MTH 208 Lecture 19: 12.3
15 May 2020
School
Department
Course
Professor
NDfset of all stochastic vectors wnentries 3
ECXxn each Xizo and X1xn 3
Recalt Jn vector wall 1s
Then if IcOn fnTJn since CK xn XtXn
Mini
Max Thm For every min real matrix It Fstochestric To't f't st
go.fm
GTA IfafTHAT
or equivalently FIFE ITAI gtfmE EfnJtAt3
Ingeneral I't J't are called MinimaxStrategy
They are strategies which minimize their own maximum gain or
equivalently maximize their own minimum gain
Prout let cOn Geom satisfy
zXo
XoJm EAf and WYo
Atty
EyoJn
JTAE
Consider
Sma AX XJm ITAI zxJyTjm C
holds tYeon
Xo 1Xo
Sme ATTEYoJn Ty
TAX Atf C
holds few
Wehave ZXEfTAL eyo wYoJn YITE yo
FromdualityThm we know ZXwYo are equal
hence each Emust be equality at 8foptimal 1h GLOP
zit XGATA yo w
Moreover since XEAT 4707FELTwhenever satisfiesGLOP
constraints
xJTALA tgeom
efyehjmhJTAH xox yx
TA.fi't
similarly NafJtAf yo yTAf
Recall apurestrategy idominates strategy jfor player Aif
strategy ialways leadsto an outcome for Awhich is at least
as goodas strategy joutcomes C
assuming all otherplayerstrategies
are holdfixed
Document Summary
D f set of all stochastic vectors w each xizo and x 1. Minimax thm for every min real matrix it. They are strategies which minimize their own maximum gain or equivalently maximize their own minimum gain. Fromdualitythm we know z hence each e must be equality at 8 f optimal 1h glop zit x. Moreover since x e at 4 7 07feltwhenever satisfiesglop constraints. Jtala t geom x fyehjmhjtah xox yxta. fi"t e similarly nafj taf yo ytaf. Recall a purestrategy i dominates strategy j for player a if strategy i always leadsto an outcome for a which is at least as good as strategy j outcomes cassuming all otherplayer strategies are holdfixed. E w xf o similarly strategy now 1 dominates in corresponding entries now 3 now 3 recall vow is trying to minimize payoff since entries uh now i.
