Tight inefficiency bounds for perception-parameterized affine congestion games

Abstract

Congestion games constitute an important class of non-cooperative games which was introduced by Rosenthal in 1973. In recent years, several extensions of these games were proposed to incorporate aspects that are not captured by the standard model. Examples of such extensions include the incorporation of risk sensitive players, the modeling of altruistic player behavior and the imposition of taxes on the resources. These extensions were studied intensively with the goal to obtain a precise understanding of the inefficiency of equilibria of these games. In this paper, we introduce a new model of congestion games that captures these extensions (and additional ones) in a unifying way. The key 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. The above mentioned extensions reduce to special cases of our model by choosing the parameters ρ and σ accordingly. As in most related works, we concentrate on congestion games with affine latency functions here. Despite the fact that we deal with a more general class of congestion games, we manage to 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 perception-parameterized congestion 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 0.625 only (instead of 1 in the standard setting) the price of anarchy reduces from 2.5 to 2.155 and this is best possible.

Price of Anarchy in Stochastic Atomic Congestion Games with Affine Costs

“…For low values of p, compared to the case of a bounded number of players, where we noted above that the PoA is close to 1, its behavior is similar to the one in nonatomic congestion games, if one allows an arbitrarily large number of players. Moreover, using recent results by Kleer and Schäfer (2019), we obtain tight bounds for the PoS as a function of p. As illustrated in Fig. 1, in this case the function is increasing and smooth, except for one kink at p = 1/4.…”

“…Finally, Example 4.3 shows that the bound is tight for p in the upper region by considering a variant of the game de ned by Awerbuch et al (2013). For the upper region, Kleer and Schäfer (2019) give an alternative sequence of symmetric instances for which the PoA converges to the announced bound.…”

“…Our model with homogeneous players and a ne costs can be seen as a special case of the perception-parametrized a ne congestion games recently studied by Kleer and Schäfer (2019). These games feature two parameters ρ and σ which a ect respectively the cost perceived by the players and the cost considered by the central planner.…”

“…Indeed, on the one hand we show that their upper bound is valid on the larger interval p ∈ [p 1 , 1], and on the other hand we also obtain tight bounds outside this interval. Kleer and Schäfer (2019) also investigated the PoS, showing that PoS(p) ≤ 1+ p/(2 + p) for all p ≥ 1/4. Naturally, this upper bound on PoS is strictly smaller than our two upper bounds for PoA in the corresponding range p ∈ [ 1 4 , 1].…”

Topological Price of Anarchy Bounds for Clustering Games on Networks

“…In the case of equal-split distribution rules, our clustering games can also be modelled as exact potential (or congestion) games [33]. The inefficiency of pure Nash equilibria in these games has received a lot of attention, see, e.g., [12,13,1,8,25,27] and references therein. However, none of these results are applicable to the clustering games considered in this work.…”