close

Вход

Забыли?

вход по аккаунту

?

Одна неантагонистическая игра поиска с распределением ресурсов.

код для вставкиСкачать
Y.LI:K 519.8
BeCTHHK Cn6ry. Cep. 10, 2006, Bbln. 1
A. 10. rap'Hae8, T. Fl. CUaJ£pmO
O,ll,HA HEAHTArOHHCTH"tJECKA5I HrPA IIOHCKA
C PACIIPE,n:EJIEHHEM PEGYPCOB *)
1. BBe,n;eHHe.
B ,lI,aHHOM CTaTbe HCCJIe,lI,OBaHa O,lI,Ha HeaHTarOH11CT11'-1eCKajI 11rpa n011CKa
06'beKTa, 0606rn;arorn;ajI 3a,lI,a'-lY, paCCMOTpeHHYro CaKarY'-l11
MMeroTCjI ,lI,Ba 11rpOKa
TO'-leK
(1
11
2),
[1].
n
pi,i E [1,nj,
11rn;yIII;Hx'HenO,lI,B11)KHbIM 06'beKT, CnpjITaHHbIM B O,lI,HOM H3
~eJIO'-l11CJIeHHOrO 11HTepBaJIa
[1,n],
np11'-1eM 11rpOK11 3HaIOT BepOjiTHOCT11
K~,n;M TO'-lKa i xapaKTep113yeTCjJ ,lI,BYMjI
Ai (Pi) TaK11M11, '-ITO eCJI11 11rpOK 1 (2) 11CnOJIb3YeT
i, TO BepOjiTHOCTb 06uapY)KeU11jI ,n;aHHoro 06'beKTa
C KOTOPbIM11 06'beKT CnpjITaH B 3T11X TO'-lKax.
nOJIO)K11TeJIbHbIM11 napaMeTpaM11 nOHCKa
T pecypcoB ,n;JIjI n011CKa 06'beKTa B TO'-lKe
np11 YCJIOB1111, '-ITO OH HaxO,lI,11TCjI B 3TOM TO'-lKe, paBHa
napaMeTpbI n011CKa
1(2)
11MeeTCjI
X(Y)
Ai, Pi, i E [1, n],
1- exp(-AiT) (1-
H3BeCTHbI 06011M HrpOKaM.
B
exp(-piT)). Bce
pacnOpjI)KeH1111 11rpOKa
pecypcoB n011CKa, KOTopble OH MO)KeT pacnpe,n;eJIjITb no TO'-lKaM. TaK)Ke
npe,n;nOJIaraeTCjI, '-ITO ocyrn;eCTBJIeU11e n011CKa CBjI3aHO C HeKOTOpbIM11 3aTpaTaM11, a 11MeHHO,
11CnOJIb30BaH11e T pecypcoB 06XO,n;11TCjI
11rpOKa paBeH
1,
C I T (C2 T)
11rpoKy
1 (2), r,n;e C I , C2 ~ O.
eCJI11 OH 06HapY)K11T 06'beKT, aero npOT11BH11K HeT.
11HTepnpeT11pOBaHa KaK ~eHHOCTb CnpjITaHHOro 06'beKTa.
HaM,n;eT 06'beKT, TO 11X BblHrpbllll paBeH
11rpOK
1
nOJIY'-laeT ql, a 11rpOK
CaKarY'-l11
,n;JIjI
k
[1]
= 1, 2, T.
2
O.
nOJIY'-laeT
3,n;ecb
1
BblHrpbIlll
MO)KeT 6bITb
ECJI11 H11 O,n;11H 113 11rpOKOB He
ECJI11 06'beKT BbljIBJIeH 06011MH 11rpOKaMH, TO
q2,
r,n;e ql
+ q2
:::; 1.
=
=0
Ck
qk
e. OH He y'-l11TbIBaJI CT011MOCTb 11CnOJIb30BaHHbIX pecypcoB, a B CJIy'-lae 06­
11CCJIe,n;OBaJI Bap11aHT ,n;aHHoM 11rpbI B npe,n;nOJIO)KeH1111, '-ITO
HapY)KeH11jI 06'beKTa C'-I11TaJI, '-ITO OH He ,n;OCTaHeTCjI H11 O,lI,HOMY 113 11rpOKOB. KpOMe Toro,
CaKarY'-l11 npe,n;nOJIaraJI, '-ITO Bce pecypcbl nOHCKa 06jI3aTeJIbHO ,n;OJI)KHbl 6bITb 11CnOJIb3Q­
BaHbI 11rpOKaM11.
