  We consider a general problem of finding a strategy that minimizes the
exponential moment of a given cost function, with an emphasis on its relation
to the more common criterion of minimization the expectation of the first
moment of the same cost function. In particular, our main result is a theorem
that gives simple sufficient conditions for a strategy to be optimum in the
exponential moment sense. This theorem may be useful in various situations, and
application examples are given. We also examine the asymptotic regime and
investigate universal asymptotically optimum strategies in light of the
aforementioned sufficient conditions, as well as phenomena of irregularities,
or phase transitions, in the behavior of the asymptotic performance, which can
be viewed and understood from a statistical-mechanical perspective. Finally, we
propose a new route for deriving lower bounds on exponential moments of certain
cost functions (like the square error in estimation problems) on the basis of
well known lower bounds on their expectations.
