##  Chapter 1 Human Behavioral Models Using Utility Theory and Prospect Theory

Vineet Jagadeesan Nair

[1]Anuradha M. Annaswamy

###### Abstract

Abstract Several examples of Cyber-physical human systems (CPHS) include real-
time decisions from humans as a necessary building block for the successful
performance of the overall system. Many of these decision-making problems
necessitate an appropriate model of human behavior. Tools from Utility Theory
have been used successfully in several problems in transportation for resource
allocation and balance of supply and demand (Ben-Akiva et al., 1985). More
recently, Prospect Theory has been demonstrated as a useful tool in behavioral
economics and cognitive psychology for deriving human behavioral models that
characterize their subjective decision making in the presence of stochastic
uncertainties and risks, as an alternative to conventional Utility Theory
(Kahneman and Tversky, 2012). These models will be described in this article.
Theoretical implications of Prospect Theory are also discussed. Examples will
be drawn from transportation use cases such as shared mobility to illustrate
these models as well as the distinctions between Utility Theory and Prospect
Theory.

Keywords: Prospect theory, Behavioral modeling, Smart Cities, Transactive
Control, Utility Theory

###  1.1 Introduction

Analysis and synthesis of large-scale systems require the understanding of
cyber physical human systems. Interactions between humans and cyber-components
that interact with the physical system are varied and depend on a variety of
factors and the goals of the large-scale systems. If the problem at hand
concerns the behavior of the CPHS under emergency conditions, the interactions
between humans and automation need to focus on a shared control architecture
(Erslan et al., 2023) with appropriate granularity of task allocation and
timeline. Typically the human in this context is an expert and when anomalies
occur, either takes over control from automation or provides close supervision
to the automation to ensure an overall safe CPHS. Under normal circumstances,
the interactions may include other architectures. The role of the human is not
necessarily that of an operator or an expert, but a user. The human may be a
component in the loop, responding to outputs from the automation, and making
decisions that serve in turn as inputs or reference signals $r$ to the
physical system (see Figure 1.1).

Typical examples of such interactions have begun to occur both in power grids
and transportation and can be grouped under the rubric of transactive control.
The transactive control concept (Chassin et al., 2004; Bejestani et al., 2014;
Annaswamy and Nudell, 2015) consists of a feedback loop resulting from
incentives provided to consumers. Introduced in the context of smart grids, a
typical transactive controller consists of an incentive signal sent to the
consumer from the infrastructure and a feedback signal received from the
consumer, and together the goal is to ensure that the underlying resources are
optimally utilized. This introduces a feedback loop, where empowered consumers
serve as actuators into an infrastructure, and transactive control represents
a feedback control design that ensures that the goals of the infrastructure
are realized, very similar to Figure 1.1. The use of transactive control in
smart grids can be traced to homeostatic control proposed in (Schweppe, 1978)
and (Schweppe et al., 1980), which suggested that demand-side assets can be
engaged using economic signals. Transactive control in this context has come
to denote market-based control mechanisms that incentivize responsive loads
and engage them in providing services to the grid where and when of great need
(as described in (Katipamula et al., 2006; Li et al., 2015; Hao et al., 2016;
Somasundaram et al., 2014; Hammerstrom et al., 2008; Widergren et al., 2014;
Melton, 2015; Kok, 2013) and (Bernards et al., 2016)), and has the ability to
close the loop by integrating customers via the right incentives to meet their
local objectives such as lowering their electric bill as well as meeting
global system objectives such as voltage and frequency regulation. Another
example of transactive control is in the context of congestion control in
transportation ((Phan et al., 2016) and (Annaswamy et al., 2018)) - where
dynamic toll pricing, determined by the controller, is used as an incentive
signal to the drivers, who then decide whether or not to enter a tolled
segment, thereby regulating traffic density and possibly alleviating
congestion. In both examples, it is clear that an understanding of the human
behavior is important, and the modeling of the overall socio-technical system
that includes the human behavior and their interaction with the physical
system is an important first step in designing the feedback controller.

This article focuses on two different tools that have been proposed for
deriving behavioral models of human as a consumer. These tools include Utility
Theory and Prospect Theory, and are described in the following sections.
Prospect Theory is a framework introduced by Nobel prize-winning behavioral
economists and psychologists that has been extensively shown to better
represent decision making under uncertainty. It builds upon Utility Theory
models by introducing additional nonlinear transformations on agents’
objective utilities to model their irrational and subjective behaviors.
Cumulative Prospect Theory (CPT) is an extension of Prospect Theory that also
considers distortion of subjective probabilities through a weighting function.
Examples are drawn from transportation to elucidate the impact of these tools,
particularly in terms of predicting mode choice probabilities of passengers.

Figure 1.1: An Example of a CPHS with human in the loop.

