Editor's evaluation: Clarifying the role of an unavailable distractor in human multiattribute choice

Abstract

Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Methods Data availability References Decision letter Author response Article and author information Metrics Abstract Decisions between two economic goods can be swayed by a third unavailable ‘decoy’ alternative, which does not compete for choice, notoriously violating the principles of rational choice theory. Although decoy effects typically depend on the decoy’s position in a multiattribute choice space, recent studies using risky prospects (i.e., varying in reward and probability) reported a novel ‘positive’ decoy effect operating on a single value dimension: the higher the ‘expected value’ (EV) of an unavailable (distractor) prospect was, the easier the discrimination between two available target prospects became, especially when their expected-value difference was small. Here, we show that this unidimensional distractor effect affords alternative interpretations: it occurred because the distractor’s EV covaried positively with the subjective utility difference between the two targets. Looking beyond this covariation, we report a modest ‘negative’ distractor effect operating on subjective utility, as well as classic multiattribute decoy effects. A normatively meaningful model (selective integration), in which subjective utilities are shaped by intra-attribute information distortion, reproduces the multiattribute decoy effects, and as an epiphenomenon, the negative unidimensional distractor effect. These findings clarify the modulatory role of an unavailable distracting option, shedding fresh light on the mechanisms that govern multiattribute decisions. Editor's evaluation This study presents an important finding on the decoy effect in multiattribute economic choices in humans. It makes a compelling case for the conclusion that the distractor effect reported in previous articles was confounded with the additive utility difference between the available alternatives. Though the contribution is somewhat narrowly focused with respect to the phenomenon that it addresses – the distractor effect in risky choice – it is important for understanding this particular phenomenon. https://doi.org/10.7554/eLife.83316.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Humans strive to make good decisions but are rarely veridical judges of the world. Instead, our judgements are swayed by seemingly irrelevant factors (Stone and Thompson, 1992) and our preferences are improvised as we go along (Summerfield and Tsetsos, 2015). For instance, echoing well-documented optical illusions, we may perceive the same doughnut as larger on a tiny plate than on a bigger one (Figure 1a). Analogous distortions are encountered when we choose among alternatives that vary in more than one dimension or attribute. For instance, in the attraction effect (Huber et al., 1982), adding a similar but less advantageous ‘decoy’ option (a disk drive with a good storage capacity but at a high price) can make a target alternative (a lower-priced disk with more storage) appear more appealing than a competing alternative (an affordable disk with lower storage). Figure 1 Download asset Open asset Context effects in decision-making. (a) Judgement about object size can be influenced by the surround. (b) Documented positive (Chau et al., 2014) or negative (Louie et al., 2013) distractor effects occur at the level of unidimensional value. A higher-valued distractor (DH), compared with a lower-valued distractor (DL), can increase the choice rate of A relative to B (positive distractor effect) but can also decrease the choice rate of A relative to B (negative distractor effect), leading to opposing distractor effects in past reports (Chau et al., 2014; Louie et al., 2013). (c) During risky choices, if people derive subjective utilities of prospects by integrating reward magnitude and probability additively (additive utility or AU), and if targets {A2, B2} are more frequently paired with distractor DH while targets {A1, B1} with DL, then a positive distractor effect could be explained by additive integration (because ΔAU1 < ΔAU2) rather than the distractor’s expected value (EV: magnitude × probability). Curves: EV indifference lines. (d) Similarly, relative preference changes for target alternatives A and B can be driven by an attraction-effect bias (Huber et al., 1982; Dumbalska et al., 2020) that depends on the position of the decoy D with respect to each target alternative. If DH appears more often closer to A while DL being more often closer to B, a positive distractor effect will ensue as a by-product of the attraction effect. In all illustrations, A-alternatives are always better than B-alternatives, with D-alternatives denoting the distractor (decoy) alternatives. The attraction effect together with related contextual preference-reversal phenomena violate the very notion of