Wardropian Cycles: Intertemporal Fairness in Routing

• Wardropian Cycles are multi-day routing schedules or closed-loop cycles that guarantee each user in an origin–destination pair attains the same average travel time while preserving system-optimal flows.
• They reconcile efficiency and fairness by reassigning users over time using permutation matrices, cyclic shifts, or convex optimization techniques in different network settings.
• Computational strategies like cycle shortening via the greatest common divisor and the Greedy Assignment Rule illustrate both NP-hard challenges and promising empirical results in reducing inequity over time.

Searching arXiv for the cited papers and closely related work on Wardropian cycles. Wardropian Cycles are multi-day routing schedules in which aggregate path flows are system-optimal each day while the assignment of individual users to those paths is rotated over time so that, after a finite cycle, every user on the same origin–destination pair has the same average travel time (Hoffmann et al., 25 Jul 2025). In the formulation introduced for compliant connected autonomous vehicles, a Wardropian Cycle is constructed from daily assignment matrices that preserve a prescribed aggregate assignment, typically the System Optimum, and whose cumulative deviation from the origin–destination average travel time is zero over the cycle (Hoffmann et al., 25 Jul 2025). The term also appears in a distinct but related sense in a closed capacitated network model of ski resorts, where users repeatedly choose directed cycles rather than origin–destination paths, and a Wardrop-type equilibrium is imposed on these cycles via a variational inequality (Yarmoshik et al., 16 Sep 2025). Across these usages, the common theme is the extension of Wardrop’s equilibrium logic beyond a single static path assignment, either to repeated intertemporal permutations of users over fixed optimal flows or to endogenous equilibrium over closed loops in capacitated cyclic networks (Hoffmann et al., 25 Jul 2025, Yarmoshik et al., 16 Sep 2025).

1. Definitions and conceptual variants

The clearest formalization of Wardropian Cycles is given in the connected-autonomous-vehicle setting. For one origin–destination pair with $Q$ drivers and $K$ candidate paths, a daily assignment is a binary matrix

$A_j \in \{0,1\}^{Q\times K}$

whose rows index drivers and columns index paths, with each driver assigned to exactly one path and each path receiving the prescribed number $Q_k$ of drivers (Hoffmann et al., 25 Jul 2025). If $t_k$ is the travel time of path $k$ and

$\hat{t} = \frac{\sum_k Q_k t_k}{Q},$

the daily deviation vector is

$D_j = A_j T - \hat{t}\mathbf{1},$

where $T=[t_1,\dots,t_K]^T$ (Hoffmann et al., 25 Jul 2025). A Wardropian Cycle of length $n$ is then a finite sequence of daily assignments $K$0 such that

$K$1

so each driver’s cumulative deviation from $K$2 is zero and therefore each driver’s average travel time over the cycle equals the origin–destination average (Hoffmann et al., 25 Jul 2025).

This definition is explicitly designed to reconcile Wardrop’s first and second principles. Each day, aggregate flows coincide with System Optimum, so daily total travel time is minimized. Over the cycle, average travel times equalize across users on the same origin–destination pair, producing a time-averaged fairness analogous to User Equilibrium (Hoffmann et al., 25 Jul 2025). The paper therefore states that Wardropian Cycles make the assignment fair on top of being optimal, which amounts to satisfaction of both Wardrop’s principles (Hoffmann et al., 25 Jul 2025).

A second usage appears in the ski-resort model "Modeling skiers flows via Wardrope equilibrium in closed capacitated networks" (Yarmoshik et al., 16 Sep 2025). There, the objects are not multi-day assignment permutations but closed directed walks in a network. A skier’s strategy is a cycle $K$3 in a directed graph, the population distribution over cycles is

$K$4

and cycle utility is defined by

$K$5

where $K$6 is the sum of slope values, $K$7 is queue-free traversal time, and $K$8 is the sum of congestion-induced waiting times on lifts used by the cycle (Yarmoshik et al., 16 Sep 2025). In that closed-network setting, “Wardropian cycles” refers to equilibrium assignments of users to feasible loops, rather than temporal permutations of users across days (Yarmoshik et al., 16 Sep 2025).

These two formulations differ sharply in ontology. In the traffic-assignment paper, the cycle is temporal: a period of days after which user-level fairness is restored exactly (Hoffmann et al., 25 Jul 2025). In the ski-resort paper, the cycle is spatial: a closed route repeatedly traversed inside a closed capacitated network (Yarmoshik et al., 16 Sep 2025). The shared Wardropian element is the no-improving-deviation logic, but the objects over which equilibrium is defined are different.

