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Abstract

We study algorithmic questions concerning a basic microeconomic congestion game in which there is a single provider that offers a service to a set of potential customers. Each customer has a particular demand of service and the behavior of the customers is determined by utility functions that are non-increasing in the congestion. Customers decide whether to join or leave the service based on the experienced congestion and the offered prices. Following standard game theory, we assume each customer behaves in the most rational way. If the prices of service are fixed, then such a customer behavior leads to a pure, not necessarily unique Nash equilibrium among the customers. In order to evaluate marketing strategies, the service provider is interested in estimating its revenue under the best and worst customer equilibria. We study the complexity of this problem under different models of information available to the provider. % \begin{itemize} \item We first consider the classical model in which the provider has perfect knowledge of the behavior of all customers. We present a complete characterization of the \emph{complexity of computing optimal pricing strategies} and of \emph{computing best and worst equilibria}. Basically, we show that most of these problems are inapproximable in the worst case but admit an ``average-case FPAS.'' Our average case analysis covers general distributions for customer demands and utility thresholds. We generalize our analysis to robust equilibria in which players change their strategies only when this promises a significant utility improvement. % \item We extend our analysis to a more realistic model in which the provider has \emph{incomplete information}. Following the game theoretic framework of Bayesian games introduced by Harsanyi, we assume that the provider is aware of probability distributions describing the behavior of the customers and aims at estimating its expected revenue under best and worst equilibria. Somewhat counterintuitive, we obtain an FPRAS for the equilibria problem in the model with imperfect information although the problem with perfect information is inapproximable under the worst case measures. In particular, the worst case complexity of the considered stochastic equilibria problems increases with the precision of the available knowledge. \end{itemize}