rational-choice explanation according to which preferences should be menu-invariant (Luce, 1977; Rieskamp et al., 2006). Instead, these phenomena suggest that preferences are menu-variant or context-sensitive, with the subjective utility (a notion of attractiveness) of an economic alternative being malleable to the properties of other available alternatives. Preference reversals have been typically reported in multiattribute choice settings, instigating a departure from ‘value-first’ utility theories (Vlaev et al., 2011)—in which attributes are integrated within each option independently—towards the development of novel theories of multiattribute choice (Busemeyer et al., 2019; Hunt et al., 2014; Tsetsos et al., 2010; Turner et al., 2018; Usher and McClelland, 2004). In contrast to multiattribute decisions, decisions over alternatives varying along a single attribute have been typically regarded as complying with the principles of rational choice theory (Bogacz et al., 2006). A plausible account for this dichotomy could be that multiattribute decisions are more complex. Recently however, a new type of context effect has been reported in unidimensional multialternative decisions. In the so-called negative distractor effect, humans become more indifferent between two unequal alternatives of high reward value when a third distracting option has a high, as opposed to a low, reward value (Louie et al., 2013; Figure 1b; although see a recent replication failure of this effect, Gluth et al., 2020a). This negative distractor effect is theoretically important because it shows that violations of rationality may not be restricted to complex multiattribute decisions (Carandini and Heeger, 2011; Louie et al., 2013). In turn, this motivates the development of a cross-domain, unified theory of decision irrationality. Interestingly, in a series of more recent experiments involving risky prospects (alternatives offering reward outcomes with some probabilities), the opposite pattern was predominantly documented (Chau et al., 2014; Chau et al., 2020; Figure 1b): the decision accuracy of choosing the best of two target prospects is particularly compromised by a low expected-value (EV), ‘unavailable’ (distractor) prospect that should have never been in contention. This disruption effect fades away as the distractor’s EV increases, leading to a positive distractor effect. However, Gluth et al., 2018 casted doubts on this positive distractor effect, claiming that it arises due to statistical artefacts (Gluth et al., 2018). Later, in return, Chau et al., 2020 defended the positive distractor effect by emphasising its robust presence in difficult trials only (it tends to reverse into a negative distractor effect in easy trials, i.e., distractor effects interact across different parts of the decision space; Chau et al., 2020). Overall, the abovementioned risky-choice studies paint a rather complex empirical picture on the ways a distractor alternative impacts decision quality. Notably, in these studies, key design aspects and analyses rested on the assumption that participants’ decisions were fully guided by the EV of the prospects. However, since Bernoulli’s seminal work, it has been well established that EV is a theoretical construct that often fails to adequately describe human decisions under uncertainty (Von Neumann and Morgenstern, 2007; Bernoulli, 1954). Instead, human decisions under uncertainty are guided by subjective utilities that are shaped by a plethora of non-normative factors (Tversky and Kahneman, 1992). Such factors—including the absolute reward magnitude of a prospect (Pratt, 1964), the riskiness of a prospect (Weber et al., 2004), or even the sum of the normalised reward and probability (Rouault et al., 2019; Stewart, 2011; Farashahi et al., 2019)—can perturb people’s valuations of risky prospects in ways that the expected-value framework does not capture (Peterson et al., 2021). We reasoned that if such factors covaried with the EV of the distractor, then the distractor effects reported previously could, partially or fully, afford alternative interpretations. It could be argued that if the experimental choice-sets are generated randomly and afresh for each participant, such covariations between the distractor’s EV and other factors are likely to be averaged out. However, we note that all previous studies reporting positive and interacting distractor effects (Chau et al., 2014; Chau et al., 2020) used the exact same set of trials for all participants. This set of trials was generated pseudo-randomly, by resampling choice-sets until the EV of the distractor was sufficiently decorrelated from the EV difference (choice difficulty) between the two targets. On the positive side, presenting the same set of trials to all participants eliminates the impact of stimulus variability on the group-level behaviour (Lu and Dosher, 2008). On the negative side, using a single set of trials in conjunction with this