# Stackelberg Routing on Parallel Transportation Networks

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## Abstract

This chapter presents a game theoretic framework for studying Stackelberg routing games on parallel transportation networks. A new class of latency functions is introduced to model congestion due to the formation of physical queues, inspired from the fundamental diagram of traffic. For this new class, some results from the classical congestion games literature (in which latency is assumed to be a nondecreasing function of the flow) do not hold. A characterization of Nash equilibria is given, and it is shown, in particular, that there may exist multiple equilibria that have different total costs. A simple polynomial-time algorithm is provided, for computing the best Nash equilibrium, i.e., the one which achieves minimal total cost. In the Stackelberg routing game, a central authority (leader) is assumed to have control over a fraction of the flow on the network (compliant flow), and the remaining flow responds selfishly. The leader seeks to route the compliant flow in order to minimize the total cost. A simple Stackelberg strategy, the non-compliant first (NCF) strategy, is introduced, which can be computed in polynomial time, and it is shown to be optimal for this new class of latency on parallel networks. This work is applied to modeling and simulating congestion mitigation on transportation networks, in which a coordinator (traffic management agency) can choose to route a fraction of compliant drivers, while the rest of the drivers choose their routes selfishly.

### Keywords

• Transportation networks
• Non-atomic routing game
• Stackelberg routing game
• Nash equilibrium
• Fundamental diagram of traffic
• Price of stability

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## Notes

1. 1.

The latency in congestion n (⋅ , 1) is defined on the open interval (0, x n max). In particular, if x n = 0 or x n = x n max then the link is always considered to be in free-flow. When the link is empty (x n = 0), it is naturally in free-flow. When it is at maximum capacity (x n = x n max) it is in fact on the boundary of the free-flow and congestion regions, and we say by convention that the link is in free-flow.

2. 2.

We note that a feasible flow assignment $$\boldsymbol{s}$$ of compliant flow may fail to induce a Nash equilibrium $$(\boldsymbol{t},\boldsymbol{m})$$ and therefore is not considered to be a valid Stackelberg strategy.

3. 3.

The essential uniqueness property states that for the class of non-decreasing latency functions, all Nash equilibria have the same total cost. See for example (Beckmann et al. 1956; Dafermos and Sparrow 1969; Roughgarden and Tardos 2002).

4. 4.

Price of anarchy is defined as the ratio between the costs of the worst Nash equilibrium and the social optimum. For the case of nondecreasing latency functions, the price of anarchy and the price of stability coincide since all Nash equilibria have the same cost by the essential uniqueness property.

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Krichene, W., Reilly, J.D., Amin, S., Bayen, A.M. (2017). Stackelberg Routing on Parallel Transportation Networks. In: Basar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-27335-8_26-1

• DOI: https://doi.org/10.1007/978-3-319-27335-8_26-1

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