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A commuter departure-time model based on cumulative prospect theory

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Abstract

With a focus on planning of departure times during peak hours for commuters, an optimal arrival-time choice is derived using cumulative prospect theory. The model is able to explain the influence of behavioral characteristics on the choice of departure time. First, optimal solutions are derived explicitly for both early and late-arrival prospects. It is shown that the optimal solution is a function of a subjective measure, namely, the gain–loss ratio (GLR), indicating that the actual arrival time of a commuter depends on his or her attitude to the deviation between gains and losses. Some properties of the optimal solution and the GLR are discussed. These properties suggest that the more that the pleasure of gain exceeds the pain of loss, the greater the correlation between actual and preferred arrival times. Finally, a sensitivity analysis of the results is performed, and the use of the model is illustrated with a numerical example based on a skew-normal distribution.

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Notes

  1. In fact, in our paper we adopt weighting function form as in (2) and (3), which is proposed by Kahneman and Tversky (1979), because we are motivated by Barberis and Huang (2008) and assume that weighing function is consistent with their assumptions. As far as we know recent studies have abandoned this form of weighting function [Eqs. (2) and (3)], but we verify that if it is replaced by the weighting function with the form \(w(p)=e^{-(-\ln p)^\gamma } (0<\gamma <1)\) or \(w(p)=e^{-\delta (-\ln p)^\gamma } (0<\gamma <1,\delta >0)\) proposed by Prelec (1998), or replaced by the linear in log odds form \(w(p)=\frac{\delta p^{\gamma }}{\delta p^{\gamma }+(1-p)^\gamma }(0<\gamma <1,\delta >0)\) introduced by Goldstein and Einhorn (1987), our main results also will be valid.

  2. Cumulative prospect theory postulates a four dimensional pattern of risk-aversion: (i) Risk-aversion for gains of high probability; (ii) Risk-seeking for losses of high probability; (iii) Risk-seeking for gains of low probability; (iv) Risk-aversion for losses of low probability.

  3. It is well-defined when \(\alpha <2 \min (\delta ,\gamma )\) and \(\beta <2 \min (\delta ,\gamma )\). This condition is not necessary for some probability distributions,such as log-normal or normal distributions. The fact that this condition ensures that both integrals are finite is proved by Barberis and Huang (2008). In the setting of Barberis and Huang (2008), \(\alpha =\beta \) and \(\delta =\gamma \), and hence the condition required is that \(\alpha <2\delta \).

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Acknowledgements

The authors gratefully acknowledge financial support by the National Science Foundation of China (NSFC) (71371049, 71771051, 71701158) and Ph.D. Program Foundation of Chinese Ministry of Education CSC201706090142.

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  1. School of Economics and Management, Southeast University, Nanjing, 211189, Jiangsu, China

    Guang Yang & Xinwang Liu

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  1. Guang Yang
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  2. Xinwang Liu
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Correspondence to Xinwang Liu.

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Yang, G., Liu, X. A commuter departure-time model based on cumulative prospect theory. Math Meth Oper Res 87, 285–307 (2018). https://doi.org/10.1007/s00186-017-0619-8

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