Research Article Dynamics in Braess Paradox with Nonimpulsive Commuters Arianna Dal Forno, 1 Ugo Merlone, 2 and Viktor Avrutin 3,4 1 Department of Economics and Statistics “Cognetti de Martiis,” University of Torino, 10153 Torino, Italy 2 Department of Psychology, University of Torino, 10124 Torino, Italy 3 DESP, University of Urbino “Carlo Bo,” 61026 Urbino, Italy 4 IST, University of Stuttgart, 70550 Stuttgart, Germany Correspondence should be addressed to Ugo Merlone; ugo.merlone@unito.it Received 16 May 2014; Accepted 27 July 2014 Academic Editor: Nikos I. Karachalios Copyright © 2015 Arianna Dal Forno et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 diference between costs. We analyze a dynamical version of the paradox with nonimpulsive commuters, who change road proportionally to the cost diference. When only two roads are available, we provide a rigorous proof of the existence of a unique fxed 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 fxed point exists, and its global attractivity is numerically evidenced, also when the fxed point is locally unstable. Our analysis adds a new insight in the understanding of dynamics in social dilemma. 1. Introduction Assume that two diferent points of a network—an origin and a destination—are connected by two possible roads only. Te Braess paradox states that, under specifc conditions, adding a third road to the network decreases the efciency of the network. Tis phenomenon is known in the transportation feld and more general scientifc literature (see [1–10]). Braess’ paradox occurs because commuters try to minimize their own travel time ignoring the efect of their decisions on other commuters on the network. As a result, the total travel time may increase following an expansion of the network; in fact, even if some commuters are better of using the new link, they contribute to increase the congestion for other commuters. Te theoretical literature of the Braess paradox is partic- ularly productive, especially in transportation, communica- tion, and computer science (far from being exhaustive, see, e.g., [10, 11]). Almost all the existing works have considered the basic network similar to the one presented in this paper, with the addition of a single link. Notably, [12] proposes a broader class of Braess graphs. A measure of the robustness of the dynamic network considering the infuence of the fow on other links when certain component (node or link) is removed can be found in [13]. Te empirical literature provides evidence in support of the paradox. For example, in [14] examples are reported that occurred on a modeled network of the city of Winnipeg, while [15] focus on a portion of the Boston road network. Te experimental works are interested in studying the occurrence of the paradox in a controlled setting (see, e.g., [16–20]), not only in basic but also in augmented networks. Tis literature provides evidence in strong support of the paradox in some cases (see [18]), while statistically signifcant, but weaker support in some other cases (see [16, 17, 19, 20]). A comparison of public versus private monitoring using the same participants is performed in [21] to investigate how the type of monitoring afects route choice. Interest in possible behaviors and composition of the population facing the basic network can be found in [22] which analyze the data gathered from the observation of an experiment with human participants, codes artifcial Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 345795, 12 pages http://dx.doi.org/10.1155/2015/345795