###  1.2 Utility Theory

The starting point for Utility Theory (Ben-Akiva et al., 1985) is the
assignment of a value to an outcome in the form of a utility function. If in
general there are $N$ different possible choices, then $U_{i}$ is utility of
outcome $i$, $i=1,\ldots,N$. The benefit of the utility function is that it
provides a substrate for modeling a human’s decision when faced with these
choices. In particular, the probability that the human will select choice
$\ell$ is determined using the utility function as:

$p^{\ell}=\frac{e^{U^{\ell}}}{\sum_{j=1}^{N}e^{U^{\ell}}}\;\ell\in\\{1,\dots,N\\}$
(1.1)

Equation 1.1 then serves as a simple behavioral model of the human which can
be appropriately utilized in the overall problem of interest. For a simple
problem with only two choices $A$ and $B$, the probability $p^{A}$ is given by

$p^{A}=\frac{1}{1+e^{-\Delta U}}$ (1.2)

where $\Delta U=U^{A}-U^{B}$. Modeling the behavior of the human in the
problem, whether as a consumer, an operator, a user, or an expert, the Utility
Theory based approach entails the characterization of all possible outcomes
and the determination of the utility of each of the outcome $i$ as $U^{i}$. A
variety of factors contributes to the utility and hence determining the
behavioral model in Equation 1.1 is nontrivial. Quantitative aspects related
to economy, qualitative aspects such as comfort, and hard to identify aspects
such as strategic behavior, negative externalities, global and network based
expectations, are all factors that may need to be simultaneously accounted
for. Nevertheless, Equation 1.1 serves as a cornerstone for many problems
where CPHS models have to be derived.

####  1.2.1 An Example

Suppose we consider a transportation example where a passenger intends to
travel from point A to point B, and has two choices for transport: a shared
ride service (SRS) and public transit. The utility function for this trip can
be determined as:

$\displaystyle u$ $\displaystyle=\mathbf{a}^{\intercal}\;\mathbf{t}+b\gamma+c$
(1.3)

where the components of $\mathbf{t}=[t_{walk},t_{wait},t_{ride}]^{\intercal}$
denote the walking, waiting, and riding times, respectively, $\gamma$ denotes
the ride tariff, $\mathbf{a}=[a_{walk},a_{wait},a_{ride}]^{\intercal}$ are
suitable weights, and $c$ denotes all other externalities that do not depend
on either travel time or ride tariff. Both the weights $\mathbf{a}$ and tariff
coefficient $b$ are assumed to be negative since these represent disutilities
to the rider arising from either longer travel times or higher prices, while
$c$ can be either positive or negative depending on the characteristics of the
given travel option. All of the parameters ${\mathbf{a}},b,c$ determine the
behavior of a rider and could vary with time, the environment, or other
factors. Equation 1.3 indicates that the choice of the rider of the SRS over
other options like public transit is determined by whether
$u(SRS)>u(\hbox{public transit})$ for a given ride.

A behavioral model of a driver as above was applied to the congestion control
problem on a highway segment to determine a dynamic tolling price strategy
(Ben-Akiva et al., 1985; Phan et al., 2016). It was shown that with the
alternative $u_{0}$ corresponding to travel on a no-toll road, the toll price
can be determined using a nonlinear PI controller using a socio-technical
model that was a cascaded system with the behavioral model as in Equation
1.2-1.3 and an accumulator model of the traffic flow. Using actual data from a
highway segment in the US city of Minneapolis, it was shown that such a model-
based dynamic toll price leads to a much more efficient congestion control
(Annaswamy et al., 2018). More recently, this approach has been extended to a
more complex highway section with multiple merges and splits, and applied to
data obtained from a highway section near Lisbon, Portugal (Lombardi et al.,
2022). Here too, a behavioral model of the driver similar to Equation 1.3 was
employed. The results obtained displayed a significant improvement compared to
existing traffic flow conditions, with minimal changes to the toll price.

###  1.3 Prospect Theory

The behavioral model based on Utility Theory have two deficiencies. The first
is that the utility function is a embedded in a stochastic environment,
causing the utility function model to be more complex than that considered in
Equation 1.3. The second is that the model as considered in Equation 1.2 may
not be adequate in capturing all aspects of decision making of a human.
Strategic decision making, adjustments based upon the framing effect, loss
aversion, and probability distortion are several key features related to
subjective decision making of individuals when facing uncertainty. It is in
this context that Prospect Theory (Kahneman and Tversky, 2012) in general, and
Cumulative Prospect Theory (CPT) (Tversky and Kahneman, 1992), in particular,
provide an alternate tool that may be more appropriate. CPT builds upon
Prospect Theory by using a probability weighting function to represent the
agent’s distortion of perceived probabilities of outcomes, and uses these
probability weights to compute the subjective utilities from subjective
values. We briefly describe this tool below.

We first introduce a stochastic component into the problem and utilize the
transportation example as the starting point. As travel times are subject to
stochasticity, $u$ becomes a random process. For simplicity, suppose we assume
that there are only two possible travel time outcomes, $\mathbf{\overline{t}}$
and $\mathbf{\underline{t}}$
($\mathbf{\underline{t}}\leq\mathbf{\overline{t}}$) having corresponding
utilities $\underline{u}$ and $\overline{u}$
($\underline{u}\leq\overline{u}$), occurring with probabilities of $p\in[0,1]$
and $1-p$ respectively. It follows that the utility function for the SRS is
given by

$\displaystyle\underline{u}$
$\displaystyle=\mathbf{a}_{sm}^{\intercal}\;\mathbf{\overline{t}}+b_{sm}\gamma_{sm}+c_{sm}$
$\displaystyle\overline{u}$
$\displaystyle=\mathbf{a}_{sm}^{\intercal}\;\mathbf{\underline{t}}+b_{sm}\gamma_{sm}+c_{sm}$
(1.4)

If these outcomes follow a Bernoulli distribution, its cumulative distribution
function (CDF) is defined on the support $[\underline{u},\overline{u}]$:

$F_{U}(u)=\begin{cases}0&\mbox{if }u<\underline{u}\\\ p&\mbox{if
}\underline{u}\leq u<\overline{u}\\\ 1&\mbox{if
}u\geq\overline{u}.\end{cases}$ (1.5)

Suppose the alternative choice has a utility function $u_{o}$. If
$u_{o}\leq\underline{u}$, the customer would always choose the SRS since it
offers strictly better outcomes and conversely if $u_{o}\geq\overline{u}$. For
all other cases, the underlying model becomes a combination of Equation 1.2
and Equation 1.5.

