Incorporating uncovered structural patterns in value functions construction
Introduction
The shape of the value function is of great importance in different areas of research in decision analysis. In multiple criteria decision making, such a shape decides upon the contribution of various performances into the comprehensive value of an alternative [23]. In decision making under risk, it captures the risk attitude of a decision maker [37]. Specifically, convex or concave curvature of a value function is usually interpreted as whether the decision maker is risk-seeking or risk-averse, respectively, when the expected utility is assumed [17]. Moreover, it provides information on decision maker’s risk preference when prospect theory is considered [2], [57]. Also, S-shaped value function, proposed in the prospect theory, relates the decision maker’s risk attitude to the type of outcomes [35]. In conjoint analysis, this shape describes consumers’ trade-off behavior [25], [26] by revealing how they would react to the changes in product’s performance levels, which, in turn, provides insights on the sensitivity of a target market to the changes in features of a product or service [19]. In the same spirit, it has implications for the customer or patient satisfaction analysis [14], [27].
The debate on the shape of value function has a long history in behavioral economics and decision making [1], [35], [38], [42], [47], [58]. Based on the psychological foundations [35], empirical evidences [17], and recent insights from neuroscience [56], several functional shapes were proposed to explain and predict consumers’ choices. For instance, neoclassical economists suggested a concave shape for the value functions based on the law of diminishing marginal utility. Alternatively, [35] proposed a shape that is determined by diminishing marginal sensitivity in the domains of both gains and losses. Furthermore, [42] considered an S-shaped value function to incorporate both reference points and loss-aversion in decision making. Also, [18] proposed a shape with two concave regions at the two ends of the performance scale, with a convex region between them. On the contrary, researchers in decision analysis emphasize that the decision maker’s value function may hold any shape, and no prior assumptions about its functional form should be made [36], [37]. While recognizing the merits of parametric studies, [1] argues that their findings might have been confounded by the particular parametric families chosen. Hence a disadvantage of such an approach is that the estimations depend critically on the assumed functional form [6].
The selection of an appropriate shape of the value function is not straightforward. According to [37], the suitability of different techniques for assessment of value function depends on the decision problem, its context, and the decision maker’s characteristics. In the context of constructing value functions, based on a series of experimental analysis and empirical research, [29] reported a significant impact of several contextual factors on the shape of value functions. [39] characterized a set of conditions, involving trade-offs under certainty and others related to choice under uncertainty when selecting a utility form for assessing preferences. [22] reported a striking amount of heterogeneity in the value function’s degree of curvature across individual decision makers. A relationship between the curvature of the individual value function and individual differences in preferred decision modes, i.e., intuitive versus deliberative, was found by [50] through an analysis of preferences of 200 students. Also, [47] made a distinction between the local shape of a value function, such as local measures of curvature, and the global shape of a value function, defined as the general shape of a value function over the entire outcome domain. Based on the analysis of preferences of 332 owner-managers, they reported that the global shape of value function reflects the manager’s strategic decision structure. Such a shape is linked to the organizational behavior (i.e., the production system employed), and is more stable than the local shape that seems to drive tactical decision making [46]. Similarly, [45] suggested a relationship between the global shape of a value function and higher-order decisions. By analysis of preferences of portfolio managers, they found that the global shape of value functions, rather than the curvature, is related to the asset allocation strategies. Moreover, they suggested that the environment in which managers operate might play a role in shaping the global shape of the decision makers’ value function.
In this paper, we introduce an analytical framework for constructing value functions of a group of decision makers or a set of individuals with no prior assumptions on their shape, while acknowledging the potential existence of structural forms that would govern the general shape of these functions. Our approach is based on the assumption, motivated by the findings from the previously highlighted research, that the general shapes of individuals’ value functions are rather stable for the problem at hand, and they share some patterns that regulate their behavior over the respective domains. We characterize such patterns to be useful for a large class of functional forms and in the presence of heterogeneous value functions. Nevertheless, the effectiveness of our model might increase when individuals are exposed to similar external stimuli or share background or socioeconomic factors. Such situations are prevalent in the real-world since decisions typically are not made in isolation. Still, people share the same environment, influence each other through advice or sharing experiences, e.g., product reviews, inspiration, or imitation.
