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Unformatted text preview: CHﬂpTEE r4 Hidl Let player I be the labor union with strategy i being to decrease its
wage demand by 10(11)‘.
Let player II he the management with strategy i being to increase
its offer by 10(i1)‘. The payoff matrix is: I‘MZ .LabgL the. products respectweug Adena B.Th.cn the strategies Por' each Tncxmw‘Yacturer are:
4— Nevmat éeuetopmemh oi? botk products 1— OMK cleutLppment 0% ‘3th A
3 Omsk deveLaPmemt of product 3 Let RPWWW J. . The pusoﬁ mtTCX then becomes. Cabana Wximltmis ‘3 ‘3‘ thn Hence, both maﬁaFaefurers shou Ld use Tiar‘ynml
' dwewpmcnt amd l with Cncrease hrs share 0F tht erét by] a; [4‘1 [LU3', Each 110.53,: has the Satme semiang set. A strateng mt
Speecﬁﬁ the ﬁrst chip chosen maﬁa: cada posstbte Fursth b3
the. oppm‘n't ,ehotces 013m an! that ochsForemmple, a
tgpceal strategg is l“Puck c: ﬁrst.113 opponent chooses “1,136:qu “AK...
IS oPpG‘hf'ht Unease: Wilma“; am! 19.11? opponent damcsbpmkhmtk;
when, isr L: 1,1,5 id.j¢,k£Z§ v0.1.3? There are Sohorces
For I. come _For each t_ 3 worms of 'covdct'ﬂomluatmtegt'es,
lemming al1 ddottl‘nct stmbegces.?a.tgo¥s are deterwhd iron
the, tobte (QUL wet PagoH's are gather“lac, O or —}Qoldep¢mdmﬁ
0% whether Pbﬂer J WL'YIS 3b£1ne5,wun5 1 fume “yd tees ltcmt IQEl as I
Strategies 4, 5 and 6 of player II are dominated by strategy 3. W or” 3 his 3 ' Strategies 4, 5 and 6 of player I are dominated by strategy 3.
Strategy 1 of player II is dominated by strategy 3. Strategy 1 of player I is dominated by strategy 3. Strategy 2 of player 11 is dominated by strategy 3. Strategy 2 of playerl is dominated by strategy 3. Therefore, the Optimal strategy is for the labor union to decrease its
demand by 204 and for management to increase its offer by 205'. A wage of $1.35 will be decided. I 5» A3
5
[a H
mm _ . _ .415 a
mm“ .“5 H 155;
l‘iLZ. (a) Strategy 3 of player I is dominated by strategy 2.
Strategy 3 of player II is dominated by strategy 1.
Strategy 1 of playerl is dominated by strategy 2.
Strategy 2 of player If is dominated by strategy 1.
Therefore, the optimal strategy is for player Ito choose strategy 2
and player [I to choose strategy 1 resulting in‘a payoff of 1 to player 1. 1+1 1'41'2‘ B Strategy 1 of player 11 is dominated by strategy 3.
Strategy 2 of player I is dominated by both strategies 1 and 3.
Strategy 2 of player 11 is dominated by strategy 3.
Strategy 1 of player I is dominated by strategy 3.
Therefore, the optimal strategy is for player I to choose strategy 3
and player 11 to choose strategy 3 resulting in a payoff of I to  player 11. {HZ3 Strategy 1 of player II is dominated by strategy 3. Strategy 4 of player II is dominated by strategy 2. Strategies 1 and 2 of player I are dominated by strategy 3.
Strategy 2 of player II is dominated by strategy 3. Therefore, the optimal strategy is for player I to choose strategy 3
andplayer II to choose strategy 3 resulting in a payoff of 1 to
player II. Para. 1: uses shaky a. The fam 35 Shaun galA
saddkpaim} (3,3) FIW 11: ms S‘h’arhiauf .7.