decorrelation approach, increases the risk of introducing unintended confounding covariations in the elected set of trials, which in turn will have consistent impact on the group-level behaviour. Here, we outline two classes of unintended covariations that could potentially explain away distractor effects in these datasets. First, the EV of the distractor could potentially covary with specific (and influential to behaviour) reward/probability regularities in the two target prospects (we call them target-related covariations). For instance, if people valuate prospects by simply adding up their payoff and probability information (hereafter additive utility or AU) (Rouault et al., 2019; Farashahi et al., 2019; Bongioanni et al., 2021), two target prospects both offering £1 with probability of 0.9 vs. 0.8, respectively (i.e., EV difference: 0.1, AU difference: 0.1), will be less easily discriminable than two other prospects both offering £0.5 with probability of 0.5 vs. 0.3, respectively (i.e., EV difference: 0.1, AU difference: 0.2; Figure 1c). Crucially, if the first choice-set (AU difference: 0.1) is more frequently associated with low-EV distractors than the second one (AU difference: 0.2), then a positive distractor effect could be attributable to additive integration rather than the distractor’s EV (Figure 1c). Second, the EV of the distractor alternative could covary with its relative position in the reward-probability space (we call these distractor-related covariations). This would influence decision accuracy because, as outlined earlier, the relative position of a decoy alternative in the multiattribute choice space can induce strong choice biases (Tsetsos et al., 2010; Turner et al., 2018). For illustration purposes only, we assume that the distractor boosts the tendency of choosing a nearby target akin to the attraction effect (Huber et al., 1982; Dumbalska et al., 2020; Figure 1d). Under this assumption, if distractors with high EVs happen to appear closer to the correct target (i.e., the target with the highest EV), a positive distractor effect could be entirely attributable to the attraction effect. The aim of this paper is to re-assess distractor effects in the relevant, previously published datasets (Chau et al., 2014; Gluth et al., 2018) while looking beyond these two classes of potentially confounding covariations (target- and distractor-related). We began with establishing that the first class of target-related covariations is indeed present in the examined datasets, with positive and interacting ‘notional’ distractor effects being evident even in matched binary trials, in which the distractor alternative was not present. Using a novel baselining approach, we asked if there are residual distractor effects when the influence of these unintended covariations is controlled for, reporting that distractor effects are eradicated. We then pinpointed these target-related confounding covariations to people’s strong propensity to integrate reward and probability additively, and not multiplicatively. Moving forward, defining the key target and distractor utility variables, not using EV, but subjective (additive) utility, revealed a modest negative distractor effect. Moving on to the second class of distractor-related covariations, we established that choice accuracy was lawfully influenced by the position of the distractor in the multiattribute space (Figure 1d), consistent with a large body of empirical work in multiattribute choice (Dumbalska et al., 2020; Tsetsos et al., 2010). This ‘decoy’ influence was most pronounced when the distractor alternative was close to the high-EV (correct) target in the multiattribute space, yielding an attraction effect (i.e., a boost in accuracy) when the distractor was inferior, and a repulsion effect (i.e., a reduction in accuracy) when the distractor was superior (in subjective utility) to the target. Of note, this decoy influence peaking around the high-EV target, essentially re-describes a negative distractor effect without evoking ad hoc ‘unidimensional distractor’ mechanisms other than those that are needed to produce classic multiattribute context effects (Tsetsos et al., 2010; Tsetsos et al., 2016). Overall, our analyses update the state-of-the-art by suggesting that, when confounding covariations are controlled for, only a modest negative distractor effect survives. Further, it is conceivable that this distractor effect mirrors asymmetric classic multiattribute context effects, which occurred robustly in the examined datasets. Results We re-analysed five datasets from published studies (Chau et al., 2014; Gluth et al., 2018; N = 144 human participants) of a speeded multiattribute decision-making task. Participants were shown choice alternatives that varied along two distinct reward attributes: reward magnitude (X) and reward probability (P), which mapped to visual features (colourful slanted bars as options in Figure 2a). After learning the