I have this matrix
a b c A 3 x 2 B 3 1-4x -1
and I need to find different values of x where x is an arbitrary number (to find the Nash equilibrium in pure or mixed strategies). I'm not sure how to proceed with this do i find the saddle point first or check or dominance first by using a random x value? EDIT: so far i have if x=0 then the matrix would look like this:
a b c A 3 0 2 B 3 1 -1how do i proceed from here?
Start noting that $a$ is strictly dominated by $c$ so it will never be played. Hence the search for Nash equilibria can be restricted to the matrix $$\left[ \begin{array}{cc} x & 2 \\ 1-4x & -1 \\ \end{array} \right]$$
Next, check that $A$ strictly dominates $B$ for $x > 1/5$ and that $b$ strictly dominated $a$ for $x>2$. Hence, this game has a unique dominance-solvable Nash equilibrium $(A,b)$ for $x>2$ and $(A,a)$ for $1/5<x<2$. (I'll leave to you the special cases where equalities holds.)
Finally, consider $x<1/5$. Check that the game has no equilibria in pure strategies. By Nash's theorem, the equilibrium is in mixed strategies. Setting up usual conditions, you find that 1 plays $A$ with probability $p=(2-4x)/(4-5x)$ and 2 plays $a$ with probability $q=3/(4-5x)$.