To discriminate between disadvantageous and advantageous IA, we, first, aggregated the choice data across all four reward distributions at the age, allocator-, recipient-, and dyad-sex level (Fig. 2).
<Figure 2 about here >
Figure 2 - Fraction of Equity choice across Age (rows) and Choice type (column). Greyscale bars indicate the 4 possible combinations of allocator- and recipient-sex (see inset for subgroup labels). To assess statistical differences in proportion of equal choices between groups inside an (age, choice type) combination, a Chi-Squared test was run. *: p<0.05 in a Chi-Square test for equal proportion (uncorrected), suggesting unequal proportions between subgroups.
Descriptively, the percentages of equity choices (i.e., inequity aversion) increased with age across all four reward distributions. The lowest proportion of equity choices was found in 3-4-year-old children in the costly disadvantageous distribution (31.30%), the highest in 7-8-year-old children in the non-costly disadvantageous distribution (87.90%). Regarding sex of the allocator and recipient, on a descriptive level, boys made more equity choices in both disadvantageous distributions when paired with a male (non-costly: 81.30%; costly: 62.50%) than when paired with a female recipient (non-costly: 72.50%; costly: 59.40%). The reversed pattern was found in the two advantageous distributions, i.e. boys preferred choices with equal outcomes when paired with a female (non-costly: 58.00%; costly: 39.10%) compared to a male recipient (non-costly: 48.40%; costly: 23.40%). Likewise, girls made more equity choices in both disadvantageous distributions when paired with a male (non-costly: 81.40%; costly: 64.30%) compared to a female recipient (non-costly: 67.10%; costly: 48.70%). In the two advantageous distributions there was no consistent choice behavior. Girls made more equity choices with female recipients in the non-costly distribution (71.10%; with male recipients: 51.40%), but more equity choices with male recipients in the costly distribution (50.00%; with female recipients: 43.40%). In all groups, there were always more equity choices in the non-costly than the costly distributions. On the whole, these results already preclude a simple efficiency maximizing choice heuristic, as both boys and girls preferred the numerically inferior option (2 vs. 5 or 6 tokens) in the disadvantageous distributions. Similarly, an always-choose-equal rule cannot explain these results, either, because of considerable within-subject variability in choosing the equal option across choice options. Moreover, in at least the costly advantageous condition, both boys and girls (averaged across ages) did not prefer the equal outcome.
As each child was either first or second in the role of allocator, we compared whether being allocator first or second influenced choices over all dilemmas. The allocator order did not significantly impact choice allocation in any of the dilemmas (Chi-square tests per dilemma, c(1,N=279) ≤ 2.79; p > 0.05).
Analysis of the choice data with mixed-effect models
We used mixed-effect models to estimate the coefficients for Age (continuous), Type of Cost (Non-costly vs. Costly), Type of Inequity (Disadvantageous vs. Advantageous), Allocator Sex, and Recipient Sex on Equity Choice. We used forward model selection, based on Akaike Information Criterion (AIC), to arrive at the most parsimonious solution (see Table 2) and ANOVA tests to compare nested models for significant increase in explained variance.
Table 2 Statistical model comparisons. GLMMs were fitted with increasing complexity and compared on AIC for parsimony. Significance between (nested) model fits was assessed with an ANOVA. Model lm2.0 was chosen as a reference, modeling interactions between age and cost type (costly/non-costly) and between age and IA type (advantageous/disadvantageous). More complex models, involving sex of allocator or recipient, were compared to this reference model. lm2.5 (omitting age) is not nested with lm2.0 and thus this comparison cannot be statistically evaluated. However, the AIC score is much worse, suggesting it does not provide a better data fit.