feature-to-attribute mapping, participants were instructed to choose the most valuable one among multiple options placed in this multiattribute space (Figure 2a) on each trial and to maximise the total reward across trials. In the ternary-choice condition, one option (the ‘distractor’, or D) was flagged unavailable for selection early in the trial. The expected value (or EV, i.e., X multiplied by P) of the higher- and lower-value available alternatives ('targets’ H and L, respectively), and of the unavailable distractor were denoted by HV, LV, and DV, respectively. Figure 2 with 3 supplements see all Download asset Open asset Re-assessing distractor effects beyond target-related covariations in ternary- and binary-choice trials. (a) Multiattribute stimuli in a previous study (Chau et al., 2014). Participants made a speeded choice between two available options (HV: high expected value; LV: low expected value) in the presence (ternary) or absence (binary) of a third distractor option (‘D’). Three example stimuli are labelled for illustration purposes only. In the experiment, D was surrounded by a purple square to show that it should not be chosen. (b, c) Relative choice accuracy (probability of H choice among all H and L choices) in ternary trials (panel b) and in binary-choice baselines (panel c) plotted as a function of both the expected value (EV) difference between the two available options (HV − LV) (y-axis) and the EV difference between D and H (x-axis). Relative choice accuracy increases when HV − LV increases (bottom to top) as well as when DV − HV increases (left to right, i.e., positive D effect). (d–f) Predicting relative accuracy in human data using regression models (Methods). Asterisks: significant effects [p < 0.05; two-sided one-sample t-tests of generalised linear model (GLM) coefficients against 0] following Holm’s sequential Bonferroni correction for multiple comparisons. n.s.: non-significant. (g) Rival hypotheses underlying the positive notional distractor effect on binary-choice accuracy. Left: EV indifference contour map; Right: additive utility (AU) indifference contour map. Utility remains constant across all points on the same indifference curve and increases across different curves in evenly spaced steps along the direction of the dashed grey line. Decision accuracy scales with Δ(utility) between H and L in binary choices. Left (hypothesis 1): Δ(EV) = (HV − LV)/(HV + LV) by virtue of divisive normalisation (DN); because HV2 − LV2 = HV1 − LV1, and HV2 + LV2 > HV1 + LV1, Δ(EV2) becomes smaller than Δ(EV1). Right (hypothesis 2): Δ(AU) = AUH − AUL; Δ(AU2) < Δ(AU1) by virtue of additive integration. (h, i) Regression analysis of model-predicted accuracy in binary choice. EV + DN model corresponds to hypothesis 1 whilst AU model corresponds to hypothesis 2 in panel g. Error bars = ± standard error of the mean (SEM) (N = 144 participants). Rational choice theory posits that an unavailable D should not in any way alter the relative choice between two available H and L targets. However, the behavioural data in this task (Chau et al., 2014) challenged this view. By examining the probability of H being chosen over L in ternary trials, that is, the relative choice accuracy, Chau et al., 2014; Chau et al., 2020 reported that a relative distractor variable ‘DV − HV’ (the EV difference between the distractor and the best available target) altered relative accuracy. We note that other recent studies, using different stimuli, quantified the distractor influence by means of an absolute distractor variable DV (Louie et al., 2013; Gluth et al., 2020a). Wherever possible, we quantify distractor effects using both the relative (DV − HV) and absolute (DV) distractor variables. To begin with, we note that, in addition to ternary trials in which D was present (but unavailable), participants encountered ‘matched’ binary trials in which only the H and L target alternatives were shown. These binary trials are ideally suited for assessing the extent to which previously reported distractor affects can be attributed to covariations between the distractor variable and target-related properties. This is because, for each ternary trial, we can derive a respective binary-choice baseline accuracy from binary trials that had the exact same targets but no distractor (Figure 2—figure supplement 1c). Given that participants never saw D in binary trials, the distractor variable is notional and should have no effect on the binary baseline accuracies. However, if D does paradoxically ‘influence’ binary accuracies, then this would signal that the distractor variable covaries with other unspecified modulators of choice accuracy. We dub any effect that D has upon binary-choice accuracy as the ‘notional distractor effect’. We emphasise here that a notional distractor effect is not a genuine empirical phenomenon but a tool to diagnose target-related covariations in the experimental design. Re-assessing distractor