Last modified: 28 March 2017

#### Abstract

**On interpreting models of risk in a random utility setting**

Modellers of travel behaviour are increasingly incorporating travellers’ responses to uncertain conditions by including a representation of attitudes towards risk. The approach is particularly relevant to commuting behaviour where travellers face travel time variability due to congestion, competition for parking spaces or the possibility of overcrowded public transport services. To date the representations of risk attitudes fall broadly into one of two camps. The first is based on a measure of variance (Jackson & Jucker, 1982; Senna, 1994) whereby an individual is said to be averse if they tend to choose alternatives with lower variability (i.e., a negative parameter on the variance attribute). This model extends to scheduling risk whereby the respondent attaches a disutility to the probability that they will be late (or early) to an appointment (Noland & Small, 1995). The second branch of models adopts the variants of non-expected utility from economic research. The transport literature has focussed on the Arrow-Pratt constant relative risk aversion (Pratt, 1964) along with Kahneman-Tversky probability perception weights (Kahneman & Tversky, 1979). However, a closer study of the econometric literature reveals that direct translation of economic risk models to stochastic (random) theories of utility is not so straight forward and that the interpretation of the risk parameter (alpha) is context dependent (Wilcox, 2011).

Following the lead from recent research into stochastic utilities of risk, this paper demonstrates that interpreting risk in a mode choice setting does not have a one-to-one correspondence with the sign or the magnitude of alpha. Using simple examples we show that determining whether an individual is risk averse based on the sign of risk parameter can lead to an incorrect inference. A number of remedies are proposed and applied to a mode and time departure stated preference survey where travel time variability is presented in the attribute space.

The following is a summary of the paper without the equations

**Risk Attitudes **

**Definition:** *Agent A is stochastically more risk averse than agent C if agent A is more likely to choose a safe alternative ** than agent C*

The same definition applies to a stochastically more risk tolerant individual,

**Definition:** *Agent D is stochastically more risk averse than agent C if Agent D is more likely to choose a risky alternative ** than agent C*

For simplicity of the following exposition we may consider *agent C to hold a risk neutral position, *in which case *agent A* is said to be risk averse and *agent D* is said to be risk seeking.

In random utility framework, the definitions above translate into the systematic utility differences (proof provided). The utilities for risk averse *agent A* indicate a stronger preference for the safe alternative when compared to risk neutral *agent C* through the comparative utility differences and similarly the utilities for risk seeking *agent D* indicate a lesser preference for the safe alternative when compared to risk neutral *agent C.*

**What is the Risky Alternative? **

To identify risky and safe alternatives, four definitions (noise, second-order dominance, weights of the tail and variance) are commonly exploited. In the case of mean-preserving spread pairs, these four definitions are consistent (Rothschild & Stiglitz, 1970). However, this is not the case when expected values differ across alternatives. This paper focusses on the second-order stochastic dominance and variance definitions for risky prospects.

**Risk behaviour in random utility models**

The paper examines the relationship between alpha and the utility differences by using a theorem of the mean twice (equation omitted) on the utility differences. In choice contexts that would be appropriate for examining the risk attitude to travel time variability the sign of the second-order derivatives of the resulting utility difference function of alpha identifies a point of inflection.

Figure 1a and 1b address the context-dependent interpretation of risk attitude with the values. When drops in the area where the R-difference value is below the horizontal axis, these values indicate risk averse attitude, otherwise risk seeking.

The paper provides further analysis on alpha the interpretation of when the definition of the safe alternative is the one with smaller variance. In this case the meaning of is also context dependent and cannot be immediately interpreted as a measure of risk attitude.

**Learnings for Discrete Choice Modelling**

Determining whether an individual is risk averse/risk seeking based on the sign and value of alpha may lead to an incorrect inference. A possible remedy is to examine the relationship of risk attitude and alpha values for each choice scenario and estimate a separate for each choice scenario. This is unlikely to be operational in a stated choice model and is inappropriate for revealed preference when all choice scenarios are also unique. The paper examines two possible approaches to examine risk in context dependent choice tasks. The first is to apply Wilcox’s (2011) utility context transformation to the attribute levels in the choice set. This shows an improvement, but is not a solution because the choice tasks used for transport are far more complex that Wilcox’s monetary gambles. The second approach broadly classifies the choice tasks and a two alpha mixed logit is estimated. This shows an improvement to fit of the data and rejects the null hypothesis that the interpretation of the a in CRRA is context independent.

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