Problems in statistical analysis, economics, and many other disciplines often involve a trade-off between rewards and additional information that could yield higher future rewards. This thesis investigates such a trade-off, using a class of problems known as bandit problems. In these problems, a reward-seeking agent makes decisions based upon his beliefs about a parameter that controls rewards. While some choices may generate higher short-term rewards, other choices may provide information that allows the agent to learn about the parameter, thereby potentially increasing future rewards. Learning occurs if the agent's subjective beliefs about the parameter converge over time to the parameter's true value. However, depending upon the environment, learning may or may not be optimal, as in the end, the agent cares about maximizing rewards and not necessarily learning the true value of the underlying parameter.