Pricing of parking games with atomic players

• A parking competition game where vehicles compete for parking spaces.
• Equilibrium assignments are described as a system of nonlinear equations.
• Optimum price vectors of parking spaces are characterized as a union of polyhedrons.
• Valid price vectors are introduced to ensure system optimum.
• Robust price vectors are designed to achieve the best worst-case outcome.

This paper considers a parking competition game where a finite number of vehicles from different origins compete for the same number of parking spaces located at various places in a downtown area to minimize their own parking costs. If one vehicle reaches a desired vacant parking space before another vehicle, it will occupy the space and the other vehicle would have to search elsewhere. We first present a system of nonlinear equations to describe the equilibrium assignment of parking spaces to vehicles, and then discuss optimal pricing schemes that steer such parking competition to a system optimum assignment of parking spaces. These schemes are characterized by a union of polyhedrons. Given that the equilibrium state of parking competition is not unique, we further introduce a valid price vector to ensure that the parking competition outcome will always be system optimum. A sufficient condition is provided for the existence of such a valid price vector. Lastly, we seek for a robust price vector that yields the best worst-case outcome of the parking competition.