Multi-Path Refund Mechanism

• Multi-path refund is a mechanism for redistributing congestion toll revenues through optimal tolling and systematic refunding to enhance efficiency and equity.
• It uses system-optimal congestion pricing with path-dependent tolls based on marginal costs, extending classical Pigovian approaches by actively refunding surplus revenues.
• The method minimizes post-refunding wealth disparity via slack transfer allocation, employing tractable LP or greedy algorithms to achieve user-favorability and equity.

A multi-path refund, formally situated within the congestion pricing and revenue refunding (CPRR) paradigm, defines a mechanism for redistributing congestion toll revenues in nonatomic multi-commodity network settings. The mechanism integrates optimal congestion pricing—implemented via path-dependent tolls—with systematic revenue refunding schemes designed to jointly maximize network efficiency and equity. In contrast to classical Pigovian tolling approaches that merely internalize congestion externalities, multi-path refund mechanisms explicitly transfer the revenue surplus back to users, addressing both aggregate efficiency (travel cost minimization) and distributional fairness (wealth inequality mitigation), while ensuring every participant is weakly better off post-intervention compared to the untolled equilibrium (Jalota et al., 2021).

1. Mathematical Framework and Fundamental Definitions

Consider a directed graph $G = (V, E)$ representing a transportation network, with user groups $g \in \mathcal{G}$, each characterized by a specified origin–destination (OD) pair $(s_g, u_g)$ and aggregate demand $d_g$. For each group $g$, the feasible path set $\mathcal{P}_g$ comprises all simple $s_g \to u_g$ paths. Flows $f$ assign nonnegative values $f_{P,g}$ to each path–group pair, subject to the constraint $\sum_{P \in \mathcal{P}_g} f_{P,g} = d_g$ for all $g \in \mathcal{G}$0. Edge loads are $g \in \mathcal{G}$1, and edge latency functions $g \in \mathcal{G}$2 are convex, continuously differentiable, and strictly increasing.

Users are infinitesimal (nonatomic), characterized by positive initial wealth $g \in \mathcal{G}$3 and value-of-time $g \in \mathcal{G}$4 (group-homogeneous as $g \in \mathcal{G}$5, $g \in \mathcal{G}$6), incurring disutility along path $g \in \mathcal{G}$7 given by $g \in \mathcal{G}$8. The baseline (untolled) equilibrium flows $g \in \mathcal{G}$9 satisfy the Wardrop principle: for all $(s_g, u_g)$0 and $(s_g, u_g)$1, if $(s_g, u_g)$2 then $(s_g, u_g)$3, with $(s_g, u_g)$4 defined as above. The total untolled cost is $(s_g, u_g)$5.

2. System-Optimal Congestion Pricing

System optimum (SO) flow $(s_g, u_g)$6 minimizes aggregate disutility:

$(s_g, u_g)$7

subject to demand conservation. The toll on edge $(s_g, u_g)$8 is set as the Pigou–marginal-cost toll:

$(s_g, u_g)$9

reflecting the group-weighted marginal congestion cost each group imposes. The corresponding SO equilibrium is characterized by generalized path costs $d_g$0 uniform across used paths for group $d_g$1.

3. Revenue Refunding and User-Favorability

The total toll revenue under SO tolling is $d_g$2. A refund function $d_g$3 determines the lump-sum transfer for group $d_g$4, with the budget-balance condition $d_g$5. The net cost for a user in $d_g$6 is thus $d_g$7. User-favorability, a central property, requires that for all groups $d_g$8,

$d_g$9

ensuring no user is worse off than in the untolled state. A closed-form refund is achieved via slack transfers $g$0 with $g$1 and

$g$2

The user-favorability constraint is strictly enforced by this prescription.

4. Equity and Efficiency Joint Optimization

Efficiency is quantified by social welfare $g$3. Post-transfer equity is measured by the discrete Gini index of ex-post group incomes $g$4:

$g$5

The joint efficiency–equity problem is a constrained (finite-dimensional, convex) program:

$g$6

with the solution “pouring” surplus first onto the poorest ex-post group, then the next, yielding a max–min optimal allocation when the Gini index is piecewise linear.

5. Structure and Properties of the Optimal Multi-Path CPRR Mechanism

The optimal CPRR mechanism is constructed as follows:

• Compute SO flows $g$7 and edge tolls $g$8 by solving the convex SO-flow problem.
• Determine slack transfers $g$9 to minimize Gini subject to surplus distribution $\mathcal{P}_g$0, using either a small LP or a greedy max–min allocation algorithm.
• Refund group $\mathcal{P}_g$1 by $\mathcal{P}_g$2.

The resulting equilibrium is Wardrop under the new (tolled) costs, satisfies budget balance, user-favorability, and achieves minimal post-refunding disparity (Gini-minimality) among all possible user-favorable refunding schemes (Jalota et al., 2021). This highlights that the mechanism attains SO flows, always makes every user weakly better off, minimizes post-refunding wealth-inequality, and remains computationally tractable.

6. Computational Algorithm and Practical Implementation

The procedure for deploying a multi-path CPRR mechanism is explicitly as follows:

1. Solve the convex SO-flow problem $\mathcal{P}_g$3 using convex-QP or Frank–Wolfe methods to obtain $\mathcal{P}_g$4 and $\mathcal{P}_g$5.
2. Assign tolls $\mathcal{P}_g$6.
3. (Optional) Compute untolled UE flow $\mathcal{P}_g$7 to establish the baseline cost $\mathcal{P}_g$8.
4. Calculate the refunding budget $\mathcal{P}_g$9.
5. Solve for $s_g \to u_g$0 minimizing $s_g \to u_g$1 given $s_g \to u_g$2 and $s_g \to u_g$3, using a small LP or max–min greedy method.
6. Disburse $s_g \to u_g$4 to each group.

Key Features

Property	Mechanism Guarantee	Achieved By
System-optimal flows	Yes	Marginal-cost tolls
User-favorability	Strong (no user worse)	Explicit refund constraint
Equity (Gini-minimality)	Optimal among feasible	Slack transfer allocation
Computation	Tractable, LP/greedy	Convex optimization structure

The combination of these features positions multi-path refund-based CPRR mechanisms as a rigorous solution for reconciling efficiency and equity within networked congestion pricing, with direct algorithmic realizations available for applied settings (Jalota et al., 2021).