View of Numerical methods in game theory (addendum). Generalized cooperative game

(1)

ß. CRSa-ICI alìci O. LÍPO\¡^N ^t) I

I.\'f

^II J ìIlrvt. I c.\

_

I ì li

\¡t,.Ij

D,

ÀN-\Ll SIt

h\L'i\IÉt

ìIeUJl

nf t)D .I.HÉO]ìIìi ì)ri L,ApptìoXtù{¿\.ftoN

L'¡TNÅI,YST] NTJ}'IÉIIIQLTE ET I,¿T'TT{ìIOITIE I}]ì tr,'ÄtsPROX}MÁTtrON

lbrne 17,

^Ì\*o

2, 1Íì¡l{}, pp. gg-ll2

9B

(iä) Iror f ^e 9(l*)

^th'e

m'appittll D

^--+

^vn :

i ^J

^d.p.

^is ^ad,ditiue

^z

1t

:" ror.(rr.), ^,E¿ì 7l¡: ^O, i' + i, ^v¡, : II¡

i:l -?

LJ

^ut

i:I

l( Il l\riIlRIC:\L i\,IDT1{ODS IN cÄi\fÐ TTÌIìOtìy

(r\DDJlNDtllI)

G

ENIìRAI .I'I E7) COOPIIRATIVtr GÄÙTE lllìNEST

Dr\NI

(Clrr j-Napoca)

(iv) Ior _f ^e ^9(1") ^wa

^lù(t'De

^lim _ls

^yD

^{-= Ho}

¿---ào

She proof follou's f::orn lemma 2 and' definil'ion

²

RE FIìIìljN (]D

S

xI]ocqalrû.,Lipovarlo.,()ltstlnlr:l.cuttlottts]rücluresonrittl¡so|sels,l'ìer'.Iìourn.

* m"tft. I'trlcs.,\ppl ,30 (198ô),2lc-220'

2. corrsta'tin ,r,,'"i.-'¿i ;"';";; \., tìlcrrcrlc tlc

t,urtlt..ít probaltilistä

si

uplicalii' inlcç¡ctlintr

I, ßull-

l¡ íIísl

ic

sul¡mcasur es' lcfiorts s2nczs 13ull'

,.23s-244'

,urli

nu I tio oq t¿¿s ì{nLÌrórnaLicâ-l)rc\r' cl'anal¡'se

'' l,

7õ

-80'

,,pnl*

'CuttttL'

ìIlrtìr' 2iì' ì)ôpaltrn'

rlo ìI¿rth' I

p*

l()ncliotÌs

tltt lriuttglr

¡tour

lcs

^cs¡ttr:rs

.'¡,"11'' ii' Rctìt"ttl'a"h' isNÀI \-ol'

^{41' Rirk-}

l. lllathemafical Fouutl¿rlion. .Ihe cooperati\/e game ma¡r þs morlellerl

stalt frorn the concepl, of cooperative

12.

All thc concepts relatecl

1,o

il, rvill 21. In this paper _ri¡e shall Íntrocluce e

gq,nte.

distribut proclucl'ion ancl it's

possibre. ir:i#;Jt?i"ft"#å

bY each _nut _ttre _extent _of

trategies appliecl ):y the coalitions.

gam,e (g.c.g.).

Ad,mitted, Intercoulitior¿. One c on a, solutional.cornplete system of co

_).

the Íntercoalition is at the same time _, _l,

sib-le._

Ilolr'er¡err it is a union of pos T callecl urlnt,ítted intercoalitioz rvllen it

then the g.c.g. is identifiecl with the tified with i,he coalition of all tho

o callecl arbitrat,íott,

gam,e.

the set of players 1 be clecom-

{'ì.cceivc¡l 15.II. 1987 I nsl iLttl

ttl

^Pol^ilelutic

Ctúedra tle À4tttctttttlícit 1900 7'irttiçoarct

lkttttitttitt

(2)

GÐNER.ALIZED ÇOOPERATTVE GAME

100 ^EFÙNESTDAÀII 2

101

called minitmu,nt, püAolÍ.'Ihe minimum payoff is,called assm'ed n'¡ini'ntu,tn

l"yoll ^rvhen the'inieicoalition i! a possible coalition component

^of

[.".u]ó. . Âs a consequence, in the case of c.g., the minimum payoff ^the is

,

^and-

in

1,he

case of a proper g.c.g. the minimurn p?yoff

min'im,unz ga.aoll. For l,he component coalitions of

^1,he

lil,ions

1,he

minimum payoff is warrantecl. The extreme

¡:ase

in

1,he

one of

^bhe

intercoalition "I consisting of all Ure players when

.I is a veritable intercoalil,ion, and all the componenl, coalitions of the s.c.s.c. have only rvarrantecl Þa)'ofÎ. Ini;ermediate arbil,ral,ion games lie

'betr¡'een the tlvo extreme

cases.

The c.g. is changed inl,o g.c.g. of clifferenb generalitv levels by enlarging the clor ain of clefinition of the probability rlector P in the definition D72.2 [1]. Tlie ^ploclucl, ^spa,ce is replacecl ]ry ploclucl, spâces inlvhich the components a

e,

no mole,llossible coalitions.

Ol course.

^'ohe

payoff achievecl in adclition is clistrilrtted among the Players of the coalil,ions ^so that the s-fabitity principle, bobl.r of the intracoalitiona[

staþility ancl also of the interooalitional onc, pcrsists.

trn 1,his

l'a¡', tlle g.c.g.

is comlral,ibÌe rvith Nash's arbitratjr-¡n scherne [], ^3, 5l' Depcnding

^o¡r

'¡he dornain of deTinition lbargairring: itomain) oÍ thi¡ plobabilit¡' \'ectotP, brere ar.e sever.al types of arbitratìol games, amons- rvhich rve ^cliscuss two in rletail.

A filst t¡'pe of o"rbitration garne ís obtained when the bargaining domain u.f '¡[re probabilil,¡r -/ector ?(f), associated- rviUr the s.c.s.c.

