Question

Find the optimum strategies for player A and player in the game represented by the following payoff matrix. Find the value of the game. What is the optimum strategy for player A? Choose the correct answer below, and fill in the answer box(es) to complete your choice. (Type integers or simplified fractions.) O A. The game is strictly determined. Player A should choose row and row 2 with probability O B . The game is not strictly determined. Player A should choose row 1 with probability

Answer 1

Solution of game theory problem using matrix method Player A Player B B1 B2 A1 1/3 A2 Solution: 1. Saddle point testing Players Player B ва В, 6 07 Player A 41 Az | 04 We apply the maximin (minimax) principle to analyze the gam Player B B1 B2 B. B Row Minimum Player A A A2 [6 07 -6 (O) (-4) (-4) Column (0) Maximum 0 Select minimum from the maximum of columns Column MaxiMin = (0) Select maximum from the minimum of rows Row MiniMax = (-4] Here, Column MaxiMin + Row MiniMax : This game has no saddle point.

Matrix size is 2 x 2, so dominance rule is not required. reduced matrix Player B B, B2 4 -6 0] Player A .Az | 04 | For this reduced matrix, calculate Padi and PCof Pa = (:] and pes+ [* ] [1 1 ] Padi Player A's optimal strategies = — [1 1] « Padj + [1] [11][***][1] - 10

Pa = – and p2 = 5, where p, and p, represent the probabilities of player A's, using his strategies A, and A, respectively. Similarly, [1 1] x P cof Player B's optimal strategies = – [1 1] *Padj* [1] [1 1116 6] 91 = and q2 = 5, where q, and qz represent the probabilities of player B's, using his strategies B, and B, respectively. Hence, Value of the game v = (Player A's optimal strategies) (Payoff matrix Pi) * (Player B's optimal strategies) [2311-6 01 5 ||0 13

Hence option B is correct with above given probabilities from player A .

I.e . row 1 with probability 2/5 and row 2 with probability 3/5.

Now next question

Solution Solution of game theory problem using matrix method Player A Player B B1 B2 3 -7 A2 -2 5 Solution: 1. Saddle point testing Players Player B B, B2 A 3 -7 Player A . ده م We apply the maximin (minimax) principle to analyze the game. Player B Row B1 B2 Minimum Player A A A2 (3) -7] [-2] 5 -7 [-2] Column (3) 5 Maximum Select minimum from the maximum of columns Column MaxiMin = (3) Select maximum from the minimum of rows Row MiniMax = [-2] Here, Column MaxiMin + Row MiniMax :: This game has no saddle point.

Matrix size is 2 x 2, so dominance rule is not required. reduced matrix Player B В. В., A137] Player A Az - 25 For this reduced matrix, calculate Padi and P. cof [5 21 and P cof 17 3] [1 1 ] Padj Player A's optimal strategies - - [1 1] «Pad * [1] [1 1) (3 3][1] - [? 10] = [1 ] 10 P1 = 1 nd P2 = 17, where p, and p, represent the probabilities of player A's, using his strategies A, and A, respectively.

Similarly, Player B's optimal strategies = – [1 1] x P Cof [1 1] x Pad • [1] [1111; }] [11][ ]][1] - [*] 91 = 1, and q2 = , where q, and qz represent the probabilities of player B's, using his strategies B, and By respectively. Hence, Value of the game v = (Player A's optimal strategies) * (Payoff matrix Pi) * (Player B's optimal strategies) 1 7 10 13 -71 5 17 11-2
hence for player A option A is correct

And for player B option B is correct with given probabilities 12/17 and 5/17

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Answer 2

Solve the following game Player - B I II Player - A 1 10 6 2 8 12