwD =@2 x}[k j@] =yxOvvmO}rwD R= xOt; CUO x@ |=yCt}k |xv}y@ pL |Q=Po}=H =@

"O};|t CUO x@ 38 |x]@=Q 'Q=Pox}=tQU OwU `@=D QO

�1 = ��� (�f2 + �f1) + �2F1 + �1F2

(�4�1�2 + �1�2)2

�(38)

"CU= xOW xOQw; CUw}B QO F1 w F2 Q}O=kt'\wQW |Q=QkQ@ CQwY QO w s} R=OQB|t Q=Pox}=tQU OwU `@=D QakD |UQQ@ x@ xt=O= QOOwU `@=D swO jDWt Qw_vt u}O@ "OwW|t x@U=Lt u; x@ \w@ Qt OwU |xv}W}@ Q=Oktx@U=Lt 39 |x]@=Q QO pw= CmQW QO |Q=Pox}=tQU OYQO ?UL Q@ Q=Pox}=tQU

"OwW Qakt OwU `@=D =D OW=@ |ivt O}=@ Q=Okt u}= "OwW|t@2�I@x2 = 2A2�2�(�21�2(4�1�2) + (2�2 + �1)2)+

2�1�2(2�1�2 � 2(�1 + �2)�1 � �21)�2+

(�1(�2 � �1)2 + �2�21)�22)=(�4�1�2 + �1�2)2 (39)

R= TB Q=Pox}=tQU OwU `@=D |=Q@ |i=m w sRq \}=QW uOw@ Q=QkQ@ ZQi =@CQwY x@ Q=Pox}=tQU O}O R= pw= CmQW QO |rO=aD |Q=Pox}=tQU OYQO 'pOt pL

"O};|t CUO x@ 40 |x]@=Q@2�I@x2 = 0! x = �1�2F3 + �2F4�2 + F5�22

A�F6(40)

"CU= xOt; CUw}B QO F6 =D F3 Q}O=ktx@ =Q CmQW Qy QO xOW |Q=Pox}=tQU |rO=aD Q}O=kt u=wD|t 3 w 2 \@=wQ ltm x@

"OQw; CUO x@ Q} R CQwY

s1 = �1�2F3 + �2F4�2 + F5�22�F6

(41)

s2 = A�F6 � (�1�2F3 + �2F4�2 + F5�22)�F6

(42)

"Ow@ Oy=wN 42 |x]@=Q CQwY x@ Q=Pox}=tQU OwU Q=Okt u}vJty�I = (�(I2 + I3 � 2�31�22I4�2 � �21�2

(I5 + I6 + I7)�22 + 2�1�2(Is + I9

�I10�42 � I11 + I12 + �1�2(I13)�22

+I14 + �1�1I15 + c1�1I16)) + 2�2�1

(�1�22I17 + 2�31�22((2�2 + �1)2

(2c2�2 � C1�1) + I18 + I19 + �21�2�2I20))=M (43)

"CU= xOW xOQw; CUw}B QO M w I20 =D I1 Q}O=kt

G}=Dv p}rLD "6"OwW|t xDN=OQB xOt; CUO x@ G}=Dv p}rLD w |UQQ@ x@ CtUk u}= QO

xm Cio u=wD|t |Q=Pox}=tQU R= TB w p@k xOvvmO}rwD Ct}k uDiQo Q_v QO =@QO Ct}k w OwW|t QHvt Ct}k Vy=m x@ |Q=Pox}=tQU OW=@ Q=QkQ@ Q} R \}=QW Qo =

16

1 1|xQ

=tW'35

�1|xQwO

'�139

8u=DU@

=D�h}QW

C}Q}Ot

w`}=vY

|UOvyt C}U=UL p}rLD "2"7

x}=tQU dr@t pm QO Q}}eD �x}=tQU pm u=R}t QO Q}}eD Q}F-=D CLD =Q |R=@ pt=wa OwU u=R}t QO Q}}eD '2 pmWQO Q=Pox}=tQU xm |}x}=tQU u=R}t QO V}=Ri= '2 pmW x@ xHwD =@ "OyO|t u=WvQO Q}F-=D u}= COW "OW Oy=wN |R=@ pt=wa s=tD OwU V}=Ri= ?Hwt 'OQ=O Q_v

