Modeling the Combined Effect of Travelers’ Contrarian Behavior, Learning and Inertia on the Day-to-Day Dynamics of Route Choice
Abstract
:1. Introduction
- Introducing inertia into the dynamic process model representing the day-to-day evolution of the network state under mixed direct and contrarian route choices;
- Providing a more general statement of the fixed point stability conditions through the analysis of the Jacobian matrix of the resulting two-dimensional nonlinear dynamical system;
- Analyzing model behavior in the case of nonlinear link cost functions.
2. Method
2.1. Model Formulation
- (i = 1, 2) is the mean perceived cost of route i at the start of day t;
- (i = 1, 2) is the average flow-dependent cost actually experienced by users on route i;
- is the probability that a user who reconsiders their previous day’s choice will choose link 1 on day t, expressed as a function of the difference between day t’s perceived costs;
- () is a parameter quantifying the extent to which the most recent travel experience (“yesterday’s trip”) contributes to the formation of the cost perceived by users at the start of day t;
- () is a parameter representing the fraction of travelers who actually reconsider their previous day’s choice.
2.2. Fixed Point Stability Analysis
2.2.1. Fixed Point Stability in the Case of Linear Link Cost Functions
2.2.2. Fixed Point Stability in the Case of Fourth-Power Link Cost Functions
2.2.3. Discussion of Fixed Point Stability Results
- An adequately balanced mix of direct and contrarian subjects within the traveling population is conducive to fixed point stability, whereas strongly homogeneous route choice behaviors tend to trigger the occurrence of instabilities;
- The range of values ensuring fixed point stability depends on the inertia (α) and memory depth (β) parameters, as well as on the degree of sensitivity of costs to flows (γ) and of route choices to costs (μ);
- The stability regions defined by conditions (54) and (64) are not affected by a swap of the values of α and β or by a swap of the values of γ and μ;
- The width of the stability regions decreases as γ and/or μ increase. Unlike the latter parameters, which appear on both sides of expressions (54) and (64), α and β affect only the lower bound of the stability region. More specifically, it is easy to check that the derivatives of the left-hand sides of (54) and (64) with respect to α and β have a positive sign. In light of expressions (1)–(3), this suggests that appropriately reduced day-to-day cost and flow updating rates have the potential to offset the destabilizing effects of steeply increasing cost–flow functions and highly cost-sensitive route choice behaviors;
- Conditions (55), (56) and (65), (66) indicate that the composition of the traveling population in terms of direct/contrarian choice behaviors may become irrelevant when stability of the fixed point is already ensured by sufficiently small values of the other model parameters;
- Comparison of expressions (54) and (64) suggests that the stability region for fourth-power cost functions is twice as large as for linear cost functions. This circumstance can be explained by observing that, at the fixed point, the derivative of the fourth-power function is equal to , which is exactly half of the derivative of the linear cost function appearing in (54).
3. Numerical Examples
3.1. Linear Link Cost Functions
3.2. Fourth-Power Link Cost Functions
4. Discussion and Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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α | β | φ Min | φ Max | |
---|---|---|---|---|
γμ = 1 | 0.1 | 0.1 | 0 | 1 |
0.5 | 0.5 | 0 | 1 | |
0.75 | 0.75 | 0 | 1 | |
0.9 | 0.9 | 0 | 1 | |
1 * | 1 * | 0 | 1 | |
γμ = 2.5 | 0.1 | 0.1 | 0 | 0.9 |
0.5 | 0.5 | 0 | 0.9 | |
0.75 | 0.75 | 0 | 0.9 | |
0.9 | 0.9 | 0 | 0.9 | |
1 * | 1 * | 0.1 | 0.9 | |
γμ = 5 | 0.1 | 0.1 | 0 | 0.7 |
0.5 | 0.5 | 0 | 0.7 | |
0.75 | 0.75 | 0 | 0.7 | |
0.9 | 0.9 | 0.2012 | 0.7 | |
1 * | 1 * | 0.3 | 0.7 | |
γμ = 10 | 0.1 | 0.1 | 0 | 0.6 |
0.5 | 0.5 | 0 | 0.6 | |
0.75 | 0.75 | 0.2222 | 0.6 | |
0.9 | 0.9 | 0.3506 | 0.6 | |
1 * | 1 * | 0.4 | 0.6 | |
γμ = 15 | 0.1 | 0.1 | 0 | 0.5667 |
0.5 | 0.5 | 0 | 0.5667 | |
0.75 | 0.75 | 0.3148 | 0.5667 | |
0.9 | 0.9 | 0.4004 | 0.5667 | |
1 * | 1 * | 0.4333 | 0.5667 |
α | β | φ Min | φ Max | |
---|---|---|---|---|
γμ = 1 | 0.1 | 0.1 | 0 | 1 |
0.5 | 0.5 | 0 | 1 | |
0.75 | 0.75 | 0 | 1 | |
0.9 | 0.9 | 0 | 1 | |
1 * | 1 * | 0 | 1 | |
γμ = 2.5 | 0.1 | 0.1 | 0 | 1 ** |
0.5 | 0.5 | 0 | 1 ** | |
0.75 | 0.75 | 0 | 1 ** | |
0.9 | 0.9 | 0 | 1 ** | |
1 * | 1 * | 0 ** | 1 ** | |
γμ = 5 | 0.1 | 0.1 | 0 | 0.9 ** |
0.5 | 0.5 | 0 | 0.9 ** | |
0.75 | 0.75 | 0 | 0.9 ** | |
0.9 | 0.9 | 0 ** | 0.9 ** | |
1 * | 1 * | 0.1 ** | 0.9 ** | |
γμ = 10 | 0.1 | 0.1 | 0 | 0.7 ** |
0.5 | 0.5 | 0 | 0.7 ** | |
0.75 | 0.75 | 0 ** | 0.7 ** | |
0.9 | 0.9 | 0.2012 ** | 0.7 ** | |
1 * | 1 * | 0.3 ** | 0.7 ** | |
γμ = 15 | 0.1 | 0.1 | 0 | 0.6333 ** |
0.5 | 0.5 | 0 | 0.6333 ** | |
0.75 | 0.75 | 0.1296 ** | 0.6333 ** | |
0.9 | 0.9 | 0.3008 ** | 0.6333 ** | |
1 * | 1 * | 0.3667 ** | 0.6333 ** |
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Meneguzzer, C. Modeling the Combined Effect of Travelers’ Contrarian Behavior, Learning and Inertia on the Day-to-Day Dynamics of Route Choice. Appl. Sci. 2023, 13, 3294. https://doi.org/10.3390/app13053294
Meneguzzer C. Modeling the Combined Effect of Travelers’ Contrarian Behavior, Learning and Inertia on the Day-to-Day Dynamics of Route Choice. Applied Sciences. 2023; 13(5):3294. https://doi.org/10.3390/app13053294
Chicago/Turabian StyleMeneguzzer, Claudio. 2023. "Modeling the Combined Effect of Travelers’ Contrarian Behavior, Learning and Inertia on the Day-to-Day Dynamics of Route Choice" Applied Sciences 13, no. 5: 3294. https://doi.org/10.3390/app13053294