A Choice Model of Utility Maximization and Regret Minimization

Abstract

Consumers often try to achieve multiple goals when purchasing products and services, where choices are sought that maximize utility and other objectives such as to minimize regret. If the performance of choice outcomes are associated with high level of uncertainty at the time of purchase, consumers worry that the alternative of interest may turn out to be not optimal, an outcome they want to avoid when making a purchase. The authors propose a new regret function and explore its properties. Then, they propose a generalized framework of multiple goal pursuit and apply it to a utility maximization and regret minimization problem. Pareto-optimal sets are an outcome of multiple goal optimization problems where there are multiple alternatives that are nondominated. The proposed framework enables the authors to generalize a dual-goal problem as a constrained optimization problem where either utility is maximized subject to the constraint on regret or regret is minimized subject to the utility constraint. The proposed model fits the data better, provides improved predictions, and offers a tractable solution to a problem of utility maximization and regret minimization.

Keywords

multiple goal pursuit performance uncertainty consideration sets

Consumer choices are often linked to multiple goals. When the performance of choice outcomes are uncertain, consumers care about not only maximizing their expected utility from a transaction but also avoiding the possibility that their choices turn out to be suboptimal. The uncertainty in the benefits produced by products requires decision models that extend beyond just taking expectations of utility because poor product performance may lead to regret. The influence of regret is easy to imagine in medical procedures where outcomes are highly uncertain, but it also enters into less catastrophic decisions where there is still interest in “playing it safe.”

This research addresses the situation in which consumers make decisions under performance uncertainty of products and services. The standard discrete choice model cannot be successfully applied to these decisions because it assumes that consumers make choices with full information on the outcomes. In the standard random utility model (Manski et al. 1981), the unobservable error term represents information known to consumers at the time of purchase but unknown to the researcher. It is a strong assumption that consumers have full information on every choice alternative. Service offerings often entail high levels of performance uncertainty even when the service features are observable to consumers. The performance of the outcomes is largely dependent on factors associated with the application context, and it is not easy to predict how the outcomes will be realized. For example, grocery shoppers during the COVID-19 pandemic worry that they might get exposed to the virus and do not know in advance whether the in-store shopping will be a success, a complete disaster, or anywhere in between, even if they know about the safety measures the stores take (i.e., observed features). In this case, every shopping alternative within consumers' choice set can potentially be optimal, and consumers try to optimize the choice outcome by evaluating not only the expected utility (i.e., benefit) of interest but also the possibility that the forgone alternatives turn out to be better. We develop a model that accounts for this choice behavior by including an error whose realization is not known to the decision maker and relating it to a concept of anticipated regret. People experience regret when their choice turns out to be suboptimal, and they may want to avoid this when making decisions that maximize utility.

One approach to modeling multiple goal pursuit is to construct an overall objective function that combines a set of subobjective functions with appropriate weights (e.g., Dellaert et al. 2018; Swait and Marley 2013; Swait et al. 2018; Wallin et al. 2018). This approach assumes that a goal-balancing objective consisting of linear combination of those goals exists and that the goals are fully exchangeable across arguments of the subfunctions, which may not be easy to represent in the presence of subfunction nonlinearities. For example, it might be that less salt is preferred to more salt, holding the taste of an item fixed, but consumers might be willing to tolerate a little more salt in dishes with exceptional taste. Likewise, consumers may not choose an alternative despite the high expected utility when they cannot tolerate even the slightest anticipated regret. More fundamentally, multiple-objective problems are formulated because it is difficult to conceive of collapsing the objectives onto a common scale. An alternative approach that does not require specification of a single objective function is known as an ε-constraint method (Haimes et al. 1971), where all but one of the subobjective functions are represented as constraints to the optimization of the remaining goal.

The advantage of the proposed method is that the constraints help identify optimal solutions for the remaining goal being optimized. There exists N possible formulations when pursuing N goals, as each can be selected as the remaining goal, and it is therefore necessary to compare which formulation fits the observed data best. Constraints are not the same as objectives, where the former leads to regions in the solution space that contain viable solutions (e.g., the budget set) and the latter is typically represented as a more continuous function with properties that conform to regularity conditions (e.g., the utility function). While it is always possible to think about reformulating a constrained-utility-maximizing problem into a cost-minimization problem, one of the two formulations will fit the data better depending on the true form of the utility and cost structures.

In this article, we propose a solution for a discrete choice model where consumers maximize utility and minimize regret associated with their choices. A regret function is formulated in terms of unanticipated choice outcomes, and its properties are discussed. The additional constraint works to identify a set of nondominated alternatives from which a final choice is made, and operates in a manner similar to a model of consideration and choice. Model estimation requires the collection of data on acceptable candidate offerings that can be used to identify multiple constraining functions. We apply our model to data on grocery shopping, pain relievers, and air travel using national samples of respondents.

The article is organized as follows: First, we develop a regret function and discuss its properties. We then propose a solution to a dual-goal problem of utility maximization and regret minimization, along with an associated expression for the model likelihood, and justify our approach. We next present model estimation procedures. The following sections then present empirical results and related discussion. Finally, we offer concluding remarks.

Model Development

We begin our model development by proposing a new regret function. Loomes and Sugden (1982) define regret as something that an individual experiences when the forgone consequence is more desirable than the consequence as a result of choice. Bell (1982) defines regret as a sense of loss a person experiences after making a decision under uncertainty and discovers that other alternatives would have been preferable. The key construct underlying the definition of regret is a sense of loss individuals experience associated with the fact that the forgone alternatives would have been the better option. According to Hetts et al. (2000), individuals can forecast counterfactual alternatives and the associated psychological costs by mentally simulating the potential outcomes. Thus, consumers can anticipate regret prior to their purchases and incorporate it into decision making (Zeelenberg et al. 1996).

Regret is pervasive in consumer decision making, as all products are associated with some level of performance uncertainty. Performance uncertainty arises because of variation in production, variation in the usage context, and unforeseen factors that affect the ability of products and services to influence our state of being. Some products, such as goods sold in a supermarket, are characterized by low levels of production uncertainty and near-constant consumption contexts. This is not the case in product categories where outcomes are less certain and features are less familiar to consumers, such as in many services, medicine, and travel decisions.

There have been a number of attempts to model consumer's dual-goal pursuit of utility maximization and regret minimization. Loomes and Sugden (1982) and Bell (1982) proposed utility models that incorporate regret components. They argued that the expected utility cannot fully explain human behavior and some people try to trade off financial return to avoid regret. Chorus et al. (2013) proposed a random regret minimization (RRM) model as an alternative to a random utility maximization model to explain human decision making. RRM takes a semicompensatory form, where anticipated regret comes from pairwise comparisons of observed attributes. Regret is subtracted from utility to arrive at an overall objective function for explaining choice. However, studies in the field of transportation research (e.g., Chorus et al. 2013) have shown that utility maximization, regret minimization, and the weighted sum of both show similar model fit.

Our generalized approach to modeling dual-goal optimization is to employ the ε-constraint method, where different goals are separately handled via a constraining function and an objective to be optimized. The method is naturally transferred to a discrete choice setting where consumers may have to target one goal and settle for the other at acceptable levels if conflict between the two goals exists.

Anticipated Regret

We begin with the source of regret and propose a new regret function. We first introduce an additional error term in the utility for a product that represents performance uncertainty. The additional error term enables us to model anticipated regret associated with making a nonoptimal choice. We then propose a regret function and explore its properties.

Performance uncertainty

In a conventional random utility model, the utility function consists of deterministic component V and random error component $η$ denoted by

U = V + η, η \sim f (\cdot) .

(1)

Here, the assumption is that while consumers are deterministic and rational utility maximizers, researchers are unable to observe all of the relevant factors that may have affected consumer choice behavior. However, the assumption that consumers have full knowledge of product performance prior to purchase is a strong one in many cases. This is especially true for goods whose performance characteristics are difficult to be quantified with certainty, such as travel and medical procedures. There may be different forms of uncertainty affecting the performance in different settings, and we denote this uncertainty as an additional error term

ξ

. Unlike

η

, which is assumed to be unknown to researchers but known to consumers,

ξ

is a component that is also unknown to consumers at the time of purchase.

