Discrete Dynamics in Nature and Society

Volume 2015, Article ID 345795, 12 pages

http://dx.doi.org/10.1155/2015/345795

## Dynamics in Braess Paradox with Nonimpulsive Commuters

^{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

Received 16 May 2014; Accepted 27 July 2014

Academic Editor: Nikos I. Karachalios

Copyright © 2015 Arianna Dal Forno et al. This 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.

#### 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.

#### 1. Introduction

Assume that two different points of a network—an origin and a destination—are connected by two possible roads only. The Braess paradox states that, under specific conditions, adding a third road to the network decreases the efficiency of the network. This phenomenon is known in the transportation field and more general scientific literature (see [1–10]). Braess’ paradox occurs because commuters try to minimize their own travel time ignoring the effect 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 off using the new link, they contribute to increase the congestion for other commuters.

The theoretical literature of the Braess paradox is particularly productive, especially in transportation, communication, 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 influence of the flow on other links when certain component (node or link) is removed can be found in [13]. The 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. The 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. This literature provides evidence in strong support of the paradox in some cases (see [18]), while statistically significant, 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 affects 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 artificial behaviors emerged by mean of grounded theory, and uses ABM simulations. For a review on ABM with special focus on spacial interactions and networks see [23].

The paradox has also been presented in a dynamic game framework. 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. This approach has been used to consider choices both in small and in large groups in [28] and also when considering the introduction of a third choice which dramatically changes the dynamics in terms of complexity (as analyzed in [29]).

Analyzing the Braess paradox in terms of a ternary choice game shows new interesting dynamical characteristics that are investigated in [30], with a particular interest in the coexistence of several equilibria. Nevertheless, as illustrated in [22, 31], considering only impulsive commuters is not sufficient to describe the dynamics observed in experiments with human participants. Furthermore, as impulsive commuters consider only the sign and not the size of difference between costs, the population dynamics does not depend on the cost functions as long as the indifference point remains the same. In this paper, we consider a different behavior suggested by Amnon Rapoport (We are grateful to him for this helpful suggestion.) and which was used in [22] for artificial commuters. Although the full analysis of a homogenous—yet different—population seems to provide a limited contribution, it may be a step forward to analyze the aggregate decision behavior in social dilemmas. In fact, in order to analyze heterogeneous populations, it is important to well understand the dynamics properties of component behaviors. In particular, we consider commuters which are concerned not only about the sign difference in payoffs but also on the relative difference. This kind of behavior is similar to the one considered in [32], where the propensity to switch choice is modulated by the difference between payoffs. 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. Intuitively, this strategy seems to be more robust. The main result of the present work is that it is in fact robust. By contrast to the strategy considered in [30], which may lead to a stable cycle of any period, the strategy considered in this paper leads to a globally attracting fixed point. Depending on the parameters, this unique fixed point may be locally stable or unstable. However, the trajectories are convergent, in a few steps, to the fixed point. This result is rigorously proved for the case of a binary choice problem. For the ternary choice problem an analogous result is evidenced as well.

The plan of the work is as follows. In Section 2 the formalization of the dynamic model is reported. Section 2.1 includes the detailed description of the case with only two roads, which is represented by a* one-dimensional piecewise smooth continuous map*. We prove the existence of a unique fixed point, which may be locally stable or unstable in one partition. We prove that in any case it is globally attracting, that is, all the trajectories are ultimately converging to the fixed point. The case extended to three roads is considered in Section 3 and it is reduced to a* two-dimensional piecewise smooth discontinuous map*. We prove the existence of a unique fixed point, which may be locally attracting or not. We have only numerical evidence that also in this more general case, the fixed point is globally attracting. This can be rigorously proved for the particular cases in which the fixed point belongs to the boundaries of the domain of interest. The last section is devoted to the conclusions.

#### 2. The Dynamic Model with Two Roads

The Braess paradox can be illustrated by Figure 1 as follows. Assume there is a unitary mass of commuters from start () to end () and there are two roads: one passing through left () and the other through right (). The cost of each road is given by the sum of the time spent along each segment. In this network, segments - and - do not depend on traffic and they cost and , respectively. On the contrary, time spent along segments - and - is proportional to the number of travellers and therefore, they cost and , where is the fraction of the population using segment -. At the Nash equilibrium commuters are distributed in such a way that both roads (-- and --) have equal cos;, that is, a fraction of commuters will choose -- and a fraction will choose --. For example, assume as in [22] that and . Then, the symmetry of costs makes the population to split exactly into two equal fractions at the Nash equilibrium (), so that the travel time for both roads is. Now we assume that a very fast road is built connecting to at cost . As we will see in Section 3, at the new Nash equilibrium commuters are distributed in such a way that the three roads (--, ---, and --) have equal cost; that is, a fraction of commuters will choose --, a fraction will choose --, and the rest will choose ---. In the numerical example, this means that the entire population will choose --- (since in the other two roads the fraction of commuter is ) with a total travel cost of . According to [19], this is considered paradoxical as it shows how adding extra capacity to a network can reduce overall performance. More correctly, it is just counterintuitive and the mathematical reason is that there is a distinction between Nash equilibria and optima.