  We present a Karchmer-Wigderson game to study the complexity of hazard-free
formulas. This new game is both a generalization of the monotone
Karchmer-Wigderson game and an analog of the classical Boolean
Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing
monotone and general games.
  Using this game, we prove hazard-free formula size and depth lower bounds
that are provably stronger than those possible by the standard technique of
transferring results from monotone complexity in a black-box fashion. For the
multiplexer function we give (1) a hazard-free formula of optimal size and (2)
an improved low-depth hazard-free formula of almost optimal size and (3) a
hazard-free formula with alternation depth $2$ that has optimal depth. We then
use our optimal constructions to obtain an improved universal worst-case
hazard-free formula size upper bound. We see our results as a significant step
towards establishing hazard-free computation as an independent missing link
between Boolean complexity and monotone complexity.