We now address the second deficiency in Utility Theory. Conventional Utility
Theory postulates that consumers choose among travel options based on their
respective expected utilities (Fishburn, 1988; Von Neumann and Morgenstern,
2007). Alternatively, random utility models are another framework within
Utility Theory that predict choice probabilities based on the utilities of
different alternatives (computed using logit models) without accounting for
risk, by assuming certain distributions for unobserved factors and error
terms. However, both of these are inadequate when there is significant
uncertainty involved. Prospect Theory (PT) is an alternative to Utility Theory
that better describes subjective human decision making in the presence of
uncertainty and risk (Tversky and Kahneman, 1992; Kahneman and Tversky, 2012),
and Cumulative Prospect Theory (CPT) is a variant of PT that weighs different
outcomes using distorted subjective probabilities as perceived by passengers.
This is needed since individuals have been shown to consistently underestimate
the likelihood of high probability outcomes while overestimating the
likelihood of less likely events (Tversky and Kahneman, 1992). To describe
CPT, we introduce a value function $V(\cdot)$ and a probability distortion
$\pi(\cdot)$ given by (Guan et al., 2019a) and (Prelec, 1998), with $\pi(0)=0$
and $\pi(1)=1$ by definition. These nonlinearities map the objective utilities
($u$) and probabilities ($p$) of each possible outcome to subjective values,
as perceived by the passengers. Note here that the probability weighting
function $\pi(\cdot)$ as described in Equation 1.7 is unique to CPT. The
graphs in Figure 1.2 show examples of how the value and probability weighting
functions may vary according to the objective utility $u$ and actual
probability $p$, respectively.

$\displaystyle V(u)$ $\displaystyle=\begin{cases}(u-R)^{\beta^{+}}&\mbox{if
}u\geq R\\\ -\lambda(R-u)^{\beta^{-}}&\mbox{if }u<R\end{cases}$ (1.6)
$\displaystyle\pi(p)$ $\displaystyle=e^{-(-ln(p))^{\alpha}}$ (1.7)

Figure 1.2: Illustrations of the CPT value function and probability weighting
functions (Guan et al., 2019a).

The CPT parameters here describe loss aversion ($\lambda$), diminishing
sensitivity in gains ($\beta^{+}$) and losses ($\beta^{\\_}$) and probability
distortion ($\alpha$). The reference $R$ is the baseline against which users
compare uncertain prospects. These can vary across individuals and also
depending on the particular set of alternatives the customer is facing.

With the above distortions in the value function, the overall utility function
for a given stochastic outcome gets modified. Noting that the utility function
$U$ is a random variable, a Utility Theory based derivation is as follows. If
$U$ takes on discrete values $u_{i}\in\mathbb{R},\forall
i\;\in\\{1,\,\,\dots,\,n\\}$ and the outcomes are in ascending order, i.e.,
$u_{1}<\cdots<u_{n}$, where $n\in{\mathbb{Z}}_{>0}$ is the number of possible
outcomes, one can determine the objective utility $U^{o}$ as the expectation
of $U$ according to Utility Theory as (Von Neumann and Morgenstern, 2007),
i.e.,

$U^{o}=\sum_{i=1}^{n}p_{i}u_{i}$ (1.8)

where $p_{i}\in(0,1)$ is the probability of outcome $u_{i}$, and
$\sum_{i=1}^{n}p_{i}=1$. In contrast, a Prospect Theory based derivation of
utility function includes $V$ rather than $u_{i}$ and the probability
distortion $\pi(.)$ applied to the probabilities $p_{i}$. Suppose we define
the corresponding utility function as $U^{s}_{R}$, which is the subjective
utility perceived by the passenger according to Cumulative Prospect Theory,
then

$U^{s}_{R}=\sum_{i=1}^{n}w_{i}V(u_{i})$ (1.9)

where $R$ denotes the reference corresponding to the framing effect mentioned
above, and $w_{i}$ denotes the weighting that represents the subjective
perception of $p_{i}$. Suppose that $k$ out of the $n$ outcomes are losses,
$0\leq k\leq n,k\in\mathbb{Z}_{\geq 0}$, and the rest are non-losses, i.e.,
$u_{i}<R$ if $1\leq i\leq k$ and $u_{i}\geq R$ if $k<i\leq n$. We can then
derive the subjective probability weights assigned by the decision maker to
each of these discrete outcomes from the cumulative distribution function of
$U$ given by $F_{U}(u)$, as follows:

$w_{i}=\begin{cases}\pi\big{[}F_{U}(u_{i})\big{]}-\pi\big{[}F_{U}(u_{i-1})\big{]},&\text{if}\;i\in[1,k]\;\text{(losses)}\\\
\pi\big{[}1-F_{U}(u_{i-1})\big{]}-\pi\big{[}1-F_{U}(u_{i})\big{]},&\text{otherwise}\;\text{(non-
losses)}\end{cases}$ (1.10)

where we have assumed $F_{U}(u_{0})=0$ for ease of notation.

