  When does society eventually learn the truth, or underlying state, via
sequential observational learning? This paper develops the interplay of
preferences satisfying single-crossing differences (SCD) and a new
informational condition, directionally unbounded beliefs (DUB). SCD preferences
and DUB information are a jointly minimal pair of sufficient conditions for
learning. When there are more than two states, DUB is weaker than unbounded
beliefs, which characterizes learning for all preferences (Smith and Sorensen,
2000). Unlike unbounded beliefs, DUB is compatible with the monotone likelihood
ratio property, and satisfied, for example, by normal information.
