The complexity of algorithms that compute
strategies or operate on them typically depends on the representation
length of the strategies involved.  One measure for the <I>size</I> of
a mixed strategy is the number of strategies in its
<I>support</I>---the set of pure strategies to which it gives positive
probability.  This paper investigates the existence of ``small'' mixed
strategies in extensive form games, and how such strategies can be
used to create more efficient algorithms.  The basic idea is that, in
an extensive form game, a mixed strategy induces a small set of
<I>realization weights</I> that completely describe its observable
behavior.  This fact can be used to show that for any mixed strategy
<I>mu</I>, there exists a realization-equivalent mixed strategy
<I>mu'</I> whose size is at most the size of the game tree.  For a
player with imperfect recall, the problem of finding such a strategy
<I>mu'</I> (given the realization weights) is NP-hard.  On the other
hand, if <I>mu</I> is a behavior strategy, <I>mu'</I> can be
constructed from <I>mu</I> in time polynomial in the size of the game
tree.  In either case, we can use the fact that mixed strategies need
never be too large for constructing efficient algorithms that search
for equilibria.  In particular, we construct the first
exponential-time algorithm for finding all equilibria of an arbitrary
two-person game in extensive form.<p>