Equilibrium Causal Games: Separation, Identification, and the Identifiability of Cyclic Latent States

2026-07-21

AI Summary

This work addresses equilibrium systems featuring cyclic feedback and unobserved measurement sensors by proposing the Equilibrium Causal Game (ECG) framework, which unifies the modeling of game rules, cyclic causal structures, latent variables, sensor mappings, and intervention mechanisms, enabling causal inference through equilibrium recomputation. Leveraging an ECG-specific separation criterion, the study establishes, for the first time, identifiability boundaries under equilibrium data, revealing the critical roles of non-Gaussianity, intervention type, and the number of targeted interventions in determining identifiability. By integrating LiNGAM, partial path identification, and multi-environment invariance analysis, the authors demonstrate that in a d-dimensional system, only d or d−1 targeted interventions are sufficient to identify observable queries, while precisely characterizing the sources of non-identifiability arising from sensor configurations and interaction structures.

Abstract

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.

Problem

• causal identification
• equilibrium systems
• latent variables
• cyclic causal models
• unknown sensing

Innovation

• Equilibrium Causal Game
• Cyclic Latent States
• Identifiability
• Unknown Sensing
• Mechanism Intervention