The multi-attribute decision making task (Chau et al., 2014; Gluth et al., 2018).

(a) On the control Two-Option Trials, participants were presented two options associated with different levels of reward magnitude and probability, each in the form of a rectangular bar surrounded by an orange box. On the experimental Distractor Trials, three options were presented. Two options were subsequently surrounded by orange boxes indicating that they could be chosen, while a third option was surrounded by a purple box indicating that it was an unchooseable distractor. (b) The association between stimulus colour and orientation to reward magnitude and probability, respectively. Participants were instructed to learn these associations prior to task performance. (c) Plots illustrating utility estimated using (left) a purely multiplicative rule and (right) a purely additive rule (here assuming equal weights for probability and magnitude). By comparing their corresponding plots, differences in the utility contours are most apparent in the bottom-left and top-right corners. This is because in the multiplicative rule a small value in either the magnitude or probability produces a small overall utility (blue colours). In contrast, in the additive rule the two attributes are independent – a small value in one attribute and a large value in another attribute can produce a moderate overall utility (green colours). (Middle) Here, we included a composite model that combines both rules. The composite model involves an integration coefficient that controls the relative contributions of the multiplicative and additive rules.

Figure 1a and b are reproduced from (Chau et al., 2014), Nature Neuroscience.

A distractor effect was absent, when the way participants combined choice attributes was ignored. Although a positive distractor effect (indicated by the DV-HV term) was present on both (a) Distractor and (b) Two-Option Trials, (c) the distractor effects in the two trial types were not significantly different when they were compared in a single analysis [indicated by the term]. Error bars indicate standard error. * p < 0.05, ** p<0.01, *** p<0.001.

A Bayesian model comparison showed that the composite model provides the best account of participants’ choice behaviour. (a) The posterior probability of each model in accounting for the behaviour of each participant’s behaviour. (b) A model comparison demonstrating that the composite model (COMP) is the best fit model, compared to the additive utility (AU), expected value (EV), and expected value and divisive normalisation (EV+DN) models. Histograms showing each participant’s fitted parameters using the composite model: (c) Values of the integration coefficient (0 ≤ η ≤ 1), (d) magnitude/probability weighing ratio (0 ≤ γ ≤ 1), and (e) inverse temperature (θ).

Participants showing a more multiplicative style of combining choice attributes into an overall value also showed a greater distractor effect. A scatterplot illustrating a positive correlation between the integration coefficient (η) and the distractor term (i.e., (DV - HV)T) obtained from GLM2 (Figure 2c). A significant positive correlation was observed [r(142) = 0.282, p = 0.000631].

Significant distractor effects were found when participants’ styles of combining choice attributes were considered. (a) A positive distractor effect, indicated by the term, was found in the Multiplicative Group, whereas a negative distractor effect was found in the Additive Group. The term was also significantly different between the two groups Plots showing all regression weighs of GLM2 when the data of the (b) Multiplicative Group and (c) Additive Group were analysed. The key term extracted in (a) is highlighted in blue and yellow for the Multiplicative Group and Additive Group, respectively. Error bars indicate standard error. * p < 0.05, ** p<0.01, *** p<0.001.

Scatterplots illustrating the correlation between the fitted parameters. (a) The Pearson’s correlation between the magnitude/probability weighting and integration coefficient was significant [r(142)=−0.259, p=0.00170]. (b) The Pearson’s correlation between the inverse temperature and integration coefficient was not significant [r(142)=−0.0301, p=0.721]. (c) The Pearson’s correlation between the inverse temperature and magnitude/probability weighting was not significant [r(142)=−0.0715, p=0.394].

A Bayesian model comparison showed that the composite model provides the best account of participants’ choice behaviour even when a model with utility curvature based on Prospect Theory (Kahneman & Tversky, 1979) with the Prelec formula for probability (Prelec, 1998) is also considered: where and represent the reward magnitude and probability (both rescaled to the interval between 0 and 1), respectively. is the weighted magnitude and is the weighted probability, while and are the corresponding distortion parameters. (a) The posterior probability of each model in accounting for the behaviour of each participant’s behaviour. (b) A model comparison demonstrating that the composite model (COMP) is the best fit model (exceedance probability = 1.000, estimated model frequency = 0.720), compared to the additive utility (AU), expected value (EV), expected value and divisive normalisation (EV+DN), and prospect theory (PT) models.

The distractor effects shown in Figure 5 remained significant after including an extra term in GLM3. GLM3 differs from GLM2 only by including an extra term: Similar to Figure 5, here (a) we observe that a positive distractor effect (i.e., positive term) was significant in the Multiplicative Group and a negative distractor effect was significant in the Additive Group. The term was also significantly different between the two groups [,]. Plots showing all regression weights of GLM2 when the data of the (b) Multiplicative Group and (c) Additive Group were analysed. The key term extracted in (a) is highlighted in blue and yellow for the Multiplicative Group and Additive Group, respectively. Error bars indicate standard error. * p < 0.05, ** p<0.01, *** p<0.001.

The distractor effect between groups were also significant following the introduction of the term to GLM3 and changing the terms to to prevent collinearity between the regressors. GLM4 is defined as: The distractor “effect’ in the Multiplicative Group remained as a positive trend, despite not reaching significance. (a) A negative distractor effect, indicated by the term, was found in the Additive Group and the effects were also significantly different between the two groups [,]. Plots showing all regression weights of GLM2 when the data of the (b) Multiplicative Group and (c) Additive Group were analysed. The key term extracted in (a) is highlighted in blue and yellow for the Multiplicative Group and Additive Group, respectively. Error bars indicate standard error. * p < 0.05, ** p<0.01, *** p<0.001.

The positive distractor effect in the Multiplicative Group shown in Figure 5 remained significant after replacing the utility function based on the normative Expected Value model with values obtained by using the composite model. The distractor “effect’ in the Additive Group remained as a negative trend, despite not reaching significance. (a) A positive distractor effect, indicated by the term, was found in the Multiplicative Group and the effects were also significantly different between the two groups [,]. Plots showing all regression weights of GLM2 when the data of the (b) Multiplicative Group and (c) Additive Group were analysed. The key term extracted in (a) is highlighted in blue and yellow for the Multiplicative Group and Additive Group, respectively. Error bars indicate standard error. * p < 0.05, ** p<0.01, *** p<0.001.