The. page. is Stable. wE‘H'i saddlapo‘m‘l' (3,2,) l<~max_ VI: ; F’Qa—E H565" sturkgy I that
flag?!“ I use; slmkﬂy 3 M’B "I , ﬂayed I 1495 Sitm»ka 3am Vizé (13) P423 [00 Strategy 1 of player II is dominated by strategy 3. Strategy 3 of player I is dominated by strategies 1 and 2. Strategy 2 of player II is dominated by strategy 3. Strategy 2 of player I is dominated'by strategy 1. Therefore, the optimal strategy is for player I to choose strategy 1
and player II to choose strategy 3 resulting in a payoff of l to player I. Wmn Maximum: 1% 0 5
. twutn Hence, u“: 0 wow» ‘Poic‘rcgcom 1 Manes Issue a and
qaoUticLom Ii ULSIth Issue 2 . (b) LS‘t pcj=?§wcnnthcj or {niches election [:OFWLcl'ucwxmlg
Then the patio“: mm: becomes 5kmth '3 01r 1’0Lc’cccxam1 chemimtesl big strategica
Stratum a. o? 'l’oLct‘L‘u‘am’l domt mica by: smteml.
Stratujies‘i and 3 0? “Po LCthcaLn’i dormnotth b3 stmtcggal.
Hemee,sumtnaic dommot’ced simmer“ acves no
Lew“ ?oL¢rccham 1. Mung Issue 1 cumJ Poutcoma» IE
moms issue a. T Kvanve,'polih;cxam IL com prevent
‘PoLcﬁqiom 1 Wm warm.qu or tunsq. (C) = i 4 I WC'YL 0" tie
0 5? T’otcttmn 11 wLLL wen Timex the pagoFF mahmx becomes ILL'1‘ I‘m7 to Min [comma mmch = o = H ' 
cm MERowmanmoﬂ _ Hence the mtntmox caterCan gums «:0 L'PchtcdanI
cannot win).’PaLcth£an\1 each use an“ Esau: onnd
?0Lci:cc£o.n 1} cam LLse ISSuc 1 or 3.HDvJ¢\1€", Mince.
Issue 1 OWers ’PoLcticLan 1 has Dial41 (Memos oF wcnm‘nq ,lme shouts! use that ama hope ’PoubLLComﬁ
which an error amd also LLScs Issue 4 1428 Advamkaﬂes: It provides the best posscblc guarantee
97" what the war5t outwme (A11 be, r‘equr‘aHess OF how sdelF—ultus the. Opponent PW}; the gameThmhm til: reduces the v‘csk 0F vexL1" undescmble, album“
50 a. Whilmmm. M1 I?“ averq Lowsarvmhve Wpr‘oaok,
amd1HﬂLNFore, lit Maw} weld Hr Hum the best ﬁﬁmties £01 0 I crI. Stratcces £0? [.4 er]:
4—1: mar1 mam. lBet on head; or tail.
 5. ’P 5 on heads, bet catsuit: m S'Er‘a’ccgies 4 armJ 3 of pager I are dommied b5
simiqu a .ELCminaicwnas the dom3nated Strategies we obtqcm the pas50F? table. 1 1 4 :1
' 1 0 5
'1 5/; 0 1+1? Column mmxcmum 5/; 5
tram MLn[coh.Lm mum] at Mmﬁlou minimal , thmkre {:here.
[.5 W0 Saddle pomrt. fc cfll’ke/ p’wer dun a pure SMW:7, '"w “Hue.— Piaf, and
“KM? ‘6 3+4ij M Inc/L « M7 a; +0 Cw: h «CW ﬂaw—+1.
war 41) wag}: 1A3 5mm.» oat mad ﬂf'mﬁjh; u: «44_ e—FP'ECM f21¥1\f1 + F21x7Y1 + {‘1‘ qul +f‘flK‘IYL /.
