Choosing and Comparing Estimators

This page is the first stop when you are deciding which EconIRL estimator to open from the menu. It assumes only the basic goal: you have observations of choices over time, and you want to estimate either a structural dynamic discrete-choice model or a reward function that explains observed behavior.

The estimators differ in what they treat as known, what they estimate, and what kind of recovery claim they can support. A method that predicts choices well is not automatically recovering the reward. Reward recovery also needs the right normalization, support, transition information, and identification argument.

Detailed math, system views, examples, and usage live on the estimator pages. Paper-number checks live in Replications. Problem-level simulation results live in Simulation Studies.

How to Read the Menu

Start from the simplest structural case and move only when a specific part of that case fails.

If the problem is small, tabular, and structurally specified, start with NFXP.
If NFXP is the right target but the repeated Bellman solves are too costly, inspect CCP, MPEC, UFXP, and NNES.
If transition-density estimation is the bottleneck, inspect TD-CCP.
If the reward itself must be learned from demonstrations, inspect the IRL estimators: MCE-IRL, Neural MCE-IRL, AIRL, AIRL-Het, and GLADIUS.
If the behavior is better described by bounded or finite-horizon planning, inspect RHIP.

The Canonical Case

The reference estimator is nested fixed point (NFXP). It estimates a small tabular structural dynamic discrete-choice model by maximum likelihood. It solves the agent’s dynamic program inside the likelihood for each candidate reward parameter.

NFXP is at its best when every one of these holds.

A single forward-looking agent in a stationary, infinite-horizon, discrete-time problem.
A small finite state space where every state can be enumerated.
A small discrete action set.
A flow payoff that is linear in known features, \(u(s,a) = \phi(s,a)^\top\theta\), with a few parameters.
A simple exogenous transition kernel, estimated in a first stage and separate from the payoff parameters.
Additive i.i.d. Type-I extreme-value shocks, which give closed-form logit choice probabilities.
A pinned normalization. An anchor action fixes the reward level, the logit scale is fixed, and the action-contrast feature rank equals the parameter count.

When this holds, the soft-Bellman operator is a contraction on a small grid. The inner fixed point solves exactly and quickly. The outer loop is full-information maximum likelihood with an exact inner solve at every candidate parameter. The result is the efficient reference estimator, with standard errors from the information matrix. The Rust (1987) bus-engine model is the canonical example.

What Breaks the Canonical Case

Each other estimator relaxes one part of the NFXP setup and pays for it somewhere else.

Source of complexity	Estimators it motivates	What changes
Large or continuous state space	CCP, MPEC, UFXP, NNES	The exact nested solve is replaced by inversion, constraints, fixed-point first-order conditions, or value approximation.
Hard-to-model transition density	TD-CCP	Estimation uses observed successor tuples instead of a transition-density model.
Unknown reward form	MCE-IRL, Neural MCE-IRL, AIRL, AIRL-Het, GLADIUS	The target shifts from structural likelihood estimation to reward recovery from demonstrations.
Latent heterogeneity	AIRL-Het	The reward is segment-specific and needs credible segment, exit-action, and absorbing-state anchors.
Bounded or finite-horizon planning	RHIP	The planning horizon becomes part of the behavioral model.

Large or continuous state spaces break the cheap exact solve. Encoded, high-dimensional, or smooth states make repeated tabular dynamic programming unattractive. CCP avoids repeated solves by using first-stage choice probabilities. MPEC and UFXP keep a structural likelihood target but change the optimization route. NNES uses a neural continuation value with finite reward parameters.

Hard transition-density problems motivate TD-CCP. The estimator still targets a finite structural reward parameter, but it estimates recursive terms from observed successor state-action pairs instead of first modeling the transition density.

Unknown rewards motivate inverse reinforcement learning. MCE-IRL recovers reward coefficients in supplied features. Neural MCE-IRL uses a neural reward map. AIRL targets the original state-only transfer claim. AIRL-Het adds anchors for segment-specific action-dependent rewards. GLADIUS learns neural Q and continuation objects, then reads reward information through anchored action contrasts.

Core Lineage

The core estimators carry the main identification stories in EconIRL.