$ξ$ is distinguished from the uncertainty addressed in dynamic models of consumer learning, where the quality uncertainty can be solved via usage and experience over time. For example, any medical procedures, despite their well-known efficacy, may lead to unexpected side effects or failure even when they are conducted over time. Flight departures are affected by external accidental factors, and consumers who travel a lot still do not know whether a flight will be delayed and for how long. These irreducible consumer errors are characterized by the choice outcomes affected by the application contexts. Table 1 briefly describes these two error terms $η$ and $ξ$ . We propose a new utility function involving two error terms:

U = V + η + ξ, η \sim f (\cdot), ξ \sim g (\cdot) .

(2)

Table 1.

Explanation of Two Error Terms.

Errors	Representation	Information Unknown To:
$η$	Private information of consumers	Researcher
$ξ$	Performance uncertainty of an alternative	Consumer and researcher

Under the proposed specification, consumers do not know all the potential outcomes and their probability of happening, but they know there will be instances when their choices turn out to be not optimal because of their uncertainty of the error realizations $ξ$ that are not yet realized. We assume that $ξ$ is independent of $η$ . If a nonzero correlation were to arise, it would have to be because the researcher's belief that the decision maker’s known private information is somehow related to their unknown information, which is difficult to justify. We therefore assume that the two error terms are independently distributed.

Regret function

Anticipated regret is a sense of concern when the chosen alternative might not be the best option. This cost would be affected by both the possibility of that happening and the magnitude of the outcome once it occurs; therefore it is innately context dependent. Consumers would compare the expected outcome of choosing a focal alternative and the best expected outcome of choosing one of other alternatives available.

Assuming that there is a choice set $A = {j : 1, 2, \dots, J}$ , the expected utility of choosing alternative j is $E_{ξ} (U_{j})$ where $U_{j}$ is the utility associated with alternative j and $E_{ξ} (\cdot)$ indicates expectation with respect to the probability distribution $ξ$ . Every alternative in the choice set $A$ has a nonzero probability of being optimal, and the expected maximum utility of not choosing j is $E_{ξ} [{max}_{\binom{n \neq j}{n \in A}} {U_{n}}]$ . The anticipated regret associated with an alternative $j \in A$ can be measured by the difference between the expected maximum utility of forgone choices and the expected utility of the selected option:

\begin{aligned} R_{j}^{A} = & E_{ξ} [max_{\binom{n \neq j}{n \in A}} {U_{n}}] - E_{ξ} (U_{j}) \\ = & E_{ξ} [max_{\binom{n \neq j}{n \in A}} {V_{n} + η_{n} + ξ_{n}}] - E_{ξ} [V_{j} + η_{j} + ξ_{j}] . \end{aligned}

(3)

The expectation is taken with respect to $ξ$ only, because $η$ is private information known to the decision maker when making a choice.

We assume $ξ_{j} \sim N (0, 1)$ . Ideally, we expect that the model can be extended to capture alternative-specific scale parameters $σ_{j}$ if data are more informative, but due to identification issues we fix the values of $σ_{j} = 1, \forall j$ . Then the regret function can be expressed as follows:

\begin{aligned} R_{j}^{A} = \sum_{n \neq j} & [\int (V_{n} + η_{n} + ξ_{n}) ϕ (ξ_{n}) \prod_{m \neq n} Φ \\ (V_{n} + η_{n} + ξ_{n} - V_{m} - η_{m}) d ξ_{n}] - V_{j} - η_{j}, \end{aligned}

(4)

where

ϕ

is the probability density function and

Φ

is the cumulative distribution function of the standard normal distribution. The derivation of Equation 4 can be found at Web Appendix A. The regret function has the following properties:

$R_{j}^{A}$ is a decreasing function of deterministic utility of focal product $j \in A$ .

\begin{aligned} \frac{\partial R_{j}^{A}}{\partial V_{j}} = \frac{\partial}{\partial V_{j}} (\sum_{n \neq j}^{} [\int_{} (V_{n} + η_{n} + ξ_{n}) \prod_{m \neq n} Φ \\ \times (V_{n} + η_{n} + ξ_{n} - V_{m} - ξ_{m}) d ξ_{n}] - V_{j} - η_{j}) \\ = - 1 < 0 \end{aligned}

(5)

Anticipated regret decreases as the utility of the focal product increases.

2. $R_{j}^{A}$ is a nondecreasing function of deterministic utility of one of forgone products $n * \in n$ .

For arbitrarily small number

Δ > 0

added to

U_{n^{*}}

\begin{aligned} R_{j Δ_{n} *}^{A} = & E_{ξ} [\max {U_{1} \dots, U_{n *} + Δ, \dots, U_{j}}] - E_{ξ} (U_{j}) \\ \geq E_{ξ} [\max {U_{1} \dots, U_{n *}, \dots, U_{j}}] - E_{ξ} (U_{j}) = R_{j}^{A} . \end{aligned}

(6)

The difference

| R_{j Δ_{n^{*}}}^{A} - R_{j}^{A} |

becomes larger when

U_{n^{*}}

increases. Intuitively, anticipated regret would be more influenced by attractive forgone alternatives. This is consistent with Zeelenberg’s (1999) suggestion that people expect regret when the most preferred alternative is not necessarily superior to other alternatives.

3. $R_{j}^{A}$ is a nondecreasing function of the number of choice alternatives in the choice set. For $A_{1} = {j : 1, 2, \dots, J}$ and $A_{2} = {j : 1, 2, \dots, J, J + 1}$ ,

\begin{aligned} R_{j}^{A_{2}} & = E_{ξ} [\max {U_{1}, U_{2}, \dots, U_{J}, U_{J + 1}}] - E_{ξ} (U_{j}) \\ = E_{ξ} [\max {U_{1}, U_{2}, \dots, max {U_{J}, U_{J + 1}}}] - E_{ξ} (U_{j}) \\ \geq E_{ξ} [\max {U_{1}, U_{2}, \dots, U_{J}}] - E_{ξ} (U_{j}) = R_{j}^{A_{1}} . \end{aligned}

(7)

The effect of adding unattractive goods to the existing choice set would be negligible on anticipated regret, but adding attractive options would make a big difference. Our regret function can explain overchoice or choice deferrals, which happens when there are a large number of alternatives in the choice set. Gourville and Soman (2005) found that product variety may not be beneficial when a nonalignable assortment increases because of the cognitive effort and the potential for regret faced by a consumer. In cases where choice deferral is an option, adding a second attractive alternative to what had been a one-alternative choice set has been shown to increase the frequency of not making a choice (e.g., Dhar 1997).

Our regret function is different from others proposed in the literature (Bell 1982; Loomes and Sugden 1982) in that it is empirically operationalizable and can be applied to a decision making that involves multinomial choice and multiattribute alternatives. Our measure of regret is determined by both the observable product features and an unobserved, and yet-to-be realized error term $ξ$ and therefore provides economic and psychological grounds as to why consumers experience regret, whereas the RRM framework proposed by Chorus et al. (2008) defines regret in terms of the observable attributes of products and is silent on the theoretical foundation of regret.

Dual-Goal Pursuit

We next consider a model of consumer decision making for individuals that simultaneously maximizes utility ( $U$ ) and minimizes regret ( $R$ ) under monetary restrictions:

\begin{aligned} \max U (x, z) \min R (x) \\ s . t . p^{'} x + z \leq E, \end{aligned}

(8)

where E is the budgetary allotment, x is a vector of inside goods, z is an outside good and p is a price vector. One way of solving this is to form a weighted sum of the two objective functions (Gass and Saaty 1955):

\begin{aligned} \max ω_{1} U (x, z) - ω_{2} R (x) \\ s . t . p^{'} x + z \leq E, ω_{1} > 0, ω_{2} > 0. \end{aligned}

(9)

This approach assumes that the two goals U and R are fully exchangeable and there exists an exchange rate

ω

that compensates one goal for another. Existing literature in the field of marketing and decision making has extensively used this approach (e.g., Dellaert et al. 2018; Swait and Marley 2013; Swait et al. 2018; Wallin et al. 2018). This approach works well when the pursuit of goals is compensatory. However, we argue that the linear rate of exchange does not always hold, and we propose a method that retains the two goals separately.

