Practical applications of EUT and non-EUT

EUT and non-EUT methods have been widely adopted in transportation research, especially during the last decade. Examples of their applications can be found in equilibrium modelling, valuation of travel time or attitudes towards travel time uncertainty and experience/learning in travel choices (Kemel and Paraschiv, 2013). The majority of these applications are based on stated preference surveys because of the level of control that they offer to the analyst. In experimental economics it is common to adopt an intermediate approach between stated and revealed preferences, i.e. incentivised laboratory experiments. Nevertheless, transport choices are more complex since they do not involve only money, but other characteristics as well (e.g.

time or comfort).

For risky choices in transport applications, EUT was often embedded with random utility models (RUM) creating an ad hoc modelling approach. Integrating these two theories provides the possibility to represent simultaneously two distinct sources of uncertainty: the uncertainty

of the decision-maker when presented with various possible outcomes (EUT) and the uncertainty of the researcher when observing the choice process of the decision-makers (RUM). The integrated RUM-EUT model is formed by adding an RUM-related error term to the value term of the EUT model of equation 4.41:

𝑢(𝑠^𝑛) = ∑ 𝑝_𝑘^𝑛[𝑣(𝑠_𝑘^𝑛)

𝐾

𝑘=1

+ 𝜀^𝑛] (4.47)

Such integration is governed by some important conceptual shortcomings that have been addressed by the framework of Liu and Polak (2007). This framework allows the explicit modelling of attitudes towards risk independent of the conventional RUM-like tastes.

It is assumed that 𝛽_𝑖 = {𝛽_𝑖𝑟;1 ≤ 𝑟 ≤ 𝑅} is a set of taste parameters associated with the vector of observable attributes 𝐴 = {𝑎_𝑚1 ≤ 𝑚 ≤ 𝑀} that affect the choice behaviour. Taste parameters are individual-specific and they are characteristic of a riskless choice. Each individual 𝑖 attaches a scalar value 𝑣_𝑖𝑘^𝑛 to the k^th outcome of prospect 𝑠^𝑛 which is a function of 𝛽_𝑖, i.e. 𝑣_𝑖𝑘^𝑛 = 𝑓(𝑠_𝑘^𝑛, 𝛽_𝑖). The scalar value for prospect 𝑠^𝑛 (𝑢_𝑖^𝑛) is a function of 𝑣_𝑖𝑘^𝑛, the probabilities associated with prospect outcomes 𝑝^𝑛 and a set of extra parameters 𝜑_𝑖 = {𝜑_𝑖𝑟; 1 ≤ 𝑟 ≤ 𝑇} that are representative of decision-making under risk. Hence, it can be denoted as 𝑢_𝑖^𝑛 = 𝑔(𝑣_𝑖1^𝑛, … , 𝑣_𝑖𝐾^𝑛𝑝₁^𝑛, … , 𝑝_𝐾^𝑛, 𝜑_𝑖).

Allowing some relaxation in EUT, each outcome 𝑠_𝑘^𝑛 instead of being evaluated based on its riskless value 𝑣_𝑖𝑘^𝑛 can be evaluated based on a non-linear function z(𝑣_𝑖𝑘^𝑛, 𝜑_𝑖). When this function is concave, then the expected utility is lower than the expected value and hence the individual is risk-averse. On the other hand, when this function is convex the individual is risk prone. A typical non-linear transformation of the value function is when the individuals are characterised by constant absolute risk aversion (CARA), i.e. the risk attitudes for a particular prospect are not affected by the value of the outcomes involved in this prospect. The functional form of CARA is 𝑧(𝑥) = (1 − 𝑒^−𝑎𝑥)/𝑎 where a positive 𝑎 expresses risk proneness while a negative 𝑎 expresses risk aversion.

Integrating EUT and RUM, the scalar value for prospect 𝑠^𝑛 can now be transformed to:

𝑢_𝑖^𝑛 = ∑ 𝑧(𝑣_𝑖𝑘^𝑛, 𝜑_𝑖)𝑝_𝑘^𝑛

𝐾

𝑘=1

+ 𝜖_𝑖^𝑛 (4.48)

where 𝜖_𝑖^𝑛 is the unobservable component⁴⁸ of the utility associated with prospect 𝑠^𝑛 and 𝑣_𝑖𝑘^𝑛 = 𝑤_𝑖𝑘^𝑛 + 𝜂_𝑖𝑘^𝑛 is the combination of observable and unobservable components of the value function associated with outcome k.

