Vickrey's model of traffic congestion discretized

Vickrey's seminal analysis of urban traffic congestion assumes a continuum of commuters acting selfishly and a continuous strategy space. We propose a discrete version of his model that assumes a finite number of commuters and a discrete strategy space. We then present an algorithm for numerically computing a symmetric mixed-strategy equilibrium solution for the discrete model and compare it with the deterministic equilibrium solution for the continuous model due to Arnott et al. [Arnott, R., de Palma, A., Lindsey, R., 1990. Economics of a bottleneck. Journal of Urban Economics 27(1), 111-130]. We report significant discrepancies in travel costs and distributions of departure time between the two solutions that slowly decrease as the number of commuters increases. We then propose and exemplify two extensions of the discrete model that allow for an uncertain number of commuters and an alternative mode of transportation not subject to congestion.