It is clear that in contrast to $U^{o}$, $U^{s}_{R}$ is centered on $R$, loss
aversion is captured by choosing $\lambda>1$, and diminishing sensitivity by
choosing $0<{\beta}^{+},{\beta}^{-}<1$. The probability distortion is
quantified by choosing $0<\alpha<1$. The extension from Equation 1.9 to the
continuous case of $U^{s}_{R}$ is

$U^{s}_{R}=\int_{-\infty}^{R}V(u)\frac{d}{du}\Big{\\{}\pi\big{[}F_{U}(u)\big{]}\Big{\\}}du+\int_{R}^{\infty}V(u)\frac{d}{du}\Big{\\{}-\pi\big{[}1-F_{U}(u)\big{]}\Big{\\}}du$
(1.11)

With the above objective evaluation of a utility function as in Equation 1.8
and subjective evaluation as in Equation 1.9, one can now determine the
probability of acceptance of an outcome as follows. As has been shown in
Equation 1.1, the evaluation of the probability of acceptance of an outcome
$\ell$ requires the utility of all alternate outcomes. Without loss of
generality, suppose there are only two alternatives, with the objective and
subjective utility of option $i\in\\{1,2\\}$ given by $U_{io}$ and
$U_{is_{R}}$ respectively. Then the objective probability of acceptance of
choice $1$ using Utility Theory is given by

$p^{o}_{1}=\frac{e^{U_{1o}}}{e^{U_{1o}}+e^{U_{2o}}}$ (1.12)

while the subjective probability of acceptance of option $1$ using Prospect
Theory is given by

$p^{s}_{1}=\frac{e^{U_{1s_{R}}}}{e^{U_{1s_{R}}}+e^{U_{2s_{R}}}}$ (1.13)

####  1.3.1 An example: CPT modeling for SRS

We illustrate the Prospect Theory model using the transportation example
considered above, extended to the case where a passenger now has three choices
$i\in\\{1,2,3\\}$, (i) public transit like buses or the subway, (ii) using a
shared ride pooling service (SRS) and (iii) another which may be an exclusive
ride hailing service (such as UberX). The SRS has greater uncertainty in pick
up, drop off and travel times when compared to the UberX and transit
alternatives, due to the possibility of more passengers being added en route.
Thus, both UberX and transit can be treated as certain prospects when compared
to the SRS option, which has uncertain outcomes. CPT can then be used to model
the passenger’s risk preferences to predict their decision making under such
uncertainty. The discussions above show that a number of parameters related to
the CPT framework have to be determined. These include
$\alpha,{\beta}^{+},{\beta}^{-},\lambda$ defined in $V(\cdot)$ and
$\pi(\cdot)$, which are in addition to the parameters $\mathbf{a},b,c$ defined
in (Equation 1.3) associated with the travel times
$t_{\text{walk}},t_{\text{wait}},t_{\text{ride}}$ and tariff coefficients, for
all three travel modes. In order to estimate these parameters, we designed and
conducted a comprehensive survey eliciting travel and passenger risk
preferences from $N=955$ respondents in the greater Boston metropolitan area
(Jagadeesan Nair, 2021). Note that the constant terms in the utility function,
$c_{UberX}$ and $c_{SRS}$ were measured relative to public transit as a
baseline, i.e. $c_{transit}=0$.

Table 1.1: Parameters describing the discrete choice logit models for SRS and UberX. Parameter | Mean | SE | SD | SE
---|---|---|---|---
$a_{walk}$ [$min^{-1}$] | -0.0586 | 0.0053 | 0.1412 | 0.0079
$a_{wait}$ [$min^{-1}$] | -0.0113 | 0.0182 | 0.1491 | 0.0356
$a_{ride,\;transit}$ [$min^{-1}$] | -0.0105 | 0.0013 | 0.0284 | 0.0017
$a_{ride,\;UberX}$ [$min^{-1}$] | -0.0086 | 0.0014 | 0.0058 | 0.0010
$a_{ride,\;SRS}$ [$min^{-1}$] | -0.0186 | 0.0013 | 0.0095 | 0.0007
$b$ [$\$^{-1}$] | -0.0518 | 0.0050 | 0.0597 | 0.0042
$c_{UberX}$ | -2.5926 | 0.1800 | 2.3034 | 0.1558
$c_{SRS}$ | -2.2230 | 0.1497 | 1.8175 | 0.1530

Table 1.1 summarizes the mean values and standard deviations of the parameters
that we estimated for the discrete mode choice model using maximum simulated
likelihood estimation, along with their standard errors. We can also use the
estimated travel time and price coefficients to determine the passengers’
value of time (VOT) spent on different modes. The value of time is defined as
the extra tariff that a person would be willing to pay or cost incurred to
save an additional unit of time, i.e., it measures the willingness to pay
(WTP) for extra time savings. In absolute terms, the VOT spent on mode $i$ can
be calculated as the ratio between the marginal utilities of travel time and
trip cost:

$\displaystyle VOT_{i}$ $\displaystyle=\frac{\frac{\partial U_{i}}{\partial
t}}{\frac{\partial U_{i}}{\partial\gamma}}=\frac{a_{i}}{b_{i}}$ (1.14)

