For the 2 times 2 game find the optimal strategy for each pl

For the 2 times 2 game, find the optimal strategy for each player. Be sure to check for saddle points before using the formulas. (2 -3 1 2) For row player R: For column player C: Find the value v of the game for row player R. Who is the game favorable to? The game is favorable to the row player. The game is favorable to the column player. This is a fair game.

Solution

Here, a = 2 , b = -3 , c = 1 , d =2

The optimal strategy for the row player is to set the probability of playing Row 1 equal to

r1 = d c/ a b c + d .

= 2 -1 / 2 - ( -3) -1 +2

r1 = 1/6

The row player’s probability of playing Row 2 is then determined as 1 r1

r2 = 1 - r1

= 1 -1/6

r2 = 5 /6

The optimal strategy for the column player is to set the probability of playing Column 1 equal to

c1 = d b / a b c + d

= 2 - ( -3) / 2 - ( -3) - 1 +2

c1 = 5 / 6

The column player’s probability of playing Column 2 is then determined as 1 c1

c2 = 1 - c1

c2 = 1 - 5/6

c2 = 1/6

The value of the game (expected payoff for the row player if both players play optimally) is given by

= ad bc / a b c + d

= ( 2 * 2 - ( -3) * 1) / 2 - ( -3) -1 + 2

v = 7 / 6