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Urgency-aware Routing in Single Origin-destination Itineraries through Artificial Currencies

Leonardo Pedroso, W. P. M. H. Heemels, Mauro Salazar

TL;DR

This work tackles misaligned incentives in mobility by introducing Karma, an artificial currency earned or spent during travel but not bought or traded, to align selfish routing with societal optima on a single origin–destination network with $n$ parallel arcs. The authors derive a closed-form best-response for individual users, model mesoscopic aggregate dynamics as aggregate Markov chains, and propose a gradient-free numerical design to set arc prices so that Wardrop equilibrium converges to the societal optimum. They show that the proposed mechanism can achieve near-optimal aggregate flows while significantly reducing users’ perceived discomfort compared to urgency-unaware policies. Numerical results in a five-arc case demonstrate convergence to $oldsymbol{x}^ ext{star}$ with minimal cost gaps and substantial comfort gains, highlighting practical potential for demand-management in mobility systems.

Abstract

Within mobility systems, the presence of self-interested users can lead to aggregate routing patterns that are far from the societal optimum which could be achieved by centrally controlling the users' choices. In this paper, we design a fair incentive mechanism to steer the selfish behavior of the users to align with the societally optimal aggregate routing. The proposed mechanism is based on an artificial currency that cannot be traded or bought, but only spent or received when traveling. Specifically, we consider a parallel-arc network with a single origin and destination node within a repeated game setting whereby each user chooses from one of the available arcs to reach their destination on a daily basis. In this framework, taking faster routes comes at a cost, whereas taking slower routes is incentivized by a reward. The users are thus playing against their future selves when choosing their present actions. To capture this complex behavior, we assume the users to be rational and to minimize an urgency-weighted combination of their immediate and future discomfort. To design the optimal pricing, we first derive a closed-form expression for the best individual response strategy. Second, we formulate the pricing design problem for each arc to achieve the societally optimal aggregate flows, and reformulate it so that it can be solved with gradient-free optimization methods. Our numerical simulations show that it is possible to achieve a near-optimal routing whilst significantly reducing the users' perceived discomfort when compared to a centralized optimal but urgency-unaware policy.

Urgency-aware Routing in Single Origin-destination Itineraries through Artificial Currencies

TL;DR

This work tackles misaligned incentives in mobility by introducing Karma, an artificial currency earned or spent during travel but not bought or traded, to align selfish routing with societal optima on a single origin–destination network with parallel arcs. The authors derive a closed-form best-response for individual users, model mesoscopic aggregate dynamics as aggregate Markov chains, and propose a gradient-free numerical design to set arc prices so that Wardrop equilibrium converges to the societal optimum. They show that the proposed mechanism can achieve near-optimal aggregate flows while significantly reducing users’ perceived discomfort compared to urgency-unaware policies. Numerical results in a five-arc case demonstrate convergence to with minimal cost gaps and substantial comfort gains, highlighting practical potential for demand-management in mobility systems.

Abstract

Within mobility systems, the presence of self-interested users can lead to aggregate routing patterns that are far from the societal optimum which could be achieved by centrally controlling the users' choices. In this paper, we design a fair incentive mechanism to steer the selfish behavior of the users to align with the societally optimal aggregate routing. The proposed mechanism is based on an artificial currency that cannot be traded or bought, but only spent or received when traveling. Specifically, we consider a parallel-arc network with a single origin and destination node within a repeated game setting whereby each user chooses from one of the available arcs to reach their destination on a daily basis. In this framework, taking faster routes comes at a cost, whereas taking slower routes is incentivized by a reward. The users are thus playing against their future selves when choosing their present actions. To capture this complex behavior, we assume the users to be rational and to minimize an urgency-weighted combination of their immediate and future discomfort. To design the optimal pricing, we first derive a closed-form expression for the best individual response strategy. Second, we formulate the pricing design problem for each arc to achieve the societally optimal aggregate flows, and reformulate it so that it can be solved with gradient-free optimization methods. Our numerical simulations show that it is possible to achieve a near-optimal routing whilst significantly reducing the users' perceived discomfort when compared to a centralized optimal but urgency-unaware policy.
Paper Structure (17 sections, 3 theorems, 31 equations, 4 figures)

This paper contains 17 sections, 3 theorems, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Central Operator's Problem
  4. Individual User's Problem
  5. Mechanism Design Problem
  6. Best Response Strategy
  7. Mesoscopic Average Behavior
  8. Aggregate Decision
  9. Stationary Markov Chain Model
  10. WE as an Aggregate Markov Chain
  11. Pricing Design Problem
  12. Numerical Design Method
  13. Numerical Results
  14. Conclusion
  15. Proof of Lemma \ref{['lem:feasibility']}
  16. ...and 2 more sections

Key Result

Lemma III.1

Consider a traveling user with Karma $k$, sensitivity $s$, Karma reference $k_\mathrm{ref}$, and prices $\mathbf{p}$. Problem prb:individual is feasible if and only if $k \geq \max(0,k_\mathrm{ref}+(\min_j\mathbf{p}_j)(T+1))$.

Figures (4)

  • Figure 1: Single origin-destination network of $n$ arcs.
  • Figure 2: Schematic representation of one time-step of the overall model.
  • Figure 3: Best response strategy of Problem \ref{['prb:individual']} for aggregate flows $\mathbf{x}^\star$, prices $\mathbf{p}^\star$, and $k_\mathrm{ref} = 0$.
  • Figure 4: Numerical simulation results.

Theorems & Definitions (7)

  • Definition II.1: Wardrop Equilibrium
  • Lemma III.1
  • proof
  • Theorem III.1
  • proof
  • Theorem III.2
  • proof