The motive cocktail in altruistic behaviors

Prosocial motives such as social equality and efficiency are key to altruistic behaviors. However, predicting the range of altruistic behaviors in varying contexts and individuals proves challenging if we limit ourselves to one or two motives. Here we demonstrate the numerous, interdependent motives in altruistic behaviors and the possibility to disentangle them through behavioral experimental data and computational modeling. In one laboratory experiment (N = 157) and one preregistered online replication (N = 1,258), across 100 different situations, we found that both third-party punishment and third-party helping behaviors (that is, an unaffected individual punishes the transgressor or helps the victim) aligned best with a model of seven socioeconomic motives, referred to as a motive cocktail. For instance, the inequality discounting motives imply that individuals, when confronted with costly interventions, behave as if the inequality between others barely exists. The motive cocktail model also provides a unified explanation for the differences in intervention willingness between second parties (victims) and third parties, and between punishment and helping.


Introduction
Many people voluntarily provide resources like shelter, food, and healthcare to refugees fleeing war-torn regions, while others advocate sanctioning responsible nations, even at the expense of personal cost.This altruistic behavior, known as third-party punishment (3PP) and helping (3PH), involves sacrificing personal interests to punish transgressors or help victims.Such behaviors have been observed both in laboratory (Fehr & Fischbacher, 2004;Henrich et al., 2006;Jordan et al., 2016;Marlowe et al., 2008) and field studies (Balafoutas et al., 2014(Balafoutas et al., , 2014;;Singh & Garfield, 2022).What, then, motivates these actions?
The desire to reduce inequality is believed to underlie both altruistic punishment and helping behaviors (Darley & Pittman, 2003).Unlike other altruistic behaviors that potentially enhance reciprocity (Gintis, 2000) or signal trustworthiness (Jordan et al., 2016), most 3PP or 3PH actions do not yield any direct benefit to the third party (Balafoutas et al., 2014), thus highlighting humans' innate aversion to inequality.Indeed, by quantifying inequality as a kind of loss in a rational framework of utility maximization, the seminal work of Fehr and Schmidt (1999) provides a unified explanation for a range of social-economic phenomena (Fehr & Fischbacher, 2004), including altruistic punishment and helping behaviors (Blain et al., 2022;Ferguson, 2021;Paulus et al., 2013;Qu et al., 2018;Stallen et al., 2018;Zhong et al., 2016).There is also neural evidence (Hsu et al., 2008;Stallen et al., 2018;Tricomi et al., 2010) that the utility of (in)equality is computed in the human brain.
The power of this normative framework (Fehr & Fischbacher, 2004), lies in its potential to integrate different motives into one utility measure to address the complexity of human altruistic behaviors.However, its potential is far from thoroughly explored, because most previous studies only focused on one or two motives and modeled them as independent and additive terms in utility calculation.The resulting utility-maximizing solution thus has difficulty explaining intricate behavioral patterns, unless accompanied by additional assumptions that are incompatible with utility maximization.For example, to explain why people may choose to punish the transgressor but not punishing enough to restore equality (Fehr & Fischbacher, 2004), a personal tendency, willingness to punish, is introduced (Stallen et al., 2018).Similarly, that people often (Batistoni et al., 2022;Dawes et al., 2007;FeldmanHall et al., 2014FeldmanHall et al., , 2018;;Ferguson et al., 2019;Singh & Garfield, 2022;van Doorn et al., 2018;Wiessner, 2020) but not always (Hu et al., 2015;Lotz et al., 2011;van Prooijen, 2009;Wang et al., 2022) prefer helping victims over punishing transgressors even when the two actions equally reduce inequality is partly attributed to empathy for the victim (Hu et al., 2015;Leliveld et al., 2012;van Prooijen, 2009).
Here we aimed to extend the normative framework of utility maximization to provide a unified explanation for a wider range of phenomena in altruistic behaviors.We constructed a series of computational models that assume altruistic behaviors are driven jointly by multiple socioeconomic motives.These "cocktail motive" models not only cover a comprehensive set of socioeconomic motives established in the literature, such as selfcentered inequality (Fehr & Fischbacher, 2004;Zhong et al., 2016), victim-centered inequality (Li et al., 2022;Qu et al., 2018;Xie et al., 2017;Zhong et al., 2016), and efficiency (Engelmann & Strobel, 2004;Hsu et al., 2008), but may also consist of "compound" motives that are non-linear combinations of more elementary motives.
