  A detection system with a single sensor and $\mathsf{K}$ detectors is
considered, where each of the terminals observes a memoryless source sequence
and the sensor sends a common message to all the detectors. The communication
of this message is assumed error-free but rate-limited. The joint probability
mass function (pmf) of the source sequences observed at the terminals depends
on an $\mathsf{M}$-ary hypothesis $(\mathsf{M} \geq \mathsf{K})$, and the goal
of the communication is that each detector can guess the underlying hypothesis.
Each detector $k$ aims to maximize the error exponent under hypothesis $k$,
while ensuring a small probability of error under all other hypotheses. This
paper presents an achievable exponents region for the case of positive
communication rate, and characterizes the optimal exponents region for the case
of zero communication rate. All results extend also to a composite hypothesis
testing scenario.
