Path deviations outperform approximate stability in heterogeneous congestion games

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 perturbed Nash flow (i.e., for the perturbed latencies) with respect to an original Nash flow. We show that for homogeneous players perturbed Nash flows coincide with approximate Nash flows and derive tight bounds on their inefficiency. In contrast, we show that for heterogeneous populations this equivalence does not hold. We derive tight bounds on the inefficiency of both perturbed and approximate Nash flows for arbitrary player sensitivity distributions. Intuitively, our results suggest that the negative impact of path deviations (e.g., caused by risk-averse behavior or latency perturbations) is less severe than approximate stability (e.g., caused by limited responsiveness or bounded rationality). We also obtain a tight bound on the inefficiency of perturbed Nash flows for matroid congestion games and homogeneous populations if the path deviations can be decomposed into edge deviations. In particular, this provides a tight bound on the Price of Risk-Aversion for matroid congestion games

Price of Anarchy in Bernoulli Congestion Games with Affine Costs

We consider an atomic congestion game in which each player participates in the game with an exogenous and known probability $p_{i}\in[0,1]$, independently of everybody else, or stays out and incurs no cost. We first prove that the resulting game is potential. Then, we compute the parameterized price of anarchy to characterize the impact of demand uncertainty on the efficiency of selfish behavior. It turns out that the price of anarchy as a function of the maximum participation probability $p=\max_{i} p_{i}$ is a nondecreasing function. The worst case is attained when players have the same participation probabilities $p_{i}\equiv p$. For the case of affine costs, we provide an analytic expression for the parameterized price of anarchy as a function of $p$. This function is continuous on $(0,1]$, is equal to $4/3$ for $0<p\leq 1/4$, and increases towards $5/2$ when $p\to 1$. Our work can be interpreted as providing a continuous transition between the price of anarchy of nonatomic and atomic games, which are the extremes of the price of anarchy function we characterize. We show that these bounds are tight and are attained on routing games -- as opposed to general congestion games -- with purely linear costs (i.e., with no constant terms).Comment: 29 pages, 6 figure

Wardrop Equilibrium in Discrete-Time Selfish Routing with Time-Varying Bounded Delays

This paper presents a multi-commodity, discrete- time, distributed and non-cooperative routing algorithm, which is proved to converge to an equilibrium in the presence of heterogeneous, unknown, time-varying but bounded delays. Under mild assumptions on the latency functions which describe the cost associated to the network paths, two algorithms are proposed: the former assumes that each commodity relies only on measurements of the latencies associated to its own paths; the latter assumes that each commodity has (at least indirectly) access to the measures of the latencies of all the network paths. Both algorithms are proven to drive the system state to an invariant set which approximates and contains the Wardrop equilibrium, defined as a network state in which no traffic flow over the network paths can improve its routing unilaterally, with the latter achieving a better reconstruction of the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach

Different Policy Objectives of the Road Pricing Problem – a Game Theory Approach

Using game theory we investigate a new approach to formulate and solve optimal tolls with a focus on different policy objectives of the road authority. The aim is to gain more insight into determining optimal tolls as well as into the behavior of users after tolls have been imposed on the network. The problem of determining optimal tolls is stated and defined using utility maximization theory, including elastic demand on the travelers’ side and different objectives for the road authority. Game theory notions are adopted regarding different games and players, rules and outcomes of the games played between travelers on the one hand and the road authority on the other. Different game concepts (Cournot, Stackelberg and monopoly game) are mathematically formulated and the relationship between players, their payoff functions and rules of the games are defined for very simplistic cases. The games are solved for different scenarios and different objectives for the road authority, using the Nash equilibrium concept. Using the Stackelberg game concept as being most realistic for road pricing, a few experiments are presented illustrating the optimal toll design problem subject to different pricing policies considering different objectives of the road authority. Results show different outcomes both in terms of optimal tolls as well as in payoffs for travelers. There exist multiple optimal solutions and objective function may have a non- continuous shape. The main contribution is the two-level separation between of the users from the road authority in terms of their objectives and influences.