2. Mathematical structure of multi-day Wardropian Cycles

The multi-day traffic formulation begins with a standard static assignment problem on a directed network $K$9 with origin–destination demands $A_j \in \{0,1\}^{Q\times K}$0, path flows $A_j \in \{0,1\}^{Q\times K}$1, link flows

$A_j \in \{0,1\}^{Q\times K}$2

and path travel times

$A_j \in \{0,1\}^{Q\times K}$3

under increasing convex link costs such as BPR-type functions (Hoffmann et al., 25 Jul 2025). System Optimum solves

$A_j \in \{0,1\}^{Q\times K}$4

while User Equilibrium is characterized by equality of travel times on all used paths for each origin–destination pair (Hoffmann et al., 25 Jul 2025). The Price of Anarchy is

$A_j \in \{0,1\}^{Q\times K}$5

with

$A_j \in \{0,1\}^{Q\times K}$6

(Hoffmann et al., 25 Jul 2025).

Wardropian Cycles operate after fixing an aggregate assignment $A_j \in \{0,1\}^{Q\times K}$7, typically a discretized System Optimum flow on an origin–destination pair. The daily assignment matrices preserve these aggregate path counts exactly: $A_j \in \{0,1\}^{Q\times K}$8 while also assigning each driver to exactly one path: $A_j \in \{0,1\}^{Q\times K}$9 (Hoffmann et al., 25 Jul 2025). The fairness requirement is expressed through the cumulative deviations $Q_k$0 and inequity measures

$Q_k$1

(Hoffmann et al., 25 Jul 2025). Exact equalization at the end of a cycle means $Q_k$2 and $Q_k$3 (Hoffmann et al., 25 Jul 2025).

The paper’s central existence construction uses permutation matrices. Starting from any valid assignment matrix $Q_k$4, let $Q_k$5 be the cyclic shift matrix and define

$Q_k$6

After $Q_k$7 days, each driver has been assigned to route $Q_k$8 exactly $Q_k$9 times, so each driver’s total travel time equals $t_k$0, and every driver’s average equals $t_k$1 (Hoffmann et al., 25 Jul 2025). This proves that Wardropian Cycles always exist and are finite for any origin–destination pair and any given aggregate assignment (Hoffmann et al., 25 Jul 2025).

The paper then shortens cycles using the greatest common divisor

$t_k$2

which reduces the cycle length from $t_k$3 to $t_k$4 by shifting by $t_k$5 positions per day (Hoffmann et al., 25 Jul 2025). A further shortening based on partitioning the multiset of route times into equal-mean groups is discussed, but the associated partition problem is NP-hard via reduction from SUBSET-SUM (Hoffmann et al., 25 Jul 2025). This places optimal cycle design in a combinatorial regime even though existence is elementary.

3. Fairness, optimality, and Cyclical User Equilibrium

The central claim for Wardropian Cycles in traffic assignment is that they eliminate the usual efficiency–fairness tension between System Optimum and User Equilibrium by shifting fairness from a one-day to a many-day horizon (Hoffmann et al., 25 Jul 2025). System Optimum is efficient but typically unfair because some users consistently receive shorter paths than others. User Equilibrium is fair within a day because all used paths for a given origin–destination pair have equal travel time, but it is inefficient and suffers from a Price of Anarchy (Hoffmann et al., 25 Jul 2025).

Wardropian Cycles preserve System Optimum every day by maintaining SO path flows, so daily total travel time remains minimal. Fairness is restored over the cycle because

$t_k$6

which means every user’s average travel time equals the origin–destination average $t_k$7 (Hoffmann et al., 25 Jul 2025). If the underlying aggregate assignment is SO, then $t_k$8 is the SO average and is typically strictly less than the UE travel time $t_k$9 for that origin–destination pair (Hoffmann et al., 25 Jul 2025). The paper describes this as satisfying SO every day and a UE-like fairness in the time average (Hoffmann et al., 25 Jul 2025).

This leads to the new equilibrium concept of Cyclical User Equilibrium. For a Wardropian Cycle $k$0, CUE requires both equalization over the cycle,

$k$1

and individual benefit relative to UE,

$k$2

(Hoffmann et al., 25 Jul 2025). This is a time-averaged stability notion: no driver can improve over the cycle by reverting to the UE baseline, and since all drivers on the same origin–destination pair have identical average travel time, no one can improve by swapping positions in the cycle (Hoffmann et al., 25 Jul 2025). The paper formalizes this with a Pareto order on cyclical assignments and states that a Wardropian Cycle based on SO is strictly better than UE, while among all CUEs it is Pareto-optimal (Hoffmann et al., 25 Jul 2025).