Lie"? '5’ buff (’th FretLL'I4f6 M‘s‘39 \I‘+ Y1 :} “OLA aﬂJJOHJHM I
cm (i). VlfrhVﬁn :25“. :éiﬂ—xL)
CW w s" m = mm»)
{31): \IILAE : Y1 fgl (:04. J71 I“ T13
"" I; h +2; {l‘hJ
1: 5' S‘
1} $1. + "q 14.91 5L2 (4—an = 5%; n) (x'L x; x;,x.‘:) = tog/3' 03,5) and W 5/3 + 3' :0? >951 => %, = => L11. L31).; (#3, 4/3,) :45 3x1+U~XO = 1%11'1 LIX, a x1+1 2;th mam. 2w)
aha V: 335
31KH1.3+\5.3L‘*1*1)=8/5 {'05 053‘4‘1 1) 1%‘3'8/5 1H5“ 4 I5; = Us
:3. Lahgilgg): Limo, 'w) 143130,) Shkay 3 0F ﬁlm’qu 1'5 dominakd by I reducing ﬁe Mk {D5 4 . 30km. daesmt awdm butterHos I.Ma»kdm$nol* swim Maﬁa
4.. 30% docs Mot swcm mastrm &.kad.oas net swim MW'
5.3ohm i085 not swim lama 31.M0wk does 1nd: Mm MSW Let {:he PWjoCF entrees be the tota1 Pawn Won an
all ﬁne; events when at gcuem Wrap smfeg “5 are used
btd the teaMS — Then We p410“: who; becomes: 4 N {L 3 Sfmfejxj 2 OP AJ. Team dominated b3 stmkagtj 1}
Sfmtegg’l OF Ga.N.Tea7n domtwated be} strategg 6’ ' “1":55) Expected 13ch OFF => :41”: g at. +43 “xt=141lx:"ol x3: 4/‘1
amd V=l2'/£ WM (a; (ma) £(¥1+4a)+~1§(—¥1+43) g 1:. 1/; Fov 04m! =91“! +131} my:
13 4.431;.12 5"; Q
‘jqﬁo ' 13:34]: Iggg 1/: Thai“ Cs) J'on shoutd Alwags Suem the backstroke
and ahould SWL'W'I the butterHal 09‘ breastamu sack wdﬂn probeMUN 1’2 4'Lso, Mark shoufd mtwaugs chm
the mtterﬂ—u}. and should swcm the backstroke 0r
breaststroke each weft; thabch‘hj 1’3. And the I'LJ'. Team can cxpect {'0 act 134/; pants an the {uhch
wants. (in) The stmtggcgs For the We teams are as £71 part (a) Let p“. 3 ‘m CF My” Fer partta};t\mj: :5,AJ.Team wens
31/; 4:55 aids {tor partta13tha£65,A.J.T¢amLoses The pm, oﬁF mamx btwmes Stratqu 51 01: AJ. Team dominq‘l'ed bg shuregg 1..
Strutqu ‘I 0? SN Team dominated lag stmiegqa. After tLt‘rnihah'Yq Hne dominated strategies the.
N II 131/; CS added to each chm4,151: OptimaL
Strategies an: unchanged.13uri‘hermore, the maﬁ Mabn‘x 0? part to.) is obtained. Hence the strategies
Swan in part at.) are stilt optimal and mama1:9; o l4.HL!Lc756nce John and Mark are the best: swcmmers cm
'lahelir mspectcue teamsJ Hug welt alum.15 Swim in
two eventa since the, team a», do no he Eter CF they} :5ow in (mtg one or m events.He.nce,cF_ewher
does not SwC‘m Eh (:hc Ferst event) the butterth,He wCLL swab} Sow ﬁne but two events.
Tbm; the strategies {‘or the AI. team are : 430hn embers the butterth amd than enters the
back5mm regardless OI yumgum Park enters I {the buﬁchUj. Liohn enters the. butterqu and than 5ow5 the
backatroke. C9 MaxK enters the EadieFELL}, but
chms breaststroke C? Mark does not. 5. John em'tm the bungm; and then swcms breast
shake. EC Mowk tankers the buﬁoqu1mI: ch'nns
{Me backst'r‘okc (J? Mark does not. 11. john ant:ch the butiorFLusaLhd then sworn5 like
breastsbr‘oke mewMess o'F wkelrhu Marke'ntcrs
We buttchLg. 5.John dogs 'hut swim the buEtUHAj ﬂhnd, the,“
enters both the bmshtmke amd the Mmm_ The skatcgtes (lor the 6..N.Teonm are, as alcove
but withthe Fotes 0? 30km amd Mark ravaged. T he 0 PF moder L1‘5 {4'}? 1119A (a) (amt) 5Wtcg¢1 5 0? 6.N.Teann domcnates out! "others.