"CU=yxOvvmO}rwD OwU R= QDW}@ Q=Pox}=tQU OwU=@ CqwYLt xm |}=yCt}k 'OW=@ QDW}@ x}=tQU pm u=R}t xJQy '3 pmW j@]u=R}t |Dt}k QF= Cra x@ Vy=m u}= "O@=}|t Vy=m 'OvwW|t x[Qa Q=R=@ x@ =yv;

"CU= |O}rwD |=yxv} Ry Vy=m QO |Q=Pox}=tQU?Hwt |Q=Pox}=tQU u=R}t V}=Ri= 'CU= xOy=Wt p@=k 4 pmW QO xm Qw] u=ty

"x}=tQU u=R}t ?UL Q@ |R=@ pt=wa OwU "2 pmW

"A C=Q}}eD ?UL Q@ Ct}k C=Q}}eD "3 pmW

"x}=tQU u=R}t QO Q}}eD ?UL Q@ =[=kD QO Q}}eD "4 pmW

O}OH Cr=L QO OwU V}=Ri= Qou=}@ xm CU= C@Ft xQ=wty R}v 51 |x]@=Q"CU=

u=wD|t � w � |=yQDt=Q=B |=Q@ xOvvmO}rwD O}O R= |Dw=iD|@ \=kv x@ xHwD =@"CiQo Q_v QO =Q Q} R u}}=B OwOL

� = ((�c1�1 + �p1(�1 + p2�2 + c1�1 � p1�1)

�c1p2�2� + (1 + �)(p01�1 + c1p01�1�p021 �1 � c1p02�2 + p01p02�2)

=(� � 1)x(�1 � p01�1 + p02�2)) (52)

� = (p01 � c1)(�1 � p01�1 + p02�2)=

(p01 � p1)�1(�c1p1 + p21 + c1p01 � p021 )

�1 + �2(c1p2 � p1p2 � c1p02 + p01p02) (53)

C}U=UL p}rLD "7?UL Q@ C}U=UL R}r=v; TBU w OwW|t |UQQ@ |OOa p=Ft =OD@= VN@ u}= QO

"OQ}o|t s=Hv= |OOa |=yxO=O

|OOa p=Ft "1"7|wQ Q@ TBU "OwW|t xDN=OQB u; pL w |OOa p=Ft |UQQ@ x@ CtUk u}= QOG=QNDU=|D} Q}OtG}=Dv w OQ}o|tCQwYC}U=ULp}rLD xr�UthrDNt|=yQDt=Q=B

"OwW|t�1 = 20000;�2 = 20000; �1 = 8;�1 = 6; �2 = 7;�2 = 5;c1 = 6000; c2 = 7000; � = 0=4; � = 0=0003;A = 40000000R= pY=L |xOW x@U=Lt \@=wQ QO jwi |=yQDt=Q=B Q}O=kt |Q=Po}=H =@

"O};|t CUO x@ 1 pwOH 2 w 1 |=yx}[k

"xr�Ut pL x@ \w@ Qt G}=Dv "1 pwOH�0 p0 p c0 si

2 0195 2248 35 64850 275 1 19083160 xOvvm O}rwD�10^7 pw=

19066 2754 6 75600 724 95 20916840 xOvvm O}rwD600 swO

� 1� x x Q=Po x}=tQU2 072678 52 3 47 7�10^7

17

"""w|Q=

PoCt}k

|=Q@|R=

@|x}Q_

vOQm}wQ

"|Q=Pox}=tQU R= TB O}rwD |xv} Ry |=R= x@ |R=@ pt=wa OwU "7 pmW

"=DrO u=R}t ?UL Q@ =[=kD "8 pmW

"OwU u=R}t |wQ Q@ w x}=tQU u=R}t u=tRsy Q}}eD "9 pmW

?} Q[ u}= xJQy '9 pmW j@] R}v Q=Pox}=tQU x@ |t}rUD OwU ?} Q[ OQwt QO"O@=}|t Vy=m =yxOvvmO}rwD OwU 'OW=@ QDW}@

u}@ 0=5 < � < 0=6 =@ OwU O=R=t xm|Dr=L QO OwW|t xOy=Wt u}vJtyQ=Pox}=tQU "OQ}o|t Q=Qk xOwOLtl} QO O=Qi= |xty OwU 'OwW s}UkD |R=@u}iQ]

