Constrained Recursive Logit (CRL) Model

• CRL is a route choice model that improves standard recursive logit by explicitly excluding infeasible paths via accumulated cost constraints.
• The model augments the state space by adding accumulated cost information, restoring the Markov property for efficient dynamic programming.
• It achieves numerical stability and behavioral realism, outperforming standard RL in scenarios with stringent travel constraints.

Constrained Recursive Logit (CRL) is a route choice model designed to fix a core behavioral flaw in the standard Recursive Logit (RL) model: RL gives positive probability to every path in the network, including paths that are infeasible or unrealistic under the traveler’s constraints. CRL retains the main advantages of RL—especially no path sampling and dynamic-programming-based estimation—but excludes infeasible paths from the universal choice set by explicitly encoding feasibility constraints into the recursive structure. In the route-choice formulation introduced in 2025, the model is initially non-Markovian because feasibility depends on accumulated path cost, and tractability is recovered by extending the state space so that standard value-iteration methods can be used on an augmented network (Tran et al., 1 Sep 2025).

1. Standard recursive logit and the motivation for constraints

In standard RL, route choice is represented as a sequence of local decisions on a directed network. If $s$ is the current state, $A(s)$ is the feasible successor set, $d$ is the destination, and the deterministic part of utility is $v(s'|s;\boldsymbol{\beta})$, then under i.i.d. type-I extreme value shocks the value function satisfies the log-sum-exp Bellman equation

$V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$

The corresponding conditional choice probabilities are multinomial-logit probabilities over successor states, and the probability of an observed path $\sigma=\{s_0,\dots,s_T\}$ factorizes as the product of those local transition probabilities (Tran et al., 19 Oct 2025).

The limitation motivating CRL is behavioral rather than merely numerical. Because the effective universal choice set is “all possible paths,” standard RL assigns nonzero probability even to routes that violate practical constraints, such as exceeding a travel-time deadline, exhausting battery energy before reaching a charging station, or violating a rental-duration limit. In that sense, infeasibility is not represented as exclusion; it is represented only indirectly through utility penalties. The explicit aim of CRL is therefore to remove infeasible paths from the choice set while preserving the recursive computation that makes RL attractive for route choice analysis (Tran et al., 1 Sep 2025).

A related numerical motivation appears in earlier work on prism-constrained RL. There, the difficulty is that standard RL estimation can become numerically unstable when parameters updated during estimation violate the feasibility condition for the value function, especially when positive network attributes are introduced. The prism-based approach addresses this by constraining the path set in a state-extended network, thereby stabilizing value-function computation during estimation (Oyama, 2022).

2. Formal CRL formulation and feasible path sets

CRL introduces a transition cost function $c(s'|s)$ together with an upper bound $\alpha$ on accumulated cost along a path. If $S_t=\{s_0,\ldots,s_t\}$ denotes the observed history up to stage $t$, the accumulated cost is

$A(s)$0

The central modeling rule is that a path is feasible only if its accumulated cost never exceeds $A(s)$1 at any intermediate step (Tran et al., 1 Sep 2025).

Because feasibility depends on the full history, CRL is initially non-Markovian on the original network. The value function is therefore defined on path-history states: $A(s)$2 If at any point $A(s)$3, then $A(s)$4, so every successor from that history receives zero probability. The paper states the implication directly: if a subpath accumulates cost greater than $A(s)$5 at any intermediate stage, then the entire path has zero probability (Tran et al., 1 Sep 2025).

The associated feasible choice set depends on the sign structure of the cost process. For nonnegative costs $A(s)$6, the restricted universal choice set is

$A(s)$7

where $A(s)$8 is the set of all origin–destination paths. For general costs, possibly including negative values, total terminal cost is insufficient; stepwise feasibility must be imposed: $A(s)$9 This distinction matters because a path can end with total cost below $d$0 while still violating the bound at an intermediate step. The paper also extends the construction to multiple constraints by replacing $d$1 and $d$2 with vector-valued quantities and requiring feasibility componentwise at every stage (Tran et al., 1 Sep 2025).

