Bayesian Wardrop Equilibrium

• Bayesian Wardrop equilibrium is a belief-dependent framework that extends classical traffic equilibrium by computing expected costs based on uncertain network states.
• It incorporates various information structures—including public, heterogeneous, and private signaling—to model how travelers choose routes under uncertainty.
• The approach underpins analyses of repeated learning, variational potential structures, and information design, ultimately informing efficient network routing strategies.

Bayesian Wardrop equilibrium is a belief-dependent extension of Wardrop’s nonatomic traffic equilibrium to environments with uncertain network states. In the formulations studied in routing and congestion games, edge or path costs depend on both congestion and an unknown state, while travelers choose routes by minimizing expected cost under a prior, posterior, or signal-contingent belief. A Bayesian Wardrop equilibrium is therefore a Wardrop equilibrium computed with respect to expected costs, not complete-information costs. Across the literature, the concept appears in repeated learning models with public Bayesian updating, in routing games with public or heterogeneous signaling, and in large anonymous Bayesian games through correlated and deterministic flow-based generalizations (Macault et al., 2020).

1. Core definition and formal structure

The baseline setting is a nonatomic routing game on a directed network with a single origin-destination pair, a feasible path-flow vector, and edge costs that are continuous and strictly increasing in edge load. In the complete-information benchmark, a Wardrop equilibrium is a feasible flow such that every path carrying positive flow has minimal path cost among all feasible paths. Under strictly increasing edge costs, the induced equilibrium load vector is unique (Macault et al., 2020).

Bayesian Wardrop equilibrium introduces an uncertain state. In one standard formulation, the state space is finite, $\Theta=\{\theta_1,\dots,\theta_m\}$, and each edge cost is state-dependent, $\ell_e(x_e,\theta)$. Given a belief $\mu\in\Delta(\Theta)$, expected edge and path costs are

$$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$

A flow $f$ is a Bayesian Wardrop equilibrium for belief $\mu$ if

$$f_p>0 \implies \ell_p(x,\mu)\le \ell_q(x,\mu),\qquad \forall q\in\mathcal P.$$

This is the stagewise definition used in repeated social-learning models of routing under uncertainty (Macault et al., 2020).

Equivalent formulations appear in signaling models. Given a belief $\varphi$, a flow $f^\varphi$ is a $\varphi$-based Wardrop equilibrium if every used path has minimal expected cost

$\ell_e(x_e,\theta)$0

Under continuous and strictly increasing costs, the equilibrium path flow need not be unique, but the equilibrium edge-flow is unique (Verbree et al., 2021). In heterogeneous-information models, the equilibrium condition is written separately for each information class or message: a route used by a population after observing a signal must minimize that population’s conditional expected cost (Wu et al., 2019).

A recurring structural assumption is identifiability: for every pair of states, at least one edge has different state-contingent cost functions. This does not, by itself, guarantee that the equilibrium process will identify the state, because equilibrium play may fail to visit informative edges or informative load levels (Macault et al., 2020).

2. Information structures and heterogeneous beliefs

The Bayesian content of the equilibrium notion is entirely encoded in the information structure. In repeated routing models, travelers observe a common public belief and choose routes myopically according to expected costs under that belief; after equilibrium play, an information system updates the public posterior using observed equilibrium loads and realized costs on used edges (Wu et al., 2019). In public-signaling models, a traffic information system observes the realized state, commits to a signaling scheme, and induces a posterior $\ell_e(x_e,\theta)$1 through Bayes’ rule; the induced routing outcome is then the corresponding posterior-Wardrop equilibrium (Verbree et al., 2021).

The literature also studies heterogeneous information across traveler populations. In a two-route model with an incident-prone route, only a fraction $\ell_e(x_e,\theta)$2 of travelers receives a noisy signal, while the remaining $\ell_e(x_e,\theta)$3 routes on the basis of the common prior. The Bayesian Wardrop equilibrium condition is population-specific: informed travelers minimize conditional expected cost given the realized signal, and uninformed travelers minimize ex ante expected cost under the prior (Wu et al., 2019). A related two-population model distinguishes a high-accuracy and a low-accuracy information service, with players conditioning on private binary incident signals; the equilibrium is again a Bayesian Wardrop equilibrium because each population type uses only routes with minimal expected cost given its own information (Liu et al., 2016).

Private signaling yields a further extension. In a fair private signal policy, users with the same OD pair face the same conditional message distribution, but distinct users can receive different messages. If users choose paths to minimize posterior expected travel time after receiving a message, the induced flow is a Bayesian Wardrop equilibrium when every used path is optimal conditional on that message. In that model, the equilibrium flow is unique for each signaling rule because the delays are strictly increasing, even if the underlying path-choice variables are not (Ambrogio et al., 3 Sep 2025).