"pw= |xOvvmO}rwD CqwYLt O}rwD |xv} Ry |=R= x@ |R=@ pt=wa OwU "5 pmW

"swO |xOvvmO}rwD CqwYLt O}rwD |xv} Ry |=R= x@ |R=@ pt=wa OwU "6 pmW

R= TB Ct}k QO xm |Wy=m u=R}t =@ xH}Dv u}= "OW Oy=wN =[=kD u=R}t V}=Ri="OQ=O s}kDUt |x]@=Q 'O};|t OwHw x@ |Q=Pox}=tQU

O}rwD |xv} Ry QO Q}}eD �pt=wa OwU 'OW=@ QDq=@ CqwYLt O}rwD |xv} Ry xJQy '6 w 5 pmW x@ xHwD =@u} Q}=U x@ C@Uv |QDW}@ |Q}PB Q}F-=D Q=Pox}=tQU OwU u}vJty "Ow@ Oy=wN QDtmpmW QO "OwW|tu}t-=D Cw=iDt\}=QW QO OwUCw=iD R= Q=Pox}=tQU OwU =Q} R "OQ=OQ=Okt c1 =OD@= QO "CU= Q}}eD p=L QO pw= |xOvvmO}rwD O}rwD |xv} Ry QO Q}}eD '5'c1 Q=OktV}=Ri= =@ "CU= QDW}@ pw= |xOvvmO}rwD OwU w OQ=O c2 x@C@Uv |QDtmC=Q}}eD R}v 6 pmW QO "O@=}|tV}=Ri=?}kQ OwU wVy=m pw= |xOvvmO}rwD OwUQO "CU= xOy=Wt p@=k 5 pmW x@=Wt OvwQ R}v Q=Owtv u}= QO &CU= Q_v Ot c2"CU= xOW Q=Pox}=tQU OwU Vy=m ?Hwt O}rwD Ct}k V}=Ri= 'Q=Owtv wO Qy

Ct}k |Wy=m ?} Q[ QO Q}}eD �OwU 'OW=@ QDW}@ O}rwD |=yxv} Ry |wQ |Qw=:vi Q}F-=D xJQy '7 pmW x@ xHwD =@"CU= QDTwULt Q=Pox}=tQU OwU |wQ Q@ Q}F-=D u}= "Ow@ Oy=wN QDW}@ |R=@ pt=wau=R}t 'OW=@ QDW}@ O}rwD |xv} Ry |wQ |Q=Pox}=tQU Q}F-=D u=R}t xJQy u}vJty

"8 pmW Ow@ Oy=wN QDq=@ |Q=Pox}=tQU uDiQo s=Hv= =@ Q=R=@ |=[=kDOQ}o|t jraD Q=Pox}=tQU x@ xm OwU R= |OYQO w x}=tQU pm dr@t QO Q}}eD Q}F-=D Qo =QDW}@ xOvvmO}rwD OwU xm OwW|t xOy=Wt 's} Q}o@ Q_v QO u=tRsy CQwY x@ =QQ=Pox}=tQU OQwt QO C}U=UL u}= xmv}= ut[ "Ovm|t Q}}eD � u=R}t x@ xHwD =@\UwD x}=tQU V}=Ri= xm OwW|t Ci=} QO ?r]t u}= u}vJty "CU= QDOwyWt

"OQ=O Q=Pox}=tQU OwN OwU |wQ Q@ |QDW}@ Q}F-=D Q=Pox}=tQU

18

1 1|xQ

=tW'35

�1|xQwO

'�139

8u=DU@

=D�h}QW

C}Q}Ot

w`}=vY

|UOvyt

"=[=kD u=R}t |wQ Q@ w x}=tQU u=R}t u=tRsy Q}}eD "10 pmW

"OW=@ xDW=O Q_v Ot =Q ?r]t u}= O}=@ |R=@ Q@yQ u=wva x@"Ovm|t|UQQ@ =[=kD u=R}t |wQ Q@ =Q � w x}=tQU u=R}t u=tRsy C=Q}}eD '10 pmW

CWwv=B1. investment

�References� `@=vt1. Islam Bidgoli, G. and Ehsham Rothi, R. \The applica-

tion of game theory in evaluating investment in stocks",Financial Knowledge Analysis of Securities, 4, pp.101-129 (2011)

2. Smit, H.T. and L.Trigeorgis \Quantifying the strategicoption value of technology investments", Montreal: 8thAnnual International Real Options Theory (2004).