We propose a more generalized method to formulate multiple goal optimization (Haimes et al. 1971). Unlike the linear-combination-of-weights formulation, the proposed method generates unique Pareto-optimal points by varying the upper bounds of multiple constraints (Chiandussi et al. 2012). For our dual-goal optimization problem, the method can represent one of the two objectives as an additional constraint, such that

\begin{aligned} \max U (x, z = E - p^{'} x) \\ s . t . R (x) \leq θ . \end{aligned}

(10)

Alternatively, we can also think of a formulation where regret is an objective and utility works as a constraint with the lower bound, such that

\begin{aligned} \min R (x) \\ s . t . U (x, z = E - p^{'} x) \geq θ . \end{aligned}

(11)

The relationship between the original dual-goal problem and the corresponding ε-constraint formulation is illustrated with a simple example for j = 2 in a discrete choice setting. The following is an example where there is no conflict between the two goals (i.e., the utility-maximizing alternative is identical to the regret-minimizing alternative):

\begin{aligned} \max U (x_{1}, x_{2}) = 2 x_{1} + x_{2} \\ \min R (x_{1}, x_{2}) = - 3 x_{1} - x_{2} \\ s . t . x_{1} + x_{2} \leq 1, x_{1}, x_{2} \in {0, 1} . \end{aligned}

(12)

Figure 1, Panel A, depicts the original dual-goal programming problem in the case of discrete choice, where only one of the alternatives is chosen. A consumer maximizes U and minimizes R within the monetary budget set (area filled with black oblique lines). The solution to this optimization problem is

(x_{1}^{*}, x_{2}^{*}) = (1, 0)

. Because the solution maximizes U and minimizes R simultaneously, it is the Pareto-optimal solution.

Figure 1.

Illustration of dual-goal programming (example 1).

We can convert the same problem to the corresponding proposed formulation (for an alternative formulation, see Web Appendix B):

\begin{aligned} max U (x_{1}, x_{2}) = 2 x_{1} + x_{2} \\ s . t . x_{1} + x_{2} \leq 1, R (x_{1}, x_{2}) \leq θ, x_{1}, x_{2} \in {0, 1} . \end{aligned}

(13)

R (x_{1}, x_{2})

, once used to be an objective function, now becomes an additional constraint represented as a thick orange line of Figure 1, Panel B, and the area filled with orange oblique lines above the constraint represents another budget set defined by R (i.e.,

R (x_{1}, x_{2}) \leq θ

). If the value of intercept −θ is smaller than or equal to 3 (shown as −θ′ in Figure 1, Panel B), a solution exists and the solution set

(x_{1}^{*}, x_{2}^{*}) = (1, 0)

is the same as the original problem. The solution

(x_{1}, x_{2}) = (0, 1)

can be also included in the feasible choice set depending on the value of −θ (shown as −θ″ in Figure 1, Panel C). This is a case where

(x_{1}, x_{2}) = (1, 0)

and

(x_{1}, x_{2}) = (0, 1)

are both feasible but

(x_{1}, x_{2}) = (1, 0)

is the most preferred. In either case, the solution set from the original problem and the corresponding ε-constraint method are identical as long as the solution exists.

Tension can exist between the two goals such that there is no unique solution that simultaneously optimizes both objective functions. Consider a second example:

\begin{aligned} \max U (x_{1}, x_{2}) = 2 x_{1} + x_{2} \\ \min R (x_{1}, x_{2}) = - x_{1} - 3 x_{2} \\ s . t . x_{1} + x_{2} \leq 1, x_{1}, x_{2} \in {0, 1} . \end{aligned}

(14)

This problem is illustrated in Figure 2, Panel A, where the two solutions

(x_{1}^{*}, x_{2}^{*}) = (1, 0)

and

(x_{1}^{*}, x_{2}^{*}) = (0, 1)

are both Pareto-optimal. That is, the solution

(x_{1}^{*}, x_{2}^{*}) = (1, 0)

is superior in terms of the first goal (

U

) and the second solution

(x_{1}^{*}, x_{2}^{*}) = (0, 1)

is superior in terms of the second goal (R).

Figure 2.

Illustration of dual-goal programming (example 2).

The proposed method solves the dual-goal problem by transforming one of the goals to a constraint and leads to a unique solution. We again focus on the formulation where R is converted into an additional constraint (for an alternative formulation, see Web Appendix B). In Figure 2, Panel B, objective R is transformed into a constraint, and the constraining value θ determines the set of feasible points with $(x_{1}^{*}, x_{2}^{*}) = (0, 1)$ optimal. Figure 2, Panel C, shows a solution set where θ takes on a different value and $(x_{1}^{*}, x_{2}^{*}) = (1, 0)$ is optimal. Feasible choice set $(x_{1}^{*}, x_{2}^{*}) = (0, 1)$ is identified by an observed consideration data. In either case, the proposed approach produces a unique solution, which naturally fits into a discrete choice setting where one choice alternative is observed. The value of θ plays an important role in determining the solution, and we treat θ as a model parameter.

The proposed method works because consumers may have to optimize one of the goals and settle for the other at an acceptable level when pursuing dual goals in a discrete choice setting. The method is flexible enough to incorporate different solution sets depending on the value of θ and captures a wide range of consumer choice behavior associated with dual-goal pursuit. The selection of which goal to retain is arbitrary, and the model likelihoods for the alternative formulations should differ. Therefore, it is important to compare the alternative formulations of the original problem when conducting analysis and to test which formulation best describes consumer decisions based on the model fit. We view this point as an advantage of the framework, because we do not have to know a priori which of the two goals consumers focus on. The posterior distribution of the threshold parameter θ is estimated in virtue of the observed consideration set data, and the remaining goal being optimized is retained as the objective function of the best-fitting model. The ε-constraint method, with its acceptable levels of performance, not only comes easily with simultaneous goal pursuit in discrete choice but also is a generalized framework that does not assume subfunction linearities. We can extend our framework to a setting where there are more than two goals (for the generalization of the framework, see Web Appendix C).

Likelihood

We assume a linear utility structure associated with models of discrete choice, and first consider the formulation where regret is modeled as a constraint. Because we focus on a discrete choice setting, the budget constraint E cancels out and we estimate the price coefficient instead. Then the probability that an alternative j is chosen is given by

\Pr (R_{j} \leq θ and U_{j} > U_{n} \forall n \neq j),

(15)

where

R_{j}

and

U_{j}

are the alternative-specific regret and utility, respectively. Alternatively, the formulation where regret is minimized and utility is used as a constraint leads to

\Pr (U_{j} \geq θ and R_{j} < R_{n} \forall n \neq j) .

(16)

For discussion purposes, we focus on the formulation in Equation 15 but examine both formulations in our empirical applications.

