Multialternative context effects obtained using an inference task

Abstract

When decision-makers are faced with a choice among multiple options that have several attributes, preferences are often influenced by how the options are related to one another. For example, consumer preferences can be influenced, and even reversed, by the context defined by available products. This article discusses three standard context effects found in the preferential-choice literature: the attraction, similarity, and compromise effects. While decision theorists have attempted to explain these three effects under single modeling accounts, it has never before been demonstrated that these effects can be obtained within the same experimental paradigm. A set of experiments is described that demonstrate the three effects in an inference task. The paradigm is completely novel, as no previous experimental work has examined the standard context effects in inference. The experiments also add to evidence that the effects are not confined to choices among options with affective value, such as consumer products. The experimental results provide evidence that these effects might be a general property of human choice behavior and bring into question explanations of the effects that are based on the concept of loss-aversion asymmetry.

A number of empirical studies have demonstrated the importance of context on choice preferences in situations involving several options with multiple attributes. Three effects—the attraction (Huber, Payne, & Puto, 1982), the similarity (Tversky, 1972), and the compromise (Simonson, 1989) effects—have been central to research on contextual sensitivity in multialternative decision-making. While decision theorists have attempted to explain these three effects under single modeling accounts, there has been no empirical evidence suggesting that the three effects can be obtained under the same experimental paradigm. This article offers the first account of the three standard context effects occurring in the same domain.

The three standard context effects appearing in the literature—the attraction, similarity, and compromise effects—arise in choices among three alternatives that have two attributes. For example, in a typical consumer goods decision task, participants might be asked to choose among a set of three cars that vary on the two attributes of price and quality. The options can be graphically represented in a two-dimensional attribute space, as is illustrated in Fig. 1. In the figure, the x- and y-axes correspond to the attribute values, and points in the space correspond to different options. Context effects arise from changes in the relative preferences for the options due to their placement in the attribute space. The attraction effect occurs when a dominated option (e.g., option R_B in Fig. 1) increases the probability of selecting the dominant alternative (e.g., B). The similarity effect arises when similar options (e.g., S_B and B) compete with one another, so that the relative preference for a dissimilar option (e.g., C) increases. The compromise effect occurs when an enhancement occurs in the probability of choosing an intermediate alternative (e.g., C) when extreme options (e.g., B and D) are included in the choice set. More details about the effects are given below within the experiments.

Fig. 1

The various options used in the inference experiments, plotted in a two-dimensional space defined by the two attribute values. The context effects were assessed by comparing ternary choice sets to other ternary choice sets. The options labeled R, F, and RF refer to attraction decoys, where R represents range decoys, F represents frequency decoys, and RF represents range–frequency decoys. Range decoys are dominated on the focal option’s weakest attribute, frequency decoys are dominated on the focal option’s strongest attribute, and range–frequency decoys are dominated on both of the focal option’s attributes. The choice sets {B, D, R_B} and {B, D, R_D} were used to assess the range attraction effect; choice sets {B, D, F_B} and {B, D, F_D} were used to assess the frequency attraction effect; and choice sets {B, D, RF_B} and {B, D, RF_D} were used to assess the range–frequency attraction effect. Options labeled S refer to similarity decoys. The choice sets {B, C, S_B}, {B, C, S_C1}, {C, D, S_C2}, and {C, D, S_D} were used to assess the similarity effect. Finally, the choice sets {A, B, C}, {B, C, D}, and {C, D, E} were used to assess the compromise effect

All three effects violate the simple scalability property (Krantz, 1964; Tversky, 1972). This property states that alternatives in a choice set, A, can be given a strength scale value, u, that is independent from the other options, and the probability of selecting a particular option is determined by the general formula ${ \Pr }\left[ {x\left| {\text{A}} \right.} \right] = F\left[ {u(x),u(y),{ }.{ }.{ }.{ },u(z)} \right]$, where F is an increasing function in the first variable and a decreasing function in the remaining variables. This property underlies most of the utility models used to study choice behavior, including Luce’s (1959) “ratio-of-strengths” model.

The experimental paradigm discussed in this article is an inference task in which participants are asked to make decisions about criminal suspects. The three context effects were tested in three separate experiments. These experiments tested how people infer which suspect out of a set of three is most likely to have committed a crime, on the basis of two separate eyewitness testimonies. In these scenarios, the suspects represent the different choice options, and the eyewitness testimonies represent the two attributes, in a manner similar to a consumer product with the attributes of quality and price.

