Saddle Points

The row with value 5 and the column with value 5 intersect in the top right entry of the payoff matrix. This entry is called the saddle point or minimax of the ... Thisgamediffersfromgame1inthatithasnodominantstrategies. Therulesareasfollows:Ifplayer1plays anickel,player2giveshim5cents.Ifplayer2playsanickeland player1playsaquarter,player1gets25cents.Ifbothplayersplay quarters,player2gets25cents.Wegetapayoffmatrixforthisgame: Player2 Nickel&nbsp Quarter&nbsp Player1&nbsp Nickel 5 5 Quarter&nbsp 25 -25 Noticethattherearenolongeranydominantstrategies.Tosolvethis game,weneedamoresophisticatedapproach.First, wecandefineloweranduppervaluesofagame.Thesespecifytheleast andmost(onaverage)thataplayercanexpecttowininthegameifboth playerplayrationally.To findthelower valueofthegame,firstlookattheminimumoftheentriesineachrow. Inourexample,thefirstrowhasminimumvalue5andthesecondhas minimum-25.Thelowervalueofthegameisthemaximumofthesenumbers, or5.Inotherwords,player1expectstowinatleastanaverageof5 centspergame.Tofindtheuppervalueofthegame,dotheopposite. Lookatthemaximumofeverycolumn.Inthiscase,thesevaluesare25 and5.Theuppervalueofthegameistheminimumofthesenumbers,or 5.So,onaverage,player1shouldwinatmost5centspergame. Nickel&nbsp Quarter&nbsp Min&nbsp Nickel 5 5 5 Quarter&nbsp 25 -25 -25 Max 25 5 Noticethat,inourexample,theupperandlowervaluesofthegameare thesame.Thisisnotalwaystrue;however,whenitis,wejustcallthis numberthepurevalueofthegame.Therowwithvalue5andthecolumn withvalue5intersectinthetoprightentryofthepayoffmatrix.This entryiscalledthesaddlepointorminimaxofthegameandisboththe smallestin itsrowandthelargestinitscolumn.Therowandcolumnthatthesaddle pointbelongstoarethebeststrategiesfortheplayers.So,inthis example,player1shouldalwaysplayanickelwhileplayer2shouldalways playaquarter.