  We analyze the Gambler's problem, a simple reinforcement learning problem
where the gambler has the chance to double or lose the bets until the target is
reached. This is an early example introduced in the reinforcement learning
textbook by Sutton and Barto (2018), where they mention an interesting pattern
of the optimal value function with high-frequency components and repeating
non-smooth points. It is however without further investigation. We provide the
exact formula for the optimal value function for both the discrete and the
continuous cases. Though simple as it might seem, the value function is
pathological: fractal, self-similar, derivative taking either zero or infinity,
and not written as elementary functions. It is in fact one of the generalized
Cantor functions, where it holds a complexity that has been uncharted thus far.
Our analyses could provide insights into improving value function
approximation, gradient-based algorithms, and Q-learning, in real applications
and implementations.