This paradigm is completely novel, as no previous experimental work has examined the three standard context effects in inference. The experiments also add to the evidence suggesting that context effects are not confined to choices among options that have affective value such as consumer products. In a choice among consumer products, personal desirability plays an important role in how the attributes are evaluated. However, in the present paradigm, the task is not to select the most desirable option, but to select the most likely one. In other words, the judge is asked to evaluate the relative likelihoods of each alternative on each attribute rather than to assess the options on the basis of personal likes and dislikes of the attributes.

Previous research has provided evidence of the similarity and attraction effects occurring in domains without affective value. In Tversky’s (1972) original demonstration of the similarity effect, he used three types of experimental stimuli: squares containing dots, candidates for a scholarship, and gambles. More recently, Choplin and Hummel (2005) found the attraction effect using unidimensional perceptual stimuli. Maylor and Roberts (2007) obtained attraction and similarity effects in an episodic memory task, and Tsetsos, Usher, and McClelland (2011) obtained the similarity effect using time-varying psychophysical stimuli. While these studies have greatly added to our understanding of context effects, they have not demonstrated the occurrence of all three effects in the same, non-consumer-goods domain.

Experiment 1: the attraction effect

The attraction effect refers to an enhancement in the choice probability of an option through the introduction of a similar but inferior alternative. Consider the choice set {B, D} and two decoys, A_B and A_D, where A_B is a similar but inferior option as compared to B, and A_D is a similar but inferior option as compared to D. The attraction effect occurs when people show a stronger preference for option B when it is presented along with its inferior comparison (A_B), and similarly for option D. Formally, the attraction effect occurs when the probability of choosing B is greater when the decoy favors B than when it favors D, and vice versa: Pr[B |{B, D, A_B}] > Pr[B | {B, D, A_D}] and Pr[D |{B, D, A_B}] < Pr[D | {B, D, A_D}]. This method of using all ternary choice sets to test the effect follows from Wedell (1991) and helps us achieve a more powerful test by increasing the effect size.

A further distinction is made by the three different types of dominated options: range, frequency, and range–frequency decoys (Huber et al., 1982). Figure 1 shows a graphical representation of these different decoys. Range decoys (denoted R_B and R_D in Fig. 1) refer to options that increase the range of the attribute dimension on which the focal alternative is the weakest. In other words, these decoys are dominated on the focal alternative’s weakest attribute. Throughout the article, the term focal is used to refer to an option that is enhanced by the addition of a third alternative. For example, B would be considered a focal option in the choice set {B, D, R_B} because it is enhanced by R_B. Frequency decoys (denoted F_B and F_D) are dominated on the focal alternative’s strongest attribute value. These decoys increase the frequency of options with attribute values similar to the focal alternative’s superior dimension. Range–frequency decoys (denoted RF_B and RF_D) combine both a range and a frequency manipulation. Thus, these decoys are dominated on both the focal’s strongest and weakest attributes. All three decoys were included in this experiment because previous findings have shown that they produce different magnitudes of the attraction effect (Huber et al., 1982).

To test the attraction effect, six different ternary choice sets were used. The six sets arose from two choice sets for each of the three types of decoys (i.e., range, frequency, and range–frequency). The two choice sets for each type of decoy were designed to compare choice probabilities when the decoy favored option B with choice probabilities when the decoy favored option D, as is shown in Fig. 1. For example, the range attraction effect occurs when Pr[B |{B, D, R_B}] > Pr[B | {B, D, R_D}] and Pr[D |{B, D, R_B}] < Pr[D | {B, D, R_D}].

The attraction effect violates the simple-scalability property because, according to this property, the inequality Pr[B |{B, D, R_B}] > Pr[B | {B, D, R_D}] implies that the strength of R_B is less than the strength of R_D. However, the inequality Pr[D |{B, D, R_B}] < Pr[D | {B, D, R_D}] implies the exact opposite—that the strength of R_D is less than the strength of R_B.

Method

A group of 47 undergraduate students from Indiana University participated for course credit. The participants were told that they would see three suspects of a crime on each trial and were instructed to select the suspect who seemed most likely to have committed the crime, on the basis of the strengths of two eyewitness testimonies. The participants were told that the strengths of the eyewitness testimonies were reported on a 0–100 scale, with 0 implying very weak evidence of guilt and 100 implying very strong evidence of guilt. Participants were also told that the testimonies of both eyewitnesses were equally valid and important, and that the strengths of the testimonies were equated. Participants did not receive any feedback during the experiment, so there were no consequences for their selections.

