## 4 Discussion and Conclusions  
The work reported in this paper provides an analytical framework for the study of contrarian behavior in repeated route choice situations, a topic that has been investigated mainly by means of empirical or simulation approaches thus far. Allowing a fraction of the network users to deliberately contradict the standard cost-minimizing route choice paradigm introduces into the modeling of day-to-day route choice a mix of behavioral attitudes which may have important implications for the long-term evolution of a traffic system. The main policy implication of this study is that explicit consideration of the existence of contrarian behavior in route choices may enhance the accuracy of traffic flow forecasts, thus providing a better basis for the design and evaluation of traffic management strategies. As indicated by the analytical and numerical findings presented in this paper, contrarian behavior may have significant impacts on key properties of traffic systems, such as the attractiveness and stability of network equilibria, as well as on system performance metrics such as average travel time. Another aspect of practical relevance of the study of contrarian route choice behavior relates in particular to the deployment of ATIS. The existence of individuals responding in a contrarian mode to the advice provided by these systems may act as a mitigating factor against the concentration of choices on the recommended routes, an adverse effect typically expected to occur as a result of ATIS deployment [42; 43; 44]. The ability of the traffic management agency to predict drivers' reaction to information, taking into account the existence of contrarian responses, may result in improved forecasts, ultimately enhancing the credibility of the system. Our findings clearly show that a balanced combination of direct and contrarian behaviors can protect the system from instabilities triggered by high sensitivities of travel costs to flows and of route choices to costs. This makes the results of our study particularly relevant to traffic systems characterized by significant levels of congestion and high ATIS market penetration, since under these circumstances an appropriate mix of direct and contrarian choice behaviors may become a key stabilizing factor. In addition, our results indicate that global network performance may benefit from a properly diversified composition of the traveling population in terms of direct and contrarian choice behaviors. Interestingly, these findings suggest that, from a traffic management standpoint, the fraction of travelers mak  
Figure 12: Time evolution of dynamical system for \(\alpha=0.5;\beta=0.5;\gamma\mu=10;\varphi=0.8;Z(0)=0.01\) (case of fourth-power link cost functions). ing contrarian route choices could be considered a control variable rather than a behavioral parameter. In practice, this could be achieved through properly designed policies for the allocation of personalized route recommendations in the context of ATIS [45], or through the definition of optimal routing strategies for connected and autonomous vehicles [46]. Inertia and memory of past travel experiences are important behavioral traits commonly found in real-world repeated route choice situations. Analysis of the effect of these characteristics in the proposed model suggests that they can counter the negative impact of low fractions of contrarian subjects on stability. This is due the smoothing effect induced by sufficiently small values of the parameters \(\alpha\) and \(\beta\) in the flow and cost updating mechanisms, which reduces the rate of adjustment governing the underlying dynamical process. It follows that the case of inertia-free and memoryless travelers shown in Tables 1 and 2, albeit unrealistic, provides an upper bound on the exposure of the system to the occurrence of instability.