## Abstract

We investigate how and to which extent one can exploit risk-aversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy (PoA) wrt. the total latency is improved. The starting point is to introduce some small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases, due to risk-aversion. We adopt the simple model o*f γ*-modifiable routing games, a variant of selfish routing games with restricted tolls. We prove that computing the best *γ*-enforceable flow is **N****P**-hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured *γ*-enforceable flow whose PoA improves fast as *γ* increases. We show that the complexity of computing such a *γ*-enforceable flow is determined by the complexity of computing a Nash flow for the original game. Moreover, we prove that the PoA of this flow is best possible in the worst-case, in the sense that there are instances where (i) the best *γ*-enforceable flow has the same PoA, and (ii) considering more flexible modifications does not lead to any improvement on the PoA.

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## Notes

- 1.
Despite our brief discussion about how such an upper bound

*γ*_{e}can be determined, we deliberately avoid getting into the details of how*γ*_{e}’s are calculated. This depends crucially (and not always in a simple way) on the particular practical application and cannot be incorporated into a theoretical model. - 2.
To simplify the model and make it easily applicable to general networks, we assume that the latency modifications (and the resulting individual costs of the players) are separable. This is a relatively standard simplifying assumption (see e.g., [16, 18]) on the structure of risk-averse individual costs in networks and only affects the extension of our results to series-parallel networks.

- 3.
Note that i

*f ℓ*_{e}(*x*) is not strictly increasing,*x*_{e}may not be uniquely defined (and it may be \(L^{\prime } = L\)). Then, for each*e*∈*E*_{1}, we let*x*_{e}be the largest value such that \(f^{\prime }_{e} + x_{e} \leq o_{e}\) and \(L^{\prime } = \ell _{e}(f^{\prime }_{e}+x_{e})\) (i.e., if \(L^{\prime } = \ell _{e}(o_{e})\),*x*_{e}becomes \(o_{e} - f^{\prime }_{e}\) so that*e*moves from*E*_{1}to*E*_{2}). For each*e*∈*E*_{3}, we let*x*_{e}be the smallest value such that \(L^{\prime } = \ell _{e}(f^{\prime }_{e}+x_{e})\). - 4.
Before rerouting, the equilibrium cost for the links in

*E*_{1}∪*E*_{3}(with traffic rate \({\sum }_{e \in E_{1} \cup E_{3}} f^{\prime }_{e}\)) is \(L < L^{\prime }\) and*ℓ*_{e}(*o*_{m}) >*L*. After we reroute*x*units of flow from link*m*to*E*_{1}∪*E*_{3}, the equilibrium cost for the links in*E*_{1}∪*E*_{3}(with traffic rate \({\sum }_{e \in E_{1} \cup E_{3}} (f^{\prime }_{e} + x_{e})\)) is \(L^{\prime } > L\) and \(\ell _{e}(o_{m}-x) < L^{\prime } < L\). Hence, by continuity and due to the parallel link structure of the network, the unique equilibrium flow has equilibrium cost \(L^{\prime \prime } \in (L, L^{\prime })\). - 5.
For that, recall that the flow that minimizes the total latency is an equilibrium for the game where the latency on every edge

*e*has been changed from*ℓ*_{e}(*x*) to (*x**ℓ*_{e}(*x*))^{′}. But since all latency functions of edges in*X*_{2}are affine and share the same constant (recall*ℓ*_{i}(*x*) = (*x*/*s*_{i}) + 4), equal latencies with respect to (*x**ℓ*_{e}(*x*))^{′}is equivalent to equal latencies with respect to*ℓ*_{e}(*x*). Thus the optimal total latency is achieved when the edges in*X*_{2}are at equilibrium, sharing the same latency. - 6.
Note that such edges will remain non-optimal, because in later steps, they may only lose more flow.

- 7.
Note that such edges will remain non-optimal, because in later steps, they may only gain more flow.

- 8.
For completeness, one can give a detailed proof of this statement by induction on the decomposition of \(G^{\mathbf {\Gamma }_{1}}_{1}\) and \(G^{\mathbf {\Gamma }_{2}}_{2}\).

- 9.
Any (unused) path

*p*with an unused edge has \(\ell _{p}(o) \geq \max \nolimits _{p: o_{p} > 0} \ell _{p}(o)\). Moreover, the perceived cost o*f p*can only increase due to edge modifications. Since the modifications corresponding to the solution of (*O*_{γ}) make the perceived cost of all used paths equal to \(\max \nolimits _{p: o_{p} > 0} \ell _{p}(o)\),*o*becomes a Nash flow of \(\mathcal {G}^{\mathbf {\Gamma }}\). - 10.
Property (d) requires that \(\mathcal {D}\) should be closed under addition of constants, as long as the resulting function remains nonnegative.

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An extended abstract of this work [8] has appeared in the 11th Conference on Web and Internet Economics (WINE 2016). Research was supported by the project Algorithmic Game Theory, co-financed by the European Union (European Social Fund) and Greek national funds, through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework -Research Funding Program: THALES, investing in knowledge society through the European Social Fund, and by grant NSF CCF 1216103. The majority of this work was done while the second author was at the Department of Informatics and Telecommunications of the University of Athens.

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Fotakis, D., Kalimeris, D. & Lianeas, T. Improving Selfish Routing for Risk-Averse Players.
*Theory Comput Syst* **64, **339–370 (2020). https://doi.org/10.1007/s00224-019-09946-8

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### Keywords

- Selfish routing
- Uncertainty
- Risk-aversion
- Restricted tolls
- Price of anarchy