On Symmetry for Characterizing Agent Heterogeneity in Multi-Agent Systems

Understanding agent heterogeneity can inform the design of shared and agent-specific policy representations, however existing criteria provide no unified structural test. We study finite-horizon POSGs and define a type-compatible transformation as a policy-evaluation symmetry when complete policy evaluation commutes with it for every decentralized behavioral policy. Policy-evaluation symmetries form a group for which agent orbits define structural agent types. This classification does not depend on the particular policy chosen or the distribution of initial states. We prove that every componentwise environment automorphism is a policy-evaluation symmetry, but that the converse does not hold. Applying the framework to cooperative Dec-POMDPs shows that agents can belong to different structural types even when they share a common reward. For prescribed weighted reward aggregation, we derive an exact value identity and sufficient conditions for the resulting Dec-POMDP to inherit the policy-evaluation symmetries of the source POSG. Finally, we give sufficient conditions under which repeated model factorizations induce policy-evaluation symmetry, and show that preserving an optimal Boltzmann policy set neither implies nor follows from this symmetry in general.