We adapt a multi-attribute value theory approach [16], [37]. Hence the proposed framework is applicable to decision problems which involve a set of alternatives (e.g., products or services) evaluated in terms of different criteria (e.g., features, benefits, or attributes with preference ordered performance scales). The input information is a set of binary choice examples, i.e., indirect preference information in the form of holistic pairwise comparisons, from each individual or decision maker. Individual value functions, one for each decision maker, are constructed simultaneously so that to assign a higher score to the alternatives preferred by the respective individual over other alternatives in the choice tasks. This setup is standard for choice-based conjoint analysis [40]. However, the most common conjoint estimation methods based on econometric models require certain assumptions on the distribution of preference parameters. In turn, our approach formulates the problem of preference estimation as an optimization problem (for an overview on the use of optimization techniques in preference estimation in conjoint analysis, see [12], [13], [54], [55]). Specifically, we formulate the underlying analytical framework in terms of convex optimization based on quadratic programming, which is computationally efficient and applicable to problems of realistic size.
When constructing the individual value functions, the proposed framework searches for regularities in the general shape of value functions by uncovering the structural patterns that regulate general behavior of value functions over different regions of the performance scales or domains of outcomes. To this objective, in a unified framework, individual value functions for all decision makers are examined within each region of a performance scale and are contrasted over pairs of different regions. Comparisons over the regions are exploited to capture structural patterns that regulate the general shape of value functions. Then, the uncovered structural patterns, if any, are incorporated into the preference estimation process.
In our model, the following two components characterize the structural patterns: i) the correlation in relative slopes of value functions (relative to the population mean) between all pairs of regions of the performance scale, and ii) the homogeneity in value functions’ slopes within a region of the performance scale. Such a simple characterization captures the general behavior of value functions in a large class of functional forms, including convex, concave, S-shape, inverse S-shape, or their combinations. For instance, a set of individuals with a mixture of concave and convex value functions might exhibit a pattern indicating that value functions with smaller slopes, relative to other individuals, at the most left region of the function’s domain, tend to demonstrate larger slopes at the most right region. Nevertheless, our model avoids making arbitrary inferences on structural patterns even if the forms of individuals’ value functions belong to such classes, e.g., when the exhibited patterns across the individuals are conflicting. For example, consider a group of individuals with S-shape, convex, or concave value functions, with half being of S-shape, the rest equally distributed between the remaining two classes and relatively similar curvatures within each class. In such a scenario, half of the population, associated with the concave and convex value functions, supports a negative correlation between the two extreme regions. In contrast, the other half, characterized by S-shape value functions, supplies evidence for a positive correlation. Hence, due to the specific assumed distribution of value functions among the three classes, no meaningful correlation between the two extreme regions of the performance scale could be observed.
For illustrative purposes, the above examples refer to a pair of extreme regions of the performance scale. Nonetheless, in our model, such comparisons are made for all pairs of regions and all individual regions in search of, respectively, systematic correlation and homogeneity. Therefore, the systematic patterns are identified between and over different regions of value functions when the individuals persistently provide evidence in their favor. Moreover, the two components of the structural patterns act in opposite directions. The first component – referring to the correlation in relative slopes – looks for evidence to make the value functions diverge from the population mean. The other component – considering homogeneity in slopes – seeks evidence to bring the value functions closer to each other. Transcendence of one component depends on the relative amount and strength of the evidence that is in its favor compared to those for the other component.
