@inproceedings{BorndoerferOmontSagnoletal.2012, author = {Bornd{\"o}rfer, Ralf and Omont, Bertrand and Sagnol, Guillaume and Swarat, Elmar}, title = {A Stackelberg game to optimize the distribution of controls in transportation networks}, volume = {105}, booktitle = {Proceedings of the 3rd International Conference on Game Theory for Networks (GAMENETS 2012)}, doi = {http://dx.doi.org/10.1007/978-3-642-35582-0_17}, pages = {224 -- 235}, year = {2012}, abstract = {We propose a game theoretic model for the spatial distribution of inspectors on a transportation network. The problem is to spread out the controls so as to enforce the payment of a transit toll. We formulate a linear program to find the control distribution which maximizes the expected toll revenue, and a mixed integer program for the problem of minimizing the number of evaders. Furthermore, we show that the problem of finding an optimal mixed strategy for a coalition of \$N\$ inspectors can be solved efficiently by a column generation procedure. Finally, we give experimental results from an application to the truck toll on German motorways.}, language = {en} } @article{BorndoerferBuwayaSagnoletal.2015, author = {Bornd{\"o}rfer, Ralf and Buwaya, Julia and Sagnol, Guillaume and Swarat, Elmar}, title = {Network spot-checking games: Theory and application to toll enforcing in transportation networks}, volume = {65}, journal = {Networks}, publisher = {Wiley Periodicals, Inc.}, doi = {10.1002/net.21596}, pages = {312 -- 328}, year = {2015}, abstract = {We introduce the class of spot-checking games (SC games). These games model problems where the goal is to distribute fare inspectors over a toll network. In an SC game, the pure strategies of network users correspond to paths in a graph, and the pure strategies of the inspectors are subset of arcs to be controlled. Although SC games are not zero-sum, we show that a Nash equilibrium can be computed by linear programming. The computation of a strong Stackelberg equilibrium (SSE) is more relevant for this problem and we give a mixed integer programming (MIP) formulation for this problem. We show that the computation of such an equilibrium is NP-hard. More generally, we prove that it is NP-hard to compute a SSE in a polymatrix game, even if the game is pairwise zero-sum. Then, we give some bounds on the price of spite, which measures how the payoff of the inspector degrades when committing to a Nash equilibrium. Finally, we report computational experiments on instances constructed from real data, for an application to the enforcement of a truck toll in Germany. These numerical results show the efficiency of the proposed methods, as well as the quality of the bounds derived in this article.}, language = {en} }