Our model requires data on the alternatives considered as viable candidates and also the most preferred alternative. Conceptually, observed consideration set data are necessary to understand which of the two objectives leads to the choice. For a given choice occasion, the probability option $j_{m}$ is chosen can be represented by the product of a marginal probability of consideration C, and the conditional probability of choice given consideration, y:

\begin{aligned} \Pr (j_{1}, \dots, j_{m} \in C, y = j_{m}) \\ = \Pr (j_{1}, \dots, j_{m} \in C) \Pr (y = j_{m} | j_{1}, \dots, j_{m} \in C) . \end{aligned}

(17)

The first factor in Equation 17 is the probability that choice alternatives

j_{1}, \dots, j_{m}

belong to a consumer's consideration set. Incorporating the proposed regret function of Equation 4, we assume that consumers only consider alternatives where anticipated regret associated with the focal product is less than a threshold θ. Alternatives whose anticipated regret exceed the threshold are screened out of a consumer's consideration set and become not feasible. Then the probability that alternatives

j_{1}, \dots, j_{m}

belong to C is

\begin{aligned} \Pr (j_{1}, \dots, j_{m} \in C) & = \Pr (R_{j_{1}} \leq θ, \dots, R_{j_{m}} \leq θ) \\ = \Pr {\begin{aligned} \sum_{n \neq j_{1}} [\int (V_{n} + η_{n} + ξ_{n}) ϕ (ξ_{n}) \prod_{m \neq n} Φ (V_{n} + η_{n} + ξ_{n} - V_{m} - η_{m}) d ξ_{n}] - V_{j_{1}} - η_{j_{1}} \leq θ, \dots, \\ \sum_{n \neq j_{m}} [\int (V_{n} + η_{n} + ξ_{n}) ϕ (ξ_{n}) \prod_{m \neq n} Φ (V_{n} + η_{n} + ξ_{n} - V_{m} - η_{m}) d ξ_{n}] - V_{j_{m}} - η_{j_{m}} \leq θ \end{aligned}} . \end{aligned}

(18)

Note that the joint probability in Equation 18 cannot be factored into the product of marginal distributions because the error term

η

of one alternative affects the anticipated regret of other alternatives.

Consumers are assumed to evaluate products up to $V + η$ , which are known to them and take expectations over $ξ$ (i.e., expected utility). The second factor in Equation 17 is the probability that product $j_{m}$ is chosen given that $j_{1}, \dots, j_{m}$ are in the consideration set computed as follows:

\begin{aligned} \Pr (y = j_{m} | j_{1}, \dots, j_{m} \in C) = \Pr (V_{j_{m}} + η_{j_{m}} > V_{j_{n}} \\ + η_{j_{n}}, \forall n \neq m | j_{m}, j_{n} \in C, η_{j_{m}}, η_{j_{n}} \in Ξ), \end{aligned}

(19)

where

Ξ = {η | R_{j} \leq θ, \forall j \in C}

. Because the final choice is made among the alternatives in the consideration set, the probability of choice must be defined conditional on the error term

η

that satisfies the first-stage inequality in Equation 18. Note that the proposed model is an integrated model of the dual-goal pursuit and the parameters will be estimated in a way to optimize the joint likelihood of consideration and choice given by Equation 15 or 16.

Model Estimation

The proposed formulation to the dual-goal optimization problem results in the likelihood that resembles the models of consideration and choice. We elaborate on how the likelihood is evaluated and address the related estimation challenges. We employ numerical methods to circumvent the computational challenges and provide the result of sensitivity analysis to guarantee the reliability of the methods. In addition, we introduce a random effect specification to incorporate consumer heterogeneity along with prior specifications, used in the estimation procedures.

Likelihood Evaluation

The likelihood of the data across I individuals and T choice occasions can be expressed as

\begin{aligned} L = \prod_{i = 1}^{I} \prod_{t = 1}^{T} [\Pr (R_{i j_{1} t} \leq θ_{i}, \dots, R_{i j_{m} t} \leq θ_{i}, R_{i j_{m + 1} t} \geq θ_{i}, \dots, R_{i j_{J} t} \geq θ_{i}) \\ \Pr (V_{ijt} + η_{ijt} > V_{i j^{'} t} + η_{i j^{'} t}, \forall j^{'} \neq j | j, j^{'} \in C, η \in Ξ)], \end{aligned}

(20)

where

j = j_{1}, \dots, j_{m}

indicates the alternatives that are in the consideration set and

j = j_{m + 1}, \dots, j_{J}

represents those that are not in the consideration set. The proposed model likelihood does not have a closed form, and we use Monte Carlo methods for the evaluation. That is, instead of analytically integrating out the error terms, we generate a large number of error draws to evaluate the likelihood by simulation. The probability of forming a consideration set that includes alternative

j = j_{1}, \dots, j_{m}

but does not include

j = j_{m + 1}, \dots, j_{J}

can be approximated by

\Pr (R_{i j_{1} t} \leq θ_{i}, \dots, R_{i j_{m} t} \leq θ_{i}, R_{i j_{m + 1} t} \geq θ_{i}, \dots, R_{i j_{J} t} \geq θ_{i}),

(21\rm a)

\begin{aligned} \approx \frac{1}{D} \sum_{d = 1}^{D} I {R_{i j_{1} t} \leq θ_{i}, \dots, R_{i j_{m} t} \leq θ_{i}, R_{i j_{m + 1} t} \geq θ_{i}, \dots, \\ R_{i j_{J} t} \geq θ_{i}}, η_{j}^{d} \overset{i . i . d .}{\sim} EV (0, 1), \end{aligned}

(21b)

where D is the number of error terms drawn. The probability that alternative j is chosen from the given consideration set C can be approximated by

\Pr (V_{ijt} + η_{ijt} > V_{i j^{'} t} + η_{i j^{'} t}, \forall j^{'} \neq j | j, j^{'} \in C, η \in Ξ),

(22\rm a)

\begin{aligned} \approx \frac{1}{D^{'}} \sum_{d^{'} = 1}^{D^{'}} I {V_{ijt} + η_{ijt}^{d^{'}} > V_{i j^{'} t} + η_{i j^{'} t}^{d^{'}}, \\ \forall j^{'} \neq j | j, j^{'} \in C, η \in Ξ}, \end{aligned}

(22\rm b)

where

D^{'} (\leq D)

is the number of error terms drawn at the first stage and satisfying the inequality in Equation 21b. Data required for the estimation of the proposed model include the observed consideration set and choice, and each type of the data is used to evaluate the likelihood given in Equations 21a and 22a, respectively. For example, if Alternatives 1 and 2 are in consumer's consideration set and Alternative 3 is not, the likelihood associated with the consideration set is

\Pr (R_{1} \leq θ, R_{2} \leq θ, R_{3} > θ)

. In addition, if Alternative 1 is finally chosen, then the likelihood associated with the choice is

\Pr (V_{1} + η_{1} > V_{2} + η_{2} | η_{1}, η_{2} \in Ξ)

, where

Ξ

is a set of error terms that satisfy

{η : R_{1} \leq θ, R_{2} \leq θ, R_{3} > θ}

condition.

Because the calculation of the regret measure is computationally intensive, we employ Clark’s (1961) approximation to estimate the regret function $R_{j} = E_{ξ} [ma x_{n \neq j} {V_{n} + η_{n} + ξ_{n}}] - E_{ξ} (V_{j} + η_{j} + ξ_{j})$ . According to Clark, when $X_{1}, X_{2}, \dots, X_{M - 1}, X_{M}$ are normally distributed random variables,

\begin{aligned} E [\max {X_{1}, \dots, X_{m}, \dots, X_{M}}] \\ = E [\max {X_{1}, \dots, X_{m}, \dots, \max (X_{M - 1}, X_{M})}] \\ \approx E [\max {X_{1}, \dots, X_{m}, \dots, X_{M - 2}, Y_{M - 1}}], \end{aligned}

(23)

where

Y_{M - 1}

is a two-moment Normal approximation to

\max (X_{M - 1}, X_{M})

. Because

Y_{M - 1}

is also approximately Normal, we can iterate this procedure until there is one term left in the Emax operator. Applying this approximation to our regret function,

X_{m} = U_{m} \sim N (V_{m} + η_{m}, 1) .

(24)

See Web Appendix D for the detail concerning implementation of the approximation algorithm.

In using Clark's (1961) approximation, we need to determine the sufficient number of draws of the error terms to guarantee stable estimates. We implement a sensitivity analysis by using D = (2,000, 3,000, 4,000) number of draws of $η$ . Simulation data are generated according to the proposed model with randomly generated variables for 600 respondents and 13 choice occasions per respondent, approximately the size of our empirical studies reported in the next section.

Table 2 reports the posterior mean and its standard deviation for each of the three cases. Figures in bold represent that the true values do not lie within the $95 %$ credible intervals. We find that 4,000 draws of $η$ are needed to obtain reliable estimates for a model that is representative of the size of models we investigate later. Web Appendix E provides Markov chain Monte Carlo (MCMC) trace plots for D = 4,000.