We designed a third-party intervention task-the Intervene-or-Watch task (Fig. 1a-b), which enables an unusually rich set of experimental conditions for testing this variety of motives that would otherwise be undistinguishable.On each trial (Fig. 1c-d), participants saw the outcomes from a dictator game, where the dictator ("transgressor") allocated more to themselves than to the receiver ("victim", e.g., 88 vs. 12 tokens).As the unaffected thirdparty, participants decided whether to accept a specific intervention offer (e.g., spending 10 tokens of their own payoff to reduce the transgressor's payoff by 15 tokens).Each participant completed 300 trials in 100 different conditions that varied in the transgressorvictim inequality as well as the scenario (punishment vs. helping), the cost and the impactto-cost ratio of the intervention offer.
We performed one laboratory experiment (N=157) and a pre-registered online experiment (N=1258), with all major findings of the former replicated in the latter.A threeway interaction of inequality × cost × impact ratio found in participants' intervention decisions suggests utility calculations that go beyond linear combinations of different motives.Indeed, participants' behavioral patterns were best fit by a "motive cocktail" model whose utility calculation involves seven socioeconomic motives, including two compound motives.We called the compound motives "inequality inattention", as if people are increasingly inattentive to the inequality between others when the intervention cost rises.Individuals' cocktail motives fall into three groups: justice warriors who have a strong intention to intervene whenever there is inequality, pragmatic helpers who are sensitive to the impact of their intervention to help the victim, and rational moralists who seek to achieve an acceptable standard of morality at the lowest cost to self-interest.Our model provides a unified explanation not only for 3PP and 3PH, but also for a wider range of phenomena in the literature, such as why the third party often but not always prefers helping than punishment, and why the harmed second party punishes more intensely than the third party does.

Results
On each trial of our Intervene-or-Watch task (Fig. 1c-d), participants saw the results of a dictator game, where an anonymous dictator ("transgressor") allocated more amounts to themselves than to an anonymous receiver ("victim"), such as 88 vs. 12 tokens.As the third-party, participants received 50 tokens in each trial and were offered an opportunity to intervene, such as spending 10 tokens (intervention cost) to reduce the transgressor's payoff by 15 tokens (impact ratio = 15/10 = 1.5).The task was to decide whether to accept this intervention offer to intervene or to keep all 50 tokens to themselves.Each trial was either in a punishment scenario (as the example above, Fig. 1a & 1c) or in a helping scenario (to increase the victim's payoff, Fig. 1b & 1d).Across trials, we varied the inequality between the transgressor and the victim (50:50, 60:40, 70:30, 80:20, or 90:10, with ±2 jitters), the intervention cost (10, 20, 30, 40, or 50), and the impact ratio (1.5 or 3.0).

Behavioral patterns in third-party punishment and helping
In Experiment 1, there were 157 participants.We first performed a generalized linear mixed model analysis (GLMM1, see Table S1) on participants' decisions (to intervene or not) to assess the effects of each independent variable and their interactions.

Interaction effects.
