# Risk Averse Second Best Toll Pricing

## Abstract

Existing second best toll pricing (SBTP) models determine optimal tolls of a subset of links in a transportation network by minimizing certain system objective, while the traffic flow pattern is assumed to follow user equilibrium (UE). We show in this paper that such toll design approach is risk prone, which tries to optimize for the best-case scenario, if the UE problem has multiple solutions. Accordingly, we propose a risk averse SBTP approach aiming to optimize for the worst-case scenario, which can be formulated as a *min-max* problem. We establish a general solution existence condition for the risk averse model and discuss in detail that such a condition may not be always satisfied in reality. In case a solution does not exist, it is possible to replace the exact UE solution set by a set of approximate solutions. This replacement guarantees the solution existence of the risk averse model. We then develop a scheme such that the solution set of an affine UE can be explicitly expressed. Using this explicit representation, an improved simplex method can be adopted to solve the risk averse SBTP model.

## Keywords

Risk Averse Nonlinear Complementarity Problem User Equilibrium Lower Level Problem Optimal Toll## Preview

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## Notes

### Acknowledgments

The authors would like to thank the four anonymous referees for their insightful comments and helpful suggestions on an earlier version of this paper. The first author also appreciates the insightful discussions with Dr. Paul Tseng at the University of Washington on the fortified descent simplex method. This work is supported in part by Air Force Office of Scientific Research Grant FA9550-07-1-0389, and National Science Foundation Grants DMI- 052 1953, DMS-0427689 and IIS-0511905.

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