^Ñ,,

S¡ :l)7(¡,

h,

e M,, t : Lt ø, is given tluoug'h the follorving ststem of restriclions:

,.,', JP{I¡J' :

1

' ' _lP{t¡IIo, r _,Yu,,,

^'lt,e

lÍ,,

where : J is a line matrix consisl,ipg of elements cclual to

^tr

; ^floo ^is

the payoff mal,rix of the coalüion I{t,, Hrr: IlI,, ^iel{¡,, ^rvriti,en

^as

a co¡'Lmn matrix with elements linearJy orctered accorcling to the lexico- graplrical old.er of situa,tions sr s: ^(sr, ...,

^su)

i Vrrris

^1,he

value of tho char.acteristic funclion collesponding to the coalilion I(,,. Solvitlg ^tho system of restrictions ^('F) ^as a linear prograrnming problem, one obl,ains

tbe basic solutions ^P(t, i), i : ^L, i,. ^With ^their ^{heþ' the} probability vectors P(Ð, t:T,rr, ^an¿l ^1'' are ciei.errninecl as in the case of the c'g.

fronr Ð L2.2 _l7).

A seconä type of arbitration ^.gârne, ^also _called _arbitration

^gume_o,f

n¿inim,ar

t1¡yte,

it obtaine¡l from the first type, replacing

^1,he

flee member

^'Øou

by the fuection V i ^l/u..îhe maximizat'ion ^{of the} funcl,ion

[z

is required, The term '(minimâx" is"motivatecl by the fact thatthe ^payoff, achievod.

ovel the assurecl payoff ¡n-ith the minimum value of some coaìitions

-I(¡,,

has the maximum possibie value.

In the caso of both types of arbitral,ion ^game, the follox.ing plopeÞ ties holcl

:

.Ihc probability vec1,or, P(l) is deterrniuecl iir such a way that

bhe eÏfective achieved payoff P(t)HKhis ^greaber ^tha'tr or nqual to ^tho

tosoro2äfrtåå1îtt""ätåi'ä;JJîi,ïìnotflJ^?:n"-^-lq,urationisrequired Døta or,aro,*'oï,'î^;"::':i"::- ^prosram ^rØcoo2 ^t Ll' ^- ^-v14¡rvu

tions given i; T^biYT.' Entry _data of type _Á. _undergo the _mociifica-

.i i

I 'I

I

l

; I

'Iablc

_I

ENTRY _DA

TA

OF TYPE A (l'Iodificct part) SYMBOL

SIGNIFICANCE ru,'v(10) Adnritted intercoalitions _:

0 -

possible coaìitions, coalitions

+0-coalitionl

ccntpotrents

of ilìe

sohttionaì compìcte systenr of

I -

^ctrsc^{o^f}_

{hc

arl_¡itraIion garnc

z -

^case^o[

^ilte

aÌbitration _gaincof rninirna_r type

proqranl Structtrre_. Tlie structure of the proerarn

_Jr

,i^H;rij ;n

^c^h^a^p^r

^cr ^v ï

"i ;;

" ;

^äp

äi

_r^r

_i, úä

¿;ö%_"

;î

""fi ^,..?l?i

_"*^^_"

äï":{

l1r "åh;1rî"iätå1",1å:ls

of the new prosram unts, incl'ded _in appe'dix _B

**,o" 1fl1ítf'"å.flfrcalling ^certai' subroutine, it uoro" rhe arblrarion matrix of the bargaining _clomaih

REDVI. ft revalu-es l,he players, payoff of everv

r

ï3;ì' h,åï

"åT"î ¡'

-*

o

r

th

õ ¡ã-i J .äi"iio"i _; i ;i

u"

ffi

^;¿

å,

fi

i,iï',î _:"

_3å

"1"i:

of ^the strategies belonging tg the _elements

ord ^of ^the. ^matrix g."_;-îã inu ,,""* of the z r N 4. * trerermin",

"""r#îiå:: ^¿ïää ;r;";_ _srMp _;

(3)

ERNEST DANI ⁴ 5

ú02 GENERALIZED COOPERATIIVE GAME ú03

Taìrlc ² .tØc10

FINÀI,

^\IEIì.TDX ⁴⁵⁶

*

^.*

457

*

458 x 459 {.

460 ^'*

467 + 462 ,r 463 'r 464

*

465

*

466 _'t 467

*

468 x 469

*

47O

*

477

*

472 a 4'.t3 x 474 a 475

*

476

*

477

*

478 479 _a'r 480 a 481

*

482

*

483 ì.

484 _{.

rÞ

485

*

_'¡

486

*

487 ,F

488

*

489 a 490

¡

491

¡

492

*

493 ,r 494 + 495 x 496

*

497

*

498 + 499 'r 500

*

501 _'h 502

*

503 + 504 ₊ 505

*

506

*

502

*

508

*

509 x 510 iÉ

511 .*

512

*

513

*

514

*

515

*

SEG]\,IEN1'A1

INITIAI, ^VERTIìX

SUBRØUTINE _GENPRøB

-

(l l,-I5,NII-I2,XX,NR.\¡,ù,rp,Lp,KNN,LNN,NA,XNØRII2,

-

ì)FrN,rpu,NuÀ,r1,rz,nrx,Mo,Nr,Nl,ñrs,ñ¿,ñs,1nj ^--

^-' INT]]GEIì

Z

10

ì,lBO(t):rwv(1)

g4LL

Wp l(1,MÍlo(1),LNN, rr,KNN, rz,À,rx) NO-N1xN2*N3xN4xlrrb

OALL

r\,I¡\'t'RCØn.Ip

_ (IZ,NO

IS,ì,IEÀ,I1,

-l,IO,N1 CALI,

NN,KNN,IPIJ,J\,IBO)

l\tP:o -

CALI, CALC.