B
CBjI311 C 3a,n;a'-laMH pacnpe,n;eJIeHlljI pecypcoB CT011T TaK)Ke ynOMjIHYTb
cTaTbro Ma3aJIOBa 11 CaKarY'-l11
[2],
B KOTOPOM 6blJI paccMoTpeH ,n;pyroM acneKT, a 11MeHHO,
OnT11MaJIbHOe MeCTOnOJIO)KeH11e pecypcoB.
KpoY'-lep
[3]
11CCJIe,lI,OBaJI aHTarOHHCT11'-1eCKYro Hrpy C pacnpe,n;eJIeH11eM pecypcoB, MO,n;e­
JI11pyrorn;yro aTaKY paKeTHblX lllaxT H HX 3aIII;HTy.
06Pa30M. B O,n;HOM 113
n
GHa MO)KeT 6blTb on11CaHa CJIe,n;yIOIII;HM
TO'-leK CnpjITaH 06'beKT, npH'-IeM H3BeCTHbI BepOjiTHOCTH
Pi,
C KOTQ­
pblM11 06'beKT CnpjITaH B TO'-lKax. MMeroTCjI ,n;Ba HrpOKa: Hrn;yrn;11M 11 npjI'-IyIII;HM. Mrn;yIII;HM ,
HMejI B CBoeM pacnOpjI)KeH11H
r,n;e
L:~=l Xi
)KeH1111
Y
X
pecypcoB, pacnpe,n;eJIjIeT 11X
= X, C ~eJIbro HaMT11 CnpjITaHHbIM 06'beKT.
pecypcoB, pacnpe,n;eJIjIeT HX
Y = (YI,'" ,Yn)
= (Xl, •.• , Xn)
x
no TO'-lKaM ,
IIpjI'-Iyrn;HM, 11MejI B CBOeM pacnopjI­
no TO'-lKaM, r,n;e L:Z:I
Yi
= Y,
'-IT06bI
npenjITCTBOBaTb 06HapY)KeH11ro 3TorO 06'beKTa. IIpe,n;noJIaraeTcjI, '-ITO BepOjiTHOCTb 06Hapy­
)KeH11jI 06'beKTa paBHa
3aIII;HTbI B TO'-lKe
L:~=l (1- exp( -AiXi))
i, i E [1, n].
YKa3aHHoro 06'beKTa.
exp( -PiYi) r,n;e
Ai
11
Pi -
napaMeTpbI nOHCKa H
IJ;eJIb Hrn,yrn;ero - MaKCHMH3a~11jI BepOjiTHOCT11 06HapY)KeHHjI
BacToH 11 rapHaeB
[4]
HCCJIe,n;OBaJI11 HeaHTarOH11CTH'-IeCK11M BapHaHT
,n;aHHoM 11rpbI, y'-lHTbIBarorn;11M CTOHMOCT11 n011CKa
CI
11 3arn;HTbI
C2,
a TaK)Ke npe,n;nOJIarM,
'-ITO HrpOK11 MoryT 11CnOJIb30BaTb TOJIbKO '-IaCTb HMerorn;11XCjI B 11X pacnOpjI)KeHH11 peCYPCOB .
B 3TOM 11rpe ~eJIbro Hrn;yrn;efo HrpOKa jIBJIjIeTCjI MaKCHM113a~HjI O)K11,n;aeMOM CTOHMOCTH HaM­
,n;eHHoro 06'beKTa (He YMaJIjIjI 06rn;HOCTH MO)KHO npe,n;nOJIaraTb, '-ITO CT011MOCTb CnpjITaHHOrO
06'beKTa palma
1
3a BbI'-IeTOM 3aTpaT), a ~eJIbro 3arn;Hrn;arorn;erOCjI 11rpOKa -
M11H11M113a~
.) Pa60Ta '!aCTH'!HO <pHHaHCHpOBaHa 1'13 cpe~CTB rpaHTa npe3H~eHTa P<l> no nOMep)f{Ke BeAY~HX Hay'!­
HbIX IlIKOJI P<l> (.N~ HIII-2174.2003.1).
