Cyclical User Equilibrium

• Cyclical user equilibrium is a traffic assignment framework that evaluates fairness and optimality over repeated cycles rather than single static trips.
• It employs methods such as shift-matrix permutations, greedy assignment rules, and variational inequalities to balance system-optimal flows with driver fairness.
• Empirical analyses, like those in Barcelona, demonstrate that while enhancing overall efficiency, cycle lengths and within-cycle inconvenience remain practical challenges.

Cyclical user equilibrium denotes a family of equilibrium concepts in which user rationality, fairness, or route optimality is evaluated over cycles rather than in a single static assignment. In one formulation, the cycle is temporal: a finite sequence of daily assignments equalizes average travel times across drivers while preserving system-optimal path flows on every day and keeping each driver strictly better off than under User Equilibrium (Hoffmann et al., 25 Jul 2025). In another formulation, the cycle is topological: users choose closed loops in a capacitated network, and equilibrium requires that all used cycles attain maximal utility under endogenous waiting times (Yarmoshik et al., 16 Sep 2025). In both cases, the concept extends Wardrop-style reasoning beyond the conventional origin–destination path setting.

1. Conceptual scope and defining variants

The term has at least two technically distinct meanings in the recent literature. The first is a multi-day assignment equilibrium for compliant connected autonomous vehicles, where fairness is defined over a repeating sequence of daily route allocations rather than day by day. The second is a closed-network cycle-choice equilibrium, where the user strategy itself is a cycle in a graph and the equilibrium object is a stationary distribution over cycles.

Formulation	Strategic object	Defining equilibrium condition
Wardropian Cycles / CUE	Sequence of daily assignment matrices $A_1,\dots,A_\zeta$	$\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$ and every driver’s cycle-average travel time is below $t^{UE}$ (Hoffmann et al., 25 Jul 2025)
Closed capacitated network equilibrium	Cycle-flow vector $\mathbf n \in \Delta^C$	All used cycles have maximal utility; equivalently a variational inequality on the simplex (Yarmoshik et al., 16 Sep 2025)

This distinction is important because adjacent literatures often use related vocabulary without defining a cyclical equilibrium in the strict sense. Static deterministic user-equilibrium approximation methods remain snapshot-based (Nikolic et al., 2021). Dynamic user equilibrium with route and departure time choice studies within-day temporal equilibrium, not a periodic or multi-day cycle (Sakai et al., 10 Aug 2025). Schedule-based transit assignment with hard vehicle capacities allows cyclic behavior in a heuristic, but the equilibrium itself is static and side-constrained rather than cyclical (Harks et al., 2024).

2. Multi-day cyclical user equilibrium and Wardropian cycles

In the multi-day formulation, an origin–destination pair with $Q$ drivers and $K$ available paths is represented by a binary daily assignment matrix

$A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$

with

$$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$

and

$$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$

Here $Q_k$ is usually taken from the System Optimum solution, so each day preserves system-optimal path flows (Hoffmann et al., 25 Jul 2025).

Let $\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$0 be the route travel-time vector and $\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$1 the OD-average travel time. The daily deviation vector is

$\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$2

and cumulative deviation up to day $\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$3 is

$\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$4

A Wardropian Cycle of length $\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$5 is a finite sequence $\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$6 such that

$\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$7

Over the full cycle, each driver’s total deviation from the OD-average travel time vanishes, so average travel times are equalized across drivers (Hoffmann et al., 25 Jul 2025).

The corresponding Cyclical User Equilibrium is defined by a sequence $\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$8 satisfying

$\sum_{j=1}^{\zeta}\mathbf D_j=\mathbf 0$9

and

$t^{UE}$0

The first condition equalizes cycle-average travel times among users. The second ensures that every driver is strictly better off than under the User Equilibrium benchmark. The formulation is explicitly intended to combine the daily efficiency of System Optimum with the long-run fairness associated with Wardrop’s first principle (Hoffmann et al., 25 Jul 2025).

This construction differs sharply from standard one-day User Equilibrium and System Optimum. In the User Equilibrium description used in the same work, all used routes for an OD pair have equal travel time and all users experience equal travel time $t^{UE}$1. System Optimum minimizes $t^{UE}$2 but can assign some users to slower routes than others. Cyclical User Equilibrium resolves that tension intertemporally: each day remains SO-like, but fairness is restored over the cycle (Hoffmann et al., 25 Jul 2025).

3. Cycle-based equilibrium in closed capacitated networks

A distinct meaning of cyclical user equilibrium appears in a closed-network model of ski resorts, where users are assigned to cycles in an oriented graph whose links are lifts and slopes. The set of admissible strategies is the set of all cycles $t^{UE}$3, and the normalized cycle-flow vector lies in

$t^{UE}$4

The network is closed because users do not enter or leave via external sinks or sources during the steady state, and capacitated because each lift $t^{UE}$5 has finite throughput $t^{UE}$6 (Yarmoshik et al., 16 Sep 2025).

