  We suggest a generalization of Karchmer-Wigderson communication games to the
multiparty setting. Our generalization turns out to be tightly connected to
circuits consisting of threshold gates. This allows us to obtain new explicit
constructions of such circuits for several functions. In particular, we provide
an explicit (polynomial-time computable) log-depth monotone formula for
Majority function, consisting only of 3-bit majority gates and variables. This
resolves a conjecture of Cohen et al. (CRYPTO 2013).