A plausible implication is that Wardropian Cycles recast Wardrop’s first principle from a pointwise statement about used routes on a single day into an intertemporal statement about average treatment over a finite horizon. That interpretation is consistent with the paper’s own claim that Cyclical User Equilibrium extends Wardrop’s principles to multi-day cycles and time-averaged travel times (Hoffmann et al., 25 Jul 2025).

The ski-resort paper yields a different fairness–stability picture. There, equilibrium means that every used cycle maximizes value-per-time: $k$3 while unused cycles have no higher utility (Yarmoshik et al., 16 Sep 2025). The equilibrium is encoded as the variational inequality

$k$4

(Yarmoshik et al., 16 Sep 2025). Here fairness is not interpersonal equalization over time but equilibrium equalization of maximal utility across the set of used spatial cycles.

4. Algorithms and computational complexity

Constructing some Wardropian Cycle is straightforward; constructing a short cycle or one with favorable interim fairness is not. The traffic paper identifies several optimization problems around cycle design and shows that many are NP-hard (Hoffmann et al., 25 Jul 2025). Beyond GCD shortening, it considers reordering daily deviation vectors to minimize the maximum cumulative deviation

$k$5

and proves NP-hardness via a reduction from SUBSET-SUM (Hoffmann et al., 25 Jul 2025). The same computational barrier appears in finding equal-mean partitions of route-time multisets to shorten cycles (Hoffmann et al., 25 Jul 2025).

Because exact optimization is hard, the paper introduces the Greedy Assignment Rule, a Markovian assignment rule that, at each day $k$6, sorts drivers by current cumulative deviation and paths by travel time, then assigns the most favored drivers to the longest routes and the least favored drivers to the shortest routes while respecting the required path flows $k$7 (Hoffmann et al., 25 Jul 2025). The assignment is encoded by a permutation matrix $k$8 built from the order of $k$9 and the path-capacity map $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$0, and the next assignment is

$\hat{t} = \frac{\sum_k Q_k t_k}{Q},$1

(Hoffmann et al., 25 Jul 2025). Proposition 2.11 states that for fixed flows and path times, this greedy assignment minimizes the next day’s inequity $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$2 among all feasible assignments, via the rearrangement inequality (Hoffmann et al., 25 Jul 2025).

The same section proves a uniform bound on cumulative deviations. If $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$3, and $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$4, $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$5 are the counts of drivers with nonnegative and negative initial deviations, then for all drivers $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$6 and days $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$7,

$\hat{t} = \frac{\sum_k Q_k t_k}{Q},$8

(Hoffmann et al., 25 Jul 2025). This implies convergence of average travel times to $\hat{t} = \frac{\sum_k Q_k t_k}{Q},$9 over time, even when the process is not forced into an exact finite cycle (Hoffmann et al., 25 Jul 2025).

The ski-resort formulation yields a different computational pipeline. Given a distribution $D_j = A_j T - \hat{t}\mathbf{1},$0 over cycles, waiting times are found by solving the convex program

$D_j = A_j T - \hat{t}\mathbf{1},$1

where $D_j = A_j T - \hat{t}\mathbf{1},$2 is the cycle–lift membership matrix and $D_j = A_j T - \hat{t}\mathbf{1},$3 the lift capacities (Yarmoshik et al., 16 Sep 2025). The dual variables are the lift waiting times $D_j = A_j T - \hat{t}\mathbf{1},$4, and strict convexity yields a unique solution $D_j = A_j T - \hat{t}\mathbf{1},$5 for each $D_j = A_j T - \hat{t}\mathbf{1},$6 (Yarmoshik et al., 16 Sep 2025). The global equilibrium is then solved via the Extragradient method on the simplex $D_j = A_j T - \hat{t}\mathbf{1},$7, with Euclidean projections and repeated evaluations of $D_j = A_j T - \hat{t}\mathbf{1},$8 via the queueing subproblem (Yarmoshik et al., 16 Sep 2025).

These two algorithmic pictures illustrate the split within the literature. Traffic Wardropian Cycles are combinatorial objects built from permutations of users over fixed optimal flows (Hoffmann et al., 25 Jul 2025). Closed-network Wardropian cycles are equilibrium objects on a finite set of spatial loops solved through convex optimization and variational inequalities (Yarmoshik et al., 16 Sep 2025).

5. Empirical results and observed behavior

The connected-autonomous-vehicle study reports large-scale experiments on static benchmark networks from the Transportation Networks repository: Barcelona, Anaheim, Eastern Massachusetts, Berlin Tiergarten, and Sioux Falls (Hoffmann et al., 25 Jul 2025). The daily aggregate assignments are computed using Frank–Wolfe, then discretized to integer path counts $D_j = A_j T - \hat{t}\mathbf{1},$9, after which exact cycles or greedy assignments are constructed at the origin–destination level (Hoffmann et al., 25 Jul 2025).

