  We study the complexity of deciding the existence of mixed equilibria for
minimization games where players use valuations other than expectation to
evaluate their costs. We consider risk-averse players seeking to minimize the
sum ${\mathsf{V}} = {\mathsf{E}} + {\mathsf{R}}$ of expectation ${\mathsf{E}}$
and a risk valuation ${\mathsf{R}}$ of their costs; ${\mathsf{R}}$ is
non-negative and vanishes exactly when the cost incurred to a player is
constant over all choices of strategies by the other players. In a
${\mathsf{V}}$-equilibrium, no player could unilaterally reduce her cost.
  Say that ${\mathsf{V}}$ has the Weak-Equilibrium-for-Expectation property if
all strategies supported in a player's best-response mixed strategy incur the
same conditional expectation of her cost. We introduce ${\mathsf{E}}$-strict
concavity and observe that every ${\mathsf{E}}$-strictly concave valuation has
the Weak-Equilibrium-for-Expectation property. We focus on a broad class of
valuations shown to have the Weak-Equilibrium-for-Expectation property, which
we exploit to prove two main complexity results, the first of their kind, for
the two simplest cases of the problem: games with two strategies, or games with
two players. For each case, we show that deciding the existence of a
${\mathsf{V}}$-equilibrium is strongly ${\mathcal{NP}}$-hard for certain
choices of significant valuations (including variance and standard deviation).