Estimator	Main question	What changes relative to NFXP
NFXP	What is the exact tabular structural likelihood estimate?	Nothing. It is the reference under the maintained DDC assumptions.
CCP	Can we avoid repeated Bellman solves?	Uses first-stage choice probabilities and Hotz-Miller inversion. NPL iterates the pseudo-likelihood route.
TD-CCP	Can we avoid transition-density estimation during parameter estimation?	Uses observed successor state-action pairs to estimate recursive terms.
MCE-IRL	Can demonstrations identify a reward?	Replaces structural likelihood with maximum causal entropy feature matching.
Neural MCE-IRL	Can the reward be nonlinear?	Replaces fixed linear reward features with a neural reward map.
AIRL	Can a state-only reward transfer across dynamics?	Separates state reward from shaping under the original AIRL assumptions.
AIRL-Het	Can anchored rewards differ across latent segments?	Adds exit-action and absorbing-state anchors, then estimates segment-specific rewards.
GLADIUS	Can high-dimensional offline DDC avoid repeated solves?	Learns Q and continuation objects, then reads projected action contrasts.

Core Estimators Side by Side

Use this table to narrow the choice before opening a method page.

Estimator	Use when	Data and transition requirement	Reward target	State scale	Avoid when	Evidence status
NFXP	You need the reference structural DDC likelihood and counterfactual policy analysis.	Discrete panel data; transitions known or estimated first.	Finite parametric structural reward.	Small or moderate tabular state-action spaces.	Repeated exact Bellman solves are too expensive or transition modeling is the main bottleneck.	Synthetic tabular simulation and the Rust (1987) Table IX replication.
CCP	You want a faster Hotz-Miller or NPL tabular structural estimate.	Discrete panel data; transitions known or estimated first; strong empirical action support.	Finite parametric structural reward.	Small or moderate tabular state-action spaces.	Many states have weak or one-action support, or you need the direct nested fixed-point likelihood.	Synthetic tabular simulation with support conditions.
TD-CCP	Transition-density modeling is hard but the reward has known finite features.	Panel trajectories with current and next state-action information; an environment is still needed for post-fit counterfactuals.	Finite linear structural reward.	Encoded or higher-dimensional discrete states.	State space is small enough for tabular likelihood methods, support is sparse, or the target is a neural reward map.	Encoded-state finite-parameter hard case with locally robust standard errors.
MCE-IRL	Demonstrations should be explained by maximum causal entropy feature matching.	Demonstrations from a discrete dynamic decision problem; transitions known or supplied.	Supplied finite reward features.	Tabular state-action spaces.	You need likelihood-based structural standard errors or reward features are unknown.	Synthetic supplied-feature simulations.
Neural MCE-IRL	Demonstrations should be explained by an unrestricted neural reward under the maximum causal entropy objective.	Demonstrations from a discrete dynamic decision problem; transitions known or supplied.	Neural reward map.	Tabular or encoded state-action spaces.	You need finite structural parameters with standard errors, or supplied reward features are enough.	Synthetic neural-reward recovery simulation.
AIRL	Adversarial recovery under the original state-only AIRL assumptions is the research object.	Demonstrations from a discrete dynamic decision problem; transitions available for validation or post-fit evaluation.	State-only reward with shaping term under a fixed normalization.	Discrete dynamic decision settings.	Reward is action-dependent, an absorbing-state normalization is central, or structural action-dependent recovery is required.	Synthetic state-only AIRL simulation.
AIRL-Het	Latent segments have different dynamic preferences and segment-specific counterfactuals matter.	Repeated user trajectories; credible exit-action and absorbing-state anchors.	Segment-specific action-dependent reward.	Encoded discrete dynamic choice settings.	Segment membership is weakly identified, no credible reward anchor exists, or a homogeneous estimator is enough.	Synthetic serialized-content simulation.
GLADIUS	You want neural Q and continuation modeling with anchor-moment reward recovery.	Dynamic discrete choices; known transitions; credible anchor action with known rewards.	Neural reward recovered from neural Q and continuation objects.	High-dimensional encoded state features.	No credible anchor action exists or you need tabular structural estimation.	Preview: projected reward diagnostics.

How the Papers Relate

The source papers are not all answering the same question. Some keep the structural DDC target and change the computation. Others switch to reward recovery from demonstrations.