The paper reports the following system-level values for total travel time and Price of Anarchy.

City	$T=[t_1,\dots,t_K]^T$0 (min)	$T=[t_1,\dots,t_K]^T$1 (min)	PoA
Barcelona	1,297,794	1,268,541	1.02
Anaheim	1,322,588	1,304,584	1.01
EMA	28,183	27,325	1.03
Berlin T.	581,509	565,388	1.03
Sioux Falls	7,480,157	7,194,761	1.04

In Barcelona, moving from UE to SO saves around 670 vehicle-hours per peak hour, and these savings are fully preserved in Wardropian Cycles because SO flows are implemented every day (Hoffmann et al., 25 Jul 2025). At the origin–destination level, about 82.5% of OD pairs have shorter travel times in SO than UE; the remaining approximately 17.5% would require special treatment or OD-fair SO to fit the CUE definition (Hoffmann et al., 25 Jul 2025).

For Barcelona, exact cycles based on the existence construction have full-shift lengths with max 777 days, mean 31 days, median 16 days, and 95th percentile around 100 days (Hoffmann et al., 25 Jul 2025). GCD-shortened cycles retain the same maximum 777 days but improve mean to 26 days, median to 11 days, and 95th percentile to about 91 days (Hoffmann et al., 25 Jul 2025). These figures support the paper’s conclusion that exact Wardropian Cycles always exist but can be impractically long for some OD pairs (Hoffmann et al., 25 Jul 2025).

The Greedy Assignment Rule yields faster practical equalization. In Barcelona, the normalized inequity $T=[t_1,\dots,t_K]^T$2 drops to about $T=[t_1,\dots,t_K]^T$3 of $T=[t_1,\dots,t_K]^T$4 after 5 days, about $T=[t_1,\dots,t_K]^T$5 after 10 days, about $T=[t_1,\dots,t_K]^T$6 after 20 days, and about $T=[t_1,\dots,t_K]^T$7 after 50 days (Hoffmann et al., 25 Jul 2025). Across Barcelona, Anaheim, and Sioux Falls, less than 7% of the initial inequity remains after 10 days (Hoffmann et al., 25 Jul 2025). Across all five networks, the normalized inequity summed over ODs decreases roughly to 10–17% after 5 days, 3–16% after 10 days, 1–13% after 20 days, and about 1–6% after 50 days (Hoffmann et al., 25 Jul 2025). This suggests that near-equalization can be achieved quickly even when exact cycles are long.

The ski-resort paper reports numerical tests on a synthetic graph with five cycles and five lifts, solving the queueing subproblem in CVXPY at every iteration of the Extragradient method (Yarmoshik et al., 16 Sep 2025). The variational-inequality gap

$T=[t_1,\dots,t_K]^T$8

decreases monotonically, and multiple random initializations converge to the same $T=[t_1,\dots,t_K]^T$9, suggesting a unique equilibrium and stable numerical behavior for the tested instances (Yarmoshik et al., 16 Sep 2025).

6. Relation to broader Wardrop dynamics and the question of cycles

The phrase “Wardropian cycles” also interacts with a different literature on whether route-adjustment dynamics can themselves cycle. The paper "Hessian Riemannian Flow For Multi-Population Wardrop Equilibrium" (Bakaryan et al., 22 Apr 2025) is particularly explicit on this point. It studies multi-population Wardrop equilibrium on generalized graphs, formulates equilibrium as a variational inequality, and introduces a continuous-time Hessian Riemannian flow

$n$0

for each population $n$1 (Bakaryan et al., 22 Apr 2025). Under continuity, local Lipschitzness, and strict monotonicity of the cost mapping, the system has a unique global solution, trajectories remain feasible and positive, and converge to the unique Wardrop equilibrium (Bakaryan et al., 22 Apr 2025).

The same paper states directly that the Lyapunov structure rules out limit cycles or other non-convergent recurrent behavior of the HRF: if there were a nontrivial periodic orbit, the Bregman-divergence Lyapunov function would have to be periodic, contradicting strict monotone decrease (Bakaryan et al., 22 Apr 2025). It further notes that although the paper does not use the phrase “Wardropian cycles,” the convergence theorem plus strict monotonicity imply that no persistent Wardropian cycles, understood as limit cycles or recurrent orbits of the adjustment dynamics, can occur under HRF (Bakaryan et al., 22 Apr 2025).