In their majority, individuals are risk-averse and in a dynamic pricing context, they tend to choose a certain price over an uncertain one (Bonsall and Shires, 2005). Moreover, it has been found that the upper end of the price distribution disproportionally affects their choice. For example, EV drivers would avoid charging their vehicle during a specific period if there was a possibility of a very high cost.

The assessment of state-dependent prices, according to empirical evidence, strongly depends on individuals’ expectations (Lindsey, 2011). The judgement of dynamic prices is affected by historical prices as well as by prices charged for similar services or under similar circumstances. These reference-dependent effects are likely to decay for increasing experience with the price mechanism, as individuals become more familiar with gains and losses.

Potentially, this is the reason dynamic pricing has become acceptable for airline tickets and other travel-related services (e.g. dynamic tolls).

Prospect theory is suitable for modelling the response to price signals because it has been observed that choices depend on how the individual perceives the monetary transaction: in a positive or negative way. For example, paying for out-of-home charging might be considered as a loss, due to the fact that this service is free for most of the respondents at the time being.

4.3.4 Empirical estimation

For the booking game of the EV-PLACE survey, as it was presented in Chapter 3, respondents had to choose between one certain and one risky alternative according to the activities and the timings of the home-based tour that they have selected earlier. The background characteristics like the location of the charging post were similar to the charging game. Also, the charging attributes, i.e. charging duration, CISDE and walking time to the destination, were randomly selected from the design levels of the previous SP exercise and were fixed across the choice situations.

Respondents are presented with a deterministic price (“BOOK NOW”, the safe option) and with a random price (“BOOK LATER”, the risky choice). Specifically, 𝐶_𝑁 is the deterministic price of the “booking now” choice, 𝐶_𝐷 is the decreased potential price of the risky choice while

48 Note that the error term 𝜖_𝑖^𝑛 applies now to the whole prospect and not to the expected term as in equation 4.47.

𝐶_𝐼 is the increased potential price of the risky choice. If the integrated RUM-EUT is treated as the basic model specification, then the utility functions for the two booking alternatives, for the riskless form of 4.48, are given by:

𝑢_𝑖^𝑁𝑂𝑊 = 𝐴𝑆𝐶^𝑁𝑂𝑊+ 𝛽_𝐶𝐶_𝑁+ 𝜀_𝑖 (4.49)

𝑢_𝑖^{𝐿𝐴𝑇𝐸𝑅}= 𝐴𝑆𝐶^{𝐿𝐴𝑇𝐸𝑅}+ 𝛽_𝐶[𝑃_𝐼𝐶_𝐼+⁽1 −𝑃_𝐼)𝐶_𝐷] + 𝜀_𝑖 (4.50) where 𝑃_𝐼 is the probability for a future increase in price, (1 − 𝑃_𝐼) is the probability for a future decrease in price, 𝜀_𝑖 is the error associated with the analyst’s observations, 𝐴𝑆𝐶^𝑁𝑂𝑊 and 𝐴𝑆𝐶^{𝐿𝐴𝑇𝐸𝑅} are the alternative specific constants and 𝛽_𝐶 is the sensitivity to price.

The estimation of the model was carried out with BIOGEME 2.2 (Bierlaire, 2003). Like for the charging choice model in subsection 4.2.4, a scale parameter is estimated in order to reduce the variance of the error term for the Panelbase respondents. The parameter estimates and the goodness-of-fit of the model are presented in Table 4.11.

The results of the RUM – EUT model are quite intuitive, with a positive constant for the “book now” option (i.e. an implicit preference for the non-risky choice, indicating a tendency towards myopic behaviour), and a statistically significant negative coefficient for the charging price, which is very close to the estimation from the charging game (𝛽_𝐶𝑃 = −0.918 in Table 4.4).