Table 2.

Sensitivity Analysis.

D	β₁	β₂	β₃	β₄	β₅	β₆	β₇	β₈	β₉	β₁₀	θ
True Value	1.5	1.5	1.4	1.3	.3	.6	1.2	.2	.7	−1.2	2.5
2,000	1.523 (.081)	1.589 (.081)	1.477 (.081)	1.414 (.079)	.343 (.043)	.571 (.045)	1.281 (.043)	.208 (.043)	.742 (.064)	−1.263 (.045)	2.443 (.055)
3,000	1.577 (.103)	1.593 (.104)	1.461 (.100)	1.299 (.106)	.317 (.043)	.588 (.044)	1.330 (.045)	.201 (.044)	.784 (.062)	−1.314 (.054)	2.390 (.054)
4,000	1.381 (.109)	1.433 (.111)	1.341 (.109)	1.163 (.110)	.282 (.041)	.626 (.045)	1.177 (.045)	.140 (.045)	.644 (.068)	−1.128 (.053)	2.431 (.056)

Notes: Standard deviations of the means are in parentheses.

Statistical inference

We incorporate consumer heterogeneity by introducing random effect specification for individual-level parameters and employing hierarchical Bayesian framework. We assume that individual-level parameters $β_{i} = [β_{i 1}, \dots, β_{iK}]^{T}$ and $θ_{i}$ follow a multivariate Normal distribution with mean $\bar{β}$ and covariance $Σ_{β}$ , such that

(\begin{matrix} β_{i} \\ θ_{i} \end{matrix}) \sim MVN (\begin{matrix} \bar{β}, & Σ_{β} \end{matrix}) .

(25)

Prior distributions are given by

\bar{β} | Σ_{β} \sim MVN (\bar{\bar{β}}, Σ_{β} \otimes A^{- 1})

and

Σ_{β} \sim IW (v, v \cdot I_{nvar})

, where nvar denotes the dimension of

β

, and

I_{nvar}

is an (nvar × nvar) identity matrix. Diffuse priors are set assuming consumers do not have prior information on the parameters of interest. We provide detailed estimation procedures in Web Appendix F.

Empirical Analysis

We apply our proposed model to three sets of data collected to examine consumer preference for grocery shopping, pain relievers, and air travel. We investigate the empirical performances of the proposed framework, along with alternative models. The grocery shopping study examines the potential COVID-19 infection of grocery shoppers and represents a choice setting where regret is potentially large and hard to predict. The pain relievers study replicates the grocery shopping study in that the side effects after taking pain relievers may lead to large regret. The air travel study examines regret associated with delayed flights, a setting where potential regret exists but is smaller and the bad outcome is not devastating.

Grocery Shopping and COVID-19

The COVID-19 outbreak has changed people's grocery shopping behavior in many ways. Consumers worry that the store visit might increase the chance of infection even when all staff members are required to take safety measures, and they want to avoid the potentially fatal outcomes. In response, the stores have begun to provide delivery and curbside pickup service and to display their current safety measures status online. We investigate how different types of delivery and COVID-related safety measures affect consumer’s grocery shopping behavior.

Data description

We collected data, via Amazon Mechanical Turk (MTurk), from a national sample of individuals who regularly grocery shop. Respondents who did not grocery shop themselves or were unwilling to try home delivery or curbside pickup were screened out. We also screened out respondents who were suspected of having provided random responses. This was implemented by fitting standard multinomial logit (MNL) and eliminating respondents whose choice likelihood values are less than .3, a slightly higher number than a random choice likelihood (Allenby et al. 2014). The analyzed data set consists of 394 respondents who were presented 14 choice tasks.

Our model requires data on the consideration set and the preferred choice alternative to identify the model parameters. Consideration set formation is a natural process consumers go through, and it is known that consumers form consideration set even in the simplest choice settings (Allenby and Ginter 1995). We therefore collect data by asking respondents to indicate the alternatives they would consider, and the alternative they prefer the most among those considered. In the choice tasks, only the alternatives that are selected in the first question are programmed to appear in the following question for reliable collection of the data. Figure 3 illustrates an example choice task while the second question is displayed in the separate subsequent page in the real task.

Figure 3.

Example choice task: grocery shopping and COVID-19.

Respondents were asked to think about the next time they grocery shop when indicating decisions. Each choice task had five alternatives, and respondents were not allowed to opt out. Attributes included delivery options, minimum order, COVID-related safety measures, and price (delivery fee). Table 3 summarizes attributes and levels used in the choice tasks; we used as a baseline a store visit with minimum order $0 and no COVID-related measures.

Table 3.

Attributes and Levels: Grocery Shopping and COVID-19.

Attributes	Level 1	Level 2	Level 3	Level 4	Level 5
Delivery options	Store visit	Home delivery (within 1 h)	Home delivery (within 4 h)	Curbside pickup (by appointment)	Curbside pickup (within 1 h)
Minimum order	$0	$10	$20	—	—
COVID-related measures	None	Masks required for all staff	Masks and gloves required for all staff	Masks, gloves, and temperature checks required for all staff	—
Price	Continuous variable ranging from $5 to $10

Model fit and parameter estimates

We estimate the model using Bayesian MCMC methods. We run 80,000 iterations and keep every 10th draw to generate the posterior mean and standard deviation, with the first half of the draws discarded as burn-in. We randomly shuffle the choice tasks for each respondent and use 12 choice tasks for model calibration and the remaining 2 tasks to evaluate predictive performance.

We compare our proposed model with a set of alternative models, with and without consideration set data incorporated. The models without consideration set data are the MNL model for utility maximization, regret minimization, and a model where the choice probabilities are driven by a weighted sum between the utility maximization and the regret minimization. This latter model corresponds to the specification described in Equation 9. The consideration set data allow estimation of the constraint θ in the proposed models of dual-goal pursuit. The first formulation is one that maximizes utility as the objective function subject to the regret constraint, and the second formulation assumes that respondents minimize regret subject to the utility constraint. For these models, we use D = 4,000 error draws.¹

In-sample fits are measured with the log-marginal density (LMD) using the Newton–Raftery (1994) approximation and the Gelfand–Dey (1994) approximation. In-sample measures for MNL models compute the LMD associated with the observed choice data and those for the proposed models compute the LMD associated with both consideration set and choice data. We also compute hit probability and Watanabe–Akaike information criterion (WAIC; Watanabe 2010) for predictive measures and report the component-wise fits (i.e., choice for MNL models; consideration set and choice given consideration set for the proposed models). Table 4 displays the summary of in-sample and predictive fit results.

Table 4.

In-Sample and Predictive Model Fit: Grocery Shopping and COVID-19.

	In-Sample Fit				Predictive Fit
	(LMD-NR)		(LMD-GD)		(Hit Probability)			(WAIC)
Model	CS + CH	CH	CS + CH	CH	CS	CH \| CS	CH	CS	CH \| CS	CH
Utility max (MNL)	—	−2,224	—	−3,722	—	—	.677	—	—	4,079
Regret min (MNL)	—	−2,085	—	−3,967	—	—	.683	—	—	4,067
Weighted sum (MNL)	—	−2,187	—	−5,851	—	—	.679	—	—	4,063
Utility max with regret constraint	−7,799	—	−11,117	—	.399	.727	—	7,597	2,594	—
Regret min with utility constraint	−7,523	—	−10,817	—	.428	.776	—	7,160	2,125	—

We find that the best-fitting model of consideration is regret minimization with utility constraint based on Newton–Raftery and Gelfand–Dey approximations to LMD. The model is also predictively the most accurate. This suggests that grocery shoppers pursuing the dual-goal focus on avoiding the regret (derived from potential COVID-19 infection) once utility (derived from the quality and price of grocery shopping options) is within the acceptable level. As a COVID-19 infection may cause significant consequences to consumers, we can interpret this result that the choice outcomes are generated from consumers’ play-it-safe decision.²

Parameter estimates of the utility-maximizing MNL model and the best-fitting model are displayed in Table 5. We report the posterior mean and standard deviation of heterogeneity (i.e., not the standard deviation of the mean). We find that the proposed model of regret minimization with utility constraint has smaller intercepts and smaller partworths for other product attributes relative to the utility maximization (MNL). As expected, we find that strong COVID-related measures have a significant effect on consumer’s grocery shopping behavior. Inferences about aspects of product value and consumer utility are relatively unaffected by our proposed model, but the predicted demand is affected by the presence of constraints, as we show in the “Discussion” section.

