Equilibrium Causal Games: Separation, Identification, and the Identifiability of Cyclic Latent States
Power grids, markets, and interacting populations, settle into feedback driven equilibria observed through unknown sensors. Our Equilibrium Causal Game (ECG) joins a game to its cyclic causal model, hidden inputs, sensor map, and rules for interventions and equilibrium selection; interventions edit declared objects and recompute equilibrium. Under stated conditions, ECG-separation is sound but incomplete in our examples. Back-door/half-trek routes identify observed queries. Yet for an untouched rotationally symmetric Gaussian block, second moments determine only a source-frame rotation, across which distinct-variable effects generically change. Unknown sensing creates a separate ambiguity. In passive stable linear models without self-effects, unknown wiring and full-rank unknown sensing leave $B$ completely unidentified for $d\ge2$. Under LiNG, non-Gaussianity removes the source rotation; mechanism interventions separate sensing from interactions. With unknown support, invariant sensing, aligned responses, and well-posed single-target interventions identify $(H,B)$ up to declared equivalence. Of $d$ targets, $d-1$ suffice exactly when the sole untargeted node directly parents all others; otherwise $d$ are needed. Acquisition probes are excluded; known wiring gives no universal count. With nonlinear sensing, isotropic Gaussian source blocks admit hidden twists within and across blocks in labelled environments preserving required radial laws. Conversely, under stated positivity, informative one-block changes, rank, and irreducibility conditions, the finest independent source-block representation is identified within the stated alternative class up to block permutation and blockwise coordinate changes, but not downstream mechanisms or the sensor/interaction split. Together, these results show which causal conclusions equilibrium data support and which require targeted experiments.