Thanks to our factorial experimental design with 4 dimensions and 100 conditions, we also identified three two-way and one three-way interaction effects that were seldom documented before.Under a higher impact-to-cost ratio, the preference for helping over punishment was stronger (scenario × ratio interaction: b = -0.39,95% CI [-0.47, -0.30], p < 0.001; Fig. 1j), and the probability of intervention changed more dramatically with the transgressor-victim inequality (inequality × ratio interaction: b = -0.08,95% CI [-0.14, -0.02], p = 0.017; Fig. 1k) and with cost (cost × ratio interaction: b = -0.08,95% CI [-0.14, -0.02], p = 0.015; Fig. 1l).According to the three-way interaction of inequality × cost × ratio (b = -0.21,95% CI [-0.27, -0.15], p < 0.001), a higher ratio also led to a stronger modulation of the intervention cost to participants' sensitivity to inequality (Fig. 1i).c-d, Time course of a trial for the punishment (c) and helping (d) scenarios.On each trial, participants first saw the outcome of a dictator game-out of 100 tokens how much the dictator ("transgressor", cartoon figure in orange shirt) allocated to themselves and to the receiver ("victim", blue shirt), such as 70 vs.30 (c) or 88 vs. 12 (d).As a third party starting with 50 tokens, participants (white shirt) were provided with an intervention offer, such as spending their own 10 tokens to reduce the transgressor's payoff by 15 tokens (c), or spending their own 20 tokens to increase the victim's payoff by 60 tokens (d).Participants' task was to decide whether to accept the intervention offer (press "yes") or do nothing (press "no").e-h, Main effects of scenario (e), transgressor-victim inequality (f), impact-to-cost ratio (g), and intervention cost (h) on the probability of accepting the intervention offer, p(yes).Participants had a higher probability to help the victim than to punish the transgressor (e), were more willing to intervene when the transgressor-victim inequality was more extreme (f), when the impact-to-cost ratio was higher (g), and when the intervention cost was lower (h).For each box plot, the bottom, middle, and top lines of the box respectively indicate the first quartile, the median, and the third quartile; the whiskers represent 1.5 times the interquartile range (IQR), which is the distance between the third quartile (Q3) and the first quartile (Q1).Data points beyond 1.5 times the IQR from the upper and lower quartiles are considered outliers and are represented by the color points.Each filled circle denotes one participant.The *** denotes p < 0.001 for the difference between adjacent conditions from the post hoc comparison, and p-values were corrected using the Bonferroni method.The black dot inside each box denotes the group mean.The gray line superimposed on the boxes denotes the prediction of the best-fitting model (i.e., the seven-motive "motive cocktail" model, described later).
i-l, Interaction effects on p(yes).The inequality × cost × ratio three-way interaction (i) shows that participants' probability of intervention increased more slowly with the transgressor-victim inequality when the intervention cost was higher, with this modulation effect more pronounced under a higher impact-tocost ratio.The two-way interactions involving the impact ratio show that under a higher ratio, the preference for helping over punishment was stronger (j), and the probability of intervention changed more dramatically with inequality (k) and with cost (l).Each dot denotes the mean across participants.Error bars denote SEM.As in e-h, the lines denote the predictions of the best-fitting model.
In sum, the behavioral patterns in our Intervene-or-Watch task not only agree with classic effects about altruistic punishment and helping, but also reveal intriguing interaction effects.These interactions pose challenges for previous decision models that assume a linear combination of simple motives.

Seven socioeconomic motives and their hypothetical effects
What socioeconomic motives may have driven the observed 3PP and 3PH behaviors?
Besides self-interest (the core of classical economic models), we considered five classes of computationally well-defined socioeconomic motives (Fig. 2a), which expand into seven motive terms in utility calculation.Five of these motives are adapted from the literature, including three variants of inequality aversion (Fehr & Fischbacher, 2004;Zhong et al., 2016), efficiency (Engelmann & Strobel, 2004;Hsu et al., 2008), and reversal preference (Li et al., 2022;Xie et al., 2017).The remaining two motives under the class of "inequality inattention" are newly defined here to capture the interaction between self-interest and inequality aversion.As unfolded below, each motive affects the utility gain from intervention relative to non-intervention (thus the tendency to intervene) in a different way (Fig. 2b).
Self-centered inequality refers to the payoff difference between self and others (Fehr & Fischbacher, 2004).Depending on whether it favors self or other, it can be further divided into disadvantageous inequality (self < other) and advantageous inequality (self > other), which lead to two variants of inequality aversion motives that are respectively controlled by parameters  and .Larger  implies a stronger aversion to receiving lower payoff than others (e.g., self 50 vs.transgressor 88), while larger  implies a stronger aversion to receiving higher payoff than others (e.g., self 50 vs.victim 12).In our experimental setting, before intervention, the participant always had lower payoff than the transgressor but higher payoff than the victim.As the result, the larger the , participants would be more motivated to penalize the transgressor to reduce their disadvantageous inequality, but less motivated to help the victim because helping would increase their disadvantageous inequality with the transgressor and even create a disadvantageous inequality with the victim (Fig. 2b, column 1).In contrast, the larger the  , participants would be more motivated to intervene in both the punishment and helping scenarios, unless the larger punishment leads to an undesirable advantageous inequality over the transgressor (Fig. 2b, column 2).
