Theory

This section collects the proof ideas behind the public core estimators. Read it as the bridge between the estimator pages and the source papers: the estimator pages say what each method does, while these pages say which mathematical restriction makes the object interpretable.

The proofs and proof sketches in this section draw from Kang (2026), Rawat and Rust (2026), and the classical sources cited inline. The notation is adapted to the EconIRL estimator pages.

How to Read This Section

Start with the soft Bellman page if the connection between dynamic discrete choice and maximum-entropy IRL is unfamiliar. Then read identification before choosing an estimator: most estimator differences are consequences of what is known, what is normalized, and whether the transition kernel must be estimated.

Topic	Main question	Estimators it supports
Soft Bellman and DDC-MaxEnt Equivalence	Why do logit DDC and entropy-regularized IRL produce the same policy equation?	NFXP, CCP, TD-CCP, MCE-IRL, AIRL, GLADIUS.
Identification and Anchors	What is recoverable from behavior alone, and what does an anchor action add?	NFXP, CCP, TD-CCP, AIRL-Het, GLADIUS.
Classical DDC Proof Routes	How do NFXP, CCP, and TD-CCP enforce the identifying equations?	NFXP, CCP, TD-CCP.
IRL Identification Boundaries	What do MCE-IRL, AIRL, GAIL, and IQ-Learn identify?	MCE-IRL, Neural MCE-IRL, AIRL, AIRL-Het, IQ-Learn.
GLADIUS and ERM	How does the ERM route avoid transition-density estimation while keeping the Bellman identification condition?	GLADIUS.
Reward Projection and Feature Rank	When does a recovered reward imply a finite parameter vector?	NFXP, CCP, TD-CCP, MCE-IRL, Neural MCE-IRL, AIRL-Het, GLADIUS.

The Common Object

All pages use the same finite-action discounted decision problem. A state is \(s\), an action is \(a\), the transition kernel is \(P(s' \mid s,a)\), the reward is \(r(s,a)\), and the discount factor is \(\beta \in (0,1)\). The choice-specific value is \(Q(s,a)\), the soft value is

\[ V_Q(s) = \log \sum_{a \in A} \exp Q(s,a), \]

and the implied policy is

\[ \pi_Q(a \mid s) = \frac{\exp Q(s,a)}{\sum_b \exp Q(s,b)}. \]

Read \(Q\) as the object behavior identifies first. The reward \(r\) is recovered only after a Bellman equation and a normalization pin down the state-wise level that the softmax cannot see.