Question

Copy of Copy of Consider the following matrix game M 5 Set up the minimization and maximization linear programming problems for obtaining mixed strategies corresponding to row player and column player, respectively. DO NOT FIND the strategies.

Answer 1

-2 5 -4 SOLUTION We ham the Pay-off matrix M = 1-3 5 -Y PLAYER (B) PLAYER(A) B B2 Row minimum A) -२ -3 Az Column maximum CLEARLY; this game has no saddle Paint Here, the value of game lies b/w - 3 and -2 So, we can add a constant K = 5 to all the Element of the Day-off marrix. Then, Pay-off matrix becomes. 5 PLAYER (B) BI B2 2 3 AL A2 10 an PLAYER CAD P2 V2 suppase, v = value of game. And, Pa, Pz → Probabilitles of Silering Strategis A, and Az suspectively. Similarly, ovi, Wy → probabilities of Sulicking Stretugis B, and B2 respectively. We know that playuls Als objectim is to maximize the expected gains, which is obtained by maximizing value of v.

Now, The Expurtud gain for player A will be giun by. QP14 10P2 V and 3pi+ P2 >V -> 2()+2014) » and 3 (f) + ( f) > 1 Let ! Now, PLAYER A has to maximize v; for this A has to minimiza then corresponding Lipp becomis. Minimum zp = t = x, +12 Subject to 2x1 + 10x271 3x1 + x2 > 1 and 1, I2 710 Answur. Also, PLAYER B's objectim is to minimimize his Losses. wwin can be oudured by minimizing v. Thus, The Expe red loss for Player B will be giun by 2v1 + 30 EV and loa, tu cv 2()+3) 1 and 101 ) + ( 062) El and Taking all = Y, 3 = 92.

Then, the corresponding L.P.P far PLAYER B is vriven by Manimum zor L - YtY2 subject to ay, +37₂ <1 loy,+ Y₂ cl Answer: v and Yo Y₂200