Victim-centered inequality refers to the payoff difference between the transgressor and the victim (Zhong et al., 2016).This variant of inequality aversion implies that participants have a distaste for the higher payoff of the transgressor over the victim.Participants with larger  would intervene more in almost all punishment and helping scenarios (Fig. 2b, column 3), unless the victim-centered disadvantageous inequality is too small (e.g., transgressor 51 vs. victim 49) to compensate for the cost of intervention.
Efficiency concern is a motive frequently used for modeling economic games (Engelmann & Strobel, 2004;Hsu et al., 2008) but seldom for 3PP or 3PH before, which assumes that people care about the overall welfare of others, that is, the sum of the transgressor's and the victim's payoffs in our case.Participants with larger  would be more likely to help the victim to increase the overall welfare, but less likely to penalize the transgressor to avoid reducing the overall welfare, regardless of the inequality between others (Fig. 2b, column 4).
Reversal preference refers to the motive that participants intend to reverse the payoff difference between the transgressor and the victim, rewarded by their payoff difference in the opposite direction (i.e., after intervention the victim would be more well-off than the transgressor).The parameter  controlling reversal preference is allowed to be either positive or negative, respectively implying a willingness or reluctance to reverse the economic status of others, thus making the term a generalized form of rank reversal aversion (Li et al., 2022;Xie et al., 2017).Individuals with more positive  would be more willing to punish or help when the impact (i.e., cost × ratio) is large enough (relative to the inequality) to yield a rank reversal between the transgressor and the victim (Fig. 2b, column 5).
Inequality inattention refers to people's tendency to underestimate the inequality between others as the intervention cost increases.We defined two types of inequality inattention motives: inaction inequality inattention (controlled by   ) and action inequality inattention (controlled by   ), which represent a diminished awareness of inequality when choosing not to intervene and when opting to intervene, respectively.Inequality inattention motives are "compounds" that reflect the modulation of self-interest to victim-centered inequality aversion.Participants with larger   would be less likely to intervene which differs from that of smaller  (victim-centered inequality aversion) in that it may cause no intervention even when the transgressor-victim inequality is high (Fig. 2b, column 6).
Conversely, participants with larger   would be more likely to intervene when the transgressor-victim inequality grows, irrespective of the scenario, as if they have an optimistic belief in the minimization of inequality following intervention (Fig. 2b, column 7).
Many of these motives would remain unidentifiable in a task involving only two parties, testing exclusively either punishment or helping scenarios, or lacking variation in cost or impact ratio.However, in our Intervene-or-Watch task, the seven motives forecast unique effects on intervention decisions, thus making them distinguishable in behavioral data.
Subsequent modeling analysis validated the discernibility of each parameter, even under simultaneous modeling (see Methods and Fig. S1).Each column is for one motive, with the parameter controlling the motive's magnitude going from small to large for the top, middle, and bottom heatmaps.For simplicity, when examining the effects of one single parameter, we set all other parameters to zero.The parameters   and   are exceptions, where the parameter  is set to 1, because their utility terms are multiplied by  .Each heatmap contains four submaps, horizontally divided by scenario (left for punishment, right for helping) and vertically by impact ratio (bottom for 1.5, top for 3.0).The x-axis of each submap denotes the severity of inequality, running from near equality (left: 50:50) to extreme inequality (right: 90:10).The y-axis of each submap denotes the intervention cost, from low cost (bottom: 10) to high cost (top: 50).Colors code △U, where more reddish (more positive △U) corresponds to a stronger preference for choosing yes while more bluish (more negative △U) a stronger preference for choosing no.For illustration purposes, the values of △U were scaled separately for each column and separately for positive and negative values.Each motive shows a distinct influence on △U and would thus lead to distinguishable effects on third-party intervention decision behaviors.

The "motive cocktail" model with all seven motives best predicts human behaviors
We assessed the seven socioeconomic motives' contribution to altruistic behavior by incrementally incorporating them into utility calculations, creating a series of increasingly complex computational models.We then compared these models' predictive power for the behavioral patterns observed in Experiment 1. Starting from a baseline coin-flipping model, which intervened at a fixed probability, and a self-interest (SI) model, we introduced five motive classes as utility terms in the following order: self-centered inequality (SCI), victimcentered inequality (VCI), efficiency concern (EC), reversal preference (RP), and inequality inattention (II).This process yielded seven different models (see Methods) with different predictions (Fig. S2).We used maximum likelihood estimation to fit each model to individual participants' decisions, and the corrected Akaike Information Criterion (AICc, Hurvich & Tsai, 1989) to evaluate each model's relative goodness-of-fit, accounting for complexity.We also computed the protected exceedance probability (PEP, Rigoux et al., 2014) to provide a group-level measure of each model's potential superiority.