© A. 10 . rapHaeB, T. n. CH~}IPTO, 2006
26
O)KM.n;aeMoM CTOMMOCTM HaM.n;eHHOrO 06'beKTa 3a_BhI'IeTOM 3aTpaT. Tor.n;a BhIMrphIlllM M~­
lU,erO
M1
M2
M 3alU,MlU,aIOlU,erO
MrpOKOB paBHhI
n
n
= LPi(1- e-AiXi)e-JliYi -
M 1(x,y)
C1 LXi,
i=l i=l
n
n
M2(X, y) =- LPi(.li=l e->'iXi)e-JliYi -
C2 LYi.
i=l
9Ta Mrpa MCrrOJIh30BanaCh BaCTOHOM M rapiIaeBhIM .n;mi Mo.n;eJIMpOBaHM.H BoeHHO-TeXHOJIO­
rM'IeCKOrO KOHq)JIMKTa «3Be3.n;lIhIe BOMHhI», KOTophIMnpOM30111eJI B 80-e rO.n;DI
CIlIA
M
2.
xx B.
Me)K.n;y
CCCP;
I1cCJIe.n;yeM CJIe.n;yIO~IO HeaHTarOHMCTM'IecKYIO Mrpy C
'll0pMynHpoBKa HrpbI.
pacnpe.n;eJIeHMeM pecypcOB.
I1MeIOTc.H .n;Ba MrpoKa
M3
n
(1
M~lU,MX HenO,n;BM)KHhIM 06'beKT, HaXO,ll,HlU,MMC.H B O.n;HOM
M 2),
[1, n], npM'IeM MrpoKM 3HaIOT BepO.HTHOCTM Pi, C
E [1, n]. KpoMe Toro, Ka)K.n;M TO'IKa i xapaKTepM3y­
TO'IeK u,eJIO'IMCJIeHHOrO MHTepBana
i
KOTOPhIMM 06'beKT Cnp.HTaH B TOqKaX
eTC.H .n;ByM.H nOJIO)KMTeJIhHhIMM napaMeTpaMM rrOMCKa
eCJIM MrpoK
1
Ai (J.Li) .n;JI.H MrpoKa 1
06'be~Ta B TO'IKe i,
(2) TaKMMM, 'ITO
(2) MCrrOJIh3yeT 7 pecypcoB .n;JI.H nOMCKa
TO BepO.HTHOCTh
ero 06HapY)KeHM.H npM YCJIOBMM, 'ITO OH HaxO.n;MTC.H B 3TOM TO'IKe, paBHa
( 1-exp(-J.Li7)).
Bce napaMeTphI
p.H)KeHMM MrpOKa 1(2) MMeeTC.H
no TO'IKaM.
Ai,J.Li, i E [1,n],
X(Y)
1-
exp( -Ai7)
M3BeCTHhI 060MM MrpoKaM.
B
pacno~
pecypcoB nOMCKa, KOTophIe OH MO)KeT pacnpe.n;eJI.HTh
I1crroJIh30BaHMe 7 pecypcoB MrpOKaMM CB.H3aHO C HeKOTophIMM qmHaHcoBhIMM
3aTpaTaMM, a MMeHHO, OHO 06Xo.n;MTC.H
C17 (C2 7)
MrpOKy
1
(2), r.n;e
C1, C2 ~ O. BhIMrphIlll
3.n;ech 1 MO)KeT 6hI;h
HrpOKa paBeH 1, eCJIM OH o6HapY)KMT 06'beKT, aero npOTMBHMK HeT.
MHTepnpeTMpoBaHa KaK u,eHHOCTh 06'beKTa. ECJIM HM O.n;MH M3 MrpOKOB He HaM.n;eT 06'beKT, TO
HX BhIMrphIlll paBeH
paseH
q1,
O.
a MrpoKa 2 -
ECJIM 06'beKT 06HapY)KeH 060MMM MrpOKaMM, TO BhIMrphIlll MrpOKa 1
Q2,
r.n;e
Q1
+ Q2 ~ 1, T. e.