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5. Wu, L.-C. and et al. \Options in technology investmentgames: the real world TFT-LCD industry case", Techno-logical Forecasting and Social Change, 79(7), pp. 1241-1253 (2012).

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7. Berg, J., Dickhaut, J. and McCabe, K. \Trust, reci-procity, and social history", Games and Economic Be-havior, 10(1), pp. 122-142 (1995).

8. Kanagaretnam, K. and et al. \The impact of social valueorientation and risk attitudes on trust and reciprocity",Journal of Economic Psychology, 30(3), pp. 368-380.(2009)

9. Buchan, N.R., Croson, R.T. and Solnick, S. \Trust andgender: An examination of behavior and beliefs in theinvestment game", Journal of Economic Behavior & Or-ganization, 68(3), pp. 466-476 (2008).

10. Gans, J.S. \Regulating private infrastructure invest-ment: optimal pricing for access to essential facilities",Journal of Regulatory Economics, 20(2), pp. 167-189(2001).

11. Ajagbe, A. and et al. \The selection criteria for �nancingyoung �rms using debt options", Frontiers of Business,Management and Economics, pp. 265-269 (2013).

12. Kong, J.J. and Kwok, Y.K. \Real options in strategicinvestment games between two asymmetric �rms Eu-ropean", Journal of Operational Research, 181(2), pp.967-985 (2007).

13. Houser, D., Schunk, D. and Winter, J. \Distinguishingtrust from risk: An anatomy of the investment game",Journal of Economic Behavior & Organization, 74(1),pp. 72-81 (2010).

14. Benaroch, M. \Managing information technology invest-ment risk: A real options perspective", Journal of Man-agement Information Systems, 19(2), pp. 43-84 (2002).

15. Nishihara, M. and Fukushima, M. \Evaluation of �rm'sloss due to incomplete information in real investmentdecision", European Journal of Operational Research,188(2), pp. 569-585 (2008).

16. Huisman, K. and et al. \Strategic investment under un-certainty: merging real options with game theory", pp.22(2003).

19

=[=kD u=R}t |wQ |QDW}@ Q}F-=D '�C=Q}}eD 'CU= OwyWt R}v pmW QO xm Qw] u=ty"OW Oy=wN =[=kD V}=Ri= ?Hwt R}v |Q=Pox}=tQU u=R}t V}=Ri= u}vJty "OQ=O

|Q}oxH}Dv "8"CU= xOvvmO}rwD wO =@ |Q=R=@ QO |Q=Pox}=tQU u=R}t |UQQ@ 'hOy xr=kt u}= QO|aU Q=DWwv u}= QO "Ov=xOW xDiQo Q_v QO u} Ro}=H |=yq=m CQwY x@ CqwYLtQ=Pox}=tQU xm |}xvwo x@ &OwW xO=iDU= xr�Ut pL |=Q@ =y|R=@ |x} Q_v R= CU= xOWu}}aD w |Q=PoCt}k |=Q@ |Q}os}tYD p=@vO x@ w OvDUy wQ}B =yxOvvmO}rwD w Q@yQw =yxOvvmO}rwD x@ \w@ Qt OwU `@=wD |UQQ@ =@ "s}DUy |Q=Pox}=tQU |xv}y@ u=R}t|Q=Pox}=tQU |xv}y@ Q=Okt w |rO=aD Ct}k Q}O=kt 'xv=o =OH CQwY x@ Q=Pox}=tQUVy=m x@ xHwD =@ "OW x@U=Lt =yv; OwU uOQm xv}W}@ |=Q@ Q=Pox}=tQU \UwDx=Qty V}=Ri= =@ Q=R=@ |=[=kD 'OQ=wt ?re= QO '|Q=Pox}=tQU R= |W=v Ct}k|R=@ pt=wa |xty OwU Q}O=kt V}=Ri= Ea=@ |Q=Pox}=tQU u}= u}vJty "OwW|t

"OwW|t=Q pt=a u}= u=wD|t "CU= xOWv xDiQo Q_v QO pwB |v=tR VRQ= xr=kt u}= QO

"OQm |UQQ@ |D; C=ar=]t QO =Q u; x@ \w@ Qt Q}F-=D w OQm ^=Lr |R=@ QO R}v

"""w|Q=

PoCt}k

|=Q@|R=

@|x}Q_

vOQm}wQ17. Dixit, A.K. and Pindyck, R.S. Investment Under Un-

certainty, Second printing, Princeton University Press,Princeton, USA. (1996).

