Game Theory 2x2 Matrix: Finding Optimal Strategy for (-3,-4), (-7,2)

[mathsuxhard]

Homework Statement

{(-3,-4),(-7,2)} Find the value of the game (basically saying find x*)

Homework Equations

E((x,1-x), 1)=E((x,1-x),2))

The Attempt at a Solution

Basically I used dot product of (x,1-x) and column 1 (-3,-7). I got 4x-7 for that, which is my E((x,1-x). Then I found the dot product of (x,1-x) and column 2 (-4,2) and got -6x+2, which is my E((x,1-x),2)). Set them equal to each other to get my x, and got x=9/10. My Y is 1-x, which is 1/10. Therefore my x*=(9/10,1/10). The back of the book says my x* is (6/10,4/10), so I don't know what the hell I did wrong. I hate this class.

 

[Päällikkö]

You've mixed up the columns and rows.

I've not studied game theory per se, but in evolutionary biology, I've come across it. There the tables are typically constructed in the way that the profit is to the player in the left column i.e.
_H D
H -3 -4
D -7 2
Meaning that if a H plays a D, H will get -4, and the D will get -7. If everyone plays H with probability p (and thus D with probability 1-p), suppose you play H, the expected winnings are:
W(H) = -3p + (-4)(1-p)
and should you opt for D:
W(D) = -7p + 2(1-p)

Now W(H) and W(D) should be equal and you get the answer you were looking for.