Bicretieria Optimization in Routing Games

Two important metrics for measuring the quality of routing paths are the maximum edge con-gestion C and maximum path length D. Here, we study bicriteria in routing games where eachplayer i selfishly selects a path that simultaneously minimizes its maximum edge congestion Ciand path length Di. We study the stability and price of anarchy of two bicriteria games:• Max games, where the social cost is max(C, D) and the player cost is max(Ci, Di). Weprove that max games are stable and convergent under best-response dynamics, and thatthe price of anarchy is bounded above by the maximum path length in the players’ strategysets. We also show that this bound is tight in worst-case scenarios. • Sum games, where the social cost is C+D and the player cost is Ci+Di. For sum games, wefirst show the negative result that there are game instances that have no Nash-equilibria.Therefore, we examine an approximate game called the sum-bucket game that is alwaysconvergent (and therefore stable). We show that the price of anarchy in sum-bucket gamesis bounded above by C∗ ·D∗/(C∗+D∗) (with a poly-log factor), where C∗ and D∗ are theoptimal coordinated congestion and path length. Thus, the sum-bucket game has typicallysuperior price of anarchy bounds than the max game. In fact, when either C∗ or D∗ issmall (e.g. constant) the social cost of the Nash-equilibria is very close to the coordinatedoptimal C∗+D∗ (within a poly-log factor). We also show that the price of anarchy boundis tight for cases where both C∗ and D∗ are large.