B .n;aHHOM cJIY'Iae MrpOKM .n;eJI.HT CTOMMOCTh
HaM.n,eHHoro 06'beKTa M npM 3TOM .n;onycTHMa BhIllJIaTa KaKOM-TO 'IaCTM CTOMMOCTM TpeTheM
CTopoHe, KOTOpM o6eCne'IMBaeT co6mo.n;eHMe npaBMJI MrphI. .
-qMcThle cTpaTerMM MrpOKa 1 onMChIBaIOTC.H BeKTopaMM
CypChI, pa3Melll,aeMhle MrpOKOM 1 B TOqKe
i,
X
­
=
(Xl, ... ,x n ),
Xi -
r.n;e
pe­
npMqeM
n
L Xi ~ X, Xi ~
i=l
0 .n;JI.H
i
= 1, •.. ,n.
'llHcThle CTpaTerMM MrpOKa 2 onMChIBaIOTC.H BeKTopaMM
pa3MelU,aeMhle MrpOKOM
2
B TOqKe
i,
Y
(1)
= (Y1, . .. ,Yn), r.n;e Yi -
pecypchI,
npMqeM
n
LYi ~ Y, Yi ~
i=1
IIycTh
H
M2 -
81
M
82
0 .n;JI.H
i
= 1, ... , n.
(2)
- MHO)KeCTBa Bcex qMCThIX CTpaTerMM MrpOKOB. KpOMe Toro, rrYCTh
M1
BhIMrphIlllM MrpOKOB npM YCJIOBMM, 'ITO OHM McnOJIh3YIOT 'IMCThle CTpaTerMM X M
y.
Tor.n;a
n
M 1(x, y)
n
= LPi(1- e-AiXi)e-JliYi + Q1 LPi(1- e->'i Xi)(1 -
i=l
n
M 2(x, y) == LPi(1 - e-JliYi)e-Ai~i
i=1
n
e- JliYi )
-
i=l
n
+ Q2 LPi(1 - e->'i X i)(1 - e- JliYi )
i=l
-
C1 LXi,
i=l
n
C2 LYi.
i=1
27
BY)J,eM MCKaTb prumoBecMe no H3IIlY, T. e. TaKYIO napy CTpaTerMH
(x"', y"') E 81
X 82, qTO
Ml (x, y"') :::;; Ml (x"', y"') )J,JUI JII06oro x E 8 1 , M 2(x"',y) :::;; M 2(x"',y"')
H
)J,JIH
JII06oro
y E 82, 3. PemeHHe HrpbI. B CMJIY BomYTocTM M l (x,y) no x npM <pMKcMpoBaHHoM y
M 2(x, -y) no y npH <pMKcHpoBaHHoM x H3 TeopeMbI KYHa-TaKKepa (CM. [5]) nOJIyqaeM
YTBep)K)J,eHHe, npe)J,CTaBJIHIOIII,ee Heo6xo)J,HMOe M )J,OCTaTOqHOe YCJIOBHe Toro, qTO napa CTpa­
TerHH (x"', y"') HBJIHeTCH pannOBeCHeM no 'H3IIlY.
IIapa cmpameeuu (x"', y"') .H.6.1/..H.emC.H. pa8'H06eCUe.M. no H;nny moeaa u
a u f3 ma~ue, "tmo
TeopeMa 1.
~oeaa cy~ecm6y1Om 'Heompu~ame.ll.'b'H'b,e
mO.l/.'b1W moeaa,
eae
{ =~ 0,0,
a
> 0,
eC.l/.u xi
eC.l/.u xi
= 0,
eC.l/.u Yi
eC.l/.u Yi
= 0,
(3)
eC.l/.u 2:1=1 x; = X,
~n Xj'" < X
eC.l/.u L.Jj=1
U
eae
f3 {~o, eC.l/.u
= 0,
eC.l/.u
°
Haw.y. IIycm'b A(a)
mo
°
> U yi = 0, mo
(ii) eC.l/.u xi
(iii) ec.l/.U xi
'"
Yi
= 0,
=
°U yi >
0, mo
1 1 Pil-'i
n B(f3)
= I-'i
(iv) eC.l/.u xi
°
> U yi > 0, mo
'" _ 1 1
2qlii2
-ql q2 - E>iiil + Oiii2
'" _ 1 1
2q2iil Yi - I-'i n -qlq2 + E>iiil - Oiii2
x· - l
28
;\
n
(4)
2:1=1 yj = Y,
2:1=1 yj < Y.