effects beyond target-related covariations We used logistic regression (generalised linear model or GLM) to quantify the effect of the distractor variable on relative choice accuracy (the probability of H choice among all H and L choices). Differently from previous studies, which focused primarily on ternary-choice trials, here we analysed two different dependent variables (Figure 2d—f): ternary-choice relative accuracies (‘T’) and their respective baseline accuracies (‘B’) (see Methods and Figure 2—figure supplement 1c). For the ternary-choice condition, our GLM reveals a significant main effect of the relative distractor variable ‘DV − HV’ (t(143) = 3.89, p < 0.001), and a significant interaction between this distractor variable and the index of decision difficulty ‘HV − LV’ on relative choice accuracy, t(143) = −3.67, p < 0.001 (Figure 2d). These results agree with previous reports when analysing ternary choice alone (Chau et al., 2014; Chau et al., 2020). Turning into the matched binary baseline accuracies, the GLM coefficients bear a striking resemblance to those of the ternary-choice GLM (see Figure 2e; also compare Figure 2b to Figure 2c for a stark resemblance between T and B accuracy patterns), with (this time) notional positive and interacting distractor effects being observed. Crucially, neither the main distractor nor the (HV − LV) × (DV – HV) interaction effects are modulated by ‘Condition’ in a GLM combining T and B together, |t(143)| < 0.92, p > 0.72 (Figure 2f). We see similar results when the relative distractor variable DV − HV is replaced by the absolute distractor variable DV or when a covariate HV + LV is additionally included: the distractor’s EV had no differential effect on relative accuracy across binary vs. ternary condition (Figure 2—figure supplements 2 and 3). These results equate the distractor effects in ternary trials (D present) with the notional distractor effects in binary trials (D absent), indicating that the former arose for reasons other than the properties of D. Specifically, the notional distractor effects (Figure 2e) in binary trials (with only H and L stimuli present) indicate that the value of D covaries with target-related properties (other than HV − LV, which was already a regressor) that modulate choice accuracy. Next, we use computational modelling to unveil these confounding target-related properties. Integrating reward and probability information additively in risky choice The original study reported a surprising phenomenon (Chau et al., 2014): decisions seem to be particularly vulnerable to disruption by a third low value, as opposed to high value, distracting alternative, that is, a positive distractor effect. The very definition of this effect hinges upon the a priori assumption that decisions rely on calculating and subsequently comparing the EVs across choice options. An alternative prominent idea is that participants eschew EV calculations (Hayden and Niv, 2021); instead, they compute the difference between alternatives within each attribute and simply sum up these differences across attributes. As mentioned in the Introduction, we refer to this class of strategies that involve the additive (independent) contributions of reward and probability as the AU strategy. Indeed, past studies in binary risky choice have reported decision rules in humans and other animals (Rouault et al., 2019; Farashahi et al., 2019; Bongioanni et al., 2021) based on a weighted sum of the attribute differences, that is, a decision variable equivalent to the AU difference between alternatives, Δ(AU) = λ(HX − LX) + (1 − λ)(HP − LP), where 0 ≤ λ ≤ 1. Although the additive combination of reward and probability may not be generalisable in all types of risky-choice tasks, it could viably govern decisions in the simple task illustrated in Figure 2a. Of note, we came to notice that the key distractor variable DV − HV positively covaries with the Δ(AU) between H and L across all choice-sets (e.g., Pearson’s r(148) = 0.314, p < 0.0001, in the case of equal weighting between X and P, λ = 0.5). This unintended covariation arose in the original study possibly due to a deliberate attempt to decorrelate two EV-based quantities (DV − HV and HV − LV) in the stimuli (Chau et al., 2014). The correlation between DV − HV and Δ(AU) is stronger in more difficult trials (r(71) = 0.52, p < 0.0001, shared variance = 27%; splitting 150 trials into difficult vs. easy by the median of HV − LV; Chau et al., 2020) than in easier trials (r(59) = 0.43, p < 0.001, shared variance = 19%), mirroring both the positive (overall positive correlation) and the interacting (change of correlation strength as a function of difficulty) notional distractor effects on binary choice. Additive integration explains notional distractor effects in binary-choice trials Notably, DV − HV also negatively covaries with HV + LV (Pearson’s r(148) = −0.78, p < 0.001), which potentially leads to another explanation of why