  The double slit experiment provides a clear demarcation between classical and
quantum theory, while multi-slit experiments demarcate quantum and higher-order
interference theories. In this work we show that these experiments pertain to a
broader class of processes, which can be formulated as information-processing
tasks, providing a clear cut between classical, quantum and higher-order
theories. The tasks involve two parties and communication between them with the
goal of winning certain parity games. We show that the order of interference is
in one-to-one correspondence with the parity order of these games. Furthermore,
we prove the order of interference to be additive under composition of systems
both in classical and quantum theory. The latter result can be used as a
(semi)device-independent witness of the number of particles in the quantum
setting. Finally, we extend our game formulation within the generalized
probabilistic framework and prove that tomographic locality implies the
additivity of the order of interference under composition. These results shed
light on the operational meaning of the order of interference and can be
important for the identification of the information-theoretic principles behind
second-order interference in quantum theory.
