Computing equilibria for a service provider game with (Im)perfect information

Rene Beier, Artur Czumaj, Piotr Krysta, Berthold Vöcking
2006 ACM Transactions on Algorithms  
We study fundamental algorithmic questions concerning the complexity of market equilibria under perfect and imperfect information by means of a basic microeconomic game. Suppose a provider offers a service to a set of potential customers. Each customer has a particular demand of service and her behavior is determined by a utility function that is non-increasing in the sum of demands that are served by the provider. Classical game theory assumes complete information, that is, the provider has
more » ... l knowledge of the behavior of all customers. We present a complete characterization of the complexity of computing optimal pricing strategies and of computing best and worst equilibria in this model. Basically, we show that most of these problems are inapproximable in the worst case but admit a fully polynomial approximation scheme in the average case. Our average case analysis covers large classes of distributions for customer utilities. We generalize our analysis to robust equilibria in which players change their strategies only when this promises a significant utility improvement. A more realistic model assumes that the provider has incomplete information. Following the game theoretic framework of Bayesian games introduced by Harsanyi, 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 counterintuitively, we obtain an fully polynomial randomized approximation scheme 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. * A preliminary extended abstract of this paper entitled "Computing equilibria for congestion games with (im)perfect information" appeared in
doi:10.1145/1198513.1198524 fatcat:pqxoww4auvhfnbzjrks77taow4