Optimum and equilibrium in a transport problem with queue effects

TL;DR

This paper analyzes how citizens choose services in an urban area considering travel and queue times, comparing a global optimal distribution with individual equilibrium strategies using optimal transport and game theory.

Contribution

It introduces a combined framework using optimal mass transportation and game theory to study global and individual service choice optimization in urban settings.

Findings

Existence of a global optimal distribution minimizing total time.

Development of time-dependent strategies converging to equilibrium.

Application of optimal transport and game theory techniques.

Abstract

Consider a distribution of citizens in an urban area in which some services (supermarkets, post offices...) are present. Each citizen, in order to use a service, spends an amount of time which is due both to the travel time to the service and to the queue time waiting in the service. The choice of the service to be used is made by every citizen in order to be served more quickly. Two types of problems can be considered: a global optimization of the total time spent by the citizens of the whole city (we define a global optimum and we study it with techniques from optimal mass transportation) and an individual optimization, in which each citizen chooses the service trying to minimize just his own time expense (we define the concept of equilibrium and we study it with techniques from game theory). In this framework we are also able to exhibit two time-dependent strategies (based on the notions of prudence and memory respectively) which converge to the equilibrium.