OnHIIleM CTPYKTYPY OnTHMaJIbHbIX CTpaTerHH HrpOKOB.
JIeMMa 1. IIpeano.l/.oaICu.M., "tmo (x"', y"') - pa8'H06eCUe no
B(f3) = f3 + C2. Toeaa
(i) eC.l/.u xi = U yi
> 0,
+ .,flJi
,
+ .,flJi' = a+Cl ,
'U
we ii
=1-
q
= A(a)/(piAi)j 9 i = B({3)/(PiJ1..i).
fli
U
,l1;oKa3aTeJIbCTBo.
(ii)
(i)
IIpe,n;noJIo)l(HM, qTO
IlOTOMY H3
(4)
(iii)
(iv) IIycTb
Tor,n;a H3
(3)
(3)
H
(4)
npH
xi = Yi = o.
BbITeKaeT, qTO
HMeeM
qTO H ,n;OKa3bIBaeT
CJIyqaii
Henocpe,n;cTBeHHo CJIe,n;yeT H3
xi > 0 H yi = o.
(ii).
MO)l(eT 6bITb ,n;OKa3aH aHaJIOmqHO
xi > 0 H yi > o.
Tor,n;a H3
(3)
H
(4)
(ii).
BbITeKaeT, qTO
r,n;e
IIo3TOMY
H
HJIH
qlii2<P;
B
CHJIY Toro, qTO
1/Ji
1/Ji > 0 H
= -qlq2 -
9iiil
<Pi
+ (q2ql
> 0,
- fliih
+ 9i iit)<Pi -
fl i q2
= O.
HMeeM
+ fliii2 + VQlQ2(qlq2_ + 2(Q2fli + Q1 9 i )) + (ii2 fl i -
iiI 9 i )2 ,
2QlQ2
29
'l/Ji
BOJIee TOrO, TaK KaK
<1H
'Pi
< 1, TO
9TO H 3aBepllIaeT ,n,oKa3aTeJIbCTBO
(iv)
H JIeMMbI.
,LI;JUI napbI iieOTpHlI,aTeJIbHbIX 9HCeJI
)KeCTBa HHTepBa.lIa
[1,
(a, 13)
onpe,n,eJIHM CJIe,n,YIOllI,He geTblpe nO,lI,MHO­
nj:
13) ={ i E [1, nj : PiAi.~ A(a), Pi!-ti ~ B(f3)},
ho = ho(a, 13) ={ i E [1, nj : A(a) < PiAi, Pi!-tiq2 + ~: ihA(a) ~ B(f3)},
100
=
100 (a,
101
=
101 (a,
13)
III = III (a, 13)
B
CHJIY TeopeMbI
1
.
Ai _
~ A(a)},
!-ti ={i E [1,nj : A(a) < PiAi, B(f3) < Pi!-ti, !-t.
A' Pi!-tiQ2 + A: (12 A (a) > B(f3) , Pi AiQ1 + !-t: iiI B (f3) > A(a)}.
={~
'
E [1,nj : B(f3)
napa cTpaTerHM
< Pi!-ti, Pi AiQ1 + -Q1B(f3)
(x*, y*),
RBJIRIOllI,aRCR KaH,lI,H,lI,aTOM Ha TO,
9T06bI
(x*(a,f3),y*(a,f3)) =
JleMMe 1 (i)-(iv) H 3aBHCR­
6bITb paBHOBeCHeM no H3llIY, MO)KeT 6bITb paCCMOTpeHa KaK napa
(x(a,f3),y(a, 13)),
x(a,f3) H y(a,f3) - <PYHKlI,HH, onHcaHHble B
a H 13, npHgeM 3HageHHR 3T11:X napaMeTpOB onpe,n,eJIRIOTCR yCJIOB11:RM11:
(1)-(4). Tor,n,a JIeMMa 1 B TepMHHax MHO)KeCTB I MO)KeT 6bITb nepe<p0pMYJIHpOBaHa.