Compared to the generalized robust conjoint method of [12], our model is based on a stronger robustness condition. Specifically, we focus on value discrimination of the most “difficult” pairs, i.e., the pairwise comparisons that are the most troublesome in terms of being reproduced by an additive value function. Moreover, the generalized robust conjoint method admits inconsistency in data, while our model assumes that there exists a value function compatible with the individual’s choices. The use of a deterministic value function, while being widely accepted in the Multiple Criteria Decision Analysis (MCDA) literature [15], [23], [31], might appear to be a restrictive assumption in the context of choice-based conjoint analysis or repeated choice [40], [43]. The importance of this restriction has been recently recognized in the MCDA literature [32]. From the perspective of random utility models additively aggregating a deterministic value function with a stochastic component, our focus is on the construction of the deterministic core component. Various random utility functions – differing in how they expand from the deterministic core component – can be derived from this [43]. Hence, throughout our analysis, we ensure the fulfillment of this assumption or check its validity in the case of, respectively, simulation studies or interpretation of data from real decision makers. Furthermore, we formulate two approaches to extend our model for dealing with inconsistent data. They are independent of making assumptions on the stochastic component or assuming a particular distribution function and do not change the component that captures the structural patterns. With these links made clear, the proposed framework can be used in a wide range of choice-based conjoint analysis problems when the robust estimation of individual-level preference parameters is desired.
Let us emphasize that the framework developed in this paper is not aimed to address the problem of group decision making. In particular, we do not assume any particular interaction process among the individuals in their evaluations. Instead, pairwise comparisons from individuals are allowed and typically are assumed to be collected independently from one another. Moreover, seeking a consensus or compromise is not among the objectives of the developed framework. Choice examples from other individuals, however, are used to augment preference information of an individual. This contributes to improving the predictive validity of the individual-level estimated preference parameters, through a mechanism of uncovering structural patterns that regulate the general shape of value functions.
Using simulation analysis and data from real decision makers, we show that accounting for structural patterns in the estimation process of preferences improves the predictive accuracy of constructed value functions considerably. Our results are confirmed by the performance tests in the view of detecting the best choice, i.e., the most preferred alternative, estimating the entire ranking list for each individual, as well as prediction accuracy (hit-rate) on the newly generated questions.
Section snippets
Construction of value functions
Consider a set of R decision makers, . The universe of alternatives consists of N alternatives {a1, a2, ⋅⋅⋅, an, ⋅⋅⋅, aN}. Alternatives are evaluated based on a family of criteria . We use the notion of “criteria” as a general concept, which can represent product attributes, decision consequences, or different points of view in assessments of decision alternatives. Each criterion evaluates each alternative according to the criterion’s evaluation
Proposed framework: accounting for structural patterns in construction of value functions
The proposed framework aims at constructing value functions for a group of decision makers or a set of individuals with no prior assumption on the functions’ shapes. In turn, it accounts for the regularities observed in the general shape of these functions captured with some structural rules governing their curvature and global shape. The framework is based on identifying the structural patterns across different regions of a value function by joint evaluation of individual value functions
Experimental analysis based on simulation
In this section, we present a comprehensive experimental analysis based on a large number of carefully designed, randomly generated decision problems covering a wide range of settings. We compare performance measures on the prediction accuracy of the joint estimation model based on uncovering structural patterns against those of independent estimations. We also relate them to the characteristics of the decision problem setting, e.g., level of heterogeneity in simulated preference parameters or
Analysis of real decision makers’ preferences
In this section, we analyze the preferences of real decision makers using the independent and joint preference estimation methods, and compare the respective performances in terms of preference recovery and predictive accuracy (hit-rate). There are two crucial differences between the analysis of preferences elicited from real decision makers and the simulated preferences that are described in Section 4.2. First, data from real decision makers is typically noisy. Second, it does not provide any
Conclusions
This paper presented a simple characterization of structural patterns that possibly regulate the general shape of value functions. The introduced model was motivated by the results from empirical research and experimental analysis in the literature that support the existence of a stable global shape of individuals’ value functions in a specific decision problem. Our characterization proved effective in a broad class of shapes of value functions and in the presence of individuals with
CRediT authorship contribution statement
Mohammad Ghaderi: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Visualization. Miłosz Kadziński: Conceptualization, Methodology, Investigation, Validation, Writing - review & editing.
Acknowledgments
We thank two anonymous referees for their constructive comments. Mohammad Ghaderi acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563). Miłosz Kadziński acknowledges financial support from the Polish National Science Center under the SONATA BIS project (grant no. DEC-2019/34/E/HS4/00045).
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