The study of context effects in decision-making — how choices can be influenced by a supposedly innocuous change to the set of possible alternatives — is a powerful lens for revealing insight into human choice behaviour. Beginning with observed violations of Luce’s axiom of Independence of Irrelevant Alternatives, the existence of context effects have led to the development of new choice theories, and in turn, choice experiments to test them (Debreu, 1960, Luce, 1959, Rieskamp et al., 2006). One noteworthy example is the asymmetric dominance effect, in which a choice set containing two alternatives x and y is expanded to include a third alternative z which is dominated on all attributes by only x. A large number of laboratory studies in humans and other species has found that this “decoy” alternative tends to influence decision-makers to choose x more often, suggesting a violation of the regularity axiom which underlies the canonical theory of rational choice (Castillo, 2020, Huber et al., 1982, Huber et al., 2014, Król and Król, 2019, Parrish et al., 2015).1

The asymmetric dominance effect and other examples of context-dependent choice behaviour have led to a general recognition across multiple disciplines that preferences over alternatives are not simply known and immutable constructs endowed to a decision-maker, but instead must be constructed at the time of choice in a given environment (Li et al., 2018, Lichtenstein and Slovic, 2006, Lieder and Griffiths, 2019, Weber and Johnson, 2006). For the asymmetric dominance effect, a number of proposed explanations describe how the attributes of a choice alternative are perceived relative to the attributes of other alternatives in the choice set.2 One early hypothesis held that such decoy effects are related to the numerical range of attribute stimuli (Huber et al., 1982). This proposal has recently been formalized by models of range normalization, in which an attribute dimension is weighted by the range of stimuli on that dimension (Bushong et al., 2020, Soltani et al., 2012).3 As the range of an attribute grows, the decision-maker under-weights that attribute. By placing an asymmetric decoy z into a choice set, this increases the observed range on the attribute for which y is best (Attribute 2 in Fig. 1). This attribute thus gets less weight and the target alternative x appears more valuable.

Alternatively, context effects like the asymmetric dominance effect might arise from diminishing sensitivity to objective stimuli — the tendency to treat a difference between small values as if it were greater than an equal-sized difference between large values. For example, Kahneman and Tversky (1984) find that subjects are often willing to drive twenty minutes to save $5 on a $15 calculator, but not to save $5 on a $125 jacket. Such phenomena can be considered choice analogs of Weber’s (1834) Law of Perception applied to the domain of numbers — increasing the intensities of two stimuli diminishes the perceptability of their difference.4 Diminishing sensitivity to symbolic number stimuli (as well as numerosity more generally) has been previously demonstrated in behaviour and is observed in the activity of cortical neurons thought to be responsible for an analog numerical perception system (Moyer and Landauer, 1967, Nieder and Dehaene, 2009).

Recently, it has been shown that a value function which exhibits diminishing sensitivity can arise from the principle of efficient coding of sensory/reward quantities (Frydman and Jin, 2021, Steverson et al., 2019). When a decision-maker faces a constraint in their ability to represent objective quantities (perhaps rooted in their neurobiology), it is optimal to compress valuations to scale with the frequency and/or importance of reward stimuli in the environment. This compression can be implemented by a simple neural computation in which the neural activity for each stimuli is divisively normalized by the sum of other stimuli in a reference set (Louie et al., 2015, Louie et al., 2011, Louie et al., 2014). While efficient, this compression still has normative consequences. Behavioural models incorporating divisive normalization have previously been used to explain context effects in choice, including choice set dependence and violations of IIA (Li et al., 2018, Louie et al., 2013, Marley et al., 2008, Webb et al., 2020). Essentially these models allow the variance of a random utility formulation to depend on the composition of the choice set, an issue particularly critical to analysing consumer choice data in both the lab and field (Louviere et al., 2002, Salisbury and Feinberg, 2010). Recent theoretical work has extended the divisive normalization model to multi-attribute choice phenomena, including the dominance and compromise effects (Dumbalska et al., 2020, Landry and Webb, 2021). Experiments reported by Dumbalska et al. (2020) and Somerville (2022) find multi-attribute choice behaviour consistent with divisive normalization.

The contribution of this paper lies along two dimensions. First, we implement a “double decoy” experiment designed to separate the two competing accounts of the asymmetric dominance effect arising from attribute normalization.5 The experiment is quite simple; we place an additional decoy alternative within the range of existing alternatives (Fig. 1) in a within-subject design.6 Since the attribute range remains constant, a range normalization model would predict a null effect on the relative choice probabilities between the target and the competitor. This is the null hypothesis we set out to test. A divisive normalization model, on the other hand, predicts a double decoy effect (almost surely). This is our alternative hypothesis. In our sample, we observe a significant decrease in the relative proportion of targets chosen (on average). We also observe considerably more variation in individual behaviour than expected under the null hypothesis of range normalization.

