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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

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