  Consider the following Simultaneous Message Passing (SMP) model for computing
a relation f subset of X x Y x Z. In this model Alice, on input x in X and Bob,
on input y in Y, send one message each to a third party Referee who then
outputs a z in Z such that (x,y,z) in f. We first show optimal 'Direct sum'
results for all relations f in this model, both in the quantum and classical
settings, in the situation where we allow shared resources (shared entanglement
in quantum protocols and public coins in classical protocols) between Alice and
Referee and Bob and Referee and no shared resource between Alice and Bob. This
implies that, in this model, the communication required to compute k
simultaneous instances of f, with constant success overall, is at least k-times
the communication required to compute one instance with constant success.
  This in particular implies an earlier Direct sum result, shown by
Chakrabarti, Shi, Wirth and Yao, 2001, for the Equality function (and a class
of other so-called robust functions), in the classical smp model with no shared
resources between any parties.
  Furthermore we investigate the gap between the smp model and the one-way
model in communication complexity and exhibit a partial function that is
exponentially more expensive in the former if quantum communication with
entanglement is allowed, compared to the latter even in the deterministic case.
