math.OCJul 21, 2026

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

Authors: Faraz Dadgostari, Neda Nazemi

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 BB completely unidentified for d≥2d\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)(H,B) up to declared equivalence. Of dd targets, d−1d-1 suffice exactly when the sole untargeted node directly parents all others; otherwise dd 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.

Explore similar work

CardsList
Jul 23, 2026stat.ME

Equilibrium Causal Digital Twins: Validation, Transport, and Identification Limits

Digital twins are often used to predict how a system would respond to an intervention. In systems with feedback, a twin must reproduce an equilibrium counterfactual, and a twin developed in one domain may fail after mechanisms change. We study when these predictions can be validated and transported. For equilibrium causal games, we give conditions on the mechanisms, equilibrium selection, and intervention design under which agreement with experimental distributions identifies the counterfactual of interest. We show why agreement of means and covariances is insufficient for distributional queries. We then introduce cyclic selection diagrams and derive criteria for direct reuse and for hybrid models that combine invariant source mechanisms with target information. An impossibility result constructs systems that agree under every experiment in a finite design but disagree on the target counterfactual, showing that validation requires structural assumptions. For linear models, we derive intervention requirements that depend on the mechanisms that changed, the observation model, and graph support. When point identification fails, we characterize the remaining range of query values. We also provide statistical tests for reconstructed means and covariances and illustrate the theory in synthetic feedback systems.
Faraz Dadgostari, Neda Nazemi
Digital TwinsEquilibriumEquivalence
AskPDF
Aug 11, 2026cs.AI

Operationalising Relative Causal Knowledge: Backbone Identifiability from Private Reports on a Shared Outcome

The Relativity of Causal Knowledge (RCK) explains how a network of agents with different structural causal models can exchange causal knowledge through a shared interventionally consistent abstraction, or backbone. We ask the prior identification question that this transport mechanism presupposes: when is that backbone determined by the agents' private causal knowledge? In the basic two-agent common-effect case, two private causes influence one shared outcome and each agent identifies only the single-cause causal marginal relevant to its own perspective. We show that, under standard compatibility, non-degeneracy, and local overlap assumptions, those local causal marginals do not identify a unique backbone. Infinitely many joint intervention kernels can induce exactly the same private reports while disagreeing on joint interventions. We then give a conditional recovery result. Additive separability removes the hidden interaction degree of freedom, but observational residual summaries remain insufficient. Identification becomes possible when agents communicate causally identified response functions. An education value-added example illustrates why this is first a communication problem, and only then a policy-composition problem.
Fabrizio Russo, Mark Somers
Causal Structure and ReasoningIdentifiabilityOutcomeReport
AskPDF
May 13, 2026stat.ML

Causal Learning with the Invariance Principle

Causal discovery, the problem of inferring the direction of causality, is generally ill-posed. We use the language of structural causal models (SCM) to show that assuming that the causal relations are acyclic and invariant across multiple environments (e.g., the way minimum wage affects employment rate is stable across different geographical regions), \textit{only} two auxiliary environments are sufficient to infer the causal graph for arbitrary nonlinear mechanisms. Moreover, we demonstrate that this implies identifiability of the SCM functional mechanisms: as a corollary, we show that \textit{two} auxiliary environments are sufficient to guarantee correct counterfactual inference. We empirically support our theoretical results on synthetic data.
Francesco Montagna, Francesco Locatello
Structural Causal ModelCausal GraphCausal Structure and ReasoningPrinciples
AskPDF