Table 1.2:  Value of time spent on different modes, obtained from the random parameters logit model. Trip leg or mode |  VOT (in $/h)
---|---
Walking | 67.8702
Waiting | 13.1480
Transit ride | 12.1703
Exclusive ride hailing | 9.9466
Pooled ride sharing | 21.5549

For the CPT model, estimated, we slightly modified the model to allow for
different probability distortion effects in the gain and loss regimes. Thus,
the modified weighting function is given by:

$\pi_{\pm}(p)=e^{-(-ln(p))^{\alpha_{\pm}}}$ (1.15)

The CPT risk parameters were estimated using the method of certainty
equivalents (Rieger et al., 2017; Wang and Zhao, 2019), by presenting surveyed
passengers with a series of chance scenarios asking them to choose a travel
mode by comparing the uncertain or risky SRS versus the certain UberX and
transit options. Nonlinear least squares curve fitting was then used to
estimate the CPT parameter values. A schematic for the overall estimation
process is shown in Figure 1.3. More details on the survey design and
estimation procedures can be found in (Jagadeesan Nair, 2021).

Figure 1.3:  Estimation of mode choice models and CPT parameters from passenger survey data. Table 1.3:  Summary of CPT parameter estimates for SRS travel preferences. | $\alpha^{+}$ | $\alpha^{-}$ | $\beta^{+}$ | $\beta^{-}$ | $\lambda$
---|---|---|---|---|---
Mean | 0.4456 | 0.1315 | 0.2166 | 0.3550 | 20.0494
Median | 0.4124 | 0.1320 | 0.2188 | 0.3649 | 11.8715
SD | 0.1828 | 0.0448 | 0.0985 | 0.1906 | 25.8554

Having estimated these parameters, we can compute the subjective or perceived
value of the SRS outcomes, and the passenger’s subjective probability of
accepting the shared ride service offer using Equation 1.6-1.13. Since the CPT
model better accounts for passengers’ irrational preferences when faced with
uncertainty and risk, this subjective acceptance probability is more accurate
than what would be predicted using conventional Utility Theory alone.

#####  1.3.1.1 Detection of CPT effects via lotteries

In addition to estimating the numerical values parametrizing the subjective
value and probability weighting functions, we also used the survey responses
to detect the key CPT effects, which are:

  * •

Framing effect: Individuals value prospects with respect to a reference point
instead of an absolute value, and perceive gains and losses differently.

  * •

Diminishing sensitivity: In both gain and loss regimes, sensitivity diminishes
when the prospect gets farther from the reference. Therefore, the perceived
value is concave in the gain regime and convex for losses, implying that
people are risk-averse in gains and risk-seeking in the loss regime.

  * •

Probability distortion: Individuals overweight small probability events and
underweight large probability events.

  * •

Loss aversion: Individuals are affected much more by losses than gains.

In order to estimate these effects, we considered simplified choice scenarios
involving monetary lotteries. Survey respondents were also asked a series of
hypothetical lottery questions after completing the SRS travel choice
scenarios. We can then test for the existence of CPT-like behaviors based on
their lottery responses, as described in (Rieger et al., 2017). More details
on the survey and lottery scenarios can be found in (Jagadeesan Nair, 2021).

Table 1.4:  Summary of key CPT effects observed from lotteries. CPT effect tested | % of valid responses
---|---
Reflection effect (framing & diminishing sensitivity) | 95.03
Probability overweighting between |
10% and 60% probability | 62.56 %
60% and 90% probability | 40.51 %
10% and 90% probability | 51.05 %
Any probability weighting | 72.44 %
Loss aversion |
Mean gain/loss ratio for mixed outcome lotteries | 3.7254
Median gain/loss ratio for mixed outcome lotteries | 1.0250

From Table 1.4, we see that the valid responses clearly display CPT effects.
The reflection or framing effect is shown by nearly all the valid respondents,
indicating that our proposed value function is likely an accurate descriptor
of how the passengers perceive their gains and losses. The probability
weighting effect is not as dominant but it is still quite significant. We find
that majority of them ($>72\%$) show at least some overweighting of
probabilities and it is also most common in the lower probability ranges
(between $10-60\%$). This agrees with CPT theory since it postulates that
people tend to overestimate the likelihood of rare events. The relatively
large value of the mean gain/loss ratio ($>1$) in the mixed lotteries
indicates a significant degree of loss aversion among the surveyed passengers.
However, the median value is quite close to 1 indicating that loss aversion
may not be as prevalent for a sizeable portion of the passengers sampled in
this study. These results from Table 1.3 and Table 1.4 demonstrate that we can
use survey responses and data on passenger choices to (i) validate CPT
behavioral effects and also (ii) estimate mathematical models and parameters
to describe their risk preferences. Using real-time data from sources like
ridesharing apps, these models can be continuously updated and improved in
order to more accurately predict passenger choice and behavior.

####  1.3.2  Theoretical implications of CPT

In the previous section, we determined mode choice and CPT behavioral models
for passengers for a large population and over many different possible travel
scenarios. In the following, we consider the case of a single trip (ride
request and offer), in order to draw a few key insights about passenger risk
preferences and travel behavior using computational experiments (Guan et al.,
2019a). Here, the passenger makes a choice between SRS and an exclusive ride-
hailing service like UberX.