Congestion is induced by a deterministic point-queue model. If $t^{UE}$7 is queue size and $t^{UE}$8 the inflow, then

$t^{UE}$9

and

$\mathbf n \in \Delta^C$0

with queue growth rate $\mathbf n \in \Delta^C$1. Waiting time at lift $\mathbf n \in \Delta^C$2 is

$\mathbf n \in \Delta^C$3

A cycle $\mathbf n \in \Delta^C$4 has value

$\mathbf n \in \Delta^C$5

and utility

$\mathbf n \in \Delta^C$6

where $\mathbf n \in \Delta^C$7 is queue-free cycle time and

$\mathbf n \in \Delta^C$8

is the total waiting time on lifts in cycle $\mathbf n \in \Delta^C$9 (Yarmoshik et al., 16 Sep 2025).

The Wardrop-style equilibrium condition is adapted directly to cycles: in equilibrium, all used cycles have maximal utility and unused cycles have no higher utility than the used ones. This is written as the variational inequality

$Q$0

The interpretation is standard: no skier can improve benefit by unilaterally switching cycles. The novelty lies in replacing origin–destination paths by recurrent loops inside a closed system (Yarmoshik et al., 16 Sep 2025).

For a fixed user distribution $Q$1, waiting times and cycle flows are recovered from a nonlinear steady-state queueing system involving cycle flows $Q$2, lift flows $Q$3, and the incidence matrix $Q$4, or equivalently from the convex program

$Q$5

The corresponding KKT conditions reproduce the queue system, and because the primal and dual are strictly convex and concave, the solution $Q$6 is unique for any fixed $Q$7 (Yarmoshik et al., 16 Sep 2025).

4. Relation to classical user equilibrium and nearby concepts

Cyclical user equilibrium remains Wardropian in spirit, but it should not be conflated with the standard User Equilibrium, dynamic user equilibrium, stochastic user equilibrium, or day-to-day learning models.

In the static deterministic setting, Wardrop’s first principle is stated as: “The travel times on all used paths between an origin and destination point are equal and less than those which would be experienced by a single vehicle on any unused path.” The k-PSA algorithm is a heuristic approximation method for this static deterministic user-equilibrium traffic assignment problem, and it does not model time-varying demand, day-to-day learning, temporal path oscillations, or cyclical equilibrium phenomena (Nikolic et al., 2021).

Within-day dynamic models are also distinct. The corridor-network study of dynamic system optimum and dynamic user equilibrium derives closed-form DSO and DUE solutions for morning and evening commutes and shows that, under certain schedule-delay conditions, the queueing delay at each bottleneck in DUE equals the optimal toll in DSO. The paper notes relevance to cyclical commuting patterns, but it does not define a separate cyclical equilibrium concept (Fu et al., 2021). Likewise, the queue replacement approach to DUE assignment with route and departure time choice works with a single assignment period $Q$8; its equilibrium queueing-delay pattern is temporal, but not cyclic in the sense of a repeating periodic equilibrium (Sakai et al., 10 Aug 2025).

Day-to-day learning models are closer in spirit but still conceptually different. The cumulative logit model CULO is a day-to-day discrete-time dynamical model that converges to User Equilibrium and, under specific initial-information conditions, to the maximum-entropy user equilibrium. It gives a behavioral interpretation of repeated route learning, but its limit object is MEUE rather than a cyclical equilibrium over recurring daily assignments (Li et al., 2024).

Stochastic and side-constrained equilibrium models modify the feasible comparison set rather than introducing cycles as the equilibrium object. The eUnit-SUE model imposes a bounded perceived travel-time interval and can exclude routes outside $Q$9, yielding a Beckmann-type convex formulation and a unique solution that approaches deterministic user equilibrium as the bound range collapses (Kitthamkesorn et al., 2024). In schedule-based transit networks with hard vehicle capacities, equilibrium is defined only against admissible deviations under boarding-capacity constraints, characterized by a quasi-variational inequality and an equivalent Bernstein–Smith equilibrium with discontinuous costs; cyclicity can appear in the heuristic update process, not in the equilibrium definition (Harks et al., 2024).

A common misconception is therefore that any model with time, queues, or iterative updates is already a cyclical user equilibrium. The literature instead separates at least three notions: multi-day cyclical fairness, closed-network cycle choice, and algorithmic or day-to-day dynamics.

5. Existence, computation, and algorithmic structure

For Wardropian cycles, existence is constructive. Given any initial assignment $K$0, let $K$1 be the shift matrix

$K$2

Then

$K$3

is a Wardropian Cycle of length $K$4. If

$K$5

the shift can be accelerated to

$K$6

which yields a cycle of length $K$7. The same work also proposes flow partitioning into subsets of equal mean travel time, but proves that dividing a set of routes with time and flow into subsets of equal mean time is NP-hard via reduction from SUBSET-SUM (Hoffmann et al., 25 Jul 2025).

