Dynamics in Braess Paradox with Nonimpulsive Commuters

Arianna Dal Forno,Viktor Avrutin,Ugo Merlone

doi:10.1155/2015/345795

Arianna Dal Forno, Viktor Avrutin + Show 1 more

Open Access

https://doi.org/10.1155/2015/345795

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Abstract

In Braess paradox the addiction of an extra resource creates a social dilemma in which the individual rationality leads to collective irrationality. In the literature, the dynamics has been analyzed when considering impulsive commuters, i.e., those who switch choice regardless of the actual difference between costs. We analyze a dynamical version of the paradox with nonimpulsive commuters, who change road proportionally to the cost difference. When only two roads are available, we provide a rigorous proof of the existence of a unique fixed point showing that it is globally attracting even if locally unstable. When a new road is added the system becomes discontinuous and two-dimensional. We prove that still a unique fixed point exists, and its global attractivity is numerically evidenced, also when the fixed point is locally unstable. Our analysis adds a new insight in the understanding of dynamics in social dilemma.

Highlights

Assume that two different points of a network—an origin and a destination—are connected by two possible roads only
The total travel time may increase following an expansion of the network; even if some commuters are better off using the new link, they contribute to increase the congestion for other commuters
In this paper we have analyzed a dynamical version of the Braess paradox with nonimpulsive commuters, who change road proportionally to the cost difference

Summary

Introduction

Assume that two different points of a network—an origin and a destination—are connected by two possible roads only. A discrete time dynamic population model with social externalities and two available choices is studied in [24] simulating an adaptive adjustment process, and in particular with impulsive commuters in [25]. Following the clinical psychology literature (see [26]), impulsive commuters have been introduced in the analysis of binary choices with externalities in [25, 27] when considering commuters whose switching rate only depends on the sign of the difference between payoffs, no matter how much they differ. Unlike the impulsive adjustment process, which makes the commuters change their choice as soon as a better choice occurs regardless of the difference, we consider a decision strategy which prescribes a change to the best available choice proportionally to the reported difference This strategy seems to be more robust.

The Dynamic Model with Two Roads

The Dynamics in the Expanded Network with Three Roads

Conclusion