A dynamic routing problem of sixteen passengers using real request data from
San Francisco was considered (see (Annaswamy et al., 2018) for details), and
the request from the $6^{\text{th}}$ passenger was used for the computational
experiments in this section. The AltMin algorithm developed in (Guan et al.,
2019b) was applied to derive the route and therefore the corresponding travel
times of the SRS. The constraints on the possible delay were set to be at most
4 minutes of extra waiting and riding, respectively. For the same request, the
travel times and price of the UberX option were retrieved from
Uber111https://www.uber.com/.

Using the travel times and prices, along with utility coefficients from Table
1.1, the objective utility of UberX $A^{o}$ and $\underline{x},\;\overline{x}$
of the SRS are calculated, using Section 1.3. Note that
$A^{o},\underline{x},\overline{x}$ are negative as they represent disutilities
due to travel times and tariffs. Using this numerical setup, we explore the
three implications via simulations: (i) fourfold pattern of risk attitudes,
(ii) strong aversion of mixed prospects, and (iii) self reference. We use the
following key properties of CPT based behavioral models for our analysis, more
details and derivations of these can be found in (Guan et al., 2019a):

The first two properties are related to static and dynamic references, and are
stated in Property 1 and 2 below. These are helpful in determining the dynamic
tariff $\gamma$ that allows $p^{s}_{R}$ to reach $p^{*}$, the desired
probability of acceptance. Let $\bar{U}=\mathbb{E}_{f_{U}}(U)$ and
$\bar{X}={\mathbb{E}}_{f_{X}}(X)$, the third and fourth property stated in
Property 3 and 4 are related to $U^{s}_{\bar{U}}$ and $p^{s}_{\bar{U}}$,
respectively.

  1. 1.

Given any static reference point $R\in\mathbb{R}$, $p^{s}_{R}$ strictly
decreases with $\gamma$.

  2. 2.

Given any dynamic reference point in the form of
$R=\tilde{x}+b\gamma,\tilde{x}\in\mathbb{R}$, $p^{s}_{R}$ strictly decreases
with $\gamma$.

  3. 3.

Given any uncertain prospect, there exists a ${\lambda}^{*}$, such that
$\forall\lambda>{\lambda}^{*}$, $U^{s}_{\bar{U}}<0$.

  4. 4.

For any uncertain prospect, given that $\lambda$ is sufficiently large such
that $U^{s}_{\bar{U}}<0$, within the price range
$\gamma\in[\underline{\gamma},\overline{\gamma})$, where $\underline{\gamma}$
satisfies $\bar{X}+b\underline{\gamma}=A^{o}$, and $\overline{\gamma}$
satisfies
${\big{[}A^{o}-(\bar{X}+b\overline{\gamma})\big{]}}^{{\beta}^{+}}-U^{s}_{\bar{U}}=A^{o}-(\bar{X}+b\overline{\gamma})$,
$p^{s}_{\bar{U}}<p^{o}$.

#####  1.3.2.1 Implication I: Fourfold Pattern of Risk Attitudes

The fourfold pattern of risk attitudes is regarded as “the most distinctive
implication of Prospect Theory” by Tversky and Kahneman (Tversky and Kahneman,
1992), which states that when facing an uncertain prospect, the risk attitudes
of individuals can be grouped into four categories:

  1. 1.

Risk averse over high probability gains.

  2. 2.

Risk seeking over high probability losses.

  3. 3.

Risk seeking over low probability gains.

  4. 4.

Risk averse over low probability losses.

These risk attitudes are often used to justify subjective decision making of
individuals for problems such as settlements of civil lawsuits, desperate
treatments of terminal illnesses, playing lotteries, and getting insurance
coverage.

We now illustrate the fourfold pattern in the SRS context using the following
scenario, which corresponds to the classic setup for the analysis of the
fourfold pattern (Tversky and Kahneman, 1992): Individuals decide between two
options, a certain prospect and an uncertain prospect with two outcomes. The
uncertain prospect is the SS, which we assume (i.e., UberX) obeys a truncated
Poisson distribution with $K=1$, i.e., the passenger is subject to at most one
delay. Therefore, the two possible outcomes of the SRS are
$(\underline{x}+b\gamma)$ and $(\overline{x}+b\gamma)$. The corresponding
probabilities can be determined using:

$f_{X}^{P}(x)=\begin{cases}\frac{1}{Z^{P}}\frac{{{(\lambda}^{P})}^{k}e^{-{\lambda}^{P}}}{k!},\,&\text{if}\,x=\overline{x}-k\frac{\overline{x}-\underline{x}}{K}\\\
0,&\text{otherwise}\end{cases}$ (1.16)

$f_{X}^{P}(\underline{x})=\frac{{\lambda}^{P}}{{\lambda}^{P}+1},\quad
f_{X}^{P}(\overline{x})=\frac{1}{{\lambda}^{P}+1}$ (1.17)

The four scenarios above are realized through suitable choices of $R$ and
${\lambda}^{P}$ as follows. A dynamic reference point $R$ is chosen to be
either $(\underline{x}+b\gamma)$ or $(\overline{x}+b\gamma)$, the SRS is a
gain if $R=\underline{x}+b\gamma$ and a loss if $R=\overline{x}+b\gamma$. The
SRS is considered high probability or low probability when the outcome that is
not regarded as the reference can be realized with a probability of
$p_{\text{NR}}$ or $(1-p_{\text{NR}})$ respectively, where $p_{\text{NR}}$ is
close to 1. In the computational experiments presented in Figure 1.4,
$p_{\text{NR}}=0.95$. Moreover, the range of the tariff is chosen as follows

$\begin{cases}\underline{x}+b\gamma<A^{o}&\text{if}\;R=\underline{x}+b\gamma\\\
\overline{x}+b\gamma>A^{o}&\text{if}\;R=\overline{x}+b\gamma\end{cases}$
(1.18)

such that the objective utility of the certain prospect, $A^{o}$, lies in the
same gain or loss regime as the SRS and therefore represents a reasonable
alternative to the SRS.

