The communication complexity of $F$ with unbounded error is the limit of the $\epsilon$-error randomized complexity of $F$ as $\epsilon\to1/2.$ Communication complexity with weakly bounded error is defined similarly but with an additive penalty term that depends on $1/2-\epsilon$. Explicit functions are known whose two-party communication complexity with unbounded error is exponentially smaller than with weakly bounded error. Chattopadhyay and Mande (ECCC Report TR16-095) recently generalize this exponential separation to the number-on-the-forehead multiparty model, using a rather technical proof from first principles.

We show how to derive such an exponential separation from known two-party work, achieving stronger parameters along the way. We present several proofs for this result, some as short as half a page. Our strongest separation is a $k$-party communication problem $F\colon(\{0,1\}^{n})^{k}\to\{0,1\}$ that has complexity $O(\log n)$ with unbounded error and $\Omega(n/4^{k})$ with weakly bounded error.