Table 5.

Posterior Mean and Standard Deviation of Random Effects: Grocery Shopping and COVID-19.

	Utility Maximization		Best Model
Partworth	Mean	SD	Mean	SD
Home delivery (within 1 h)	2.30	5.99	1.22	4.66
	(1.52, 3.09)	(5.07, 6.93)	(.69, 1.74)	(4.20, 5.15)
Home delivery (within 4 h)	1.53	5.71	.95	4.44
	(.78, 2.26)	(4.87, 6.61)	(.43, 1.45)	(3.99, 4.92)
Curbside pickup (by appointment)	.76	5.77	.01	4.57
	(−.04, 1.53)	(4.85, 6.79)	(−.52, .51)	(4.13, 5.05)
Curbside pickup (within 1 h)	.70	5.79	.16	4.46
	(−.06, 1.43)	(4.85, 6.71)	(−.35, .65)	(4.03, 4.92)
Minimum order: $10	−.74	.84	−.46	.73
	(−1.02, −.46)	(.66, 1.05)	(−.57, −.35)	(.62, .84)
Minimum order: $20	−.84	1.02	−.61	.99
	(−1.11, −.58)	(.78, 1.28)	(−.75, −.48)	(.86, 1.14)
Measures: masks	3.32	2.99	2.42	2.70
	(2.82, 3.87)	(2.52, 3.52)	(2.10, 2.76)	(2.43, 3.00)
Measures: masks + gloves	4.02	3.88	3.18	3.35
	(3.45, 4.64)	(3.34, 4.48)	(2.82, 3.58)	(3.03, 3.70)
Measures: masks + gloves + temperature	4.98	4.55	3.76	3.83
	(4.35, 5.66)	(3.98, 5.20)	(3.35, 4.20)	(3.48, 4.22)
Price	−.55	.56	−.44	.53
	(−.63, −.47)	(.50, .63)	(−.50, −.38)	(.48, .58)
Threshold (θ)	—	—	.95	5.00
			(.41, 1.47)	(4.54, 5.48)

Notes: 95% credible intervals are in parentheses.

Pain Relievers

Pain relievers such as ibuprofen and acetaminophen are over-the-counter drugs that people use to relieve pain. Taking pain relievers might cause side effects, as is the case with other medications, and consumers have concerns of suffering potentially serious ones. We investigate how uncertain outcomes from taking pain relievers affect consumers’ purchase decisions.

Data description

We collected data, via MTurk, from a national sample of individuals who take pain relievers. Respondents who had never taken pain relievers or who were unwilling to take pain relievers were screened out. The analyzed data set consists of 381 respondents who were presented 12 choice tasks.

Respondents were asked to think about the next time they have pain and are looking for pain relievers when indicating decisions. Each choice task has four alternatives and a no-choice option. Attributes include type, form, dosage, probability of suffering side effects, and price. Price per 100 count ranges from $8 to $12, and the probability of having side effects such as severe skin reactions and sensitive stomach ranges from .1% to 5%. Manipulation of the probability of side effects is intended to induce consumers’ concern about unexpected outcomes associated with pain reliever choice. Table 6 summarizes attributes and levels used in the choice tasks.

Table 6.

Attributes and Levels: Pain Relievers.

Attributes	Level 1	Level 2	Level 3	Level 4
Type	Aspirin	Ibuprofen	Naproxen	Acetaminophen
Form	Tablet pills	Capsules	—	—
Dosage	Regular strength	Extra strength	—	—
Probability of side effects	Continuous variable ranging from .1% to 5%
Price (per 100 count)	Continuous variable ranging from $8 to $12

Model fit and parameter estimates

We estimate the model using Bayesian MCMC methods. We run 80,000 iterations and keep every 10th draw to generate the posterior mean and standard deviation, with the first half of the draws discarded as burn-in. We use 10 choice tasks for model calibration and the remaining 2 tasks to evaluate predictive performance.

Table 7 shows the in-sample and predictive fits of different models. The MNL models for utility maximization, regret minimization, and the weighted sum of the two show almost identical performance, and we find that the best-fitting model is regret minimization with the utility constraint, which is also the most predictively accurate. Consumers interested in taking pain relievers focus on avoiding the unexpected outcomes once a certain level of effectiveness (i.e., utility) is achieved. This result is consistent with the grocery shopping data where consumers are concerned with possible COVID-19 infections and replicates the case where regret minimization plays a large role. We can interpret this result as consumers caring about potentially serious side effects and wanting to avoid negative consequences.

Table 7.

In-Sample and Predictive Model Fit: Pain Relievers.

	In-Sample Fit				Predictive Fit
	(LMD-NR)		(LMD-GD)		(Hit Probability)			(WAIC)
Model	CS + CH	CH	CS + CH	CH	CS	CH \| CS	CH	CS	CH \| CS	CH
Utility max (MNL)	—	−1,616	—	−3,644	—	—	.614	—	—	4,207
Regret min (MNL)	—	−1,466	—	−3,575	—	—	.625	—	—	4,175
Weighted sum (MNL)	—	−1,582	—	−3,577	—	—	.617	—	—	4,194
Utility max with regret constraint	−6,022	—	−9,278	—	.408	.746	—	7,592	2,403	—
Regret min with utility constraint	−5,276	—	−8,462	—	.429	.752	—	7,129	2,434	—

Table 8 displays parameter estimates of the utility-maximizing MNL model and the best-fitting model. The probability of side effects in all models are scaled by $\frac{1}{100}$ ; therefore 1% is coded as 1. Ibuprofen is most preferred in both models, and the probability of side effects seems to be an important factor, as expected. Extra strength is, in general, more preferred than the regular strength pain relievers. We find that the proposed model of regret minimization with the utility constraint has smaller intercepts and smaller partworths relative to utility maximization except for capsules, which suggests that the preference for capsules over tablet pills is underestimated in the utility-maximization model.

Table 8.

Posterior Mean and Standard Deviation of Random Effects: Pain Relievers.

	Utility Maximization		Best Model
Partworth	Mean	SD	Mean	SD
Aspirin	11.51	6.81	8.28	5.30
	(10.24, 12.81)	(5.65, 8.07)	(7.34, 9.15)	(4.47, 6.18)
Ibuprofen	13.65	7.13	9.97	5.14
	(12.28, 15.03)	(5.91, 8.43)	(9.04, 10.87)	(4.37, 5.98)
Naproxen	11.51	7.54	8.03	5.91
	(10.19, 12.87)	(6.36, 8.78)	(7.06, 9.01)	(5.13, 6.78)
Acetaminophen	12.10	7.26	8.80	5.63
	(10.79, 13.47)	(6.06, 8.56)	(7.86, 9.73)	(4.85, 6.48)
Capsules	−.08	1.36	.20	.89
	(−.32, .14)	(1.13, 1.62)	(.07, .31)	(.78, 1.00)
Extra strength	1.28	1.57	.52	1.27
	(1.02, 1.54)	(1.28, 1.86)	(.37, .67)	(1.13, 1.41)
Pr(side effects)	−2.41	2.25	−1.29	1.34
	(−2.76, −2.10)	(1.94, 2.58)	(−1.45, −1.13)	(1.20, 1.51)
Price	−.67	.68	−.30	.45
	(−.78, −.57)	(.59, .78)	(−.36, −.25)	(.41, .50)
Threshold (θ)	—	—	5.63	2.93
			(4.91, 6.35)	(2.50, 3.45)

Notes: 95% credible intervals are in parentheses.