With the uncertain and the certain prospect defined in the SRS context above,
we illustrate the fourfold pattern in Figure 1.4 using four quadrants.
According to the fourfold pattern (a)-(d) in Figure 1.4, the diagonal
quadrants should correspond to risk averse behavior while the off-diagonal
ones are risk seeking. In each quadrant, we plot a metric defined as
$\text{RA}=(U^{o}-A^{o})-(U^{s}_{R}-A^{s}_{R})$ with respect to the tariff
$\gamma$. This metric captures the Relative Attractiveness that the uncertain
prospect has over the certain prospect for rational individuals versus
individuals modeled with CPT. This follows since according to Equation 1.12
and Equation 1.13, $\text{RA}>0\Rightarrow p^{o}>p^{s}_{R}$. In Figure 1.4, we
note that $\text{RA}>0$ corresponds to all regions where the blue curve is
above zero and indicates risk averse attitudes, as rational individuals have
higher probability to accept the uncertain prospect than irrational ones.
Similarly, $\text{RA}<0$ corresponds to the blue line being below zero and
denotes risk seeking attitudes. In each quadrant, two subplots are provided,
where the subplot on the right corresponds to a specific set of parameters
$\beta^{+}=\beta^{-}=\lambda=1$ which completely removes the role of
$V(\cdot)$, while the subplot on the left corresponds to standard CPT
parameters chosen in the range $0<\beta^{+},\beta^{-}<1,\;\lambda>1$, and
therefore a general CPT model. And as explained before, each quadrant
corresponds to a specific choice of $R$ and ${\lambda}^{P}$, which together
determine if an outcome is a gain or loss, and whether with high or low
probability.

Figure 1.4: Illustration of the fourfold pattern of risk attitudes in the SRS
context.

The most important observation from Figure 1.4 comes from the differences
between the left and right subplots in each of the four quadrants. For
example, from Figure 1.4(a), all risk attitudes in the right subplot
correspond to $\text{RA}>0$ and therefore risk averse, while those on the left
are only risk averse for a certain price range. That is, the four fold pattern
is violated in the left subplot. The same trend is exhibited in all four
quadrants. This is because, the fourfold pattern is due to the interplay
between $\pi(\cdot)$ and $V(\cdot)$ and is valid only when the magnitude of
$\pi(\cdot)$ is sufficiently large relative to that of $V(\cdot)$, such that
probability distortion dominates (Harbaugh et al., 2009). This corresponds to
the right subplots222The subplot on the right in each quadrant corresponds to
the case where individuals are risk neutral in the gain or loss regimes
separately, and loss neutral, then $\pi(\cdot)$ alone is sufficient to
generate the fourfold pattern. as well as the left subplots within certain
price ranges.

The implication that we obtain from the analysis of the fourfold pattern of
risk attitudes is that the resulting four categories can suitably inform the
dynamic pricing strategy in the SRS, through the left subplots. That is, it
allows a quantification of two qualitative statements (1) the presence of risk
seeking passengers gives flexibility in increasing tariffs, and (2) the
presence of risk averse passengers requires additional constraints or bounds
on reasonable tariffs.

#####  1.3.2.2 Implication II: Strong Risk Aversion over Mixed Prospects

The other implication of the CPT framework is strong risk aversion over mixed
prospects. A mixed prospect is defined as an uncertain prospect whose
portfolio of possible outcomes involves both gains and losses (Kahneman and
Tversky, 2012; Abdellaoui et al., 2008). Clearly, the uncertain prospect is
always mixed when the reference point $R$ corresponds to its expectation
(i.e., the expected value of its outcomes). The strong risk aversion of mixed
prospects stems from loss aversion, as the impact of the loss component often
dominates its gain counterpart. This implication will be illustrated below in
the SRS context using two different interpretations.

The first interpretation follows from Property 3, which essentially states
that when $R=\bar{U}$, the subjective utility is strictly negative for a
sufficiently large $\lambda$. Therefore, with $R=\bar{U}$ and such a
$\lambda$, the uncertain prospect is subjectively perceived as a strict loss.
This has been verified numerically with ${\lambda}>1$. Since the objective
utility relative to the expectation is neutral, hence strong aversion is
exhibited.

The second interpretation follows from Property 4, which essentially states
that when Property 3 holds, within the tariff range
$[\underline{{\lambda}},\overline{{\lambda}})$, the uncertain prospect is less
likely to be accepted by the CPT inclined passengers compared with the
rational ones, as $p^{s}_{\bar{U}}<p^{o}$.

Figure 1.5: Comparison of $p^{s}_{\bar{U}}$ and $p^{o}$. For fair comparison,
the tariff range of $\gamma\geq\frac{A^{o}-\bar{X}}{b}$ is plotted, where the
alternative is non-loss.