This stands in contrast to the destination-preserving LWR network model "A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks" (Cristiani et al., 2014), which discusses three behavioral regimes—basic, rational, and highly rational—and observes numerically that in the highly rational regime the fixed-point iteration may oscillate between two distinct solutions on the same network (Cristiani et al., 2014). The paper states that it is unclear under which assumptions the highly rational model admits a solution, and that numerical tests have shown the algorithm oscillates between two non-equilibrium solutions, even if a small perturbation of an equilibrium solution is used as initial guess (Cristiani et al., 2014). This suggests that in dynamic route-choice settings, “Wardropian cycles” can also refer to oscillatory or cyclic equilibrium-seeking behavior rather than the finite fairness cycles of the CAV paper.

A related negative result about persistent cycling appears in "Stability Analysis of Transportation Networks with Multiscale Driver Decisions" (Como et al., 2011). That paper studies coupled fast traffic dynamics and slow perturbed best-response path-preference updates on an acyclic network, and shows that if path-preference updates are sufficiently slow, the state of the network approaches a neighborhood of the Wardrop equilibrium, with the neighborhood shrinking as the time-scale separation grows and decision noise vanishes (Como et al., 2011). Under its assumptions, persistent oscillations around equilibrium do not occur (Como et al., 2011).

These neighboring papers matter because they delimit the semantic range of “Wardropian cycles.” In the CAV setting, the cycle is a designed, desirable periodic schedule that enforces fairness (Hoffmann et al., 25 Jul 2025). In closed ski networks, it is a spatial loop chosen in equilibrium (Yarmoshik et al., 16 Sep 2025). In the dynamics literature, “cycle” may refer to undesirable recurrent behavior of the adjustment process, which some algorithms exclude and others may exhibit (Bakaryan et al., 22 Apr 2025, Cristiani et al., 2014, Como et al., 2011).

7. Applications, limitations, and open directions

The explicit application domain of Wardropian Cycles in the traffic-assignment sense is centrally routed compliant connected autonomous vehicles. The paper assumes that a central operator can assign daily routes, that users are willing to accept day-to-day variability if long-run average travel time is fair and strictly better than under UE, and that compliance is full or high (Hoffmann et al., 25 Jul 2025). It also sketches how a platform could maintain each user’s cumulative deviation history and assign routes using the greedy rule, effectively implementing intertemporal fairness accounting (Hoffmann et al., 25 Jul 2025).

The main limitations identified by the paper are structural. Exact cycle design can yield long periods, with some OD pairs requiring more than 90 days even after GCD shortening (Hoffmann et al., 25 Jul 2025). Optimal cycle shortening and interim-fairness optimization are NP-hard (Hoffmann et al., 25 Jul 2025). The core theory uses static demand and deterministic path travel times, while extensions to dynamic, stochastic, or SUE settings are left for future work (Hoffmann et al., 25 Jul 2025). The basic model also assumes homogeneous preferences within each OD, and the authors note that mixed traffic with only partial controllability remains an open direction (Hoffmann et al., 25 Jul 2025).

The ski-resort framework is more abstract and could apply to other closed systems in which users circulate through capacitated service stations and “enjoyment” links, such as amusement parks or closed multi-server networks (Yarmoshik et al., 16 Sep 2025). It also suggests extensions to multi-class users with class-specific slope values and to data-driven attractiveness models for cycles (Yarmoshik et al., 16 Sep 2025). Its limitations include static steady-state modeling, deterministic point queues, and the assumption that all relevant cycles are known and finite in number (Yarmoshik et al., 16 Sep 2025).

A plausible implication across both papers is that Wardropian Cycles are most natural in environments with either strong centralized assignment capabilities or naturally closed circulation patterns. In open, decentralized route-choice systems with incomplete compliance, the fairness-cycle interpretation becomes substantially harder to enforce, while the spatial-cycle interpretation depends on the network actually being closed and repeatedly traversed.

From a conceptual standpoint, the 2025 literature separates three distinct research programs under adjacent vocabulary. One concerns intertemporal fairness schedules that preserve SO and eliminate the Price of Anarchy in the long-run average (Hoffmann et al., 25 Jul 2025). A second concerns equilibrium on closed loops in closed capacitated networks (Yarmoshik et al., 16 Sep 2025). A third concerns whether equilibrium-seeking dynamics converge or cycle (Bakaryan et al., 22 Apr 2025, Cristiani et al., 2014, Como et al., 2011). The term “Wardropian Cycles” is therefore context-sensitive, but in all cases it denotes an attempt to extend Wardropian equilibrium reasoning beyond the single-shot static path assignment that dominates classical traffic theory.