  We consider 2-players, 2-values minimization games where the players' costs
take on two values, $a,b$, $a<b$. The players play mixed strategies and their
costs are evaluated by unimodal valuations. This broad class of valuations
includes all concave, one-parameter functions $\mathsf{F}: [0,1]\rightarrow
\mathbb{R}$ with a unique maximum point. Our main result is an impossibility
result stating that: If the maximum is obtained in $(0,1)$ and
$\mathsf{F}\left(\frac{1}{2}\right)\ne b$, then there exists a 2-players,
2-values game without $\mathsf{F}$-equilibrium.
  The counterexample game used for the impossibility result belongs to a new
class of very sparse 2-players, 2-values bimatrix games which we call normal
games. In an attempt to investigate the remaining case
$\mathsf{F}\left(\frac{1}{2}\right) = b$, we show that:
  - Every normal, $n$-strategies game has an ${\mathsf{F}}$-equilibrium when
${\mathsf{F}}\left( \frac{1}{2} \right) = b$. We present a linear time
algorithm for computing such an equilibrium.
  - For 2-players, 2-values games with 3 strategies we have that if
$\mathsf{F}\left(\frac{1}{2}\right) \le b$, then every 2-players, 2-values,
3-strategies game has an $\mathsf{F}$-equilibrium; if
$\mathsf{F}\left(\frac{1}{2}\right) > b$, then there exists a normal 2-players,
2-values, 3-strategies game without $\mathsf{F}$-equilibrium.
  To the best of our knowledge, this work is the first to provide an (almost
complete) answer on whether there is, for a given concave function
$\mathsf{F}$, a counterexample game without $\mathsf{F}$-equilibrium.
