Katrine Hjorth (DTU) Stefan Flügel, Farideh Ramjerdi (TØI) An application of cumulative prospect theory to travel time variability Sixth workshop on discrete choice models at EPFL August 19-21, 2010 Page 1
Travel time variability (TTV) is increasingly acknowledged to be an important concern for both the users and the providers of transport services. The correct measurement of the perception of reliability and its value for the users are important in the design of transport policies It has been the getting more attention in research. The focus of this paper is on the the perception of reliability and its value for the users Page 2
Outline Travel time variability (TTV) Risk and TTV Rank Dependent Expected Utility (RDEU) Data Theoretical model Estimation results Summary / further work 26/08/10 Page 3 Institute of Transport Economics
Two competing approaches exist on travel time variability TTV Mean-Variance approach Scheduling approach Page 4
Risk and TTV Travel time variability is associated with risk Risk and its perception should be reflected in the valuation of reliability Bonsal (2004) de Palma, et al (2008) de Lapport (2009) Page 5
Rank Dependent Expected Utility (RDEU) Expected value maximisation (EVM) RDEU v(.) is a probability weighting function is a value function defined with respect to a reference point, typically concave for gains and convex for losses Page 6
Rank Dependent Expected Utility (RDEU) Descriptive theories of decision under risk depart from EVM in three essential ways: 1. the transformation of outcomes: Different functional forms to capture concave for gains and convex for losses 2. the transformation of probabilities: Examples: power, Quiggin, Perlec 3. the composition rule that combines the two transformations. Page 7
Data The New Norwegian Value of Time Study (2009) Large scale national study Self-administrated web SP survey Modes Long distance: Car, Rail, Bus and Air Short distance: Car and Public transportation (PT) 26/08/10 Page 8 Institute of Transport Economics
Presentation of TTV 2 alternative trips that differ in cost and a distinct travel time distribution Alternatives pivoted around a reported reference trip 6 choices for each respondent Page 9
Theoretical model Travel time distribution and Cost Relative to the reference travel time: negative values are interpreted as gains and positive values as losses Respondents choose the alternative that generates highest value Alternative 1 is chosen whenever Page 10
Following Tversky and Kahneman (1992)
Value functions Where :
In case of diminishing sensitivity Weights for gains are Weights for losses are If Loss aversion for If Loss aversion for
Decision weight Probability weights: Where and are weights, and Weighting functions: Prelec3, Prelec2, TK2, TK1, and No weights
PRELEC3, Prelec (1998) 3 parameters (convex, concave S-shaped, or inversely S-shaped) Page 15
PRELEC2 2 parameters (convex, concave S-shaped, or inversely S-shaped) Same shape for loss and gain Page 16
TK2 (Tvresky and Kahneman,1992) 2 parameters (Inversely S-shaped if and S-shaped if ) Page 17
TK1 As TK2, but the same shape for gain and loss Page 18
The model where Page 19
Assumptions Error term logistic with scale parameter independent across choices, including choices within individual binary logit Normalize cost parameter results from different models) to 1 (to directly compare The full model not identifies or very poorly identified The interactions of and or identification of Hence we imposed the restriction Page 20
Summary of statistics of the sample Segment Description Sample size Reference cost Reference time in min (Min,Mean,Max) in NOK (Min,Mean,Max) Car short Car trips less than 100km. 1597 (10; 26,7;195) (8,4; 49,0; 396) PT short Public transport trips less than 100km 194 (10; 28,5; 90) (10; 33,1; 144) Car long Car trips longer than 100km. 603 (60; 184,9; 1045) (70; 457,1; 4430) Air Plane trips 809 (80; 192,5; 600) (150; 1289,1; 7500) Bus long Bus trips longer than 100km 443 (15; 247,9; 1439) (50; 279,6; 5000) Train long Train trips longer than 100km 551 (40; 252,5; 1319) (62; 336,7; 3598) Page 21
Distribution of outcomes on evaluation of w Car long Air Bus long Train long Car short PT short Gains Losses Gains Losses Gains Losses Gains Losses Gains Losses Gains Losses (0.2) - (0) 6644 6945 8441 9334 4690 5104 5820 6331 15747 17943 1928 2192 (0.4) - (0) 170 18 (0.4) - (0.2) 4726 5367 5895 6804 3223 3952 4013 4851 11257 12981 1385 1598 (0.6) - (0.4) 1498 1230 2022 1627 1089 918 1381 1077 3999 3187 492 404 (0.8) - (0) 104 9 (0.8) - (0.2) 304 586 407 816 217 434 285 563 690 1601 85 174 (0.8) - (0.6) 43 616 327 1416 30 781 49 917 1140 2484 107 330 (1) - (0.8) 257 446 410 1298 200 589 266 761 719 2281 76 309 Total (gains,losses) 13472 15190 17502 21295 9449 11778 11814 14500 33826 40477 4100 5007 Total 28662 38797 21227 26314 74303 9107 Page 22
Estimation results, Prelec3 Page 23
Estimation results, Prelec2 Page 24
Estimation results, TK2 Page 25
Estimation results, TK1 Page 26
Estimation results, No weights Page 27
Estimated ration of to The ratio is between 1.4-10.2. Ignoring PT the ratio is 1.4-5.9 (Horowitz & McConnell, 2002) Lowest ratio is for TK2 and highest is for no weight Car short PT short Car long Air Bus long Train long Prelec3 2.2 *** 4.4 ** 1.9 *** 1.5 ** 1.2 2.1 *** Prelec2 2.6 *** 4.4 *** 1.8 *** 1.7 *** 1.9 *** 2.4 *** TK2 1.4 *** 0.8 1.8 *** 1.4 ** 1.0 1.1 TK1 2.9 *** 8.9 *** 2.3 *** 1.6 *** 3.5 *** 4.6 *** No weighting 2.8 *** 10.2 *** 2.6 *** 1.8 *** 4.3 *** 5.9 *** Page 28
Value function for time, Perlec3 Page 29
Value function for cost, Perlec3 Page 30
Value function for time, TK2 Page 31
Value function for cost, TK2 Page 32
Value function for time, No weight Page 33
Value function for cost, No weight Page 34
Weight function for time for gains, Prelec3 Page 35
Weight function for losses, Prelec3 Page 36
Weight function, Prelec2 Page 37
Value function for gains, TK2 Page 38
Value function for losses, TK2 Page 39
Value function, TK Page 40
Conclusions is the curvature parameter for value function for time diminishing sensitivity to time changes. is lowest for Prelec3 &2 more convex(concave) functions in the gain (loss) regions and are almost invariant across weighting schemes is always smaller than and similar in size Insufficient data to identify the high end of weigh functions Page 41
Conclusions Overall results consistent across the 6 databases Significant loss aversion with respect to travel time (1.4-5.9) No significant loss aversion with respect to cost The w(.) produce significant behavioral improvements Considerable differences in the shape of probability weighting function. Low probabilities are over-weighted for losses The problem with our data set. Majority of our observations (80%) involve w evaluated at 0.2 and 0.4. Hence the weigh function is estimated based on low probability data Page 42
Conclusions It is possible to estimate VTT by this approach. The estimated VTT should capture the value of travel time variability In the first experiment in the Norwegian study time has only one value, i.e., there is no variability of time. The difference between these two VTT should be related to the value of travel time variability. Page 43