Machine learning of game strategies has often depended on {\em
competitive} methods that continually develop new strategies capable
of defeating previous ones.  We use a very inclusive definition of
{\em game} and consider a framework within which a {\em competitive
algorithm} makes repeated use of a {\em strategy learning} component
that can learn strategies which defeat a given set of opponents.  We
describe game learning in terms of sets $\cal{H}$ and $\cal{X}$ of
first and second player strategies, and connect the model with more
familiar models of concept learning.  We show the importance of the
ideas of teaching set \cite{teaching} and specification number
\cite{specification} $k$ in this new context.  The performance of
several competitive algorithms is investigated, using both worst-case
and randomized strategy learning algorithms.  Our central result
(Theorem~\ref{th:main}) is a competitive algorithm that solves games
in a total number of strategies polynomial in $\lg(|{\cal H}|)$,
$\lg(|{\cal X}|)$, and $k$.  Its use is demonstrated, including an
application in concept learning with a new kind of counterexample
oracle.  We conclude with a complexity analysis of game learning, and
list a number of new questions arising from this work.