Air Travel

According to the Bureau of Transportation Statistics (2019), the percentage of delayed flights for the major carriers such as American Airlines, Delta Airlines, and United Airlines for the last 10 years is about 20%, and those for the low-cost carriers are even higher. Thus, the probability of on-time arrival is one of the main concerns air travelers have and the source of performance uncertainty that cannot be fully solved.

Data description

We collected data, via MTurk, from a national sample of individuals planning air travel. We excluded from the sample straightliners (respondents who answered the same way for every question) and unqualified respondents who have never traveled by plane before. The analyzed data set consists of 391 respondents who were presented 15 choice tasks.

Respondents were instructed to assume that they are purchasing one-way airline tickets from New York City to Los Angeles. Each choice task has four alternatives and a no-choice option. Attributes include brand, price, probability of on-time arrival, and two additional features air travelers should take into account when purchasing the airline tickets. The price ranges from $150 to $230, which reflects the actual cost, and the probability of on-time arrival from 30% to 90%. Table 9 summarizes attributes and levels used for randomized design in air travel data.

Table 9.

Attributes and Levels: Air Travel.

Attributes	Level 1	Level 2	Level 3	Level 4
Airline	American	Delta	United	Southwest
Departure time	6 a.m.	9 a.m.	12 p.m.	—
Flight duration	6 h, nonstop	7 h, one stop	8 h, one stop	—
Pr(on-time arrival)	Continuous variable ranging from 30% to 90%
Price	Continuous variable ranging from $150 to $230

Model fit and parameter estimates

We use Bayesian MCMC for model estimation. We run 80,000 iterations and keep every 10th draw to generate the posterior mean and standard deviation, with the first half of the draws discarded as burn-in. We use 13 choice tasks for model calibration and the remaining 2 tasks to evaluate predictive performance.

Table 10 displays the summary of in-sample and predictive fit results. We find that the best-fitting model of consideration is utility maximization with the regret constraint, which is also the most predictively accurate. Unlike the grocery shopping and pain relievers data, consumers interested in air travel take anticipated regret into account along with the expected utility but aim at maximizing utility once the anticipated regret is within the acceptable level. We speculate that this result may be due to the nature of airline choice, where choosing the nonoptimal alternatives should not necessarily lead to a disastrous outcome. Benefits generated from air travel options are relatively easy to predict, and consumers may not be very concerned about the unexpected outcomes.

Table 10.

In-Sample and Predictive Model Fit: Air Travel.

Model	In-Sample Fit				Predictive Fit
	(LMD-NR)		(LMD-GD)		(Hit Probability)			(WAIC)
	CS + CH	CH	CS + CH	CH	CS	CH \| CS	CH	CS	CH \| CS	CH
Utility max (MNL)	—	−2,765	—	−4,995	—	—	.587	—	—	4,449
Regret min (MNL)	—	−2,615	—	−5,018	—	—	.593	—	—	4,419
Weighted sum (MNL)	—	−2,710	—	−5,251	—	—	.593	—	—	4,384
Utility max with regret constraint	−9,184	—	−12,010	—	.338	.731	—	8,924	2,549	—
Regret min with utility constraint	−9,599	—	−13,302	—	.296	.733	—	1,0043	2,566	—

Table 11 displays parameter estimates of the utility-maximizing MNL model and the best-fitting model. The units of price in all models is scaled by 100 so that $100 is coded as 1. We report the posterior mean and standard deviation of random effects. We find that our proposed model of utility maximization with regret constraint has larger brand intercepts and smaller partworths for other product attributes relative to the utility-maximizing MNL. The monetized utility (i.e., $β_{k} / β_{p}$ ) is fairly constant across the models, but the threshold parameter θ allows alternatives to be either considered or not depending on the consumer's willingness to accept anticipated regret under the proposed model, as we discuss next.

Table 11.

Posterior Mean and Standard Deviation of Random Effects: Air Travel.

	Utility Maximization		Best Model
Partworth	Mean	SD	Mean	SD
American Airlines	8.91	11.41	12.03	11.09
	(7.49, 10.46)	(10.17, 12.78)	(10.12, 14.85)	(9.47, 13.29)
Delta Airlines	8.54	11.79	12.06	11.31
	(7.08, 10.10)	(10.53, 13.16)	(10.13, 14.91)	(9.68, 13.45)
United Airlines	8.82	11.26	12.18	11.03
	(7.41, 10.37)	(10.04, 12.64)	(10.28, 14.99)	(9.41, 13.19)
Southwest Airlines	8.52	11.84	12.10	11.24
	(7.07, 10.10)	(10.56, 13.25)	(10.18, 14.94)	(9.58, 13.40)
Departure: 6 a.m.	−.50	2.00	−.23	1.46
	(−.80, −.20)	(1.72, 2.28)	(−.41, −.05)	(1.30, 1.63)
Departure: 9 a.m.	1.07	1.55	.54	.93
	(.83, 1.32)	(1.27, 1.82)	(.40, .68)	(.81, 1.05)
Duration: 6 h, nonstop	3.43	2.85	2.24	1.97
	(3.03, 3.83)	(2.49, 3.25)	(2.02, 2.48)	(1.77, 2.18)
Duration: 7 h, one stop	.88	1.10	.81	.83
	(.65, 1.11)	(.90, 1.34)	(.68, .94)	(.72, .94)
Pr(on-time arrival)	8.65	6.18	6.46	5.35
	(7.75, 9.66)	(5.30, 7.13)	(5.85, 7.08)	(4.83, 5.93)
Price	−5.93	3.88	−4.49	3.55
	(−6.55, −5.37)	(3.38, 4.48)	(−4.91, −4.08)	(3.21, 3.93)
Threshold (θ)	—	—	2.10	1.33
			(1.96, 2.25)	(1.22, 1.46)

Notes: 95% credible intervals are in parentheses.

Discussion

We propose a model of dual-goal pursuit where consumers are assumed to maximize utility and minimize regret and discuss how to reformulate a multiple-goal problem in a discrete choice setting using the proposed method. We apply this framework to the context of grocery shopping, pain relievers, and air travel and find that a formulation where either utility or regret serves as a constraint fits the data better than the other across different contexts.

The proposed method enables us to explore different regions of a Pareto-optimal solution space by isolating regions where one goal is optimized and the remaining goals are within acceptable levels. This approach is sensitive to the value of the constraining values θ, which we estimate from data on alternative consideration. The proposed method generalizes the notion of a consideration set, where brands are considered if certain features are present (Gilbride and Allenby 2004), to one that allows for more complex functions of model parameters (i.e., performance measures), leading to the consideration or nonconsideration of an alternative. The proposed model of dual-goal pursuit, therefore, can be viewed as a model of consideration where the evaluation of an alternative is affected by the other alternatives in the choice set.

A strength and limitation of the MNL model is that it can represent a demand system for N goods using N−1 intercepts and one price coefficient. In our analysis, we investigate more complicated demand systems using additional features associated with aspects of grocery shopping, pain relievers, and air travel. The MNL model typically allows for only one price coefficient, so it is limited in its ability to provide insights into aspects of demand afforded by more complex models. For example, the source of volume due to a price increase is subject to the independence-of-irrelevant-alternative property of the MNL model (Manski et al. 1981), where changes in share due to a price decrease are proportional to the choice share of the alternatives (Dotson et al. 2018). The presence of regret as an additional objective in decision making allows for a richer understanding of competitive effects. In addition, the presence of a considered set of brands has implications for understanding the effects of price on demand. We explore both issues next.

Competitive Effect of Anticipated Regret

Anticipated regret associated with an offering is affected by the deterministic utility of the offering and configurations of other offerings in the choice set, as shown in the properties of the proposed regret function in the “Model Development” section. The measure of anticipated regret is context dependent by nature, and we investigate the effect of product assortments on the consideration of an alternative by comparing three choice scenarios for a representative decision maker purchasing a flight option. The representative individual has parameter values equal to the mean of the random-effects distribution, as reported in Table 11.