Addressing the variation in the double decoy effect across subjects is our second contribution. It is well-recognized that subject/consumer-level heterogeneity is an important issue in discrete choice analysis (Fiebig et al., 2010, Salisbury and Feinberg, 2010). While much of the existing literature on the dominance effect has studied the presence of a robust average effect across subjects, there is also compelling evidence of substantial variation across subjects in both the presence of a dominance effect (Rooderkerk et al., 2011, Sharpe et al., 2008), as well as its correlation with other context-dependent choice behaviour like the compromise and similarity effect (Liew et al., 2016, Noguchi and Stewart, 2018, Trueblood et al., 2015). This considerable heterogeneity in decoy effects requires both a theoretical account as well as empirical methods for addressing it. Taking guidance from both the neuroscience literature on perception as well as the discrete choice econometrics literature on estimating heterogeneity, we provide an empirical extension of the pairwise normalization model for multi-attribute choice of Landry and Webb (2021). This model allows heterogeneity in its parameterization, in particular the baseline by which a participant might normalize alternatives, allowing the model to capture the variation in behaviour that we observe in the double decoy experiment.

Finally, a compelling choice model should not only provide a source of variation in individual-level behaviour, it should also be adaptable to choice sets that do not contain decoys. After all, most choice sets typically encountered by human decision-makers are not constructed by behavioural researchers or prescient marketers, or at least, this has not typically been the norm for most of human history. In addition to addressing the seemingly esoteric edge-cases that are fruitful ground for testing choice theories, a choice model should also be useful in standard empirical settings. Therefore we also compare the range and divisive normalization models to hierarchical versions of the Logit model in a standard discrete choice experimental design previously used by Fiebig et al. (2010). Using Bayesian econometric methods, we find that the pairwise normalization model outperforms both the range normalization model and the Logit model in both of our datasets. These results suggest that a choice model which incorporates diminishing sensitivity via divisive normalization provides both a theoretical account of multi-attribute choice as well as a useful empirical model for researchers in applied settings.

Consider two choice alternatives, x=(x1,x2) and y=(y1,y2), each with two objective attributes x1>y1 and x2<y2 so that the choice is non-trivial. A violation of the regularity axiom is defined to occur if the introduction of a decoy z=(z1,z2)≪x to the choice set leads to an increase in the probability that x is chosen from that set, Px(x,y,z)>Px(x,y). In a typical experiment, the decoy alternative is chosen with some small probability, therefore we also examine the weaker condition that the relative probability of x is chosen more than y in the presence of the decoy, Px(x,y,z)Px(x,y,z)+Py(x,y,z)>Px(x,y)Px(x,y)+Py(x,y).

Now consider a second decoy z′≫z. When adding z′ to the choice set that already contains x, y, and z, we define a double decoy effect to be a second shift in the relative choice proportions, beyond that of the initial decoy.

Definition 1 Double Decoy Effect

Suppose a single decoy effect such that Px(x,y,z)>Px(x,y). A double decoy effect occurs if Px(x,y,z,z′)Px(x,y,z,z′)+Py(x,y,z,z′)≠Px(x,y,z)Px(x,y,z)+Py(x,y,z).

Therefore a double decoy effect can occur in one of two ways, by increasing the relative probability that either x or y is chosen from compared to the choice set .

We now show that a range normalization value function cannot predict a double decoy effect. Let X denote the choice set and denote the set Xn as the attribute values of all alternatives for a given attribute n. Consider a value function in which each attribute value xn is scaled by the range of all values of that attribute and then aggregated, Vx(X)=∑nxnmax−min,and a mapping to choice probabilities Px(V(X)):R→[0,1] given by the Luce model, Px(V(X))=Vx(X)∑y∈XVy(X).This latter assumption requires that any change in relative choice probabilities is due to changes in Vx(X) as the choice set changes.7

Proposition 1

If the value function is given by Range Normalization (1), the relative choice probabilities from the choice sets and are identical.

Proof

The proof is constructed via contradiction. First, note that the relative choice probabilities for any choice set X are given by Px(X)Px(X)+Py(X)=Vx(X)Vx(X)+Vy(X).Suppose Definition 1 holds, then Vx(x,y,z,z′)Vx(x,y,z,z′)+Vy(x,y,z,z′)≠Vx(x,y,z)Vx(x,y,z)+Vy(x,y,z).However this implies that Vx(x,y,z,z′)Vy(x,y,z,z′)≠Vx(x,y,z)Vy(x,y,z) which would require Vx(X) or Vy(X) to be influenced by z′. This is impossible given Eq. (1) since the introduction of z′ does not change max−min,∀n. □

Therefore the Range Normalization model gives us our null hypothesis of no change in the relative choice probabilities in the presence of a second decoy z′.