Fig. 1.5 illustrates Property 4 with $f_{X}(x)$ obeying a Normal distribution,
with the tariff range $\underline{\gamma}$ and $\overline{\gamma}$
approximated using the numerical setup. It is clear from the left subplot that
within this price range, passengers exhibit strong risk aversion over the SRS,
as the orange curve is strictly above the blue one. It is interesting to note
that when ${\beta}^{+}=1$, which corresponds to the case when passengers are
risk neutral in the gain regime, the maximum possible tariff
$\overline{\gamma}\rightarrow\infty$ (see Figure 1.5(b)).

The implication regarding strong risk aversion over mixed prospects is as
follows: As the SRS has significant uncertainty, for passengers who regard the
expected service quality as the reference, and when the alternative is
relatively a non-loss prospect, strong risk aversion is exhibited. Hence the
SRS is strictly less attractive to these passengers when compared to rational
ones. Therefore, the dynamic tariffs may need to be suitably designed by the
SRS server so as to compensate for these perceived losses. Rebates and
subsidies may be a few typical examples.

#####  1.3.2.3 Implication III: Self Reference

In this section, we compare $p^{s}_{\bar{U}}$ with $p^{s}_{A^{o}}$. Four
different probability distributions are considered. In each case, how these
two probabilities vary with the tariff $\gamma$ were evaluated. The results
are shown in Figure 1.6.

Figure 1.6: Comparison of $p^{s}_{\bar{U}}$ with $p^{s}_{A^{o}}$ using four
different PDFs $f_{X}(x)$. In the truncated Poisson distribution, the
parameters are set as ${\lambda}^{P}=4$ and $K=5$.

Fig. 1.6 illustrates that for all four distributions, $p^{s}_{\bar{U}}\geq
p^{s}_{A^{o}},\forall\;\gamma$, which implies that the SRS is always more
attractive when the reference is the expectation of itself, rather than the
alternative. $p^{s}_{\bar{U}}=p^{s}_{A^{o}}$ when
$\gamma=\frac{A^{o}-\bar{X}}{b}$ therefore $\bar{U}=A^{o}$ hence the two
reference points coincide.

The following summarizes the third implication inferred from Figure 1.6:
$\bar{U}$ is essentially the rational counterpart of the uncertain prospect.
Therefore, it could be argued that, when deciding between two prospects, the
chance to accept one prospect is always higher if this prospect itself is
regarded as the reference, compared with the case where the alternative is
considered as the reference. This is due to loss aversion, i.e., $\lambda>1$,
and can be explained thus: When one prospect is regarded as the reference, by
definition, it would never be perceived as a loss and therefore not experience
the magnified perception out of losses, whereas the alternative may be subject
to being regarded as a loss and therefore can experience this skewed
perception. In contrast, if the alternative is chosen as the reference, the
roles are reversed333Other effects of CPT due to
$\alpha,{\beta}^{+},{\beta}^{-}<1$ may result in complicated nonlinearities
which might alleviate loss aversion. Therefore, this statement is valid when
$\lambda$ is sufficiently large, such that loss aversion dominates.. Moreover,
the statement is in fact intuitive as those passengers who regard the
expectation as the reference have in some sense already subscribed to the SRS,
hence are naturally inclined to exhibit a higher probability of acceptance and
therefore have higher willingness to pay. This partially explains the reason
why converting customers from competitors is typically more difficult than
maintaining the current customer base. The last observation from Figure 1.6 is
the invariance of the comparison with the underlying probability
distributions, which implies that the above implications on self reference are
fairly general.

### Summary and conclusions

Several examples of CPHS include real-time decisions from humans as a
necessary building block for the successful performance of the overall system.
Many of these problems require behavioral models of humans that lead to these
decisions. In this article, we describe two different tools that may be
suitable for determining these behavioral models, which include Utility Theory
and Prospect Theory. Tools from Utility Theory have been used successfully in
several problems in transportation such as resource allocation and balance of
supply and demand. This theory is described in Section 1.2 and illustrated
using a transportation example that consists of a shared mobility problem
where human riders are presented with the choice of different travel modes. We
then show how these models can be used to address and mitigate traffic
congestion. Cumulative Prospect Theory, an extension of Prospect Theory, is a
modeling tool widely used in behavioral economics and cognitive psychology
that captures subjective decision making of individuals under risk or
uncertainty. CPT is described in Section 1.3, with the same transportation
example used to illustrate its application potential. Results from a survey
conducted with about 1000 respondents are used to derive a CPT model and
estimate its parameters. Lottery questions were also included in the survey to
illustrate CPT effects, the results of which are described in this section as
well. Finally, a few theoretical implications of CPT are presented that
provide an overall quantitative structure to the qualitative behavior of
humans in overall decision making scenarios.

Human-in-the-loop behavioral models can be applied to several other
applications beyond ridesharing, both within the transportation and mobility
sector, and other domains that involve humans interacting with cyber-physical
systems. One such example includes demand response in power grids. Future
research involves the development of more accurate models of human operators
for which more quantitative data is required with human decision makers as
integral components of the overall infrastructure. For instance, these could
include more sophisticated utility functions for different modes that also
take into account factors other than price and travel time which could affect
passengers’ choices. In addition, more realistic data sets on ridesharing,
mode choice and dynamic pricing could be obtained either by conducting larger,
representative surveys or through pilot studies and field trials performed in
conjunction with ridesharing companies or transit authorities.

### Acknowledgments

This work was supported by the Ford-MIT Alliance.

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