In Problems 47 and 48, derive the formulas of Theorem 4 for the solution of any 2 × 2 non strictly determined matrix game by rewriting and analyzing E P , Q = a p 1 q 2 + b p 1 q 2 + c p 2 q 1 + d p 2 q 2 (4) (See the solution of the two-finger Morra game on pages G9-G11.) (A) Let p 2 = 1 − p 1 and q 2 = 1 − q 1 and simplify (4) to show that E P , Q = D p 1 − d − c q 1 + b − d p 1 + d where D = a + d − b + c . (B) Show that if p 1 is chosen so that D p 1 − d − c = 0 , then v = a d − b c D , regardless of the value of p 1 .
In Problems 47 and 48, derive the formulas of Theorem 4 for the solution of any 2 × 2 non strictly determined matrix game by rewriting and analyzing E P , Q = a p 1 q 2 + b p 1 q 2 + c p 2 q 1 + d p 2 q 2 (4) (See the solution of the two-finger Morra game on pages G9-G11.) (A) Let p 2 = 1 − p 1 and q 2 = 1 − q 1 and simplify (4) to show that E P , Q = D p 1 − d − c q 1 + b − d p 1 + d where D = a + d − b + c . (B) Show that if p 1 is chosen so that D p 1 − d − c = 0 , then v = a d − b c D , regardless of the value of p 1 .
Solution Summary: The author explains the theorem for 2times 2 non-strictly determined matrix game.
Q5) Using graphical method to solve the following matrix game
B1
B2
B3
A1
1
3
12
A2
8
6
2
Consider the matrix game
5
1-2
-3 -1 6
The game does not have a saddle point.
O Row 2 column 1 is a saddle point.
Row 2 column 2 is a saddle point.
O Row 2 column 3 is a saddle point.
Row 1 column 1 is a saddle point.
1. Consider the following matrix games.
(a)
2
-3
4
-4
-3 1 -1
3
3
2
-1
3
0
2 2
1
2
(b)
1
1
9
8
7
0
-6
-4
2
6
-9
1
-8 -6
5 -10 3
4
3
2
(c)
1
1
0
0
1
0 -1
-1 -2
6 -3
1
5
wóŃwo
-2
-8 -6
0
3
(i) Reduce the payoff matrices by repeatedly deleting strictly dominated rows and columns.
(ii) Find the saddle point if it exists.
(iii) If the reduced payoff matrix found in (i) has two rows or two columns, find the optimal
strategies of both players using the geometric method and complementary slackness.
(iv) Find the value of each game and state whether the game is fair, or in favor of the row player
or column player.
Chapter 11 Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
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