The eyewitness strengths were determined by selecting a pair of points from the two-dimensional eyewitness strength space illustrated in Fig. 1. For example, suspects associated with location B in Fig. 1 were drawn from a bivariate normal distribution with mean (35, 65) and a variance equal to 1 on each dimension, and no correlation. Allowing for noise in the eyewitness strength values helped introduce variation in the task. The eyewitness strengths for other suspects (e.g., locations D and R_B in Fig. 1) were determined in a similar manner. The suspects and eyewitness strengths were presented in a table format with different suspects in different rows. The row locations of the suspects were randomized across the experiment. Table 1 gives sample values for the range attraction effect and is formatted similarly to the experimental presentation. Suspect initials were included to reinforce the use of a different set of suspects on each trial.

Table 1 Sample values for the range attraction effect, in a format similar to the experimental presentation

Each participant completed 240 randomized trials that were divided into 40 range trials, 40 frequency trials, 40 range–frequency trials, and 120 filler trials. The 40 trials for each type of decoy were further divided so that the decoy was placed near one alternative for half of the trials and near the other alternative for the remaining trials. The filler trials also used ternary choice sets and always contained one alternative that was clearly superior. These trials were used to assess accuracy throughout the experiment.

Results and discussion

Figure 2a shows the mean choice probability for the focal alternative as compared to the mean choice probabilities for the nonfocal alternative and decoy, collapsed across both possible positions of the decoy (i.e., favoring B vs. favoring D). Figure 2b shows a scatterplot of individual choice probabilities for the range, frequency, and range–frequency decoys. In the figure, each point represents an individual participant’s choice probabilities for the focal alternative plotted on the x-axis, and the nonfocal alternative plotted on the y-axis. Again, the choice probabilities for focal and nonfocal options are collapsed across both possible positions of the decoy.

Fig. 2

Experimental results for the attraction effect. (a) Mean choice probabilities for focal and nonfocal options with range, range–frequency, and frequency decoys. Error bars show the standard errors of the means. (b) Individual choice probabilities for focal and nonfocal options with range, frequency, and range–frequency decoys. Choice probabilities for the focal option are plotted along the x-axis, and choice probabilities for the nonfocal option are plotted along the y-axis. Individual choice probabilities for a particular alternative were calculated by counting the number of times a participant selected that alternative in all of the trials of a particular type. Points that fall below the diagonal line indicate participants who demonstrated the effect, because these points represent individuals who selected focal options more often than nonfocal options. The individual choice probabilities fall mostly along the negative diagonal because the decoy alternative was rarely selected

For data analyses, two participants were removed because their accuracy was two standard deviations lower than the average accuracy on the filler trials. Across the three types of decoys, the choice probability for the focal alternative was significantly larger than the choice probability for the nonfocal alternative (t = 2.631, p = .012). The three decoys were also analyzed individually by comparing the range choice sets {B, D, R_B} and {B, D, R_D}, the frequency choice sets {B, D, F_B} and {B, D, F_D}, and the range–frequency choice sets {B, D, RF_B} and {B, D, RF_D}. The range decoy produced the largest difference in choice probabilities between the focal (M = .56) and nonfocal (M = .39) options (t = 2.819, p = .007). The frequency decoy produced the second largest difference between the focal (M = .52) and nonfocal (M = .39) options (t = 2.390, p = .021). The range–frequency decoy produced the smallest effect (M = .52 for focal vs. M = .41 for nonfocal), but this difference was still significant (t = 2.216, p = .032).

The results support previous evidence that the attraction effect can be generalized to domains in which the options do not have affective value. Furthermore, as in the consumer preference experiments of Huber et al. (1982), the range decoy produced the largest effect out of the three types of decoys.

Experiment 2: the similarity effect

The similarity effect occurs when a competitive option similar to one of the existing alternatives is added to the choice set and the probability of selecting the dissimilar option increases. Consider the choice set {B, C} and two decoys, S_B and S_C1, where S_B is similar to B and S_C1 is similar to C, as is illustrated in Fig. 1. The similarity effect occurs when the probability of choosing B is greater when the decoy is similar to C than when it is similar to B, and vice versa: Pr[B |{B, C, S_B}] < Pr[B | {B, C, S_C1}] and Pr[C | {B, C, S_B}] > Pr[C | {B, C, S_C1}]. To test the similarity effect, four ternary choice sets were used. In Fig. 1, these sets are {B, C, S_B}, {B, C, S_C1}, {C, D, S_C2}, and {C, D, S_D}. In all of the sets, the decoy (i.e., S_B, S_C1, S_C2, or S_D) is a more extreme option, in the sense that it has more extreme attribute values than the similar alternative.

Like the attraction effect, the similarity effect also violates the simple-scalability property. According to this property, the inequality Pr[B |{B, C, S_B}] < Pr[B | {B, C, S_C1}] implies that the strength of S_B is greater than the strength of S_C1. Yet the inequality Pr[C | {B, C, S_B}] > Pr[C | {B, C, S_C1}] implies the opposite.