binary-choice accuracy seems lower as the matched DV − HV variable decreases. This explanation appeals to the divisive normalisation (DN) model based on EV (Louie et al., 2013; Pirrone et al., 2022). Imagine choosing between two prospects, H and L, in two different choice-sets, {H1 vs. L1} and {H2 vs. L2}, with the EV difference between H and L being the same across the two choice-sets (Figure 2g, left panel). DN applied to EVs (‘EV + DN’, hereafter) will shrink the EV difference between H and L more aggressively for set 2 than for set 1 because of the larger denominator in set 2, rendering set 2 a harder choice problem (lower accuracy). It is important to note that, in line with previous analyses (Chau et al., 2014; Gluth et al., 2018), adding the HV + LV covariate to the GLMs does not explain away the interacting notional distractor effects in binary trials (Figure 2—figure supplement 3). Qualitatively, the above two hypotheses (‘AU’ vs. ‘EV + DN’) both predict a positive main notional distractor effect on binary-choice accuracy, but their predictions for the (HV − LV) × (DV − HV) interaction could differ. For instance, DN might even predict a slightly positive, rather than negative, interacting notional distractor effect on binary accuracy—in this stimulus set, the divisively normalised Δ(EV) happens to be more positively correlated with DV − HV in easier trials (shared variance = 39.8%) than in harder trials (shared variance = 36.7%). It is also important to note that AU on its own can approximate this EV sum effect by nearly tripling the utility difference between H and L in set 1 compared with that in set 2, which also renders set 2 more difficult (Figure 2g, right panel). These two models can thus mimic each other. To better distinguish between these candidate hypotheses, we fit a model with a softmax decision rule to each participant’s binary-choice probabilities (Methods). As expected, both the EV + DN model and the AU model (with a free λ parameter; mean λ estimate: 0.46, SE: 0.017) predict a positive ‘notional distractor’ main effect on binary accuracy (Figure 2h, i). However, the AU model, but not the EV + DN model, reproduces a negative notional (HV − LV) × (DV − HV) interaction effect on accuracy (Figure 2i), mirroring the human data (Figure 2e). The AU model thus qualitatively provides a parsimonious account of why notional distractor effects occurred in binary trials. Next, we used formal Bayesian model comparison to quantitatively identify the simplest possible model that can reproduce the binary-choice behaviour. For completeness, we added in the comparison a naive expected-value (EV) model, and a recently proposed dual-route model, which relies on EV and flexibly arbitrates between DN and mutual-inhibition (MI) processes (Chau et al., 2020). A vanilla DN model can be viewed as a nested model within the dual-route model. All models were fitted to each participant’s binary choices and reaction times (RTs) by optimising a joint likelihood function (Methods). Qualitatively, Figure 3a shows that the naive EV model fundamentally deviates from the human choice data: the model predicts a vertical gradient of choice accuracy constrained by HV − LV. When comparing these models head-to-head using a cross-validation scheme, we find that the AU model wins over any other model (Figure 3b; protected exceedance probability Ppexc > 0.99; Methods). This result still holds robustly when including, in all models, subjective non-linear distortions of attributes (Zhang and Maloney, 2012; ‘Non-linear’ in Figure 3b, c right panels; Methods). Moreover, the RTs predicted by the DN model mismatch the human RTs (Figure 3a: ‘EV + DN’). But this is not the sole reason why DN fails in the model comparisons. DN still fails to win over the AU model even when we consider static versions of the models fitted to the choice probabilities alone while ignoring the RTs (Figure 3c, ‘Static models’; Methods). These systematic model comparisons thus quantitatively support the AU model as a remarkably simple account of the ‘notional distractor’ effects in binary trials (Figure 2i). Additional model comparisons corroborate that the AU model is favoured over specific and popular instantiations of the non-linear multiplicative model (i.e., expected utility [Von Neumann and Morgenstern, 2007] and prospect theory [Kahneman and Tversky, 1979]; see Figure 3—figure supplement 1). Figure 3 with 1 supplement see all Download asset Open asset Modelling binary-choice behaviour. (a) Human vs. model-predicted choice accuracy and reaction time (RT) patterns. (b, c) Bayesian model comparison based on cross-validated log-likelihood. Linear and non-linear: different psychometric transduction functions for reward magnitude and probability (see Methods). Linear multiplicative: EV; linear additive: AU. Dynamic model: optimisation based on a joint likelihood of choice probability and RT; Static model: optimisation based on binomial likeli