What is clear from these nested model comparisons is that “Age” is the most important single factor in determining Equity Choice (all models including “Age” outperform the 4-way interactive model omitting Age – lm2.5, on AIC scores), and that the interaction of Cost and Inequity does not significantly outperform an additive model. Indeed, when we look down the list, the strongest models are those that allow for two parallel interactions: between age, sex, and cost, and between age, sex, and inequity. Of these, the model that includes recipient sex (lm2.4) is stronger than the model including allocator sex (lm2.3), suggesting that children’s equity preferences differ not only by age and choice type (lm2.0) but also by the sex of the allocator and even more by the sex of the recipient (cf. Fig.2; compare FF to FM in choice 1 for 5-6yo or MF vs. MM in choice 4 for 7-8yo). However, a 5-way model including all possible interactions between all factors failed to converge. Hence, even though our mixed effect model analysis suggests that the factors age, inequity, allocator sex and recipient sex matter for fairness preferences, we cannot know exactly how all factors interact. We thus decided to continue with a model-based analysis, taking advantage of the 2x2 setup of the choice problems (cost type x inequity type) to assess statistical difference between subgroups (combinations of Age, Allocator Sex, Recipient Sex) in the α and β parameters describing their choice patterns across the four dilemmas.
Model-based analysis of sex-dyad dependent inequity aversion
In order to get a better understanding of the social preferences underlying these choice patterns, especially regarding the interaction of age, type of inequity, allocator sex and recipient sex, we fitted the Fehr-Schmidt model of IA (Fehr & Schmidt, 1999) to the raw choice data. The Fehr-Schmidt-model reduces the model complexity and yields quantitative parameter estimates for α (disadvantageous IA) and β (advantageous IA, see methods).
Briefly, to estimate variability in α and β for a given subgroup, we applied a bootstrap approach with resampling, essentially repeating the modelling step for a randomized subsection of the original group and aggregating the obtained α and β values in a distribution from which we report the mean and variance (see methods for details). Subgroup scores were assessed for significance in comparison to confidence intervals on a reference population acquired similarly through bootstrap resampling of the original complete dataset. The 95th, 99th, and 99.9th percentile confidence intervals on these distributions are represented by dashed grey lines in the figure panels below (Figs 3+4). Comparisons between subgroups were assessed for significance through bootstrap permutation analyses.
Table 3: All a- and b-value estimates of the Fehr-Schmidt model fits
Age Group
|
Alpha
|
Sex
|
Alpha
|
Dyad
|
Alpha
|
3-4 year
|
0.10±0.04
|
F: allocator
|
0.14±0.05
|
F->F
|
0.02±0.03
|
|
|
F: recipient
|
-0.03±0.04
|
F->M
|
0.30±0.09
|
|
|
M: allocator
|
0.01±0.09
|
M->F
|
-0.27±0.13
|
|
|
M: recipient
|
0.26±0.08
|
M->M
|
0.21±0.08
|
|
|
|
|
|
|
|
|
|
|
|
|
5-6 year
|
0.66±0.11
|
F: allocator
|
0.40±0.12
|
F->F
|
0.18±0.11
|
|
|
F: recipient
|
0.34±0.12
|
F->M
|
0.76±0.16
|
|
|
M: allocator
|
0.81±0.20
|
M->F
|
0.50±0.14
|
|
|
M: recipient
|
1.00±0.21
|
M->M
|
1.22±0.25
|
|
|
|
|
|
|
|
|
|
|
|
|
7-8 year
|
1.08±0.14
|
F: allocator
|
1.05±0.27
|
F->F
|
0.64±0.05
|
|
|
F: recipient
|
0.88±0.16
|
F->M
|
1.90±0.34
|
|
|
M: allocator
|
1.08±0.25
|
M->F
|
1.59±0.40
|
|
|
M: recipient
|
1.30±0.33
|
M->M
|
0.75±0.06
|
|
|
|
|
|
|
|
Beta
|
Sex
|
Beta
|
Dyad
|
Beta
|
3-4 year
|
0.16±0.07
|
F: allocator
|
0.35±0.06
|
F->F
|
0.41±0.03
|
|
|
F: recipient
|
0.28±0.07
|
F->M
|
0.24±0.12
|
|
|
M: allocator
|
-0.22±0.21
|
M->F
|
-0.19±0.22
