Tom chooses an integer in {1,2,3} and Bob chooses an integer in {2,3,4}. If the chosen numbers are the same, no money changes hands If the numbers are different the person who picks the bigger number gets 1 dollar, unless the numbers differ by 1 for which the person picking the smaller number gets 1 dollar.
I used the simplex method here and formulated it into the following problem:
$$ max\ x_1+x_2+x_3\\ subject\ to\ x_2+2x_3\leq 1\\ 2x_1+x_3\leq 1\\ 2x_1+2x_2\leq 1\\ x\geq 1 $$
I solved the simplex tableau and got it down to reduced form but what do I do after in order to find the value of the game and the optimal strategy?
Let $x_1, x_2, x_3$ be the fractions of time Tom chooses 1, 2, 3. The correct simplex formulation corresponds to "find values of $\{ x_i \}$ the maximize V by , subject to V not exceeding expectation if Bob chooses 2, nor if Bob chooses 3, nor if Bob chooses 4". That simplex is $$ \begin{array}{ll} \mbox{Maximize} V: & \\ & x_1+x_2+x_3 = 1 \\ V \leq & x_1 - x_3 \\ V \leq & -x_1 + x_2 \\ V \leq & -x_1 - x_2 +x_3 \\ \end{array} $$ The solution is $x_1 = 3/9, x_2 = 2/9, x_3 = 3/9$ and the value is $V = -1/9$.