Method

A group of 51 undergraduate students from Indiana University participated for course credit. The participants received the same instructions as in the attraction experiment.

Each participant completed 240 randomized trials that were divided into 60 trials using options B and C, 60 trials using options C and D, and 120 filler trials. The trials testing the similarity effect were further divided so that the decoy was a similar, competing option that was placed near one alternative for half of the trials, and near the other alternative for the remaining trials. The filler trials were the same as before.

Results and discussion

Figure 3a shows the mean choice probability of the focal alternative as compared to the mean choice probabilities of the nonfocal alternative and decoy, collapsed across the two different trial types (i.e., trials using options B and C and trials using options C and D) and both possible positions of the decoy. Here, the term focal refers to the dissimilar alternative because this is the alternative that is enhanced by the decoy. Figure 3b shows a scatterplot of the individual participants’ choice probabilities for the focal and nonfocal alternatives, collapsed across the two trial types and both possible positions of the decoy.

Fig. 3

Experimental results for the similarity effect. (a) Mean choice probabilities, with error bars showing the standard errors of the means. (b) Individual choice probabilities. Choice probabilities for the focal option are plotted along the x-axis, and choice probabilities for the nonfocal option are plotted along the y-axis. Points that fall below the diagonal line indicate participants who demonstrated the effect

For the data analyses, three participants were removed because their accuracy was two standard deviations lower than the average accuracy on the filler trials. Across the two trial types, the choice probability for the focal alternative (M = .51) was significantly larger than the choice probability for the nonfocal alternative (M = .30) (t = 4.743, p < .001). Analyzing the two trial types separately, both the trials using options B and C (t = 5.701, p < .001) and the trials using options C and D (t = 3.673, p < .001) produced significant similarity effects.

The results support previous evidence that the similarity effect arises in a number of domains (Maylor & Roberts, 2007; Tsetsos et al. 2011; Tversky, 1972).

Experiment 3: the compromise effect

The compromise effect occurs when an option is selected more often when it appears to be a compromise with respect to the other alternatives in the choice set than when it appears to be an extreme. Specifically, suppose that there are two ternary choice sets {A, B, C}, with A and C being extremes, and {B, C, D}, now with B and D as extremes. The compromise effect occurs when B is preferred more often in the first set than in the second set, so that Pr[B | {A, B, C}] > Pr[B | {B, C, D}], and when C is preferred more often in the second set than in the first set, so that Pr[C | {A, B, C}] < Pr[C | {B, C, D}].

Following the experimental design of Simonson (1989), the compromise effect was tested using three ternary choice sets. In Fig. 1, these sets are {A, B, C}, {B, C, D}, and {C, D, E}. The alternatives B, C, and D are each a compromise alternative in exactly one set and an extreme alternative in one or two of the other sets. To test for the effect, the probability of selecting these alternatives when they were compromise options was compared to the probability of selecting them when they were extremes. Because C appears as an extreme option in two choice sets, a total of four comparisons can be made: B in sets {A, B, C} and {B, C, D}; C in sets {A, B, C} and {B, C, D}; C in sets {B, C, D} and {C, D, E}; and D in sets {B, C, D} and {C, D, E}.

As with the other two effects, the compromise effect also violates the simple-scalability property. The property implies that if Pr[B | {A, B, C}] > Pr[B | {B, C, D}], then the strength of A is less than the strength of D. However, the property also implies that if Pr[C | {A, B, C}] < Pr[C | {B, C, D}], then the strength of A is greater than the strength of D.

Method

A group of 52 undergraduate students from Indiana University participated for course credit. The participants received the same instructions as in the attraction experiment.

Each participant completed 240 randomized trials that were divided into 40 trials with the {A, B, C} choice set, 40 trials with the {B, C, D} choice set, 40 trials with the {C, D, E} choice set, and 120 filler trials. The filler trials were the same as before.

Results and discussion

Figure 4a shows the mean choice probability of the compromise alternatives as compared to the mean choice probabilities of the extreme and decoy alternatives collapsed across the three choice sets. Figure 4b shows a scatterplot of individual participants’ choice probabilities for the compromise alternatives and the extreme alternatives, collapsed across the three choice sets.

