Direct Sum Fails for Zero-Error Average Communication

We show that in the model of zero-error communication complexity, direct sum fails for average communication complexity as well as for external information complexity. Our example also refutes a version of a conjecture by Braverman et al. that in the zero-error case amortized communication complexity equals external information complexity. In our examples the underlying distributions do not have full support. One interpretation of a distribution of non full support is as a promise given to the players (the players have a guarantee on their inputs). This brings up the issue of promise versus non-promise problems in this context.