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                                                                                                     Submittedto:-PROF.SONALI SINGH                                         Submitted by:AHMER HASSAN (JN170173)                                                                                                                     MANAGERIAL SCIENCE:- 2ndtrimester{2017}                                                    ASSIGNMENT ON GAME THEORYQ. Two companies are competing for the sameproduct. To improve its market share, company A decides to launch the followingstrategies. A1 = give discount coupons A2 = home delivery services A3 = free giftsThe company B decides to use mediaadvertising to promote its product. B1 = internet B2 = newspaper B3 = magazine                                    Company B   Company A   B1 B2 B3 A1 6 -5 3 A2 2 -3 -7 A3     -2       7                  4 Uselinear programming to determine the best strategies for both the companies. solution.

          Company B             Minimum Company A   B1 B2 B3   A1 6 -5 3 -5 A2 2 -3 -7 -7 A3 -2 7 4 -2 Maximum   6 7 4    Minimax=-2 Maximin = 4This game has no saddle point. Sothe value of the game lies between –2 and +4. It is possible that the value ofgame may be negative or zero.

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Thus, a constant k is added to all the elementsof pay-off matrix. Let k = 4, then the given pay-off matrix becomes:   Company B Company A   B1 B2 B3 A1 10 -1 7 A2 6 1 -3 A3 2 11 8 LetV = value of the gamep1, p2 & p3 = probabilities of selectingstrategies A1, A2 & A3 respectively.q1, q2 & q3 = probabilities of selectingstrategies B1, B2 & B3 respectively.

              Company B           Probability Company A   B1 B2 B3   A1 10 -1 7 p1 A2 6 1 -3 p2 A3 2 11 8 p3 Probability   q1 q2 q3   Company A’s objective is to maximize theexpected gains, which can be achieved by maximizing V, i.e., it might gain morethan V if company B adopts a poor strategy. Hence, the expected gain for companyA will be as follows: 10p1+6p2+2p3?V-p1+p2+11p3?V7p1-3p2+8p3?Vp1+p2+p3=1and p1, p2, p3 ? 0Dividing the above constraints by V, we get10p1/V + 6p2/V + 2p3/V ? 1-p1/V + p2/V + 11p3/V ? 17p1/V – 3p2/V + 8p3/V ? 1p1/V + p2/V + p3/V = 1/VTo simplify the problem, we put p1/V = x1, p2/V =x2, p3/V = x3In order to maximize V, company A canMinimize 1/V = x1+ x2+ x3subject to;10×1 +6×2 + 2×3 ? 1-x1 +x2 + 11×3 ? 17×1 -3×2 + 8×3 ? 1and x1, x2, x3 ? 0                    Company B’sobjective is to minimize its expected losses, which can be reduced byminimizing V, i.

e., company A adopts a poor strategy. Hence, the expected lossfor company B will be as follows:10q1 – q2 + 7q3 ? V6q1 +q2 -3q3 ? V2q1 + 11q2 + 8q3 ? Vq1 + q2 + q3 = 1               and q1,q2, q3 ? 0Dividing the above constraints by V, we get10q1/V – q2/V + 7q3/V ? 16q1/V +q2/V – 3q3/V ? 12q1/V +11q2/V + 8q3/V ? 1q1/V + q2/V + q3/V = 1/V To simplify the problem, we put q1/V = y1, q2/V =y2, q3/V = y3 In order to minimize V, company B can  Maximize 1/V = y1+ y2+ y3subject to10y1 – y2 + 7y3 ? 16y1 +y2 – 3y3 ? 12y1 +11y2 + 8y3 ? 1      and y1, y2, y3 ? 0 

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