O,

cALL

^-\\:pt(1, t{B o (1),L NN, I I, I{NN, Iz, t\,IX)

IìIìTUIIN

I]ND S]ì GÀ,iEN]' AO,Ä1

S T]I]1ìØ U

TINE

I,IATRCøI\,IP

- _Ì19:M2,N3,N4,N5,r}')

⁽^I^{Z,NO, Ip}^LI,NlJt,I1,,\.IBO.Ib, r\{ S. ì'IEìI1,

INTEGI]R

_Z

10 MEn41(r):0 DØ 20

J:7,M8 Dø 20 I:I,ME

20

A(r,J):o

DØ N:O

50

J1:1,¡1

DØ 50

J2:1,N2 DØ

5O

J3:1,N3 DØ

50

J4:1,\{

DØ 50

J5=1,N5

PØ23

NTJ[,1GI]N1 GENPIìØ]]

\¡,trvll

GENPIIØB

WPI

MAl'RCØNII]

DøI[

CALCl

l

A{A'IIìCØ}.TP

\\¡TI

\\¡'llÌ ]]IN

ZIN4

\vTl)

cAI.C1 sIMP(SIt\'IP1)

'IRÀNSX

REE\T1

PøZS'fIIX

DRUNIIì]]I]O EDI'I'3 WPI

\\r1'I

\\rTl)

f-ìE11V1

\\rfD

RIN PIìØDSCJ

PØZSTRX W1'f)

cAN'l' SUBARB DRUil,IIìEDI DIVAIìC

I)Øi\,7

\vTl)

REDil,IAl'R

sIN,IP(SIN',IP1)

REDMATR

w'l'r

w'l'D

TRANSX WTD

(4)

104 6

516 517 irl 8 519 a)20 r¡27

\')t

¿24 525 526 528 l-r29 5:10 5Íl1 l¡'.ì2 il.tJ í',J4 5;15 5:ì6 Ðót ir3 8 5:l

I

540 54L 542 it43 544 i¡45 546 547 548 549 550 I'r51 552 i)J.t 554

ERNEST DANÍ

.TQJCIO

GENEIì-A,LI.ZED COOPE:RA'III\/E G"q}/fE

Jtðcl0

105

*

,ß

'*,k

!F

*

,k {ê 'k

*

'Ë

*

+

*

:i(

*

,ß

* *

t

*<

*

t

>i.

*

* *

*

* *

N:N-l

¹

lI: ^I Â(N,ù'I):1 l)ø

⁴⁰

I{:1,rPU

r,

:

NtJì'I1(I()

IF(L.EQ.o)Gø

^',l'c,⁴⁰ ìfltn'I1

(nI): Ii

r,r _ tf ^_l_¹

CALL IIIN(K'l\'Io'NlI)

^(N,M):o s:FC(K)-\Yc)

1lØ 30

J:1,NIO

'I'ø

il0

R. ISD(12). ^Fl

Q.1)S:S-

^FCO(J)

(J, J1, J2,.' 11, J4,,J5) 3O CøNTINI]E

A(NO -l-

1,II)-

S

40 CØNTINUIÌ 50

CØNTINI]I'

^(No+1'1):1 MS: NI-1 NOO:NO NOOO:NOO+1

i\{OO: n[S -l 1

I4OOO:l{OOl1 IF(IZ.rjQ.1)GØ'.rØ

⁷⁰

NOO:Nof1

NOOO:NOo*1 A(Noo'1):o

A(NOOO'1):1 DØ 60

1\I:2,ùIoo

A(NOOO,Nf)

:À(l'iOO'lÐ

60

A(Noo'Nl): -

¹

i0

⁰⁾

4), Il ^S.1, [,IS,1

' Il D Ñl 1) (5), ùI E, NIE,

l{ooo, \'looo,

A)

IlE'I'URN

ENI)

*

SEGMENT AO,A1'42

-.No'

_M1)

INTEGER

B'Z

ñøuern

^PREQISIØN

Fc,{"T,vv,zz'Fco,xNØRM

o,

575 'È

576 ^tß 57'l ,i 578

*

á79

*

580

*

581

*

582

*

583

*

584

*

585

*

586

*

587

i

58B

*

589

*

590

*

õ91

*

5Ð2 a 59:t

*

594

*

595

*

596

* lg7 *

I'r98

*

599 f.

000

*

601 ¡.

602

*

603

*

604 x 605

*

606

*

6O7 ,r 608 x 609 + ô10 _'F 011 '¡

672

*

013 x 614 'r,

615

*

616

*

ß17

*

lj18 _s 619 ,.

{ì20 a ô21

¡

¡J22

*

623 _a Ê24 a 625

*

(126

*

627

*

628

*

62r

₊ 630

¡

631

*

632

*

033 {.

634 .!

-

l"DN(50, 32),I IO (5, 1 00),WÌìT, Xù.rNO(r,). X:1.(b, 32) DIMENS

IØN

XX(I5,M H2), KNø p.rvr2(MHj¿ ₎ I) IMENS

IøN PIIIN(NAI{)

J) n\{ENS

rØN

tf ^rJO(IPU) DIÀ{ENSIøN MIìùI1(i\rS)

D rlvrE_NSIØN _Q(100),rR(l00),tlrØMØ(32),pA(100)

N:NOl-MS

rF(rz.

rìQ.2)N:N-f-1 'Wo:ç5112¡

NR:O CALL

SrMP(&20)

GØ'rØ

750

20 FICTIVE:1

CAI,I,'TRANSX(IZ,MX,NO,MS,IPU,MRO)

]F(KNN.E

_Q.2.ANr).NR.

cE.NRV)GØ

^.1Ør5o IF(KNN. E _Q.2. AND. rND(14). EQ.

MØtrø(NR+1))

-GØ IF(KNN.ItQ.2,r\ND. ^TØ

^SO IND(14).

Nlt.

MøMØ(Nrì+1))

-GØ'îø _NRO:¡Pa1

¹⁴⁰

IF(NR. _GE._Mrr2)lÃrRrTE(l08,160) IF(NR.

cE.MI:I2)GØ TØ

75O

CAI,L REEVl

-

⁽IZ,ì,IO,N,X, I5,XX(1,NRO),XT(1, Il), ñ{S,MEMl, -TPU.MUO)

Irì(NR.lìQ. o)GØ TØ

SO

DØ

40

.lu:l,N[ì

Dø 30 t:1,\{O

IF(xx(I,NRO).

GT.