A key structural result is that CRL is equivalent to a multinomial logit model over the feasible path set. For nonnegative costs, the equivalence is to MNL over $d$3; with possibly negative costs, it is to MNL over $d$4. Thus CRL preserves the random-utility interpretation of RL, but the universal choice set is replaced by a restricted universal path choice set consisting only of feasible alternatives (Tran et al., 1 Sep 2025).

3. State-space extension, Bellman equations, and estimation

The non-Markovian dependence on accumulated cost is handled by augmenting the state space. The extended state is

$d$5

where $d$6 is the original node or link state and $d$7 is accumulated cost so far. The extended successor set is

$d$8

This restores the Markov property because the current extended state contains all information needed to determine future feasibility (Tran et al., 1 Sep 2025).

On the extended state space, the value function satisfies a standard RL-style recursion: $d$9 with $v(s'|s;\boldsymbol{\beta})$0. The paper denotes the Markovian extended-space model by ERL and proves ERL is equivalent to CRL (Tran et al., 1 Sep 2025).

For estimation, the exponential transformation

$v(s'|s;\boldsymbol{\beta})$1

yields the linear system

$v(s'|s;\boldsymbol{\beta})$2

or equivalently

$v(s'|s;\boldsymbol{\beta})$3

A central proposition establishes that if $v(s'|s;\boldsymbol{\beta})$4 for all transitions, then $v(s'|s;\boldsymbol{\beta})$5 is invertible and the linear system has a unique solution. The reason is structural: strictly positive costs imply that accumulated cost strictly increases along any path, so in the extended space there can be no cycles; after suitable ordering, $v(s'|s;\boldsymbol{\beta})$6 becomes strictly upper block-triangular and $v(s'|s;\boldsymbol{\beta})$7 (Tran et al., 1 Sep 2025).

The estimation workflow remains recognizably recursive-logit-like. One specifies utilities and constraints, constructs the extended state space, builds feasible transitions, solves the Bellman equations on the augmented network, computes extended-space choice probabilities, evaluates the likelihood of observed paths, and maximizes the log-likelihood by nested fixed-point (NFXP) estimation. The computational attraction is that infeasible routes are excluded automatically by the recursion, rather than by path enumeration or exogenous alternative sampling (Tran et al., 1 Sep 2025).

4. Prism-based constrained recursive logit as a precursor

Before the explicit CRL model of 2025, a closely related constrained recursive-logit formulation appeared in prism-based work. The prism-constrained RL model, analyzed in detail in 2022 and originally proposed by Oyama and Hato (2019), restricts the feasible path set through a prism defined on a state-extended network. In that construction, a state is $v(s'|s;\boldsymbol{\beta})$8, where $v(s'|s;\boldsymbol{\beta})$9 is the choice stage and $V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$0 is a node or link, and the traveler must reach destination $V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$1 by stage $V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$2, a hyperparameter called the choice stage constraint (Oyama, 2022).

The prism is defined using shortest-step distances to the destination. A state exists only if it can still reach the destination within the remaining stages, and state-to-state transitions are pruned accordingly. The standard RL fixed point is then replaced by a stage-dependent recursion

$V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$3

which is computed by backward recursion from terminal conditions rather than by solving $V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$4. Because the horizon $V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$5 is finite and transitions are pruned by the prism, the value vector is always bounded. This is the core reason Prism-RL is numerically stable (Oyama, 2022).

The substantive role of the prism is to remove paths that require too many steps, create large detours, or form excessive loops or cycles. Empirically, the paper reports that Prism-RL is robust to starting values and true parameter values, can estimate positive attributes such as street green safely, and can achieve better fit and prediction performance than unrestricted RL by implicitly restricting behaviorally implausible paths. At the same time, the estimator does not retain the consistency property, because parameter estimates depend on the hyperparameter $V(s)= \begin{cases} 0, & s=d,\[0.5em] \mu \ln\!\left(\sum_{s'\in A(s)} \exp\!\left(\frac{v(s'|s)+V(s')}{\mu}\right)\right), & s\neq d. \end{cases}$6 (Oyama, 2022).

This suggests two distinct constrained recursive-logit strategies in the literature. One constrains path feasibility by accumulated resource-like costs and explicit thresholds, as in CRL proper; the other constrains the state space by a finite-horizon prism motivated by observed detour rates, path lengths, or behavioral plausibility. Both preserve recursive implicit enumeration, but both replace the unrestricted universal path set of standard RL with a restricted one (Tran et al., 1 Sep 2025).