Table 4.11: RUM – EUT model, booking game – base specification

Variables Coefficient Std error t-test p-value

ASCNOW 1.18 0.139 8.45 0.00^**

In order to capture systematic heterogeneity and the effect of socio-demographics and other attributes on the booking choice, equation 4.49 is transformed to:

𝑢_𝑖^𝑁𝑂𝑊 = 𝐴𝑆𝐶^𝑁𝑂𝑊+ 𝛽_𝐶𝐶_𝑁+𝛽_𝑋^′𝑿 + 𝜀 (4.51) where 𝑿 is the vector of personal attributes and 𝛽_𝑋^′ is the associated vector of parameters to be estimated. Several specifications have been tested and the majority of the variables that enter the utility function are similar with the MNL model for the charging game. Additional (or modified) explanatory variables are presented below:

 Age group (Over 60, Less than 60)

 Travel profile frequency (Undertake the selected travel profile every day, undertake the selected travel profile less frequently)

 Schedule Flexibility (Combined Likert scale values of the indicators presented in subsection 4.2.4)

 Charging frequency (Charging the EV more than once a day, Charging the EV less than once a day, Non-EV drivers)

 Daily mileage with EV (Driving more than 40 miles a day with EV, Driving less than 40 miles a day with EV, Non-EV drivers)

The estimates for the full specification, after accounting for systematic heterogeneity, are shown in Table 4.12.

Following the framework of Liu and Polak (2007), the CARA functional form has been adopted for the non-linear transformation of the value function⁴⁹. The non-linear formulation allows the researchers to investigate the decision maker’s attitude towards risk in the dynamic pricing environment. The utility function for the risky alternative is now:

𝑢_𝑖^{𝐿𝐴𝑇𝐸𝑅} = 𝐴𝑆𝐶^{𝐿𝐴𝑇𝐸𝑅}+ 𝛽_𝐶 [𝑃_𝐼(1 − 𝑒^{−𝑎𝐶𝐼})/𝑎 +(1 −𝑃_𝐼)(1 − 𝑒^{−𝑎𝐶𝐷})/𝑎)] + 𝜀_𝑖 (4.52) The estimation results from the non-linear transformation are presented along with the riskless value function in Table 4.12.

The goodness-of-fit for the full specification of the linear model is significantly improved compared to the base specification (𝜌̅ =0.232 vs 𝜌̅ =0.178). According to the estimated parameters, older individuals are more likely to choose the safe option compared to younger individuals. Similar are the findings for employed individuals. Under a different risky context, Daina (2014) has also found that older groups and people with full-time employment

49 The constant relative risk-aversion (CRRA) transformation has been also tested with the empirical data but the

demonstrate a higher risk aversion, reflected by their increased sensitivity to a latent construct for range anxiety.

On the other hand, those that have children and a higher education level tend to exhibit a more strategic behaviour. Work-based tours are associated with a more conservative response to dynamic pricing, i.e. an increased preference for the “booking now” option. Finally, people that own or lease an EV are more risk-prone and are willing to wait for a better price.

Table 4.12: RUM – EUT model, booking game – full specification accounting for systematic heterogeneity

Variables Linear function Non-linear function

Coefficient Std error Coefficient Std error

ASCNOW 1.49^* 0.805 3.27^* 1.94

Education: University Graduate -0.964^** 0.320 -0.977^** 0.323

Electric vehicle access -0.847^* 0.509 -0.861^* 0.511

Number of daily activities -0.272 0.229 -0.274 0.230

Number of profile searches 0.324^* 0.196 0.323^* 0.198

Travel profile – Every day 1.02 0.720 1.03 0.722

Travel day – Weekday 0.542 0.370 0.550 0.371

Work based tour 0.924^** 0.387 0.929^** 0.388

Schedule flexibility -0.0376 0.0320 -0.0377 0.0321

Charge EV more than once a day 0.908^* 0.548 0.920^* 0.552

Charging EV cost – free 5.04^** 2.39 5.10^** 2.38

Driving EV more than a year -1.01^** 0.323 -1.02^** 0.324

EV loyal enthusiast 0.792^** 0.322 0.800^** 0.324

Daily mileage with EV – more than 40 miles -1.07^** 0.414 -1.09^** 0.417

Scale for recruitment channel (𝜼) 0.416^** 0.0771 0.415^** 0.0762

Risk attitude parameter (𝜶) - - 0.155 0.128

EV drivers who stated that they charge their vehicle more than once a day demonstrated a myopic behaviour, which could be attributed to the planning burden associated with monitoring dynamic prices every time they charge. Furthermore, individuals that have been

possibly because they have a lower willingness to take the risk for an increased charging price.

Experienced EV drivers have a lower likelihood of being myopic while those that are labelled as “EV loyal enthusiasts” have a lower likelihood of being strategic. Finally, individuals that have been regularly driving long distances with their electric vehicle prefer the risky choice.

A potential explanation for their behaviour is that they are more familiar with risky situations, due to the fact that they repeatedly strain the limits of their battery range.

