Fig 1.
Choice task and attribute space of stimuli.
Choice task (a, b). Trials started with a central fixation cross for 1.5 s. Next, three gambles were presented. Participants made a choice by button press, without time limit. Eye movements were recorded during the choice phase. Finally, brief feedback on the choice (but not on the gamble outcome) was displayed. Note that in each trial, the vertical layout of probability and outcome attributes was determined randomly for each alternative. Choice sets in attraction trials (a) contained core options A and B and an asymmetrically dominated decoy DA (shown) or DB. Compromise trials (b) included A, B, and a compromise decoy CA or CB (shown). (c) Attribute space. Each gamble is described by two attributes: Probability p and outcome m. The core options A and B were presented in every trial, along with one of four decoy alternatives: Asymmetrically dominated decoys (DA or DB) and compromise decoys (CA or CB) are expected to elicit attraction and compromise effects, respectively. Dashed lines indicate possible stimulus placements. (d) Exact positions for CB, A and CA were calibrated for pairwise indifference in separate binary choice estimation blocks for each individual. Differences in outcome between neighbouring alternatives was not less than 2 EUR. Dominated decoys DA and DB were always 2% and 1 EUR worse than A and B respectively.
Fig 2.
Context effects are present in choices and relative dwell time.
Participants’ choices were influenced by asymmetrically dominated decoys and, to a lesser extent, by extreme compromise-making decoys. (a, d) Ternary plots of individual relative choice frequencies for target (lower left, teal), competitor (top, pink), and decoy (lower right, yellow) alternatives in attraction (a) and compromise (d) trials. Each dot represents one participant. The position on the simplex indicates relative choice frequencies for alternatives. Straight lines from the centre indicate equal frequencies for two alternatives. The red "x" indicates the group average. (b, e) Relative choice frequencies in attraction (b) and compromise (e) trials. In attraction trials, some participants strongly favoured the target alternative and almost no decoy choices were made. While target alternatives are still chosen more frequently than competitors in compromise trials, the effect is less pronounced, and extreme decoys are still chosen frequently. (c, f) Relative dwell time towards alternatives. In both, attraction (c) and compromise (f) trials, target alternatives received greater relative dwell times than competitors. d denotes Cohen’s d from paired BEST analysis with HDI95 given in brackets. Violin plots show kernel density estimates of distributions of individual values. Box plots mark lower and upper quartiles and median. Whiskers extend from first and last datum within 1.5 times the interquartile range from lower and upper quartiles, respectively. Values outside this range are indicated by open circles.
Table 1.
Relative choice frequencies across participants in the four trial types.
Across the group, target options were chosen more frequently than competitors in both types of attraction (DA and DB) and compromise trials (CA, CB). Dominated decoys were almost never chosen. In compromise trials including the high-outcome decoy CB, the decoy was chosen more frequently than core options.
Fig 3.
Model comparison and predictions.
(a) The gaze-dependent leaky accumulator (GLA) provided the best fit (lowest mean BIC) across participants, followed by the dynamic gaze baseline model, MDFT, PT, and the static gaze baseline model. The dashed line indicates the BIC of the random choice baseline model. Violin plots show kernel density estimates of distributions of individual values. Box plots mark lower and upper quartiles and median. Whiskers extend from first and last datum within 1.5 times the interquartile range from lower and upper quartiles, respectively. Values outside this range are indicated by open circles. (b) The GLA fitted most (36 of 40, 90%) participants best, with a protected exceedance probability of 1 (inset). (c-h) Observed and model-predicted probability of choosing the target (c, d), competitor (e, f), or decoy (g, h) alternatives in attraction (c, e, g) and compromise trials (d, f, h) as a function of relative dwell time advantage. Relative dwell time advantage was computed as relative dwell time towards an alternative minus the mean relative dwell time to all other alternatives. White bars and error bars show mean ± s.e. observed data from even-numbered trials. Model predictions (coloured lines) are based on 50 simulations of each odd-numbered trial. (i, j) Observed and predicted RST of the best-fitting GLA for attraction (i) and compromise (j) trials. Each circle represents one participant. The winning model’s predicted context effect sizes correlated significantly with the observed ones. Strong context effects, however, were underestimated, as indicated by the reduced slopes. GLA: Gaze-dependent Leaky Accumulator. MDFT: Multialternative Decision Field Theory. PT: Prospect Theory. GBstat: Static Gaze Baseline. GBdyn: Dynamic Gaze Baseline.
Fig 4.
(a) Overview of general switchboard framework and individual switches from which individual variants are constructed by setting the switches to a set of levels. Gaze-dependent switch levels are shaded blue. Attributes can be discounted based on gaze (lower level) and are integrated into alternative values (middle level). Alternative values can be discounted based on gaze, compared, and integrated (upper level) with leak and inhibition. (b) Average model fit associated with each switch’s levels. Each bar shows the average BIC for all model variants that had the respective switch set to this level (e.g., first panel, top bar: average BIC of all variants with gaze-dependent inhibition). Gaze-dependent inhibition and leak, independent evidence accumulation, alternative-wise gaze-discount, multiplicative attribute integration, and no attribute-wise gaze-discount yielded lower BIC on average. (c) Overview of mean BIC for each of 192 model variants. More yellow colours indicate lower BIC and better model fit. The variant with the lowest BIC is identical to the GLA (alternative-wise, no attribute-wise gaze-discount, multiplicative attribute integration, constant leak, and no inhibition) and is outlined in white. The hybrid variant which described 9 participants, mostly with strong attraction effects, best (see main text and Fig 5) has a dotted outline. Note that some variants were mathematically equivalent (see main text and Methods) including the variant with lowest BIC, which is therefore highlighted twice.
Table 2.
Overview of individually best fitting model variants.
N indicates the number of participants best described by the variant described in the row. The top variant (A) coincided with the GLA model. Note that all individually best fitting models had some form of gaze-dependence (blue shaded cells, mostly alternative-wise gaze-discount). "n.d." denotes variants where comparison mechanisms were not distinguishable by the analysis. The last two columns show observed mean (± s.d. if applicable) RST of participants best described by each variant, for attraction and compromise trials, respectively.
Fig 5.
(a, b) Number of participants better described by the hybrid variant (pink) or the GLA (grey), dependent on strength of attraction (a) and compromise (b) effects. Participants with strong attraction effects were better described by the hybrid variant. (c, d) Individual observed and predicted RST in attraction (c) and compromise trials (d). Compared to GLA (Fig 3I and 3J), the hybrid model better predicted strong attraction effects for some participants. Predictions of compromise effects are similar. (e, f) Observed and model-predicted probability of choosing the target alternative, depending on the target’s relative dwell time advantage. Like other gaze-dependent models (Fig 3), the hybrid variant generally captured the positive association between gaze and choice. In contrast to GLA, however, it predicted an overall higher probability of choosing the target in attraction trials (e). Predictions in compromise trials (f) are similar to GLA. White bars and error bars indicate mean ± s.e. observed data from even-numbered trials. Model predictions are based on 50 simulations for each odd-numbered trial.