The negation of the Braess paradox as demand increases: The wisdom of crowds in transportation networks

In the well-known Braess paradox [D. Braess, Unternehmenforschung 12, 258 (1968)], the addition of a new route in a specific congested transportation network made all the travelers worse off in terms of their individual travel cost (time). In this paper, we consider the hypothesis that, in congested networks, the Braess Paradox may “disappear” under higher demands, and we prove this hypothesis by deriving a formula that provides the increase in demand that will guarantee that the addition of that new route will no longer increase travel cost since the new path will no longer be used. This result is established for any network in which the Braess Paradox originally occurs. This suggests that, in the case of congested, noncooperative networks, of which transportation networks are a prime example, a higher demand will negate the counterintuitive phenomenon known as the Braess Paradox. At the same time, this result demonstrates that extreme caution should be taken in the design of network infrastructure, including transportation networks, since at higher demands, new routes/pathways may not even be used! Introduction. – Congestion is a fundamental problem in a variety of network systems, ranging from urban transportation networks to electric power generation and distribution networks and the Internet [1]-[16]. Congestion leads to increases in travel time, wear and tear on our infrastructure, higher emissions due to vehicular idling, as well as to losses in productivity. Congested networks are flow-dependent with induced flows being the result of the behavior of the users of the particular network. Historically, there have been two principles of travel behavior, dating to Wardrop [8]-[10], corresponding to user-optimizing (U-O) behavior, in which travelers select their optimal routes of travel individually and unilaterally, leading to an equilibrium, and systemoptimizing (S-O) behavior, in which a central controller routes or assigns the flows to particular paths in the network so that the total cost is minimized. The Braess Paradox (cf. [1] and [11] for the translation of the article from German to English) in which the cost on used paths increases for all after the addition of a new route occurs only under user-optimizing or selfish behavior. Such behavior, however, is characteristic of commuting behavior, decentralized routing on the Internet, as well as the behavior of a spectrum of network systems, including electric power generation and distribution networks, in which decision-makers act independently and noncooperatively [2]-[5]. The recognition of the existence of the Braess paradox has led, in practice, to major policy decisions such as road closures in Seoul, Korea; in Stuttgart, Germany, as well as in New York City [6]. Interestingly, it was established that, for the specific Braess network [1], the paradox no longer occurred as the demand for travel increased [5]-[7]. This leads us to the hypothesis that, under a higher demand, the Braess Paradox is negated in that the new route, which resulted in increased travel time at a particular demand, will no longer be used. We establish this result through the derivation of a formula that is applicable to any network of general topology in which the Braess paradox originally occurs. For definiteness, we consider transportation networks in which the travel cost on each link is an increasing linear function of the flows (volume of traffic on the links) and we do not limit the analysis to separable functions as was the case in the original Braess Paradox network(s). We first briefly review the U-O transportation model, referred to, henceforth, as the traffic network equilibrium