These formulations share a common logic: the nonatomic best response is taken against aggregate flow, while beliefs about the state enter only through expected route costs. This is why several papers stress that the notion is not a finite-player Bayesian Nash equilibrium with strategic type interaction; it is a nonatomic equilibrium in which uncertainty enters through priors, posteriors, or signals (Verbree et al., 2021).

3. Repeated play, Bayesian updating, and learning

In repeated routing games with unknown persistent state, Bayesian Wardrop equilibrium becomes the stage game of a stochastic learning process. At each period, a demand is realized, a Bayesian Wardrop equilibrium is played for the current public belief, the equilibrium load vector and realized costs on used edges are publicly observed, and the belief is updated by Bayes’ rule (Macault et al., 2020). A related model assumes fixed total demand, Gaussian observation noise on realized edge costs, and a public information system that broadcasts the posterior after observing aggregate loads and realized costs on used edges only (Wu et al., 2019).

A central mathematical fact in both frameworks is that the belief process is a bounded martingale, so it converges almost surely. Because equilibrium loads depend continuously on beliefs, the induced flow process converges as well (Wu et al., 2019). The limit need not correspond to complete-information routing. The literature distinguishes between learning the state and learning enough for correct routing. In one formulation, strong learning means $\ell_e(x_e,\theta)$4 almost surely, while weak learning means that the long-run equilibrium flow coincides with the complete-information flow, $\ell_e(x_e,\theta)$5 almost surely; strong learning implies weak learning, but not conversely (Macault et al., 2020).

The possibility of incomplete learning is fundamental. Since only costs on used edges are observed, travelers may learn the true costs of all used edges while remaining wrong about unused edges. The limiting object is then a rest point: the flow is a Wardrop equilibrium for the limiting belief, and every state that retains positive posterior probability agrees with the true state on the used edges, but the limiting flow can still differ from the complete-information equilibrium (Wu et al., 2019).

Random demand plays a special role in the learning analysis. Variation in demand creates variation in equilibrium loads, and that variation can reveal the state if the equilibrium is forced to sample sufficiently informative load levels. This suggests that learning depends jointly on observability and on the endogenous exploration generated by equilibrium congestion (Macault et al., 2020).

4. Topology, variational structure, and large-game generalizations

The strongest network-topology result in the repeated-learning literature is that series-parallel structure governs whether Bayesian Wardrop equilibrium generates enough endogenous exploration for learning. If the network is series-parallel and each edge cost is unbounded as load approaches capacity, then full support of demand on the entire feasible range implies strong learning, while the weaker condition that demand can get arbitrarily close to capacity implies weak learning (Macault et al., 2020). The converse is equally sharp: if the network is not series-parallel, there exist capacities and uncertain unbounded costs such that weak learning fails for every demand distribution. The Wheatstone network provides the canonical obstruction, since one can construct an informative edge that is never used in equilibrium and therefore never sampled (Macault et al., 2020).

Beyond repeated learning, several papers emphasize the variational and potential structure of belief-dependent Wardrop equilibria. In public-signaling routing games, a $\ell_e(x_e,\theta)$6-based Wardrop equilibrium is equivalent to a variational inequality $\ell_e(x_e,\theta)$7, and uniqueness is pinned down at the edge-flow level under strictly increasing costs (Verbree et al., 2021). In private-signaling models with state-dependent link delays, Bayesian Wardrop equilibrium is characterized as the minimizer of a convex potential

$\ell_e(x_e,\theta)$8

with uniqueness of the induced equilibrium flow for each signaling rule (Ambrogio et al., 3 Sep 2025).

A broader continuum-player extension is Bayes correlated Wardrop equilibrium. Here the outcome is not a single deterministic flow under each state but a distribution over flow profiles conditional on the state, $\ell_e(x_e,\theta)$9. The obedience condition is

$\mu\in\Delta(\Theta)$0

for every population $\mu\in\Delta(\Theta)$1 and every pair of actions $\mu\in\Delta(\Theta)$2. A Bayes deterministic Wardrop equilibrium is the degenerate special case in which $\mu\in\Delta(\Theta)$3 is concentrated on a single flow $\mu\in\Delta(\Theta)$4, and under complete information it reduces to the standard Wardrop equilibrium (Koessler et al., 2021).

For nonatomic games with complete information admitting a convex potential, the equilibrium structure is especially rigid: $\mu\in\Delta(\Theta)$5 Thus every correlated or coarse correlated Wardrop equilibrium is simply a distribution over Wardrop equilibria, and all equilibrium outcomes have the same costs. The same paper shows that Bayes correlated Wardrop equilibria arise as limits of Bayes correlated equilibria in finite games as the maximum player weight vanishes, and that no-regret sequences have empirical distributions converging to distributions over Wardrop equilibria in convex potential games (Koessler et al., 2021).

