Fig 1.
From binary allocation tasks to the Lambda Slider.
(A) A binary allocation task, where the threshold of for switching between Options A and B is
. The arrows point in the direction of the gradient of the utility function (Eq (1)) for a given
. (B) Adding a third option C to create a triple-dominance task, which is equivalent to two binary allocation tasks with
, but requires only one response. The shaded areas are all the locations where Option C can be placed in order for the task to be triple-dominance. C′ is a hypothetical third option that does not form a triple-dominance task with A and B because they do not fall along a strictly concave function
. (C) An illustration of a hypothetical “septuple-dominance task” in which each option would be preferred for some range of
. (D) A possible option space for a Lambda Slider, where a participant can choose any point on the curve (via a slider; Fig 2A). Each point on the curve corresponds to a unique
whose corresponding utility gradient is perpendicular to the tangent of the curve at that point.
Fig 2.
The (quadratic) Lambda Slider.
(A) The interface of the slider. The payoff to oneself (red bar) and payoff to the target (blue bar) change continuously as the participant moves the slider. (B) The payoff functions of the quadratic Lambda Slider used in Experiment 1 (a = 11.25, ,
,
and
). If we plot
and
against each other, we get a parabolic curve similar to Fig 1D. (C) Examples of the participant’s utility function for different
s. The slider position that maximizes each utility function is marked, which is an identity function of the participant’s
.
Fig 3.
The SVO Slider Measure [18].
(A) The payoff functions of the 6 primary items of the measure (black lines). Each segment represents the linear relationship between and
on one of the items, and they are labeled in the same order as in [18]. The red arc and point
provide an intuitive explanation (but not formal justification) for the calculation of SVO∘. ( B) The theoretical step function (green curve) between the output of the measure (SVO∘) and
. The labeled vertical segments correspond to the
s (the thresholds of
at which a utility-maximizing participant switches from one end to the other on the sliders) of the 6 items. The theoretical response on the circular Lambda Slider (arctan
; Eq (15) in S2 Appendix) is also plotted for comparison.
Fig 4.
Test-retest reliability of (A) the quadratic Lambda Slider and (B) the SVO Slider Measure, and (C) convergent validity between the two measures.
In (A) and (B), each data point represents one participant-target combination. In (C), for each participant-target combination, there are two data points representing the firstλpaired with the first SVO°, and the second λ paired with the second SVO°. The green line is the theoretical relationship betweenλ and SVO°, same as Fig 3B. Data points on the boundaries, which are treated as censored data, are represented as crosses (same for all figures below). The ellipses indicate the 1-σ and 2 - σ iso-density loci of the fitted bivariate normal distributions with parameters set to their posterior medians.
Fig 5.
Relationship between and social distance, for the quadratic Lambda Slider (A) and the SVO Slider Measure (B).
Each raw data point is one of the two measurements of a participant–target combination. Black points and ranges represent the means and standard errors of data in each group. Blue lines and ranges represent the conditional effects (also called marginal effects; [25]) of the social distance ranking as a monotonic predictor, with 95% credible intervals.
Fig 6.
Payoff functions of the three quadratic Lambda Sliders in Experiment 2.
The ranges on the x axes reflect the ranges of the sliders.
Fig 7.
(A) Empirical distributions of raw slider positions () for the base, positive-shift and negative-shift sliders. The black rhombuses represent means and standard errors, and the gray lines represent standard deviations. For reference, the red and blue bars mark predictions of
(Eqs (21) and (22)), respectively, given the mean on the base slider. (B and C) Responses compared to predictions of
. For each participant–target combination, there are two raw data points in each panel representing the two measurements on either slider. The diagonal lines indicate the predictions of the two hypotheses without noise. The ellipses indicate the bivariate normal distributions representing the two fitted models (see S4 Appendix).
Fig 8.
Payoff functions of the three quadratic Lambda Sliders in Experiment 3.
The ranges on the x axes reflect the ranges of the sliders.
Fig 9.
Inequity aversion in Experiment 3.
(A) Posterior distributions of for each participant sorted by posterior median. The dots indicate the posterior medians and the lines indicate the 95% credible intervals. Three participants are highlighted, whose raw responses are plotted in (B)–(D). Targets 1–5 are the targets listed by the participants in the List stage, in increasing order of social distance. The target “M” is the Maui wildfires victim. For each participant–target combination, the horizontal line represents the posterior median of
. For each participant–target–slider combination, the cross represents the predicted utility-maximizing response given the posterior medians of
and
, while the two dots are the actual responses.