This table looks simple but I don't know how to calculate the mixed strategies and the value of the game. I tried to write down the linear program but find unable to solve it. Is there any hint or thoughts? One thing to check first, it does not have a pure strategy, right?
Hint
If you assume the optimal strategy of the chooser of the rows assigns positive weight to every one of his pure strategies, and the value of the game to that player is $\ v\ $, then the optimal mixed strategy $\ (p_1,p_2,p_3)\ $ must satisfy \begin{align} v&=ap_1=bp_2=cp_3\ \ \text{, and}\\ 1&=p_1+p_2+p_3 \end{align}