Route	Paper comparator	What to carry into EconIRL
NFXP / Rust	Earlier empirical replacement models.	Treat NFXP as the exact small-tabular benchmark. Later methods usually criticize its cost, not its target.
CCP / Hotz-Miller and NPL	NFXP and maximum likelihood.	CCP is another route to the same finite DDC target when support is strong.
TD-CCP / Adusumilli-Eckardt	NFXP, CCP, and transition-density-based DDC.	TD-CCP is transition-density-free for estimation. Counterfactuals still need an environment.
MCE-IRL / Ziebart	Apprenticeship learning and non-causal MaxEnt IRL.	MCE-IRL changes the estimand. It recovers reward only in the supplied feature span and normalization.
Neural MCE-IRL / DeepIRL	Linear MaxEnt IRL.	Neural MCE-IRL is a nonlinear reward-map extension of MCE-IRL. Raw weights are not the estimand.
AIRL / Fu-Luo-Levine	GAIL and shaped adversarial rewards.	Use AIRL for the original state-only reward-transfer claim.
AIRL-Het / Lee-Sudhir-Wang	Homogeneous AIRL and pooled dynamic choice.	Exit and absorbing-state anchors do the identification work for action-dependent and segment-specific rewards.
GLADIUS / Kang-Yoganarasimhan-Jain	NFXP, CCP, TD-CCP, offline MaxEnt IRL, and Bellman-loss methods.	GLADIUS is the high-dimensional offline bridge. Its safest structural object is the projected action contrast.

Paper MDP Shapes

Use this table to match an estimator to the kind of decision process that motivated it.

Estimator	Source or showcase setting	Natural problem shape
NFXP	Rust bus-engine replacement.	Small tabular keep-or-replace panels with a full transition model.
CCP	Hotz-Miller dynamic-choice inversion.	Finite panels with reliable choice probabilities in each state.
TD-CCP	Transition-density-free DDC panels.	Current and successor state-action tuples, with finite reward features.
MCE-IRL	Taxi route preference with road features.	Demonstrations in a known controlled process with credible reward features.
Neural MCE-IRL	DeepIRL grid maps.	Nonlinear state or state-action reward maps with known transitions.
AIRL	State-only transfer MDP.	State-only reward transfer under the Fu-Luo-Levine assumptions.
AIRL-Het	Serialized-content choice with latent types.	Anchored action-dependent rewards and persistent latent segments.
GLADIUS	High-dimensional offline dynamic choice.	Offline panels where tabular dynamic programming is too costly and action contrasts are enough.

Main Axes

State Scale

State setting	Natural estimators	Why
Small tabular state space	NFXP, CCP, MCE-IRL, AIRL	The full grid can be enumerated.
Small or moderate tabular space with speed pressure	CCP, MPEC, UFXP	The structural target remains tabular, but the computation changes.
Encoded or higher-dimensional state space with finite reward parameters	TD-CCP, NNES	The reward is still finite-dimensional, but transition or value objects become harder.
Nonlinear reward over tabular or encoded states	Neural MCE-IRL	The reward map is neural, but planning still uses supplied transitions.
Repeated choices with latent segments	AIRL-Het	It estimates segment-specific rewards and policies.
High-dimensional offline state features	GLADIUS	It learns Q and continuation objects instead of repeated tabular solves.

Reward Form

Reward target	Core estimators	Main caution
Finite linear structural reward	NFXP, CCP, TD-CCP	Needs action-contrast feature rank and a fixed normalization.
Linear IRL reward basis	MCE-IRL	Identified only inside the supplied feature basis.
Neural reward map	Neural MCE-IRL	The reward matrix is the object. The raw weights are not.
State-only transferable reward	AIRL	Matches the original AIRL claim only under its state-only assumptions.
Segment-specific action-dependent reward	AIRL-Het	Needs credible exit-action and absorbing-state anchors, persistent segments, and enough trajectory support per segment.
Projected action contrast	GLADIUS	Stronger than raw Bellman reward levels in the package route.

Transition Information

Transition input	Estimators	Meaning
Explicit transition tensor	NFXP, CCP, MCE-IRL, Neural MCE-IRL, AIRL, AIRL-Het	The estimator or policy update uses a transition model.
Observed successor pairs for estimation	TD-CCP	Estimation uses successor tuples instead of a transition-density model.
Offline next states	GLADIUS	Training uses sampled next states and learned continuation objects.