JIeMMa 2. ilycrn'b (x*(a,f3),y*(a,f3)) .R.6.1t.R.ernC.R. pa6'H06eCUe.M no H3UtYI rno2aa
(i) eC.ltu i E 1001 rno xi = yi = 0 1
(ii) eC.ltu i E ho I rno xi > 0 = yi I
(iii) eC.ltu i E 1011 rno Yi > 0 = xi I
(iv) eC.ltu i E 1111 rnoxi > 0 u Yi >0.
r,n,e
Ill,He OT napaMeTpOB
0geB11:,n,Ho, 9TO OTpe30K
100 , ho,
[1, nj pa36HBaeTcR Ha geTblpe HenepeCeKaIOllI,H:X:cR nO,lI,MHO)KeCTB
101 11: Ill, Ha KaM,lI,OM 11:3 KOTOPbIX KOMnOHeHTbI CTpaTerMM
onpe,lI,eJIeHbI e,lI,11:HCTBeHHbIM 06Pa30M.
napbI
(a,f3)
B
C11:CTeMe KOOp,lI,HHaT
(pA,pV)
(x*(a;f3),y*(a,f3))
,lI,JIR <pHKC11:pOBaHHoM
3T11: nO,lI,MHO)KeCTBa MorYT 6bITb npe,lI,CTaBJIeHbI, KaK nOKa3aHO Ha p11:C.
1.
I1YCTb
CJIe,n,yIOllI,aR TeopeMa HenOcpe,n,CTBeHHO CJIe,n,yeT 11:3 TeopeMbI
1
11: JIeMM
1
H
2,
npe,lI,CTaB­
JIRIOIl~HX KOHCTPYKT11:BHOe OnHCaH11:e paBHoBecHR no H3llIY.
TeopeMa 2. ilycrn'b (x*, y*) .R.6.1tXernC.R. pa6'H06eCUe.M nOH3Uty.
T02aa (x*, y*) =
(x(a,f3),y(a,f3))1 2ae a u 13 onpeae.lt.R.wrnc.R. 6 (1), (2) 11:3 yCJIOB11:R, 9TO K(a,f3) ~ X 11:
H(a,f3) ~ Y.
113 TeopeMbl 2 nOJIY9aeTcR CJIe,lI,CTBHe, rOBOpRllI,ee 0 TOM, 9TO eCJIH CT011:MOCTb n011:CKa
,lI,OCTaT09HO BeJI11:Ka, TO 11:rpOKY He HMeeTCMbICJI OCYllI,eCTBJIRTb nOHCK.
CJIe,n;CTBHe 1.
EC.ltu max{p'iAi} ~ GIl rn02aa a
max{pi!-td ~ G21 rno2aa 13 = 0 u y* = (0, ... ,0).
30
=
0
u
x*
=
(0, ... ,0); ec.ltu
PI.!
pAq, +Aq ,B(f3)/I.!=A (a)
B(~) I------------:.~
I.!q2 +I.!ij~( a)/A=B(f3)
pA
A(a)
Puc. 1.
4.
IIo.n,MHo)f{ecTBa
100,110,101
Ii
111.
IIpHMepbI. PaCCMOTpl1M l.:Il1CJIeHHbIe npl1MepbI HaXO)K,l1;eHl15I OnTl1MaJIbHblX cTpaTe­
ruH.
TIYCTb
A = J1- = (0,1; 0, 4; 0, 3), X = Y = 10, P = (0,3; 0,4; 0, 3)
11
C1 = C2 = 0,
T. e. l1rpOKl1 06JIa,lJ,aIOT paBHbIMl1 B03MO)KHOCT5IMl1 11 OTCYTcTByeT Heo6xo)J,l1MOCTb nJIaTl1Tb
= 0,2 11 q2 = 0,8, T. e. l1rpOKl1 MoryT paCC'Il1TbIBaTb Ha pa3HbIe ,lJ,OJIl1
a = 0,01715, f3 = 0,02382 c · BeKTopOM BbIl1rpbIllieH (0,2968; 0,4766)
= (4,35; 2,87; 2, 78), y* = (1,58; 4, 39; 4, 03) .HBJI5IIOTC5I OnTl1MaJIbHbIMl1 CTpaTerl15IMl1.