Xx(I,JU)+WO)cø TØ

40

30

CØNTINUI]

GØ'1ø

r40 40 _CØN1'INIJE

JU:O

5O

JU-JIJ*1

DØ

60

I:I,MO

rF(xx(r,JU). cT.xx(t,NRo)+wo)cØ _Tø

80

60 CøNTINUE

DØ ìO JJ-JU,NR

lf ø l\,Iø(J.J): MØMø (J J +

t) DØ

70 I:1,1U0

7o

XX(I,JJ):xx(r, JJ+1) NR:NR-1

80 IF(JU.Ll'.Nrì)Gø Tø

50 90

NR:NR+1

MØMØ(NR):rND(14)

rF(LNN.EQ.1.AND.Lp.E Q.I)Gø Tø

t4o

rF(KNN.EQ.7)cØ TØ

tAO

IF(XNØRM2(NR).EQ.O)GØ

Tø

t4o CALL PØZSTRX

-

(NO,x,KO,B,NA,M1,M2,M3,IR, _Q,

-

M,NN,MM,L I,NJ,NO1,NO2, IPU, MBO,

-

I s_D(4), I SD(5), r SD(6), r SD(7), I SD(S))

Dø

170

L:1,N,4.

PA(L):O

_ rF(M1(L),EQ.O)PA(I-):P(L)

110 CØNTINUE

DØ

72O

L:1.N4

120

P^(L):PA(L'+ _0(L)

cALL

DRU-\{REDO(KO,NA, B.F-{)

DØ

730

K-1,N4

555

*

556 ^e.

557

*

558

r

559 x 560

*

561

*

562

*

563

*

5fi4

*

565

*

566

*

567

*

568

*

569 _'h 570 ^tl.

571

*

572 4 5'13 ^JÊ 574

*

(5)

GDNERALIZED COOPDRATIVE GAME t07

ERNEST D.{NI

I

106

.lØ(i10

Jlðc1.0 691 +,

692

*

693 i, 694

*

695 r,

696

t

697 ^..e 698

*

699 x 700 ,::

701 x 7O2 a 703

*

7O4

*

705 s 706 '¡

'107

*

708 _*.

709

*

710 f.

711 ìk

712 rf

71:l

*

774

*

775 ¿,

716

*

718 x 719 x 72O :3

'121

*

722

t

723 _a 724 ,t

725

*

72ß

*

728 *,

729

-

730

-

731

*

732 x r;tir * 734

*

/óÐ rk

/JD * 738

*

739

*

74O

*

741

*

742

*

743 *"

'144 x 745 a 746

*

747 x 748 _a 749

*

Dø

10

I(:l,NA

635

t

6116

*

637 x 6:ì8 _'ts

6Í:11) _'F

640 ⁺ 641 ^,Ë 642 x 643 r.

644 'r 645

*

64(ì

*

647 ^,r 648

*

649 ^,'r 650

*

651

*

652

*

653

r

*

654 _'F

6¡5 *

65(i _'t 657

*

6ó8

*

659

*

660 ⁺ 661

*

662

*

663

*

664 ;r 665

*

666

*

667 ,r 668 x 669 ^,r ô70 't 671

*

672 _'F 673 _'r

t74 *

675

*

676 'r 6?7 ^cF 678 '*

679

*

680 _'1,

¡:tr1 li)

-

^l).\(l(

)'r

X j\iO I{11( ^I

I)

xXNOIì il12(NIl)

prìNiI():I'Ir

^I

N(l() l-P¡\(li)

10 Q(I():o N:O

130

I)Ø

70 .T1:1,N1

DØ 70

J2:1,N2

DØ 70 J3:,1,NS

Dø

70 J4:-1,N4 DØ 70

J5:1,N5

N:Nf1

NII(1):.I1

N-E(2)-J2

Nlì(S):

Jl3

NIt(4):

J4

Nri(5):,J5

llì(r, (N). L'r.W) GaJ'r

ø

⁷o .t)Ø 20

L:1,Ì\I

L.r(r,):o

20 r.

r(L):o

DØ 40

,I:1,NJ L1:NO1(J) L2:¡9215¡

r(1:L2-L

r + ¹

lSo:NE(.I)

¹

cALr,

C'ANT(rSO,I{l,NN(r,1),r.I(L1)) DØ

iJOL:L7,L2

30

r,J(r.):Lr(L)-F1

40 CØN'IINUE

cÀLL

SUBARÉ(NÀ,t\f ,i\{2,1\{3,L ^I,Ì\{N.{,rR) l)Ø 50

I(:1,N4

J:T,I1(K)

rF(J.EQ.o)IP.(K):1

50 OøN1'INIJ-E

C.A.LL i

)tìl

IùIP.EI) I(I{O,N^, A, II'.) Dø 60

K:1,N4

J:M1(K)

rF(.t.EQ.o)Gø'rø

60

Q(I():

Q(I{)

} IIì(K)*P(N)

60 CØN'I'INUI:I

70 CØNTINIJ-E

cALL

D MIìC(I{O,NA,NJ,A,I\,Í I, Q,l\{N,W)

cALr,

wTD(16,r{Bo(16),NA,1,NA,1, Q) P.E'I'URN

END SEGI\{ENT ..\1

srJBrìØUTINE ZIN4(r,r,N)

INTEGEIì

Z

I)øUBLF, PRECISIØN A,WS,X,D,\^/V cør'rn{øN I At I zí6),A(32,32),WS(16),X(50),D

-

NA1" IND(35), INDX(32), INDy(32),WV(1 6)

z(3):¡

z(4):t

z(5)-_-o

z(ti):¡a

z('t):t

z(12):o z(13):

eeeee

IND(29):1

IìI]TT]I-ÌN END

IlìI):3

C,lll, ettttll

^I

-

(l(O, N,\, N.I, lDD, ^Iì, ì r,NR, ll, ÀIl,P;\' tIO, -7,1,7,1,1,3,7,7,7\

I4r)

(l.A,LL Sllfì)1ict2it.¡

150

FrC',r'IYE:1

RE'fUIìN

160 FØRN'IAT(1OX,IØW]]IìFLøW

]N

CÀLC1') ENI)

SUBRø

- (Iz,tto,

^PU,r{Bo)

òpluBl-

^s

DIi\'IIIN

l) II,IENS IØN ì'IDI\I1(l{s) DI]\,IENSIØN ù'IRO(lPU) l)llvtENSIØN Nj\l(5)

(),ÀLI. N'fD(l3,ì',lßo(13),N,1.N'1'x)

NO:N-NIS

DØ

20 I:1,\tS

CALL

^I]IN(ilII'ÙI1 (I),I'IO,NÀ'I)

(IALL

PRøD ScJ(NIì,]\',lo,Nl't) tFt rz.EO.

