Second thought: The problem is underconstrained and so probably not very useful to solve using LP. All of our constraints involve a different set of the variables. (All lines for one particular rank.) So none of the hyperplanes defined by those contraints will intersect.
Third thought: Something seems fishy. We know that the optimal solution appears at one of the vertices. But (from the second thought) we know that the only vertices are where the hyperplane for a particular rank intersects the x >= 0 constaint. That is, the optimal solution consists of just zeros and ones, never of a mixed strategy.
Which would be good news if I believed it. But we could recast the problem in terms of individual cards instead of individual ranks and get a different problem--- would the solution to that LP problem specify the exact same strategy? I find it hard to believe that the optimal bluffing frequency, for example, exactly corresponds to always or never bluffing with a particular card.