Pure Strategy Nash Equilibrium in 2-Contestant Generalized Lottery Colonel Blotto Games

In this paper, we prove the existence of pure strategy Nash Equilibrium for 2-contestant Colonel Blotto games with asymmetric values, Tullock contest success functions with r <= 1, asymmetric lobbying effectiveness and use-it-or-lose-it budgets. We show that the solution to the pure strategy Nash Equilibrium can be equivalent to the problem of solving the roots of an equation. We give the lower bound and the upper bound of these roots. We also give some sufficient conditions, with which the uniqueness of pure strategy Nash Equilibrium can be proved.