3a nOl1CK. TIYCTb
Ilpl1 ,lJ,eJIe)Ke.
U
x*
ql
Tor,lJ,a
TaKl1M 06Pa30M, npl1 pa3HbIX ,lJ,OJI5IX l1rpOKl1 .llbITaIOTC5I ,lJ,eHCTBOBaTb aCl1MMeTpWfHO. TIYCTb
ql
= q2 = 1/2, T.
e. l1rpOKl1 MoryT paCC'Il1TbIBaTb Ha O,lJ,l1HaKOBble ,lJ,OJIl1 B cJIy'Iae ,lJ,eJIe)Ka.
= f3 = 0,0206 C BeKTopOM
= y* = (2,55; 3, 87; 3, 58).
= J1- = (0,1;0,4;0,3), p =
Tor,lJ,a l1X onTl1MaJIbHble cTpaTerl1l1 Cl1MMeTpl1'IHbI, a l1MeHHO,
IlJIaTe)KeH
(0,3835; 0, 3835)
11 OnTl1MaJIbHbIMl1 cTpaTerl15IMl1
TIYCTb l1rpOKl1 l1MeIOT pa3Hble pecypcbI,
(0,3; 0, 4; 0, 3),
a
l1MeHHO,
Q
x*
A
2 MO)KeT paCC'Il1TbIBaTb Ha 60JIbllIYio ,lJ,OJIIO, HO l1rpOK 1 l1MeeT
60JIbllIl1e pecypCbI. Hanpl1Mep, ql = 0,2 11 q2 = 0,8, X = 40 11 Y = 10. Tor,lJ,a a = 0,00259,
,B = 0,021386 C BeKTopOM BbIl1rpbIllieH (0,1444; 0, 8214) 11 OnTl1MaJIbHbIMl1 CTpaTem5IMl1
x* = (23,4; 7, 55; 9,05) 11 y* = (1,4; 4, 5; 4,1). STOT y)J,l1Bl1TeJIbHbIH <paKT, 'ITO l1rpOK 1,
HeCMOTp5I Ha 60JIbllIl1e pecypcbI, nOJIY'IaeT MeHbllIe, BbI3Bau TeM, 'ITO l1rpOK 2 l1MeeT onpe­
.n,eJIeHHOe
TOr,lJ,a l1rpOK
npel1My~ecTBo,
HeCMOTp5I ua MeUbllIl1e pecypcbI, a l1MeHHO OH BbIl1rpbIBaeT B ,lJ,Byx
CJIY'Ia5Ix - l1JIl1 caM uaH,lJ,eT 06'beKT, l1JIl1 06'beKT HaH,ll,YT o6a, B TO BpeM5I KaK BbIl1rpbIlliHOH
cUTyaIIl1eH ,ll,JI5I l1rpOKa
5.
1
3aKJIIOQeHHe.
5IBJI5IeTC5I Cl1Tyalll15I, npl1 KOTOPOH TOJIbKO OH o6uapy)Kl1T 06'beKT.
CaKarY'Il1 [lll1ccJIe,ll,oBaJI HeaHTarOHl1CTl1'IecKYIO l1rpy nOl1CKa C
pacnpe,ll,eJIeHl1eM pecypcoB B npe,ll,llOJIO)KeHl1l1, 'ITO
ql = q2 = Cl = C2 = 0.
Ou nOKa3aJI,
'ITO OnTl1MaJIbUble CTpaTerl1l1 Onpe;IJ;eJIHIOTC5I CJIe,ll,yIOIIIl1Ml1 COOTUOllIeHl15IMl1:
(i) eCJIl1 i E 100 , TO xi
tii) eCJIl1 i E ho U 111,
= yi = 0, TO
°
xi > = Yi, 31
pJ.l
aJ(pA)=f3/(PJ.l)
pA
Puc. 2.