|
|
|
M: recipient
|
0.03±0.16
|
M->M
|
-0.22±0.20
|
5-6 year
|
-0.23±0.16
|
F: allocator
|
-0.04±0.22
|
F->F
|
0.04±0.20
|
|
|
F: recipient
|
0.05±0.19
|
F->M
|
-0.19±0.26
|
|
|
M: allocator
|
-0.21±0.24
|
M->F
|
0.09±0.17
|
|
|
M: recipient
|
-0.48±0.28
|
M->M
|
-0.62±0.31
|
7-8 year
|
0.60±0.07
|
F: allocator
|
0.77±0.22
|
F->F
|
0.56±0.05
|
|
|
F: recipient
|
0.61±0.10
|
F->M
|
1.01±0.21
|
|
|
M: allocator
|
0.39±0.10
|
M->F
|
0.64±0.17
|
|
|
M: recipient
|
0.51±0.14
|
M->M
|
0.25±0.06
|
Inequity aversion increases with age
As a first step in our model-based analysis, we confirmed our descriptive report above that IA increases with age in young children, and that disadvantageous IA develops earlier than advantageous IA, replicating previous evidence (Fehr et al., 2008; Blake & McAuliffe, 2011; all results of our model-based analysis can be found in Table 3). Our results blend with those studies in that a increased significantly from levels below the reference distribution (p<.001 in a 1-sample test) in younger to average levels in middle-aged children (p<.001, permutation test with Bonferroni correction for the pairwise comparison, Figure 2A). Descriptively, a increased even further to levels larger than the reference distribution (p<.01 in a 1-sample test) in older children, but the increase from the middle age to eldest age group was not significant. We found b in younger children to be inside the reference distribution, while, in middle aged children, b was found to be significantly lower than average (p<.001 in a 1-sample test). This effect was already found in other studies (Sheskin et al., 2014) and might be due to the fact that the development of advantageous IA (b) includes overcoming an initial spiteful preference for diminishing others’ relative payoff. While b-levels did not significantly differ between younger and middle-aged children, they increased significantly from middle-aged to older children (p<.001, Figure 2B) to levels significantly higher than average (p<.001 in a 1-sample test).
< Figure 3 about here >
Figure 3 Development of α- (3A) and β-parameters (3B) with age (means ± standard error of the mean, s.e.m.). Between-group differences were assessed for significance using a pairwise permutation approach, with significance levels determined from the empirical permutation distributions and adjusted with Bonferroni-correction for multiple comparisons. One-sample tests for significance reflect comparisons to the confidence intervals of the global resampled parameter distributions.
Sex and dyad-sex-dependent differences in inequity aversion
To investigate sex-differences in IA, we re-organized our sample according to the biological sex of the allocator (child making the decision) and the recipient (partner). We found that female and male allocators did not differ in a (Figure 4A) and showed a-levels falling within the reference distribution. However, female allocators had higher b-values (higher advantageous inequity aversion; p<.05), and male allocators had lower b-values (less advantageous inequity aversion; p<.05) than average, and a pairwise comparison revealed significantly higher b-values in female than male allocators (p<.01, Figure 4B). When the recipients were male, a-levels were higher than average (p<.05), but when recipients were female, a-levels were lower than average (p<.05), suggesting that disadvantageous unequal reward distributions were less tolerated when recipients were male than female. This conclusion was supported by the permutation test which revealed significantly higher a-levels with male recipients than female recipients (p<.01, Figure 4A). b-scores broken down by recipient fell within the reference distribution; here, the opposite trend emerged with descriptively higher b-values towards female recipients (p=.06, Figure 4B).
