Question

1 -1 1 1/2 4. For the matrix game 5), compute the value, the optimal 1 - 1 - 1 strategy for the row player, and two optimal strategies for the column player.

Answer 1

dues 4 -! ž matrix game - This Matrix game 1 -1 - is also called pay off Matrix at player B Bi B, B3 Mine imcem at row S Now A / Player a -1 -Ź / -- Maximum of Column I I ť ME Maxemin) = Max (-1,-1) = -1 Min (max) = min (1) Ź) = Ź - Maxemin) t min (max) principles are This Maximen f minemax abblicable 80 This game has no saddle paint not Now, to solve the game, firstly we reduce into 2x2 game by dominance rule or graphical Method: clearly Any sow is not dominated by other now e any column is not dominated by other 80 Dominance rule is not applicable

graphical method So we will se for this game bay off Matrix af yd Playera Player 13 S BI Ba B3 MAAI/ ai ţ , 2 Az II - -Ź 2 92 93 let A plays with strategies CAL,A2) whose probabilities are bi, bz f b, 7P2=1 & B play es with strategies (B1, B2, B3,) whose Probabilities are ai, 92, 93 Now Expected pay aff of A E, = Ez = -1.6, + 1$2 = .8, + 1366-6,) 1.b. - 1.62 = lobi - 1.61-612 Ez = £ b - I pq = ţ bp mţ Clape). Now we plot the above expected pay aff lines

Ź hest petent lowermana region Highest paint of lower gegeon is in tersection af E, 4E2 so abone Pay Matrix is reduced into Bi rol This is 2x2 game 2 92 Þ, = 2ga - Q21 - (all +922) - (ang tag, ) 1 (111) - (HU) - -2 dhi ร -2-2

I f > Pa = 1-b1 = 1-%= } value of game (0) = a11 922 - 012 921 (all +922) - (a12 +991) = (-1).(-1) - 1x1 (-1-1) - (1+1) 0 922-912 = 44 = Ź (a11+992)-(921 +012) 9,= -4 =) 9g = 1-9, = -= t. Hence solution of the game is - optimal strategy for prayer. A is (ģit) f optimal strategy for player 3 is (titros value of game to GU le = 0 valu a12 A