(iii)
(iv)
100, 1 10 , 101 HIll.
E 101 U 111 , TO Yi > 0 = Xi,
i E 111 , TO Xi > 0 11
> 0,
eCJUfi
eCJII1
IIo,1l;MHo)KecTBa
Y;
r,1l;e
= {i E [l,n]: Pi ~ a/Ai, Pi ~ {3/J.1.i},
ho = {i E [1, n] : a < Pi! Ai, J.1.i a ~ Ai{3},
101 = {i E [1, n] : {3 < PiJ.1.i, Ai{3 ~ J.1.ia },
111 = {i E [l,n]: a/(piAi) = (3/(PiJ.1.i) < 1}
100
11
x!
= ~ In PiAi
IIpl1
i E 110 ,
y.*
t
=
IIpH
~
a
-J.1.i1 1n PiJ.1.i
-{3
Ai
t
\ *
AiXi
x!
t
I1JII1
B
CHCTeMe KOOp,1l;HHaT
11
111
T
.L01,
* 1 Pi Ai 1 Pi J.1.i
+ J.1.iYi=
n -;- = n T
I1JII1
101
.E
=~
In PiAi,
Ai
a
* 0
Xi = ,
(PA,pV)
*
Yi
y:. = 0,
· PiJ.1.i
= J.1.i1 1nT
.
IIpH
~ E
,1l;JI.H <pI1KCI1POBaHHOM IIapbI
(a,{3)
HMeIOT BH,1l;, IIOKa3aHHbIM Ha p11C.
2.
I
11·
IIO,1l;MHO)KeCTBa
100 ,110 ,
111
O~eBH,1l;HO, ~TO B3TOM cJIy~ae MHO)KeCTBO
BblpO)K,1l;aeTc.H B JIHHI1IO, Ha KOTOPOM OIITI1MaJIbHble cTpaTerl111 OIIpe,1l;eJI.HIOTC.H HeO,1l;H03Ha~HO.
32
KaK rrOKa3aHO B ,ll;aHHOM. CTaTbe, BBe,ll;emle eCTeCTBeHHOrO o606r.u;aIOr.u;ero rrpe~oJIo)Ke­
BIUI, pa3pelllaIOIIl;erO l1rpOKaM ,ll;eJIl1Tb Bbll1rpbIIIl rrpl1 ero O,ll;HOBpeMeHHOM 06HapY)KeHl1l1,
U03BOJIHeT 1136e)KaTb HeOrrpe,ll;eJIeHHOCTl1 B orrpe,ll;eJIeHllll OrrTllMaJIbHbIX CTpaTerllM., KOTOPYIO
ilbI Bll,ll;l1M Y CaKarY'ill.
Summary
Garnaev A : Y., Szigyarto T. P. A non-zero sum resource allocation search game.
In this paper a non-zero sum allocation game is investigated. Two players look for an object
. . dden at the integer interval. The value of the object and the search cost are given. In the case of
founding of the object by both players they share its value. The aim of the players is to maximize
m e expected found value minus search cost.
,J UrrepaTypa
1. Sakaguchi M. A two-sided resource allocation game in search for a stationary object / / Math.
Japonica. 1987. Vol. 32. P. 979-991.
2. Mazalov V., Sakaguchi M. Location game on the plane / / Intern. Game Theory Rev. 2003.
'01. 5. P. 13-25.
3. Croucher J.S. Application of the fundamental theorem of game to an example concerning
antiballistic missle defence / / Naval Research Logistics Quarterly. 1975. Vol. 22. P. 197-203.
4. Baston V. J., Garnaev A. Y. A search game with a protector / / Naval Research Logistics.
2000. Vol. 47. P. 85-96.
5. Mangasarian O. L. Nonlinear programming. New York: McGraw-Hill, 1969. 220 p.
CTaTbH nocTynl1JIa B peAaK~I1IQ 24 HOH6pH 2005 r.
Документ
Категория
Без категории
Просмотров
9
Размер файла
3 345 Кб
Теги
неантагонистические, игра, поиск, распределение, одна, ресурсов
1/--страниц
Пожаловаться на содержимое документа