TD-CCP still needs an environment for counterfactuals. GLADIUS still needs a credible anchor to support reward interpretation. AIRL’s unanchored state-action discriminator should be read as behavior-fitting evidence, not as identified structural reward.

Recovery Conditions

Here “recovers” means population-level recovery of the stated reward object after the required normalization. Matching choices is weaker than recovering the reward.

Estimator or version	Reward object	Can recover it?	Conditions that matter most
NFXP	Finite linear \(R(s,a)\)	Yes.	Correct DDC model, Markov state, exogenous transitions, fixed discount and logit scale, enough support, and global likelihood optimum.
CCP one-step	Same finite target as NFXP.	Yes, in population.	Same structural conditions as NFXP, plus reliable first-stage CCPs and no zero-support cells.
CCP NPL	Same finite target as NFXP.	Yes, in population.	Same as one-step CCP, plus convergence to the relevant NPL fixed point.
TD-CCP semigradient	Finite reward parameters.	Yes, in population.	Successor tuples, consistent CCPs, recursive terms in the projection span, support, and correct normalization.
TD-CCP neural	Same finite reward parameters.	Conditional.	Same target as semigradient TD-CCP, with enough data and capacity to learn the recursive terms.
MCE-IRL	Reward coefficients in supplied features.	Yes, in population.	Known transitions, true reward in the feature span, full-rank moments, support, and fixed normalization.
Neural MCE-IRL	Anchored reward matrix.	Conditional.	Known transitions, representable reward, sufficient occupancy, and exact optimization under a fixed anchor.
AIRL	State-only reward up to a constant.	Yes under the original AIRL assumptions.	State-only reward, decomposable dynamics, sufficient expert and learner samples, and adversarial equilibrium.
AIRL-Het	Segment-specific anchored \(R_k(s,a)\).	Conditional.	Exit-action reward anchor, absorbing-state value anchor, correct segment count, segment separation, support, fixed discount, and exact policy solution.
GLADIUS dual anchor-moment	Projected action contrasts.	Conditional.	Credible anchor, learned Q and continuation objects, action-contrast rank, and support.
GLADIUS `q_only`	Full reward.	No.	Useful as a diagnostic mode, but not enough for reward recovery.

Other Estimators

The non-core estimator pages are useful when a specific computational or behavioral complication is the main reason not to use the reference route.

Estimator	Use when	Current role
NNES	The value object is too large or encoded for repeated exact dynamic programming.	Neural value approximation with finite structural parameters.
MPEC	You want a constrained-optimization check on the DDC likelihood.	Secondary structural check; overlaps with NFXP/CCP and has higher solver complexity.
UFXP	You want maximum-likelihood-grade structural estimates without nesting any fixed point in the search.	Secondary structural speed and first-order-condition variant.
RHIP	Route choice or graph planning needs a horizon-scaled entropy IRL method.	Horizon-parameterized entropy IRL for route graphs.
f-IRL	The study question is state-marginal matching under an f-divergence.	Narrower state-marginal method.
IQ-Learn	Inverse soft-Q learning or imitation diagnostics are the estimator of interest.	Preview diagnostic.

On a small, well-specified tabular problem, these alternatives do not answer a better question than NFXP. Their value appears when one of the complications on this page is real.

Linear Reading Guide

If you are reading the docs in order, use this route.

Read NFXP first. It defines the structural benchmark and the normalization issues that recur across the docs.
Read CCP next if you want the same finite DDC target without repeated nested solves.
Read TD-CCP if transition-density modeling is the estimation bottleneck.
Read MCE-IRL when demonstrations define the problem and reward features are supplied.
Read Neural MCE-IRL when the reward is nonlinear and transitions are known.
Read AIRL when state-only reward transfer is the object.
Read AIRL-Het when anchored latent heterogeneity is the object.
Read GLADIUS when high-dimensional offline state features make repeated dynamic-programming solves unattractive and projected action contrasts are enough.
Read Other Estimators when the complication is value approximation, constrained structural optimization, finite-horizon planning, f-divergence matching, or inverse soft-Q diagnostics.