5. Information design, prior inference, and welfare effects

Bayesian Wardrop equilibrium is also the equilibrium constraint in information-design problems. In public signaling, a traffic information system commits to a signaling scheme $\mu\in\Delta(\Theta)$6, users update their common prior $\mu\in\Delta(\Theta)$7 to a posterior $\mu\in\Delta(\Theta)$8, and equilibrium flows under the induced posterior are observed. A central observation is that the Wardrop equalities and inequalities generated by an observed equilibrium flow impose linear constraints on the prior. The set of priors consistent with all signal-induced equilibria is denoted $\mu\in\Delta(\Theta)$9, and a signaling scheme is $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$0-identifying if $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$1 (Verbree et al., 2021).

Under the condition that there exist two states with different equilibria, $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$2, there exists a public signaling scheme with $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$3 messages that is $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$4-identifying. The same work provides an iterative algorithm that terminates in finitely many steps and yields such a scheme, together with a robustness claim: if identification occurs through equality constraints alone, the scheme remains identifying for priors in a sufficiently small neighborhood of $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$5 (Verbree et al., 2021).

In a two-route Bayesian persuasion model, the authority chooses an information structure to minimize average spillover on one route relative to a threshold $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$6, where spillover is

$$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$7

The equilibrium notion is a Bayesian Wardrop equilibrium with informed and uninformed populations. The paper characterizes the optimal signaling rule analytically and finds that minimum spillover can be achieved so long as the fraction of travelers receiving the signal exceeds a threshold $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$8 (Wu et al., 2019).

Information heterogeneity does not uniformly improve welfare. In a two-population Bayesian congestion game with high-accuracy and low-accuracy information, the equilibrium is fully characterized under the special case in which the highly informed population is perfectly informed and the low-information population is effectively uninformed. The analysis yields threshold effects: below one threshold, both populations experience a reduction in individual cost, with the highly informed receiving a greater reduction; above another threshold, both populations realize the same equilibrium cost; and above a further threshold, increasing the fraction of highly informed players does not reduce expected social cost (Liu et al., 2016). This supports a recurrent conclusion in the information-design literature on routing: more information can become ineffective or even socially harmful once the induced coordination effect is strong enough.

Private signaling addresses the same design problem from a different direction. Under affine delays, a single OD pair, injective link-path incidence, and full-support system-optimal path flow, a fair private signaling rule can implement the system-optimal flow as a Bayesian Wardrop equilibrium if and only if an explicit obedience inequality holds for the candidate recommendation rule. More interpretable sufficient conditions involve equal expected free-flow path delays, diagonal covariance, and a Metzler condition on the matrix $$\ell_e(x_e,\mu)=\int_\Theta \ell_e(x_e,\theta)\,\mu(d\theta),\qquad \ell_p(x,\mu)=\sum_{e\in p}\ell_e(x_e,\mu).$$9 (Ambrogio et al., 3 Sep 2025).

6. Relation to classical Wardrop equilibrium and recurring misconceptions

Bayesian Wardrop equilibrium is best understood as a family of belief-indexed nonatomic equilibria rather than as a single fixed model. When the belief is degenerate on the true state, it reduces to the standard complete-information Wardrop equilibrium (Macault et al., 2020). In signaling models, when no useful information is sent, the equilibrium is simply the prior-based Wardrop equilibrium (Verbree et al., 2021). In correlated large-population models, Bayes deterministic Wardrop equilibrium reduces to Wardrop equilibrium under complete information (Koessler et al., 2021).

One common misconception is that observing an edge in equilibrium necessarily reveals its state-dependent cost function. The repeated-learning literature rejects this: two states may generate identical costs on the load levels actually visited in equilibrium, so observing used edges may fail to distinguish them. Learning therefore requires exploration at informative load levels, not merely traversal of informative edges (Macault et al., 2020).

A second misconception is that convergence of beliefs and flows implies complete learning. In repeated routing with public Bayesian updating, convergence may end at a self-confirming rest point in which travelers correctly learn the true expected costs of used edges but remain incorrect about unused edges, so the limiting flow differs from the complete-information equilibrium (Wu et al., 2019).

A third misconception is that Bayesian Wardrop equilibrium is simply Bayesian Nash equilibrium in a large game. The anonymous-game and signaling formulations treat each traveler as negligible and define equilibrium over aggregate flows, so the relevant obedience or optimality condition is Wardrop-style best response against a given flow, conditioned on beliefs or signals, rather than strategic best response to individually influential opponents (Verbree et al., 2021).

A final point of contrast concerns neighboring non-Bayesian theories. Dynamic assignment models establish existence of equilibrium in continuum-user route-and-departure-time problems under perfect information and minimal continuity, causality, and FIFO assumptions, but they do not formulate Bayesian learning or state uncertainty (0810.2486). Likewise, the Wardrop–mean-field-game correspondence on networks provides a variational bridge between flow equilibria and stationary first-order MFGs, yet it is not itself a Bayesian equilibrium theory (Saleh et al., 2022). This suggests that Bayesian Wardrop equilibrium occupies a specific position at the intersection of nonatomic routing, incomplete information, learning, and information design.