The goodness-of-fit for the non-linear transformation is slightly lower than the linear function and there is no significant difference in the parameter estimates. As with the case of mixed logit in 4.2.4.3, there could be an issue of overfitting, and cross-validation is required in order to identify it. The alpha parameter is not significant but this can be explained by the fact that risk aversion is captured by the coefficient for the alternative specific constant of the “booking now” option, which is positive and significant.

The non-EUT approaches presented in subsection 4.3.2 are also applied here, in order to identify misconceptions, biases and errors in the choice process of individuals.

First, the utility for the risky alternative is transformed based on the RDEU model as follows:

𝑢_𝑖^{𝐿𝐴𝑇𝐸𝑅}= 𝐴𝑆𝐶^{𝐿𝐴𝑇𝐸𝑅}+ 𝛽_𝐶 [𝑤(𝑃_𝐼)(1 − 𝑒^{−𝑎𝐶𝐼})/𝑎 + (1 − 𝑤(𝑃_𝐼))(1 − 𝑒^{−𝑎𝐶𝐷})/𝑎)] + 𝜀_𝑖 (4.53) where w(.) is a function that reflects the individual weights towards risky outcomes and the outcomes are ranked in an increasing preference order (i.e. the increased price is ranked first and the decreased price is ranked second). This function is given by:

𝑤(𝑠_𝑘^𝑛) = {𝜋(𝑝₁^𝑛, 𝑝₂^𝑛) − 𝜋(𝑝₂^𝑛) 𝑖𝑓 𝑘 = 1

𝜋(𝑝₂^𝑛) 𝑖𝑓 𝑘 = 2 (4.54) and 𝜋(. ) is the increasing weighting function of probability 𝑝_𝑘^𝑛 with 𝜋(0) = 0 and 𝜋(1) = 1.

Also, 𝜋(𝑝₁^𝑛, 𝑝₂^𝑛) is the weight associated with obtaining outcome 1 or better than 1.

The weighting function w(.) that is employed is this of equation 4.43, which results into an inverse-S shaped curve. The results from the estimation of the linear value function and the non-linear transformation under the RDEU model are presented in Table 4.13.

The model fit for the non-EUT models has not improved relative to the previous specifications.

The signs and magnitudes of the estimates remain similar to the EUT model and like before, the parameter for the attitude towards risk is insignificant.

Table 4.13: RUM – RDEU model, booking game - full specification accounting for systematic heterogeneity

Variables Linear function Non-linear function

Weighting function (𝑝_𝑘^𝑛) = (𝑝_𝑘^𝑛)^𝛾

((𝑝_𝑘^𝑛)^𝛾+ (1 − (𝑝_𝑘^𝑛)^𝛾))^1𝛾

) (𝑝_𝑘^𝑛) = (𝑝_𝑘^𝑛)^𝛾 ((𝑝_𝑘^𝑛)^𝛾+ (1 − (𝑝_𝑘^𝑛)^𝛾))^1𝛾

)

Coefficient Std error Coefficient Std error

ASCNOW 3.07^* 1.68 6.23^* 3.49

Education: University Graduate -0.982^** 0.325 -0.999^** 0.329

Electric vehicle access -0.911^* 0.534 -0.931^* 0.537

Number of daily activities -0.272 0.233 -0.274 0.235

Number of profile searches 0.330^* 0.198 0.330^* 0.200

Travel profile – Every day 1.02 0.739 1.04 0.743

Travel day – Weekday 0.537 0.382 0.547 0.383

Work-based tour 0.963^** 0.401 0.972^** 0.402

Schedule flexibility -0.0380 0.0327 -0.0382 0.0329

Charge EV more than once a day 0.885 0.555 0.898 0.560

Charging EV cost – free 5.18^** 2.51 5.25^** 2.51

Driving EV more than a year -1.02^** 0.327 -1.03^** 0.330

EV loyal enthusiast 0.802^** 0.327 0.813^** 0.330

Daily mileage with EV – more than 40 miles -1.06^** 0.420 -1.09^** 0.424

Scale for recruitment channel (𝜼) 0.396^** 0.0758 0.395^** 0.0742

Risk attitude parameter (𝜶) - - 0.198 0.140

The parameter 𝛾 is statistically significant for both specifications. As a result, it contains information about the individuals’ perceptions of the probabilities for risky outcomes. The value of 𝛾 is close to 1 so the distortion of the objective probabilities is small. Nevertheless, instead of the commonly encountered inverse S-shape, this weighting function has the opposite effect. This means that individuals slightly underweight low probabilities and overweight high probabilities. The crossover point for the inverse-S curve is around 0.5. The relationship between probabilities and weighted probabilities for the linear value function (it is almost the