18. He, P., Ding, H. and Hua, Z. \Strategic choice of exible production technology using game theory ap-proach", Robotics and Computer-Integrated Manufactur-ing, 28(3), pp. 416-424 (2012).

19. Almfraji, M.A. and Almsa�r, M.K. \Foreign direct in-vestment and economic growth literature review from1994 to 2012", Procedia-Social and Behavioral Sciences,129, pp. 206-213 (2014).

20. charness,G, Cobo-Reyes, R.and Jim enez, N. \An invest-ment game with third-party intervention", Journal ofEconomic Behavior & Organization, 68(1), pp. 18-28(2008).

21. \Inequality-seeking punishment", Economics Houser, D.and Xiao, E. Letters, 109(1), pp. 20-23 (2010).

22. N�oldeke, G. and Samuelson, L. \Investment and com-petitive matching", Econometrica, 83(3), pp. 835-896(2015).

23. De Borger, B., Dunkerley, F. and Proost, S. \Strategicinvestment and pricing decisions in a congested trans-port corridor", Journal of Urban Economics, 62(2), pp.294-316 (2007).

24. Kau�man, R.J. and Li, X. \Technology competition andoptimal investment timing: a real options perspective",IEEE Transactions on Engineering Management, 52(1),pp. 15-29 (2005).

25. Liu, C. and Berry, R.A. \The impact of investmenttiming and uncertainty on competition in unlicensedspectrum. in modeling and optimization in mobile, AdHoc, and wireless networks (WiOpt)", 14th InternationalSymposium on. IEEE (2016).

I3 = �8c21�51�32�21 � 4c21�41�42�21

+16c1c2�41�42�21 � 4c22�41�42�21

�4c21�41�32�31 + 4c1c2�41�32�31 � c21�41�22�41

I4 = 8(c1 � c2)�1�22(c1�1 � c2�2)+

4�2I1�1 + 2(c1 � c2)�2(�5c1�1 + 6c2�2)

�21 � 2(2c21�1 + (c21 � 4c1c2 + c22)�2)

�31 + c1(�c1 + c2)�41

I5 = 4�1�32((c21 � 4c1c2 + c22)�1 + 2c22�2)

�4(c1 � c2)�1�22(6c1�1 � 5c2�2)�1

I6 = �2(24c21�21�(33A2�2 + 8c21 � 66A�c2 + 8c22

+c1(34c2 � 66A�))�1�2 + 24c22�22)�21

I7 = 2(c1 � c2)�2(5c1�1 � 6c2�2)

�31 + (2c21�1 + (c21 � 4c1c2 + c22)�2)�41

I8 = ((c1 � c2)c2�21�32 + 2�1�22

((c21 � 4c1c2 + c22)�1 + 2c22�2)�1

�(c1 � c2)�1�2(6c1�1 � 5c2�2)�21

I9 = (4c21�21 � (5A2�2 + c21 � 10A�c2

+c22 + c1(�10A� + 6c2))�1�2 + 4c22�22)

�31 + (c1 � c2)(c1�1 � c2�2)�41)�32

I10 = c22�22(�1(�2 � �1) + �21)2 + �1�21

(c21�1(�2 � �1)2 �A2�2�2�21 + 2A�c1�2�21)

+2c2�1�2�1(A��31 + c1(�2 � �1)(�1(�2 � �1) + �21))

CUw}BF1 =

�1�1 + �1(2�2 � 2(s2� + c2)�2 + c01�1)

+(s2� + c2)�1�2

!2

F2 =

2(�1 � c01�1)�2+(�2 + (s2� + c2)�2 + c01�1)�2

!2

F3 = 4�1�2(��1 + �2 + c1�1 + (A� � c2)�2)

+2(�2�1 + (�1 + ((A� � c2) + c1)�1)�2)

