A shady character has a sports betting operation. Each month he either makes one more monetary unit, or else he is closed down by the police and all profits are confiscated. The latter occurs with probability \(1 / 8\). His desired profit level is 6 monetary units; if that is reached he will abandon the operation and retire to Florida. But he realizes that it may be better for him to quit early, since he risks the loss of all money.

(a) Formulate the problem as an optimal stopping problem.

(b) Write the system of inequalities \(V \geq T \cdot V\) for the value function \(V\) of the problem.

(c) Use expressions (1) and (2) to find the optimal stopping policy.