Figure 4.10: Objective and subjective probabilities of the risky outcomes

Since the probabilities of future prices are based on an orthogonal design, the aggregated probabilities for both outcomes throughout the choice experiment are 0.5 and, as a result, their decision weights are equal. The results suggest that when the probability of an increased price is small (i.e. 0.2 and w(0.2)<0.2 and 1-w(0.2)>0.8) then an increased price weights less than proportionally compared to the objective probability, reflecting optimism for the individuals.

However, this optimism is small because 𝛾 is close to 1. On the other hand, when the probability of a decreased price is small, following the same logic, individuals show some pessimism and tend to be risk-averse.

Following the Prospect Theory approach, the utility function for the “booking later” can be expressed as follows:

𝑢_𝑖^{𝐿𝐴𝑇𝐸𝑅}= 𝐴𝑆𝐶^{𝐿𝐴𝑇𝐸𝑅}+ 𝛽𝐶(𝑔𝑎𝑖𝑛)(1 − 𝑃_𝐼)(𝐶_𝑅𝑒𝑓− 𝐶𝐷)^𝑎+ 𝛽𝐶(𝑙𝑜𝑠𝑠)𝑃𝐼(𝐶𝐼− 𝐶𝑅𝑒𝑓)^𝛽+ 𝜀𝑖 (4.55) where 𝐶_𝑅𝑒𝑓 is the reference price, which due to the nature of the problem is assumed to be equal to the price for the safe choice, i.e. £2.50⁵⁰. The cost coefficient is now divided into gain 𝛽_{𝐶(𝑔𝑎𝑖𝑛)} and loss 𝛽_{𝐶(𝑙𝑜𝑠𝑠)} based on the relative location of the outcome with respect to the reference price 𝐶_𝑅𝑒𝑓. Finally, the parameters 𝑎 and 𝛽 reflect the degree of diminishing sensitivity.

50 It has to be noted here that this price level might be non-representative of the existing recharging cost for EV drivers. In this case, it wouldn’t coincide with the reference price from the revealed preferences of the individuals.

Nevertheless, it is undoubtedly the “reference point” for the hypothetical choice of the booking game, based on

Two PT models were estimated: one where the 𝑎 and 𝛽 parameters are fixed to one in order to capture only reference dependence, and one that allows the estimation of diminishing sensitivity. The estimation results are presented in Table 4.14.

Table 4.14: RUM – PT model, booking game - full specification accounting for systematic heterogeneity

Variables Reference dependence Diminishing sensitivity

Coefficient Std error Coefficient Std error

ASCNOW 2.14^** 0.927 2.76^** 1.03

Education: University Graduate -0.981^** 0.324 -1.00^** 0.329

Electric vehicle access -0.866^* 0.512 -0.922^* 0.533

Number of daily activities -0.274 0.231 -0.276 0.235

Number of profile searches 0.323^* 0.198 0.330^* 0.200

Travel profile – Every day 1.04 0.724 1.04 0.743

Travel day – Weekday 0.553 0.372 0.552 0.383

Work-based tour 0.933^** 0.388 0.971 ^** 0.401

Schedule flexibility -0.0379 0.0322 -0.0386 0.0329

Charge EV more than once a day 0.923^* 0.553 0.906 0.560

Charging EV cost – free 5.11^** 2.38 5.23^** 2.50

Driving EV more than a year -1.02^** 0.325 -1.04^** 0.330

EV loyal enthusiast 0.804^** 0.325 0.818^** 0.330

Daily mileage with EV – more than 40 miles -1.09^** 0.418 -1.09^** 0.424

Scale for recruitment channel (𝜼) 0.414^** 0.0756 0.397^** 0.0738

𝜶 - - 0.960^** 0.249

The model fit is similar to the RDEU model and the results agree with the a priori expectations, i.e. the price coefficient is positive when the outcome is framed as a gain and negative when it is framed as a loss. While the positive and significant parameter for the constant of the “book now” option indicates a risk-aversion for both models, the absolute ratio of the cost coefficients

In document Efficient operation of recharging infrastructure for the accommodation of electric vehicles: a demand driven approach (Page 191-200)