The main computational difficulty in the multi-day setting is not existence but optimization of cycle length and within-cycle inconvenience. A reordering result states that one can reorder the cycle so that, for every driver $K$8 and every prefix length $K$9,

$A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$0

To approximate optimal cycles, the paper introduces a Greedy Assignment Rule: at day $A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$1, drivers with the largest positive cumulative deviation are paired with longer routes, and drivers with the most negative deviation are paired with shorter routes. By the Rearrangement Inequality, this minimizes next-day quadratic inequity. The same analysis proves bounded cumulative deviations under the greedy rule (Hoffmann et al., 25 Jul 2025).

For the closed-network cycle-choice model, computation has two nested layers. For fixed $A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$2, the queue optimization subproblem is solved via CVXPY using an interior-point solver. Then the variational inequality is solved with the Extragradient method: $A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$3

$A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$4

Under monotonicity and Lipschitz continuity of $A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$5, convergence is

$A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$6

for some constant $A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$7. Convergence is monitored through the gap function

$A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$8

which is zero at equilibrium (Yarmoshik et al., 16 Sep 2025).

These two computational paradigms are structurally different. The multi-day CUE problem is combinatorial, centered on permutations and fairness over time. The closed-network equilibrium is continuous, centered on convex queue subproblems and a simplex-constrained variational inequality. This suggests that “cyclical user equilibrium” is not a single algorithmic class, but a family of equilibrium constructions linked by the role of cycles.

6. Empirical behavior, fairness, and practical limitations

The empirical case for multi-day CUE is developed through large-scale simulations. In Barcelona, switching from UE to SO improves the system by about $A_j \in \mathcal{M}^{Q \times K}(\{0,1\}),$9, corresponding to about 670 vehicle-hours saved per day, but 17.5\% of OD pairs are worse off under SO than under UE; this is the fairness problem the cyclical construction is meant to address (Hoffmann et al., 25 Jul 2025). For Barcelona OD pairs, full shift-matrix cycles have max 777 days, mean 31.37, median 16, standard deviation 52.06, 75th percentile 34, and 95th percentile 100.3. GCD-shortened cycles retain max 777 but reduce mean to 25.94, median to 11, standard deviation to 47.81, 75th percentile to 28, and 95th percentile to 91. The same study notes that many cycles remain too long for practical acceptance (Hoffmann et al., 25 Jul 2025).

The reported path-time spreads help interpret that limitation. For 80\% of Barcelona OD pairs, the difference between shortest and longest path in SO is below 1 minute; only a small number exceed 6 minutes, with a maximum just below 10 minutes. Under the greedy assignment rule, inequity falls quickly: after 5 days, total inequity is more than halved; after 20 days, inequity can be reduced by up to 88\%; after 50 days, the remaining inequity is very small. The normalized inequity $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$0 shrinks from an initial mean around 0.1085 at day 1 to 0.0031 by day 50, while the median drops from 0.0143 to 0.00002 (Hoffmann et al., 25 Jul 2025).

Cross-city experiments report positive price of anarchy in every tested network: Barcelona UE $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$1, SO $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$2, PoA $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$3; Anaheim UE $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$4, SO $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$5, PoA $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$6; EMA UE $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$7, SO $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$8, PoA $$\sum_{k=1}^K [A_j]_{i,k} = 1 \quad \text{for every } i \in \{1,\dots,Q\},$$9; Berlin T. UE $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$0, SO $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$1, PoA $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$2; and Sioux Falls UE $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$3, SO $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$4, PoA $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$5. The greedy assignment reduces inequity after 5 days to around 0.10–0.17 of initial inequity depending on city, and after 50 days to around 0.01 in Anaheim, Berlin, and Sioux Falls, with Barcelona reaching about 0.04 (Hoffmann et al., 25 Jul 2025).

The closed-network equilibrium evidence is smaller in scale but methodologically clearer. Numerical experiments on a synthetic five-cycle, five-lift network show convergence to a stable equilibrium distribution for different attractiveness parameters $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$6. The gap function decreases monotonically over iterations, and multiple random initializations converge to the same solution $$\sum_{i=1}^Q [A_j]_{i,k} = Q_k \quad \text{for every } k \in \{1,\dots,K\}.$$7, suggesting uniqueness and favorable operator behavior. On the two-lift toy network, the CVXPY implementation “was always finding the solution to \eqref{eq:queue_system} with high accuracy” (Yarmoshik et al., 16 Sep 2025).

The main controversy is therefore not existence. In both principal formulations, existence or computability is established. The contested issue is operational interpretation. In the multi-day setting, long cycle lengths and within-cycle inconvenience directly affect acceptability. In the closed-network setting, the model is tailored to systems where users remain inside the network and repeatedly traverse cycles, which is structurally different from ordinary commuter traffic. Cyclical user equilibrium is thus best understood not as a universal replacement for User Equilibrium, but as a specialized extension for settings in which fairness or optimality is naturally assessed over recurring cycles rather than isolated trips.