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Utility maximisation and regret minimisation: A mixture of a generalisation Stephane Hess – Caspar Chorus Summary • Growing interest in decision rule heterogeneity across individual respondents but also across attributes • This paper brings these two issues together • Latent class approach which not only allows for different decision rules across classes, but also differences in the decision rules used across attributes within a given class • Focus on the specific case of random utility maximisation (RUM) and random regret minimisation (RRM) • Put forward the use of a generalised random regret minimisation (G-RRM) model within individual classes Data • SC route choice behaviour among commuters who travel from home to work by car, 9 tasks per person • 550 people sampled from an internet panel maintained in April 2011 Model structure • A general specification of a model allowing for different decision rules within a latent class framework is given by: = =1 , , , where LC n is the contribution to the likelihood function of the observed choices for respondent n • Individual classes use G-RRM models, allowing not just for mixtures between RUM and RRM, but also mixed RUM-RRM classes and classes with intermediate specifications • Base structure is a G-RRM model, with random regret given by: = ≠ ln + exp ∙ − + where is IID EV1, and is the regret-weight for attribute x m • Role of regret-weight: top panel shows the effect on the attribute regret function of a step-wise variation in γ, and the bottom panel shows the effect of a continuous change in γ ln( ϒ + exp(β m ·(x jm -x im ))) x jm -x im ϒ = 0 ϒ = 0 ϒ = 0.01 ϒ = 0.1 ϒ = 0.5 ϒ =1 ϒ = 0.25 x jm -x im ϒ ln( ϒ + exp(β m ·(x jm -x im ))) 1 Route A Route B Route C Average travel time (minutes) 45 60 75 Percentage of travel time in congestion (%) 10% 25% 40% Travel time variability (minutes) ±5 ±15 ±25 Travel costs (Euros) €12,5 €9 €5,5 YOUR CHOICE □ □ □ Findings and conclusions • Allowing for different decision rules across classes and differences in decision rules across attributes would lead to very large number of different possible combinations • Put forward GRRM mixture as an alternative • Allows optimal specification in terms of split between RUM and RRM within a given class to be revealed by the data during estimation, rather than needing to be imposed by the analyst • Initial findings are promising and show a rich pattern of taste heterogeneity and decision rule heterogeneity across respondents and attributes Model A: 2x RUM Model B: 2x RRM Model C: 1x RUM, 1xRRM Model D: 2x G-RRM Log-likelihood -2,431.59 -2,416.78 -2,412.92 -2,412.83 parameters 9 9 9 10 adj ρ2 0.3671 0.3709 0.3719 0.3717 BIC 4,920.11 4,890.49 4,882.77 4,888.91 est. rob. t-rat. est. rob. t-rat. est. rob. t-rat. est. rob. t-rat. β 1 (trav. time) -0.0559 -10.15 -0.1582 -5.44 -0.0559 -9.71 -0.0558 -9.77 β 1 (% cong.) -0.0025 -1.44 -0.0052 -1.35 -0.0030 -1.65 -0.0034 -1.91 β 1 (tt var) -0.0261 -6.33 -0.0510 -4.54 -0.0260 -6.20 -0.0259 -6.22 β 1 (cost ) -0.0437 -4.63 -0.0864 -4.36 -0.0404 -4.16 -0.0808 -4.16 β 2 (trav. time) -0.0146 -12.81 -0.0314 -11.23 -0.0310 -12.23 -0.0309 -12.19 β 2 (% cong.) -0.0131 -13.38 -0.0266 -12.37 -0.0275 -12.74 -0.0276 -12.71 β 2 (tt var) -0.0088 -7.92 -0.0182 -8.27 -0.0180 -7.88 -0.0180 -7.87 β 2 (cost ) -0.0725 -15.72 -0.1451 -14.05 -0.1495 -13.75 -0.1496 -13.76 δ 1 (trav. time) - inf (fixed a priori) + inf (fixed a priori) - inf (fixed a priori) -0.6918 -4.10 δ 1 (% cong.) - inf (fixed a priori) + inf (fixed a priori) - inf (fixed a priori) - inf (fixed) δ 1 (tt var) - inf (fixed a priori) + inf (fixed a priori) - inf (fixed a priori) - inf (fixed) δ 1 (cost ) - inf (fixed a priori) + inf (fixed a priori) - inf (fixed a priori) + inf (fixed) γ 1 (trav. time) 0 1 0 0.33 γ 1 (% cong.) 0 1 0 0.00 γ 1 (tt var) 0 1 0 0.00 γ 1 (cost ) 0 1 0 1.00 δ 2 (trav. time) - inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed) δ 2 (% cong.) - inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed) δ 2 (tt var) - inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed) δ 2 (cost ) - inf (fixed a priori) + inf (fixed a priori) + inf (fixed a priori) + inf (fixed) γ 2 (trav. time) 0 1 1 1.00 γ 2 (% cong.) 0 1 1 1.00 γ 2 (tt var) 0 1 1 1.00 γ 2 (cost ) 0 1 1 1.00 π 1 33.36% 31.54% 33.28% 33.36% π 2 66.64% 68.46% 66.72% 66.64%
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