�1 + (�1 + c1�1)�21

F4 = 2�1(��1 + ((A� � c2) + c1)�1)�2

+2�1(�1 � 2(c1�1 + (A� � c2)�2))

�1 � (�1 + ((A� � c2) + c1)�1)�21

�2�2�1(�1 + �1)

F5 = (A� � c2)�2�21 + �2�1(�1 � �2)

+�1(�2 � �1)((A� � c2)�2 � c1�1)

F6 = �21�2(4�1�2 + (2�2 + �1)2)+

2�1�2(2�1�2 � 2(�1 + �2)�1 � �21)

�2 + (�1(�2 � �1)2 + �2�21)�22

I1 = �4c21�21 +

5A2�2 + c21 � 10A�c2 + c22+c1(�10A� + 6c2)

!�1�2 � 4c22�22

I2 = �16c21�61�42 + 16A2�2�51�52

�32A�c1�51�52 � 32A�c2�51�52

+32c1c2�51�52 � 16c22�41�62�16c21�51�42�1 + 16c1c2�51�42�1

+16c1c2�41�52�1 � 16c22�41�52�1

20

1 1|xQ

=tW'35

�1|xQwO

'�139

8u=DU@

=D�h}QW

C}Q}Ot

w`}=vY

|UOvyt I16 = (�1(4�1�2 � �1(2�2 + �1)) + (2�1(�2 � �1) + �21)

�2)(4�21�22 + �1�2(�1�2 � �2(�1 + �2))+

�1�2(�1(�1 � 4�2) + 2�2(�1 + �2))

I17 = 2�22�21(�c1�1 + 2c2(3�2 + �1)) + �2�1(2c2�22

�3(�3A� + c1 + c2)�2�1 � (A� � 3c1 + c2)�21)�2�(�2 � �1)2(c2�2 + c1�1)�22

I18 = ((8A� � 4c1)�22 + 12(�A� + c1)�2�1

+5c1�21)�2 � 2((�2c1 + c2)�2 + 3c1�1)�22

I19 = 8�41�32(2(A� � c1)�2 + c1(��1 + �2))��2�21�32(A�1�2 + c2(��1�2 + �2(2�1 + �2)))

I20 = 3A��2�1(�8�2 + 3�1)�2 � 2c2�2(�1(2�2 + �1)

(6�2 + �1)� �1(2�2 + �1)�2 + 2(�2 � �1)�22)

+c1(�1(�1 � 2�2) + 2�2(�1 + �2))

(�3�1�2 + �2(4�1 + �2)))

M = (�4�1�2 + �1�2)2(�21�2)(4�1�2 + (2�2 + �1)2)

+2�1�2(2�1�2 � 2(�1 + �2)�1 � �21)�2 + �1(�2 � �1)2

+�2�21)�22)

E1 = �s1��1 � p01�1 + s1�p01�1 � c1p01�1+

p021 �1 � s1�p022�2 + c1p0�2 � p01p02�2

E2 = 2�2 � 2(s2� + c2)�2 + (c01)�1

I11 = �21�22(�1(4�1�2 � �1

(2�2 + �1)) + (2�1(�2 � �1)

+�21)�2)2

+a22�21

2�1�2(2�2 + �1)� 2�2(�1 + �1)�2 + (��2 + �1)�22

!2

I12 = 2�1�2(2�31�22 c2�1(2�2 + �1)2+

4�1�2(2(A� � c2)�2 + (2A� � c2)�1)

!

��21�2

0BB@ 24A��1�2�1(�2 + �1) + c2 8�1�22(�1 + �2)� 24�1�22�1+2�2(�5�1 + 6�2)�21 + 8�2�31 + �41

!1CCA�2

I13 = 9A��1�21(�2 + �1) + c2 8�21�2(��2 + �1) + 2�31(3�2 + �1)+�1(2�2 � 3�1)�1(5�2 + �1)

!I14 = ��32

A��1�31(�2 + �1) + c2�2

(�1(�2 � �1) + �21)(2�1(�2 � �1) + �21)

!I15 = (�1(4�1�2 � �1(2�2 + �1))

+(2�1(�2 � �1) + �21)�2)(2�1�2(2�2 + �1)

�2�2(�1 + �1)�2 + (�1 � �2)�22)

21