1)S:x(NO

^ì-

l)/Nll

ipì rz.no.z)s : (x(No)-l-x(No

^l-

l))iNIì

P2¡ 16

J:1,\fO

r,F(Nl,I(.r).1ì Q.O)Gø TØ ¹⁰

xx(J)-z(r)l-

^s

10 CøNI'INUE 20 (]øNTINT,IE

-"

<iÃr-r,

tvÍD

(14,r'IBo (14),1'to,1,ÙIo,1,xx)

IìE'It]IìN

ENI)

,IR,Q, ,IPU,MBO,

DØUBLE PRECISIØN- ^P,Q,W DI\4ENSIØN A(KO)

D IIVIENSIØN NO1(NJ)'NO2(NJ)

D IMENSIøN M1(NA),M2(NA),M3(NA)'IR(NA)' Q(NA)

DIMBNSIøN

NN(M),MM(M),LI(M)

DIN4ENSIØN P(NR) DINIENSIØN ^MBO(IPU)

D INII:

NSIøN

^LJ(100),NE(5),I'IN(5)

DATA W, MN/1.D-4,5*1/

CAI,L

WTD(15,MBO(15),NR,1,NR'1'P) 681

*

682

*

683

*

684

*

685

*

686 + 687 + 688

*

689

*

690 _'þ

(6)

108 ER'NEST D.A.I\II

JØQ7O

l-o

f1

810

*

GENERALIZED COOPERAT,IVE GAME

JØC7O

tr09,

750 x 751 x 752 .r 755 ^,þ 754

t

755 ⁺

?56 ^¿r 757

*

758

*

759

*

?60

*

76I

^,ts 762

*

763

*

764

*

765 _'F 766

*

767

*

768

*

769 _'*

770

* 77t *

772

*

773 ^å, 774

*

775

*

?76 + 778

t

'179

*

?80

*

7BI

*

782

*

783 + 784

*

785 '¡

786 i.

787

*

788

*

7Bg

*

79O

*

791 _'F 792 ^¡r 793 _'i.

794 ^,r 795 x

?96

*

797

*

798 x 799

*

800

*\

801 + 802

*

B03

*

804

*

805

*

806

*

807

*

808

*

809 iß

+

^SEGMENT^À1

SUBRØUTINE DØI¡T(IZ,MX,NO,MS,LNN,I{NN, IPU,MBO)

INT]]GER

Z

DØUBLE PRECIS IØN A,'WS,X,D,WV DØUBI-E PRECISIØN TAB

cØx4tvrøN

/A1

,ws(16),X(50),D,

-\r+lI¡NP(gÐ,

Y(32),wv('B),îwî(16) DI]VIENSIøN

DIt{ENSrØN

TAB(32,7)

DÀTA

\Í8,t7132,7 _I tF(Ì\,rx.NE.O)RETURN rF(rND (35).8 _Q.1)RETURN JO:n{S

*

¹

JOO:JO*1 IOO:1.{Ol1

IF(IZ.!l

_{e. 2)IOO}:rOO

*

¹

CALL

WTD(17,n,rBO(17),M8, JOO,IOO, JOO,A)

NOO:NO

Ín1lz.nq.z¡Noo:Noof

1

DØ 70

J :7,1?

DØ 10

I:1,I{E

trO

TAB(r,J):A(r,J)

rF(r-NN.E Q.2.øR.KNN.E Q .2)GØ Tø 50 wRIT.E(108,60)

DØ 20 N,I:1,ME

20 ]NDX(M):1

z(6):MS I75:z(75) z(15):1

Dø 40

IG:1,NO CALL

REDN4ATR

-

(IG,ù{E,I7,NO, _IO,JO,T,A,B,A, INDX, IPU,MBO)

Z(3):IO

cALr_ srMP(&30) wRrTE(108,70)rc GØTØ 40

30

FICTI\/E:1 INDX(IG):0

40 CØNTINUE

z(15):r15

50

FICTIVE:1 IG:NO-l-3 Noo:NO*2

CALL

REDMATR

-

(IG,ME,17,NOO,IO, JOO,TAB,A, INDX,IpU,MBO).

IO:IO-1

tF(rz.E

Q.1)ro:ro-1 z(3):LO

IOO:IO-11

CALL

WTD(1B,MBO(18),ME,JOO,IOO,JOO,A) RETURN

60 FøRMAT(//15X,'NøT

DøMINATBD

COLUMNS"

-'ØF THE ARBITRATION

GAME:,/

_15X,'(LP

_PRØBLEM

øF THE DøMINATIøN IS' -'INCØMPATTBLE)'//)

70 FØRMAT(IIì+,50X,,NR.CØL.

:

"I5) END

SUBRØUTINE REDMATR

-

(rG,ME,I7,NO, IO, JO,TAB,.A_, INDX, IpU,MBO) DØUBLE PRECISIØN TAB,A,WO

811 ,r

812 '¡

813 x 814

*

815

¡

816

*

817 'r 818.F 819

*

820 + 821 _'F 822

*

823 _'È 824 _'¡

825 '¡

826 x 827

*

828 _'r 829 's

830 x 831

*

832

'*

^'¡

833 ,¡

DIÌVIENSIØN ME),INDX(À{E)

DATA \\rol1

CALL

IO:0

^{l^/T}^r(1^9,MBO⁽¹^9),NO,^1,NO,^1,

^INI)X)

ÐØ

20 t:1.¡6

rF( [NDX(r).8 Q._o)GØTØ20

IO:IO*1

Dø

10

J:l.Jo l0 A(ro,J):TAB(I.J)

20 CØNT]NI]E

IOO:IO *

¹

A(roo,1):1

DØ 30

J:2.Jo

3o

{Ço.Q,!):iae1rc,.r¡+ wo

DØ 40

J:1,I4E

DØ 40

I:l,r\{tt

.^'rF(t.c'f.roo.ØR.J.G'f.Jo)A(r,J):O

40

CøNTINI]E

CAII

W'fD(20,nliBC(2O),n{E, _{JO, IO,}JO,A) IìETURN

ENI)

DINIENSìØN TAB(n{E,r?),A(i\rE,

I,rilo(rPu)

D-3i

834 'r 835

*

836 'k

837.k 888

*

839 'F

840 _{.