5. Adjacent estimation developments

A second line of development concerns estimation algorithms rather than behavioral constraints. Standard RL maximum likelihood estimation has traditionally been solved by nested fixed-point methods, which are computationally expensive and can be numerically unstable. An equilibrium-constrained reformulation treats the structural parameters and the value functions as joint decision variables, imposes the Bellman equations as constraints, and shows that the resulting problem for standard RL can be exactly convexified and written as an exponential-cone program solvable by MOSEK (Tran et al., 19 Oct 2025).

That formulation is important for CRL for two reasons. First, it clarifies that recursive-logit estimation can be written as optimization with equilibrium constraints rather than only as an inner–outer NFXP procedure. Second, it separates what is currently exact from what remains model-specific. The 2025 equilibrium-constrained paper is explicit that its exact convexification applies to standard RL, and that nested RL and related correlated-utility extensions generally do not admit the same convexification. A plausible implication is that analogous exact convexification results for CRL remain a nontrivial open problem rather than an automatic consequence of the standard RL case (Tran et al., 19 Oct 2025).

Another adjacent issue is incomplete trip observation. In recursive route choice models, successive observed links or nodes may be unconnected, so the probability of an observed “jump” must sum over all feasible hidden subpaths between them. A 2022 paper shows that these missing-transition probabilities can be computed exactly by solving linear systems for recursive hitting probabilities, and introduces a decomposition-composition algorithm that reduces the number of systems that must be solved. The methods are stated to work with most recursive route choice models proposed in the literature, including RL, NRL, or discounted recursive models, and they are described as directly relevant to CRL-style recursive formulations because they avoid enumerating hidden constrained subpaths (Mai et al., 2022).

6. Empirical behavior, scope conditions, and acronym ambiguity

The empirical evidence reported for CRL is concentrated in synthetic and real route-choice networks. On synthetic DAGs with travel-time upper bounds or rechargeable-vehicle energy constraints, CRL consistently gives better in-sample and out-of-sample log-likelihood than RL when constraints matter. As constraints loosen, CRL converges to RL. On the cyclic Sioux Falls network, RL estimation often fails or becomes numerically unstable, whereas CRL remains estimable and stable. On the Borlänge network, CRL and RL parameter estimates are close, but CRL achieves slightly higher log-likelihood and better predictive performance, with the improvement described as modest because constraints are not strongly binding; CRL is also computationally more expensive (Tran et al., 1 Sep 2025).

Earlier prism-based evidence points in the same direction on estimation stability. In the Sioux Falls simulation with a positive attribute, the original RL model failed to converge for all samples, while Prism-RL estimated successfully for all samples and reproduced the true parameters closely. In the Yokohama pedestrian application, RL failed under many initial values when street green was included as a positive network attribute, whereas Prism-RL estimated successfully and yielded a positive and significant green coefficient. The prism-based models also had better fit and better cross-validated prediction than the corresponding unconstrained RL or NRL models (Oyama, 2022).

A common misconception is to read “CRL” as constrained reinforcement learning rather than constrained recursive logit. Several arXiv papers use the acronym in the constrained-MDP or policy-gradient sense, including work on deterministic policies for constrained reinforcement learning, situational-constrained sequential resource allocation, policy gradients in constrained Markov decision processes, and recursive constraints for safety in constrained reinforcement learning (McMahan, 2024, Zhang et al., 17 Jun 2025, Montenegro et al., 6 Jun 2025, Lee et al., 2022). Those papers are structurally related only at a high level through recursion and constraints; they do not develop recursive logit route-choice models.

Within recursive route choice, the main conceptual distinction is between the unrestricted universal path set of standard RL and the restricted feasible path sets used by constrained variants. CRL proper makes feasibility explicit through accumulated-cost thresholds and zero probability for violating paths; prism-based approaches make feasibility explicit through a state-extended finite-horizon prism. In both cases, the gain in behavioral realism and numerical stability is obtained by abandoning the assumption that every syntactically possible path is a behaviorally admissible alternative (Tran et al., 1 Sep 2025).