Pieter Kleer

Potential Function Minimizers of Combinatorial Congestion Games: Efficiency and Computation

We study the inefficiency and computation of pure Nash equilibria in unweighted congestion games, where the strategies of each player i are given implicitly by the binary vectors of a polytope

Path Deviations Outperform Approximate Stability in Heterogeneous Congestion Games

P_i

Tight inefficiency bounds for perception-parameterized affine congestion games

. Given these polytopes, a strategy profile naturally corresponds to an integral vector in the aggregation polytope PN = ∑i Pi. We identify two general properties of the aggregation polytope

Tight Inefficiency Bounds for Perception-Parameterized Affine Congestion Games

P_N

The Impact of Worst-Case Deviations in Non-Atomic Network Routing Games

that are sufficient for our results to go through, namely the integer decomposition property (IDP) and the box-totally dual integrality property (box-TDI). Intuitively, the IDP is needed to decompose a load profile in PN into a respective strategy profile of the players, and box-TDI ensures that the intersection of a polytope with an arbitrary integer box is an integral polytope. Examples of polytopal congestion games which satisfy IDP and box-TDI include common source network congestion games, symmetric totally unimodular congestion games, non-symmetric matroid congestion games and symmetric matroid intersection congestion games (in particular, r-arborescences and strongly base-orderable matroids). Our main contributions for polytopal congestion games satisfying IDP and box-TDI are as follows: We derive tight bounds on the price of stability for these games. This extends a result of Fotakis (2010) on the price of stability for symmetric network congestion games to the larger class of polytopal congestion games. Our bounds improve upon the ones for general polynomial congestion games obtained by Christodoulou and Gairing (2016). We show that pure Nash equilibria can be computed in strongly polynomial time for these games. To this aim, we generalize a recent aggregation/decomposition framework by Del Pia et al. (2017) for symmetric totally unimodular and non-symmetric matroid congestion games, both being a special case of our polytopal congestion games. Finally, we generalize and extend results on the computation of strong equilibria in bottleneck congestion games studied by Harks, Hoefer, Klimm and Skopalik (2013). In particular, we show that strong equilibria can be computed efficiently for symmetric totally unimodular bottleneck congestion games. In general, our results reveal that the combination of IDP and box-TDI gives rise to an efficient approach to compute a pure Nash equilibrium whose inefficiency is better than in general congestion games.

On the Switch Markov Chain for Strongly Stable Degree Sequences

We consider non-atomic network congestion games with heterogeneous players where the latencies of the paths are subject to some bounded deviations. This model encompasses several well-studied extensions of the classical Wardrop model which incorporate, for example, risk-aversion, altruism or travel time delays. Our main goal is to analyze the worst-case deterioration in social cost of a deviated Nash flow (i.e., for the perturbed latencies) with respect to an original Nash flow.

Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence.

We introduce a new model of congestion games that captures several extensions of the classical congestion game introduced by Rosenthal in 1973. The idea here is to parameterize both the perceived cost of each player and the social cost function of the system designer. Intuitively, each player perceives the load induced by the other players by an extent of ρ≥0, while the system designer estimates that each player perceives the load of all others by an extent of σ≥0. For specific choices of ρ and σ we obtain extensions such as altruistic player behavior, risk sensitive players and the imposition of taxes on the resources. We derive tight bounds on the price of anarchy and the price of stability for a large range of parameters. Our bounds provide a complete picture of the inefficiency of equilibria for these games. As a result, we obtain tight bounds on the price of anarchy and the price of stability for the above mentioned extensions. Our results also reveal how one should “design” the cost functions of the players in order to reduce the price of anarchy. Somewhat counterintuitively, if each player cares about all other players to the extent of ρ=0.625 (instead of 1 in the standard setting) the price of anarchy reduces from 2.5 to 2.155 and this is best possible.

Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences and 2-Class Joint Degree Matrices.

We introduce a new model of congestion games that captures several extensions of the classical congestion game introduced by Rosenthal in 1973. The idea here is to parameterize both the perceived cost of each player and the social cost function of the system designer. Intuitively, each player perceives the load induced by the other players by an extent of \(\rho \ge 0\), while the system designer estimates that each player perceives the load of all others by an extent of \(\sigma \ge 0\). For specific choices of \(\rho \) and \(\sigma \), we obtain extensions such as altruistic player behavior, risk sensitive players and the imposition of taxes on the resources. We derive tight bounds on the price of anarchy and the price of stability for a large range of parameters. Our bounds provide a complete picture of the inefficiency of equilibria for these games. As a result, we obtain tight bounds on the price of anarchy and the price of stability for the above mentioned extensions. Our results also reveal how one should “design” the cost functions of the players in order to reduce the price of anarchy. Somewhat counterintuitively, if each player cares about all other players to the extent of \(\rho = 0.625\) (instead of 1 in the standard setting) the price of anarchy reduces from 2.5 to 2.155 and this is best possible.

Rapid Mixing of the Switch Markov Chain for Strongly Stable Degree Sequences.

We introduce a unifying model to study the impact of worst-case latency deviations in non-atomic selfish routing games. In our model, latencies are subject to (bounded) deviations which are taken into account by the players. The quality deterioration caused by such deviations is assessed by the Deviation Ratio, i.e., the worst case ratio of the cost of a Nash flow with respect to deviated latencies and the cost of a Nash flow with respect to the unaltered latencies. This notion is inspired by the Price of Risk Aversion recently studied by Nikolova and Stier-Moses (Nikolova and Stier-Moses 2015). Here we generalize their model and results. In particular, we derive tight bounds on the Deviation Ratio for multi-commodity instances with a common source and arbitrary non-negative and non-decreasing latency functions. These bounds exhibit a linear dependency on the size of the network (besides other parameters). In contrast, we show that for general multi-commodity networks an exponential dependency is inevitable. We also improve recent smoothness results to bound the Price of Risk Aversion.