841 x 842 ,r 843

*

844

*

845 '¡

846

*

847 x 848 *, 849 .r 850 r"

'851

*

852 +s

853

*

854 '*

'855

se

856

*

857

*

858 x 859

*

860

*

SEGI4ENT A1

sUBIìØU1.INE .TRANSX(IZ,À{X,NO,t{s,

Ipu,t\{lto)

L.r-'fEGEtì _Z

srxrDr.Wv ,ws(16),X(50),D, Y(32),WV(B),rwv(16) iF(MX.NE.O)RETURN

rF(rND(35).E _Q.1)RETURN

IO:213¡

IOO:IQf

lltg

CALL WTD(21,MBO(21),IOO,1,IOO,1,X)

NOl:lr{Qf I

Noo:NO*MS

IF(IZ.Ee.2)NoO-NOO*

₁ DØ 10

Ì=NOI,NOO

10

rNDX(r):1

K:IOOf

1

Dø

30

I:1,NOO J:NOO- I*

1

rF(rNDX(J).E

Q.O)cø

^TØ20

K:K-1 x(J):X(K)

Gø TØ 30 20

x(J):o

30 CøN.TINUE

CALL WTD(22,MBO(22),NOO,1,NOO, 1,X), RETURN

END

(7)

tr10 ER,NEST DANI ú2

. .)oB

n'tøI)o2,AN:22EX,PN ^:CI'I

. Iìt'N ^FN:tsIBLIØ

%

øPNr,lti

A,LN :l)ISPI-AY,IìT :SØU,DV :RD1,GN :1,\IN ^:1

%

EDI'l'

ØL :'{,ØF :JØCOO,NLS'r',DllL

o/o

\tØD

^1O4,7O4

CALL

Pø23(À{P,NP,r,P, 1100,I( I SI),1\,IBO,L O,'r'E) g'o

Møt)

Ir2,11.2

1Íi0

^(-l¡\LLPØ-Z:](l\'IP,NP,LP, 1100,K ISD,ì'IBO'LO'TE) o/o'ÍaD(r _k)1,:A,Øß :.IØCO9,NLS'f

,l)ljl-

% Nrør)

^14,14

-

^Nr\-I,^IND(1ì5),^INDX^(32),INDY(l:ì2),\\¡\' (8)' I\\¡V(16) yo

\røD

21,22

I)^TÄ

^\,III1,MFI2,\\¡/50,32,1.D-4/

I F(I \\lv ⁽¹1). E _Q.1. Ølì. I\\¡Y( ^{1 1}).11 Q. 2)

tìl]'IUlìN

yo

ljDI'I

ØL :4,ØF :JøCIO,NL S1',DEÍ- yo M.ØD ^2,2

suB I ìØ uT

INll

Pø23(NIP,NP,T,S, I100,I{ISD,1\'{lJO,r,O,TE) yo

\,1øD

23,23

i ) rù,r rìNS

røN xillo

(1100),'r'E(LO) yo

^lØD

I ^25,26

)A'l'A

l'II-11,1'IFI2,Nù{/50,1}2,5 * 1i

rr.(r\\. v (10). E _Q.1. Ø rì.

Nr\'

⁽¹^0).Iì Q. 2) Gø' r Ø Mo

% Ì\rØr)

[ì4,84

140

IPU--:r1 yo

\fØD

^112,113

2ll0 ^(i¡\Lr, ßIN(INDll(Il),LÌ'Il,NUÙ{1) rF(r\\¡\'(10),Nrì.o)Gø't'Ø

245

% ìr[)r)

120,120

24'¿

rIi(I{NN.EQ.1)GØ

Tø

2óO

yo NIØD 123,724

2l¡O

NIO:,ISI)(3)

rF(r\\rv(10).EQ.o)CAr,r, NUI\'IGlllNl yo

\tØD

127,130

rr(I\\,

v(10).Nrì.o)cìALL GtrNPRØR

-

⁽^I],I5,MI.I2,XX,N R,NTP,I. S,I(NN,I,NN,NA',

-

xNØRÌ\f2,PF2,IPU,Ntl r'I1,

I\.r(10),IND(35),

-

^[f^{o, rsD}^(4),

rsD(5),ISD

(0),ISD (7),ISD (B),'rE) yo

MØD

1:J4,1:t4

rrr(I(NN.E Q.2)GØ'rat ²⁷⁰ yo

NtøD

390,390

rF( rE D. Iì _Q.3. AND. I 100.N8. 1 )\YR ^rTE⁽^108,¹⁵⁰⁾ yo

EDll' Øl'

:A,ØF :JøC15,NLST,DEL

yo

LI:øt) t,4

suRt'.Øf.lr

INE

PØ23(n'{P,NP,r- S, Il00,I{ IS[),MBO,

-LO,'lD)

D IIIENS

IØN

tiBO( r100),TE (LO)

rF(l\,rP'l-NP-1 I{ISD.E Q.'fE(1).øP..r,s.E Q.x{BO(1))A: ¹

%

^ENDI,IIJ

"

^fiØ.\

ÐØM. tsy callir-rg ceftâin subroutinesr it builcLs the sullmâtrix ilssocia,ted with the rnatrix of the arbil,ration ^gamer eliminating ^the

colum-ns strictty dominated by linear combinations of the columns of the initial ma,trix;

ÈÛDMA'IB. It perfolms a pâ,ri of the submatlix builòing'up mâ,de by tho subroutino DØlU;

TII,ÁNSX. It l¡úlcts the solutions relating ^1,o the inil,ial mal,r'ix

of the arbitration ^game on tho l:asis of tho ^solution ^relatinE¡ to the

^subl-

maúrix ot¡tained. by the subroutino DØM.

rr GENERALIZìjD coorJEtìATIVIì Cì1lM]j

r11

flemarks

:

n ganÌe rvith the _lrrograD. _{J ØC002}

se ol exceecling the cliinension

¿ùssig_

ìnatri_\ game arbitr,atio.n _schcmrì- (8) -f;he valuc 1, tìrat i* f,o- ¿rcaì

vc forrn.