Fig. 4

Experimental results for the compromise effect. (a) Mean choice probabilities, with error bars showing the standard errors of the means. The term decoy is used here to refer to options that always appear as extremes and never as compromises (i.e., options A and E in Fig. 1). (b) Individual choice probabilities. Choice probabilities for the compromise option are plotted along the x-axis, and choice probabilities for the extreme option are plotted along the y-axis. Points that fall below the diagonal line indicate participants who demonstrated the effect

For the data analyses, one participant was removed because of accuracy two standard deviations lower than the average accuracy on the filler trials. Across the three choice sets, the choice probability for the compromise alternative (M = .48) was significantly larger than the choice probability for the extreme alternative (M = .38) (t = 3.796, p < .001). In analyzing the four possible comparisons separately, three of the four comparisons produced significant results. Specifically, the probability of selecting C was significantly larger in set {B, C, D} than in set {A, B, C} (t = 4.469, p < .001); the probability of selecting C was larger in set {B, C, D} than in set {C, D, E} (t = 2.587, p = .013); and the probability of selecting D was larger in set {C, D, E} than in set {B, C, D} (t = 4.404, p < .001). However, there was no significant difference in the probability of selecting B in set {A, B, C} as compared to set {B, C, D} (t = 1.224, p = .227). Even though there was not a significant difference between the choice probabilities for option B, the simple-scalability property was still violated by the other comparisons.

The results provide the first evidence that, as with the attraction and similarity effects, the compromise effect is not limited to situations involving options that have affective value.

General discussion

Because most utility models cannot be used to explain the three context effects due to violations of simple scalability, developing a single theoretical framework to model all three effects is an ongoing problem of interest for cognitive modelers. Currently, two models account for the three effects: multialternative decision field theory (MDFT; Roe, Busemeyer, & Townsend, 2001) and the leaky competing accumulators (LCA) model (Usher & McClelland, 2004). Both models are part of a class of models called sequential-sampling models. They assume that information is accumulated stochastically over time, and a decision is elicited after the accumulated information reaches a certain threshold. These models also capitalize on Tversky’s (1972) elimination-by-aspects heuristic by incorporating the sequential scanning of attributes. Because the models assume that a single set of cognitive mechanisms produces the three effects, demonstrating the effects in the same experimental paradigm provides a crucial test of this assumption. The present experiments represent the first attempt at performing this test.

Furthermore, the existence of the three effects in a domain in which the options do not have affective value poses a challenge to accounts of context effects that are based on the concept of loss-aversion asymmetry (Tversky & Simonson, 1993). Because the effects have customarily been demonstrated in tasks related to consumer preferences, where there are obvious possible losses, it has been difficult to determine whether or not loss aversion is a prerequisite concept for context effects. The present research takes the first step toward addressing whether or not loss aversion is necessary for explaining the effects by providing evidence for them in inference, where the notion of gains and losses is less clear, or even altogether absent.

While the MDFT and the LCA models have much in common, and even provide the same explanation for the similarity effect, they offer strikingly different explanations for the attraction and compromise effects. The MDFT approach models the attraction and compromise effects with a distance function that compares options along dominance and indifference dimensions (Hotaling, Busemeyer, & Li, 2010). On the other hand, the LCA model accounts for these effects by assuming that alternatives are compared to one another via an asymmetric value function that is consistent with Tversky and Kahneman’s (1991) and Tversky and Simonson’s (1993) loss-aversion function.

In multialternative decisions in which people are not given an explicit reference option, it is believed that the options are evaluated in relation to one another (Tversky & Simonson, 1993). In the LCA model, when an option is being considered, an individual evaluates the advantages and disadvantages of that option along each attribute with respect to the other alternatives in the choice set. By weighting disadvantages more than advantages, as described in Tversky and Kahneman (1991), the LCA model produces the attraction and compromise effects. The asymmetric weighting of advantages and disadvantages follows from the assumption that people exhibit more aversion for losses than attraction to gains (Tversky & Kahneman, 1991; Tversky & Simonson, 1993). In a task in which someone must select the most desirable option, as in a choice among consumer products, an asymmetric weighting of gains and losses could be reasonable. However, in the present experiments, it is not clear why an individual would weight differences in eyewitness testimonies favoring one suspect more than differences favoring another suspect. While it is true that an individual might feel loss if the wrong suspect is selected, this is a loss related to the correctness of the choice rather than due to trade-offs among attributes.

While the experiments bring into question loss–gain asymmetry, further experiments and tests are needed to completely rule out the use of other asymmetric functions. It could be possible to reformulate the asymmetric value function in the LCA model in terms of attention to both positive and negative differences rather than to gains and losses. Both MDFT and the LCA model make rich dynamic predictions about the time course of choice preferences, and it could be possible to discriminate the two models on this basis, as is discussed by Tsetsos, Usher, and Chater (2010). Because the present experimental design is nondynamic, future experimental paradigms will be needed to address these issues.