< Figure 4 about here >
Figure 4 Main effects for allocator and recipient sex and for a (disadvantageous IA; 4A) and b (advantageous IA; 4B), and differences in a (4C) and b (4D) split by sex of allocator and recipient. Bars indicate the mean of the bootstrapped population scores (N=5000 draws). Error bars indicate the standard deviation of the bootstrapped population and thus act as s.e.m. Significance between groups was assessed through bootstrapped permutation analyses; significance of 1-sample tests (inside bars) reflects comparisons against the global reference population.
Unpacking these main effects of sex, we found that allocators did not treat all recipients equally. Lower a-scores, i.e. increased generosity towards female recipients when they would be better off than the allocators (Figure 4A), originated from female allocators only: while male allocators showed similar and average a-levels with recipients of both sexes, female allocators had markedly lower a-levels with female recipients (p<.001, one-sample test), and higher a-values with male recipients (p<.05, one-sample test) compared to the reference distribution, suggesting that female allocators tolerated being worse off than the recipients better when the recipients were female than when they were male. In addition, they discriminated significantly between recipients based on their sex (p<.001 permutation test, Figure 3C). Thus, girls showed an envy bias as they were selectively more generous (lower a-levels) with other girls than with boys, aligning their a-level with that of their recipient. Conversely, notably girls were unconditionally inequity averse against advantageous IA with recipients of all sexes: they showed significantly elevated levels of b-levels, compared to the reference distribution, independent of the recipient´s sex (p<.01 towards girls, p<.05 towards boys, one-sample tests, Fig. 4D), suggesting that they always aimed to reduce inequity if the other was worse off, independent of the recipient’s sex. By contrast, the same analysis with boys revealed that they showed average b-levels when the recipient was female but negative b-levels (which can be labelled spite, or negative compassion, Sheskin et al., 2014) when the recipient was male (p<.001 one-sample test, Fig. 4D). Thus, boys were sensitive to girls being worse off than them, but insensitive, or they even placed positive value, on other boys being worse off. Indeed, the pairwise comparison of b-levels towards male recipients supports this notion and shows significantly higher b-levels in female allocators, as expected from the allocator sex main effect (p<.05, one-sided). In sum, while boys aligned their levels of advantageous IA (or spite respectively), to the recipient’s sex, girls surprisingly had unconditional advantageous IA with recipients of all sexes, resulting in a mismatch between female and male advantageous IA in cross-sex interactions.
Development of dyad-sex-dependent differences in fairness preferences
Finally, we asked how these diverging patterns in egalitarianism relative to allocator- and recipient sex developed across the age groups. We focused on the strongest effects from the analyses that so far excluded age: the envy-bias, i.e., the girls’ tendency to be less tolerant toward recipients being better off when recipients were male than female, and the spite gap, i.e. girls disliking advantageous IA when the recipient was male, in contrast to boys, who were indifferent, or even placed positive value (spite) on being better off when the recipient was male. Indeed, we found that the envy bias in female allocators against cross-sex recipients became stronger with age. While we observed no envy bias in the youngest children, it appeared in middle-aged (p<.05) and peaked in older children (p<.01, Figure 5A). In a similar vein, the sex spite gap of boys against other boys, in comparison with female recipients, statistically manifested significantly only in the oldest age group (p<.05, Figure 5B).
< Figure 5 about here >
Figure 5 Development of envy bias and spite gap. Panels show the difference in parameter estimates for female minus male recipients (A) or allocators (B). Female allocators exhibited lower levels of disadvantageous IA (alpha-parameter against other female than male recipients (5A). This envy bias became gradually more pronounced with age. Spite gap, the difference in advantageous IA between female vs. male allocators towards male recipients, manifested only in the oldest children (5B). Scores indicate the difference between two bootstrap populations with respect to recipient sex (5A) or allocator sex (5B). Error bars indicate the standard deviation of the difference between these contrasted populations across bootstraps. Significance levels were assessed with a permutation analysis at the level of allocator/recipient/age as indicated.