The first choice scenario is a baseline scenario that involves three choice options, including flights from Delta, American, and United, and the second and third scenarios have four choice options, where Delta introduces a new flight option in addition to the baseline. Table 12 summarizes the three choice scenarios and the corresponding anticipated regret associated with the alternatives based on the proposed model. The difference of the newly introduced options in the second and third scenarios lies in the flight duration and the price. Long flight duration and high price are associated with negative coefficient estimates, which makes the new alternative in the second scenario (New Alt₁) less attractive than the existing alternatives. In contrast, short flight duration and low price are associated with positive coefficient estimates, which makes the new alternative in the third scenario (New Alt₂) more attractive than the other alternatives in the choice set.

Table 12.

Anticipated Regret of Choice: Air Travel.

Attributes	Alt₁	Alt₂	Alt₃	New Alt₁	New Alt₂
Airline	Delta	American	United	Delta	Delta
Departure	12 p.m.	6 a.m.	12 p.m.	9 a.m.	9 a.m.
Duration	6h, nonstop	6h, nonstop	6h, nonstop	8h, one stop	6h, nonstop
Pr(on-time arrival)	50%	50%	50%	50%	50%
Price	$200	$210	$230	$230	$180

Utility	8.55	7.84	7.32	5.50	9.99
Scenario 1 (baseline)	−.37	.86	1.52	—	—
Scenario 2	−.36	.86	1.51	3.40	—
Scenario 3	1.49	2.27	2.80	—	−1.09

Notes: Bold numbers indicate alternatives in the consideration set. Threshold estimate θ = 2.10.

The utility of Alt₁, Alt₂, and Alt₃ does not change across the different scenarios, because there is no change in their profiles. Anticipated regret for those existing alternatives changes little in the second scenario even after a new alternative is introduced, as the new alternative is considered to be an inferior alternative. Anticipated regret for the existing alternatives in the third scenario, however, increases because the newly introduced alternative is likely to be superior and the consumer is prone to experience regret by choosing the other existing alternatives. The magnitude of increase in anticipated regret is the largest in Alt₁ (from −.37 to 1.49) and the smallest in Alt₃ (from 1.52 to 2.80), because Alt₁ was the most attractive option before the new alternative (New Alt₂) is introduced and therefore is affected the most.

Product evaluations based on the anticipated regret provide implications for competition. A threshold value of 2.10, as reported in Table 11, results in Alt₂ and Alt₃ being screened out of consideration in the third scenario, which suggests that the same alternatives can be either considered or not considered depending on the configuration of the local choice set. Even when a firm is regarded as a leader in terms of market share (i.e., Delta in Scenario 1), it still has its motivation to introduce another attractive option (i.e., New Alt₂) in the product line to gain substantial share at the expense of competing offerings. As the number of attractive alternatives increases in the marketplace, consumers will not pay attention to the seemingly inferior options.

Price Elasticities and Expected Demand

Tables 13 and 14 report on the average price elasticity of alternative models based on the grocery shopping data and the pain relievers data, respectively. We simulated parameter estimates for 1,000 respondents based on estimates of the distribution of random effects in Table 5, calculated the percent changes in choice probability in response to the 10% increase in price for options one at a time, and averaged across all the choices and individuals. The estimates of price elasticity are significantly different across the different models (Table 13). A 1% increase in price leads to an estimated 2.89% decrease in choice probabilities for the utility-maximizing MNL model and a .49% decrease in our proposed model (the best-fitting model). We can observe a similar pattern in the pain relievers data (Table 14). The estimates of price elasticity of our proposed models are smaller because of the effect of consideration sets, where price responsiveness is zero unless an alternative is considered. It suggests that the demand response to a price change is overestimated under MNL models.

Table 13.

Aggregate Price Elasticity: Grocery Shopping and COVID-19.

Model	Elasticity	95% Credible Interval
Utility maximization (MNL)	−2.89	(−3.32, −2.46)
Regret minimization (MNL)	−2.60	(−3.00, −2.19)
Weighted sum (MNL)	−2.90	(−3.38, −2.44)
Utility maximization with regret constraint	−1.58	(−1.89, −1.27)
Regret minimization with utility constraint	−.49	(−.58, −.38)

Table 14.

Aggregate Price Elasticity: Pain Relievers.

Model	Elasticity	95% Credible Interval
Utility maximization (MNL)	−4.90	(−5.61, −4.19)
Regret minimization (MNL)	−4.39	(−5.02, −3.75)
Weighted sum (MNL)	−4.94	(−5.68, −4.23)
Utility maximization with regret constraint	−2.77	(−3.23, −2.31)
Regret minimization with utility constraint	−2.75	(−3.23, −2.29)

We investigate the expected demand curves in response to a price change for plausible market scenarios. For the pain relievers data, we assume a market where competitors provide products with an average probability of side effects (1%) offered at an average price ($10). Figure 4, Panel A, displays the expected market share of aspirin under the utility-maximization (MNL) model (blue curve) and the proposed model (red curve). The expected demand curve under the proposed model is more gradual in response to price changes, which is consistent with the price elasticity results. The dotted lines indicate 95% credible intervals.

Figure 4.

Expected demand curves: pain relievers.

We also compute the expected demand in response to the probability of side effects varying from 0% to 5%. Figure 4, Panel B, represents the expected demand curves of aspirin under both models, and we can observe that the expected demand curve under the proposed model is more gradual than that under the utility-maximization model. The intuition from the expected demand curves is that drastic demand changes may not occur even when the product features are improved if there exist additional constraints that form consumers' consideration set.

Concluding Remarks

This article contributes to the literature on consumer decision making in two ways. First, we proposed a new model of anticipated regret and explored its properties. Anticipated regret increases when nonfocal alternatives are more attractive, choice options are more equally valued, and the number of choice alternatives increases. We developed the formulation by introducing an additional error term into a model of random utility that represents consumer uncertainty in the performance of choice alternatives, which provides a theoretical foundation as to why consumers experience regret. When regret minimization is coupled with utility maximization, the resulting ε-constraint formulation of dual-goal optimization provides a generalized model of consideration and choice where constraints are based on multiple product attributes and model parameters. In particular, regret-based constraints suggest a new model of consideration that depends on configurations of the local choice set, which provides implications for product competition.

Second, we proposed a generalized method to address dual-goal optimization in the context of discrete choices, which can be extended to multigoal problems. Multi-objective optimization problems are common and occur when it is difficult to balance conflicting objectives. Examples include consumers deciding on the type of vehicle they want to purchase (e.g., sports car vs. minivan), patients deciding on treatment options for diseases with adverse side effects, and firms trying to maximize profits while being socially responsible. The proposed method transforms the original multi-objective problem with Pareto-optimal alternatives into a series of alternative formulations involving one objective and additional constraints on choice. The alternative formulations can be compared empirically to identify the best representation of decision making.

There are many avenues for future research. First, our model of multiple-goal pursuit assumes that the different goals are known. The discovery of alternative goals can be investigated by integrating alternative forms of data, such as text, into an extended model of choice. Identifying the goals consumers reportedly pursue would enrich the interpretation of model parameters and help provide implications for product development. Second, we can extend our model to the context of demand models for variety. Consumers may purchase different alternatives because of the tastes of different family members, not because they satiate on buying the same alternative. A third avenue for future research is in the development of a learning model that accounts for regret before and after product use.

Finally, we acknowledge the following limitation of our model. Our proposed framework depends on the observed consideration set, which can be difficult to observe in many contexts, for identification. This may be seen not so much a limitation as a necessity. We found that the consideration set data are reliable when collected via conjoint studies. In addition, now that we can have more access to observed consideration set (e.g., the “compare products” feature on Best Buy's website) than ever before, we expect that this limitation will be less important in the future.

Supplemental Material

sj-pdf-1-mrj-10.1177_00222437221094824 - Supplemental material for A Choice Model of Utility Maximization and Regret Minimization

Supplemental material, sj-pdf-1-mrj-10.1177_00222437221094824 for A Choice Model of Utility Maximization and Regret Minimization by Taegyu Hur and Greg M. Allenby in Journal of Marketing Research

Footnotes

Associate Editor

Eric Bradlow

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Online supplement:

ORCID iDs

Taegyu Hur

Greg M. Allenby

Notes

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