51020

+lklt:k;F:l')F:ßr8*:t>¡:ii+**:F*;*;t+***+:¡<*,k*:lc;k*ìk**;i*;k:Ft;È,3*;r***:¡.;F;¡r**;t;t*;i<****ì-:.*,1

.***,r,l.T I

-1.825, I.473, 1.356, 1.9?51 :

P¿m+(S+) _1.57

_71, : 0.?5 LO.2g2j -0.959,

respecúively, Therefore,

(8)

"r72 ^IÙE,NES:T^DAI\TI ^1,4 ^ÀfS1'l IE

_{]ì.I. DE}

ì.f A'r' ICA.

'TIIÉORIE I)]1 - ^l'.E\rUE

^D,I,'ÄPPIìOXINIA'TION^:\NÀ^L^YS

^E

^NUIIÉ^{Iì IQ}

^UII

,L';IN.{LYSB NUmúIìIQUII BT LA THÉORIE ÐE L'Apptrt0XIMATION Tome 17, Nio 2, lfl88, pp. ll3-124

tor the arbitlation

^ga

of uavoff. onc obtains

-

o.g"6o,

i.+rz,

^2.07711

of the c.g., considered

chapter VI t1l).

RI'FERENCES Ä NIìW ^CLÂSS OF I:INEAR, POSITIYE OITEF,ATOIiS

OF RT)RNSTEIN TYPE

L l)

^¡rrr

l, Iì.,

^ùlelodenun'rcrice

ln

^Leorìaiocurilor (Nanterical I'Ielll.,tts

in

Game 1'ltcotg)'

litlittrla

^Dacia,Cluj-Napocn, ^1983'

2.

}) a u

i, Jì',

Numerical l|4ctltotls

in

Ganrc .Iheor4 ('!clcLety4y:9:Gcncralized Cìlassical Coope-

'

,1,,i¡r,

Go^r, St,t¿i"

^Univ.Brbeg-Bolyai, Oeconornicî'

YIiXIJI' e'

^1938'

3.ìDjtrJrin,G.N.,S"r¿or,'v.G.,-Yuetlenieupriklactnuiulcoriiui¡¡r,Iz.dNauka'

J\[oslcva, 1981.

4. OdobIe ja, St.,

Psyclloloç1ie consonanlis[c,

I, II,

Libr:air'ie x.Ialoirre, Par'is, 1938, ^1939.

ó. owcrr, G.,

L-eoriaioeurilor (GameTlrcory), SetiaBazcleuratematicealeccrcetãriiopcla-

!iorrale,

Iìditru'a

1'ohnicir,

Ilucuregti,

^1974'

Iìeceived

10.III'1986

[JniuetsiLalett Clttj-Napoca Facullatect de lVlatemrtticd

çt

^ltizicå

Str. IhgãIniceanu,

No.

7

3400 Clui-NaPocø Românla

BIANCAMARIA DELLA

VECCHIA (NaPoli)

Somrnario. In ^questo lavoro si introduce e si stuðia una nuova, olasse di operatori ìins¿1i e po$itivi di tipo Bernstein. Si ovidenziano xìumeÌose proprieta- e si dimostrano alcuni teoremi d.i convergenza.

Abstract. fn ^this ^paper

^a,

^new class of lùrear positive operators ot Bernstein t¡re ^is introd.uced _and studied. several propertios and" some convergence theorems are given.

1. Iutroductiou. Let _;l' be a, oontiruous fuletion on f : _10, l-l (/e Co(f ) ^and ^denote by (SiÍ)@) the corresponding Sta,ncu polynomiálof

degree ¿. It is well known that

4 p?,,r(n) llf ,-ø)

lç

(,Si/Xø) : _8Í(/; n) : _En _r )' _neN and. øe

.fl-F

0 m

whero

pi,r@): ⁿ

k ntn'-,)(L - n)lø-e,-ø!, #

^e

I

and

ütk,-a) u(n I ⁿ⁾

^.

^. ^. ⁽ⁿ -l (k - ^1)ø).

This operator was introducecl in [16] anrl ^ßtudied in _tg _-11-rL7 _-201.

.- {oregver, from Bi operator onc can obtai i [17], aslimiting ^cases, the following two operators

View of Numerical methods in game theory (addendum). Generalized cooperative game

I.\'f

_

\¡t,.Ij

ÀN-\Ll SIt

ìIeUJl

nf t)D .I.HÉO]ìIìi ì)ri L,ApptìoXtù{¿\.ftoN

L'¡TNÅI,YST] NTJ}'IÉIIIQLTE ET I,¿T'TT{ìIOITIE I}]ì tr,'ÄtsPROX}MÁTtrON

lbrne 17,

2, 1Íì¡l{}, pp. gg-ll2

(iä) Iror f e 9(l*)

m'appittll D

vn :

i J

is ad,ditiue

:" ror.(rr.), ,E¿ì 7l¡: O, i' + i, v¡, : II¡

LJ

l( Il l\riIlRIC:\L i\,IDT1{ODS IN cÄi\fÐ TTÌIìOtìy

(r\DDJlNDtllI)

ENIìRAI .I'I E7) COOPIIRATIVtr GÄÙTE lllìNEST

(iv) Ior f e 9(1") wa

lim ls

-= Ho

She proof follou's f::orn lemma 2 and' definil'ion

RE FIìIìljN (]D

xI]ocqalrû.,Lipovarlo.,()ltstlnlr:l.cuttlottts]rücluresonrittl¡so|sels,l'ìer'.Iìourn.

* m"tft. I'trlcs.,\ppl ,30 (198ô),2lc-220'

2. corrsta'tin ,r,,'"i.-'¿i ;"';";; \., tìlcrrcrlc tlc

si

I, ßull-

ic

,.23s-244'

,urli

'' l,

-80'

'CuttttL'

ìIlrtìr' 2iì' ì)ôpaltrn'

p*

tltt lriuttglr

lcs

.'¡,"11'' ii' Rctìt"ttl'a"h' isNÀI \-ol'

l. lllathemafical Fouutl¿rlion. .Ihe cooperati\/e game ma¡r þs morlellerl

stalt frorn the concepl, of cooperative

All thc concepts relatecl

il, rvill 21. In this paper ri¡e shall Íntrocluce e

distribut proclucl'ion ancl it's

possibre. ir:i#;Jt?i"ft"#å

bY each nut ttre extent of

trategies appliecl ):y the coalitions.

gam,e (g.c.g.).