SL'nce {he {Esau£7143 page“ Mairm vs. 3 . _ 3»
t?— G.N Team uses Shrdegg .3»J LC MIN 3
\N‘C‘h , regardtess 053 H1: shutqu 1; Chosen be} an: LII. Team. (at) SWRQH 2 o? AITeam dominates shutegces {,3,4I.
I Thusch the coach Por Has GN Tea'rn' MM} choose
0mg 0? lnt‘s stmkﬁces at whalch the coach Cor
Hue AJ.Teo.m should clacose u’ldncr atmth 9 th ‘54 The pave: mm: becomgs (apter epwm‘nah‘ng {the Wdhafed straitslits 0? Al Team).
A._  1 a 5 t, ' _ Thus,cF khe mm For L—he AlTeaim knows that the Wher coadn. has a tandem»; ho 311er Mark Ch
butterR“ and backswokc mom 0H1!“ Hanna bust— Stroke) that mums coLmnn { 95 More Mikel»; {a be
(Jensen than colMTnh ’1. Thergﬁore‘ the coach for“ Hat 45.73am should (1400.5: Strateglj 5L. HH H54 Aéclbng 5 to the Entries 0; tabLe 19.6 we obtmm the
Paulo’r‘F tau; 1 A a 3 1 3 ‘1 5
.‘L 5 3' O
The, mew Unan qummmcng moéel For pLagerI 1:5:
Moxémnc X5 '
subject ho SKA3545 30
x1 «via:3 30 x.* “a *1 X‘ I. Thc, new Lenmr programequ modeL For pmger‘jl 55:
chtmﬁzc lja
subject to bgﬁ ggﬁgrlﬂ £0
3%. + 4L5;  by, é 0
‘31 * ‘11 '* ‘13 2 i
1‘5" '1'” ‘3‘! ‘5‘! ’}0
Based om the ch¥ormakion given En Sectma 13.5, the.
optimat soLutLOhs $91» these newmodeb am:
(xfmgmgh (1/44, Hm , 55/40 amo‘ 05?.152,3:,3:).(0Mc.‘m,5%00
Note 'lzhOLt x3": aﬁ6 (LL30 542%: =v+5 where V
is the orcacml game vatuc. 2. Mud man X1.
subjectto 5&*1&*3?‘3"¥H?0 5x.+3x, mac L
11 +axa+lmﬂwo 5
K14: 11+ X3 SJ. 1. x1 x5 “"30 Optimal Solution
1 : J olve Automatically by the Simplex Method: VaLue of the
Objective Function: Z = 2.36842105 Variable
X1 0.05263
X2 0.?3684
X3 0.21053
X4 2.36842
{4J1 Solve Automatically by the Simplex Method: [1153 73’ rhsuﬁe. 21; 30 Add 3 1'2: Lac}:
) en‘H‘Ll; 03C Hue whim139.. 1’9
SHLP‘EC'E f0: ¥Xr 42)“!  If 5x,+3xz + 61:3 wt, a0 L) Optimal Solution Value of the
Objective Function: Z = 3 .79166567 1115“! 97,0 insane 955 ea add ‘r’v‘o édclx
{"479 0F He fable. Solve Automatically by the Simplex Method: I) Optimal Solution WIMIJR
.\ : 4 _
juk’ecﬁ‘ +0 53f,+éi¢z 41'; 1530 Value of the
I,  1x; +S’x3 4qu *%30 Objective “nation: Z = 338101266
. a ‘
3“ .0 $31, 4 9'21 +313 + 2X9 *5 3‘9 Variable
£9" t"22:3,'1',5 2" {fix 4— x + £9 and ~x5 20 X1 0 '
. z 3 X2 0.26582
x5 3.98101 mm. yq
L1" 471’*1‘{1’ +Y‘r 1°
L" “M *Jy; Wﬁ 2" [0‘ 5:; =13, 64.23