Ad,mitted, Intercoulitior¿. One c on a, solutional.cornplete system of co

the Íntercoalition is at the same time , l,

Ilolr'er¡err it is a union of pos T callecl urlnt,ítted intercoalitioz rvllen it

then the g.c.g. is identifiecl with the tified with i,he coalition of all tho

o callecl arbitrat,íott,

the set of players 1 be clecom-

ttl

called minitmu,nt, püAolÍ.'Ihe minimum payoff is,called assm'ed n'¡ini'ntu,tn

l"yoll rvhen the'inieicoalition i! a possible coalition component

[.".u]ó. . Âs a consequence, in the case of c.g., the minimum payoff ^the is

,

in

case of a proper g.c.g. the minimurn p?yoff

min'im,unz ga.aoll. For l,he component coalitions of

lil,ions

minimum payoff is warrantecl. The extreme

in

one of

intercoalition "I consisting of all Ure players when

.I is a veritable intercoalil,ion, and all the componenl, coalitions of the s.c.s.c. have only rvarrantecl Þa)'ofÎ. Ini;ermediate arbil,ral,ion games lie

'betr¡'een the tlvo extreme

The c.g. is changed inl,o g.c.g. of clifferenb generalitv levels by enlarging the clor ain of clefinition of the probability rlector P in the definition D72.2 [1]. Tlie ploclucl, spa,ce is replacecl ]ry ploclucl, spâces inlvhich the components a

no mole,llossible coalitions.

Ol course.

payoff achievecl in adclition is clistrilrtted among the Players of the coalil,ions so that the s-fabitity principle, bobl.r of the intracoalitiona[

staþility ancl also of the interooalitional onc, pcrsists.

l'a¡', tlle g.c.g.

is comlral,ibÌe rvith Nash's arbitratjr-¡n scherne [], 3, 5l' Depcnding

'¡he dornain of deTinition lbargairring: itomain) oÍ thi¡ plobabilit¡' \'ectotP, brere ar.e sever.al types of arbitratìol games, amons- rvhich rve cliscuss two in rletail.

A filst t¡'pe of o"rbitration garne ís obtained when the bargaining domain u.f '¡[re probabilil,¡r -/ector ?(f), associated- rviUr the s.c.s.c.

(iä) Iror f ^e 9(l*)

^vn :

i ^J

^is ^ad,ditiue

:" ror.(rr.), ^,E¿ì 7l¡: ^O, i' + i, ^v¡, : II¡

(iv) Ior _f ^e ^9(1") ^wa

^lim _ls

^{-= Ho}

il, rvill 21. In this paper _ri¡e shall Íntrocluce e

bY each _nut _ttre _extent _of

the Íntercoalition is at the same time _, _l,

l"yoll ^rvhen the'inieicoalition i! a possible coalition component

The c.g. is changed inl,o g.c.g. of clifferenb generalitv levels by enlarging the clor ain of clefinition of the probability rlector P in the definition D72.2 [1]. Tlie ^ploclucl, ^spa,ce is replacecl ]ry ploclucl, spâces inlvhich the components a

payoff achievecl in adclition is clistrilrtted among the Players of the coalil,ions ^so that the s-fabitity principle, bobl.r of the intracoalitiona[

is comlral,ibÌe rvith Nash's arbitratjr-¡n scherne [], ^3, 5l' Depcnding

'¡he dornain of deTinition lbargairring: itomain) oÍ thi¡ plobabilit¡' \'ectotP, brere ar.e sever.al types of arbitratìol games, amons- rvhich rve ^cliscuss two in rletail.

' ' _lP{t¡IIo, r _,Yu,,,

; ^floo ^is

the payoff mal,rix of the coalüion I{t,, Hrr: IlI,, ^iel{¡,, ^rvriti,en

a co¡'Lmn matrix with elements linearJy orctered accorcling to the lexico- graplrical old.er of situa,tions sr s: ^(sr, ...,

value of tho char.acteristic funclion collesponding to the coalilion I(,,. Solvitlg ^tho system of restrictions ^('F) ^as a linear prograrnming problem, one obl,ains

tbe basic solutions ^P(t, i), i : ^L, i,. ^With ^their ^{heþ' the} probability vectors P(Ð, t:T,rr, ^an¿l ^1'' are ciei.errninecl as in the case of the c'g.

fronr Ð L2.2 _l7).

A seconä type of arbitration ^.gârne, ^also _called _arbitration

by the fuection V i ^l/u..îhe maximizat'ion ^{of the} funcl,ion

is required, The term '(minimâx" is"motivatecl by the fact thatthe ^payoff, achievod.

In the caso of both types of arbitral,ion ^game, the follox.ing plopeÞ ties holcl

bhe eÏfective achieved payoff P(t)HKhis ^greaber ^tha'tr or nqual to ^tho

tosoro2äfrtåå1îtt""ätåi'ä;JJîi,ïìnotflJ^?:n"-^-lq,urationisrequired Døta or,aro,*'oï,'î^;"::':i"::- ^prosram ^rØcoo2 ^t Ll' ^- ^-v14¡rvu

tions given i; T^biYT.' Entry _data of type _Á. _undergo the _mociifica-

^ilte

^cr ^v ï

_i, úä

""fi ^,..?l?i

of the new prosram unts, incl'ded _in appe'dix _B

**,o" 1fl1ítf'"å.flfrcalling ^certai' subroutine, it uoro" rhe arblrarion matrix of the bargaining _clomaih

õ ¡ã-i J .äi"iio"i _; i ;i

i,iï',î _:"

of ^the strategies belonging tg the _elements

ord ^of ^the. ^matrix g."_;-îã inu ,,""* of the z r N 4. * trerermin",

"""r#îiå:: ^¿ïää ;r;";_ _srMp _;