mm EV" EV}! : ._) SKLJJJJ'WHBJ Inning! {in 3.111% PJRy’crﬁ Pmbm' There Fore the. 9805651.: ream»; M be aLermfmﬂy éescx‘dbeé bu, '— 79: Lx‘ 3/5 5’. x, 1:. 33/3
The. matrcctcons maul be rewnittch as:
Xjé'5xw5 3/5éx4$9v'3
x35(,x1+# 3/93,‘ x1 53/3
x3, 5 5xr3 35/50:.I 41/3 ThemPore the Olgc5mic. expressﬁon Go» the‘ maxemc
2015 Value 0? x5 For a..th pocht in Hue. lusche region cs: x3 = E 5x13 30v 3/5 {x15 ﬂy"
‘65‘14'11 {:Ov 1/41$><1<3/3
H6100: , the optn'qu SoLuh'ox LB '
X: 21%! Xi: {1744 1: 11/4”
X; a 5 GAO.3 .. 3/41 (4H ILLS‘S 11010101111: SIMPLE): "£1000: FINAL TABLEAU Baleq Coefficient of  Right
VarNo 21 x1 x2 x3 11!. x5 116 11? 110 11:9 3110  side
_HJ._!%H._—I    1H 111 ‘IM 111 
2  0 1 0 0 0 0 0.455 0.545 0 0.45 0.55 0.182  0.152
x2 1 0 0 1 0 0 0.09 0.091 0 0.091 0.09 0364  0.3.5:.
Kat 2 0 0 0 0 1 4091 0.09 1 0.900 0.091 1.636  1.636
x1 3 0 1 0 0 0 0.091 0.09 0 0.09 0.091 0.636  0.636
x3 4 0 0 0 1 0 0455 0.545 0 0.45 0.55 0.182  0.152 Til1e rfrnjlal solujicm rs (“InKL) :(01336) 0.3M!) wm‘ a lam/OFF: Odaz The {Ph‘nnJ data! 5oht+r0u ,5 (y'Jyzjys) = (o) aqng 0.545) “1.5? 105mm saddle points can 6: Found From the LCY'IEQI
programming «Carma Lemon 0’? ﬁne ﬁﬂm€,parf'£a) Fouows Prom part (b).
Lb)Oorhscder Hue Linear pmgmmmcng ﬁmMahbn a; an; problern ’Fw T’Luqu II. Th; 1555' and K915 conshnini's are P£1H1+Pu%a* " +’Pcn‘jn ‘5 “jun W131." Yua‘sﬂ ' ‘ *?kh‘5n { ﬁnn
It: rowk. weOLKUg éomtmtes Kiwiwan p“ Wt" * ' ‘ +'Pim‘ﬁn 4 if)” Kan   + 13K.“ ‘3." cww 15,)...3‘
Thad: '55, the. litl" constraint cs Néuhda'nt 51:th cl: (:5
(implied bus the, kﬂmmmcnf. Hmce, eliminah'“q
Hekaus Ami naieé pure strutgaces 3?" Plaﬂerfmmsfnds
[:0 eLCmle‘E'Yts redunéant Oonsw‘atnst C'n E‘ne (mnem
Pmﬁmm p0" pin“sir 11:. SCMLLaylg, eLCmimHnol weakLg
éomcnal'ed pure stvnteﬁms 901“ NM" 1'1. correspond;
‘70 cLCWrmh’ng redundant ounshMn'ls Ch the Lyme“...
pmajra'm Pn— ‘PECMScrI.
56mg We process cannot eLCanalt Feasible 50M
h‘cms or create “rewoth a.“ optimaJ shatcases
Mhhof be ett’mﬁnaf‘d Mad. 'hew ones cannot be Outed. 14—15 ...
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This note was uploaded on 02/15/2010 for the course IE 220 taught by Professor Storer during the Spring '07 term at Lehigh University .
 Spring '07
 Storer
 Operations Research

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