Integrating each motive class (SI, SCI, VCI, EC, RP, & II) into our models led to considerable improvements in their fits (as indicated by lower AICc values in Fig. 3a).The full "motive cocktail" model that includes all the motives best predicted participants' decisions (lowest AICc, PEP > 99.99% among the seven models).A model recovery analysis (see Methods) further confirmed that the superiority of the full model was real and could not attribute to model misidentification: Among the 700 synthetic datasets generated by the six alternative models, none was misidentified as the full model (Fig. 3b).
The full model closely mirrored changes in participants' intervention probabilities across the 100 experimental conditions (Fig. 3c), successfully predicting the main and interaction effects of different variables (lines in Fig. 1e-1l).In contrast, alternative models failed to replicate certain patterns within the data (Fig. S2).To conclude, participants' thirdparty intervention decisions were jointly driven by self-interest and the seven socioeconomic motives, including the two inequality inattention terms.is plotted against the inequality (from 50:50 to 90:10).Different colors code different levels of intervention cost (from 10 to 50, darker color for higher cost).Each sub-panel corresponds to one scenario and impact ratio condition.The dots and error bars respectively denote the mean and SEM across participants.The solid lines denote the predictions of the full model.

Individual differences: justice warriors, pragmatic helpers, and rational moralists
Our Intervene-or-Watch task, with its 100 factorially-designed conditions, yielded a rich, multifaceted profile that captured not only the collective behavioral tendencies but also the nuanced 3PP and 3PH behaviors of individual participants.A clustering analysis of the behavioral patterns of the 157 participants revealed that they were best summarized by three distinct clusters (see Methods and Fig. 4a-b).Among them, the "justice warriors" (35% of participants) had an overall high probability to intervene, especially when the transgressor-victim inequality was high, and the cost was relatively low (Fig. 4j).The "pragmatic helpers" (18%) also had a high probability to intervene, but were insensitive to inequality or cost, and preferred helping over punishment (Fig. 4k).The "rational moralists" (47%) barely intervened unless their intervention cost was minimal (Fig. 4l).The full "motive cocktail" model accurately predicted not only the average behavior (Fig. 4i) but also the behavioral patterns specific to each individual cluster (Fig. 4j-l).
These dramatic individual differences were associated with different combinations of motive parameters (Fig. 4c-e).Kruskal-Wallis tests and the following post-hoc tests revealed significant differences across the three clusters for three out of the seven motive parameters (Fig. 4f-h & Fig. S3): action inequality inattention   (H(2) = 22.18, p < 0.001), reversal preference κ (H(2) = 15.57,p < 0.001) and inaction inequality inattention   (H(2) = 9.71, p = 0.008).The highest values of   , κ and   respectively occurred at justice warriors, pragmatic helpers, and rational moralists.To unravel the relationship of these parameters to the observed individual differences, we carried out a series of correlation analyses between individuals' parameter value and their sensitivity to different variables in the group level (multiple comparisons corrected for each parameter using FDR; see Fig. S4), where participants' sensitivity to a variable was defined as the normalized intervention probability difference after the corresponding variable was dichotomized.The observed behavioral differences across clusters coincide with the correlational effects of these parameters (Fig. 4m-r) and agreed with the insights we obtained through simulation (Fig. 2).For example, higher   implies increased tendency to perceive one's action as effective in reducing inequality, irrespective of the actual impact, when the intervention cost is high.Indeed, individuals with higher   were less sensitive to the impact ratio.Justice warriors, those who had the highest   among the three clusters, were least sensitive to the impact ratio (Fig. 4n).and used the silhouette value as the metric for the goodness of clustering, where a higher value indicates a larger ratio of the between-cluster to within-cluster distance.c-e, The median value of the motive parameters for each cluster.The outer contour of the spider plot indicates the highest median value for a specific parameter summarized across all clusters.f-h, The values of the action inequality inattention parameter   (f), reversal preference parameter  (g) and inaction inequality inattention parameter   (h) compared across the three clusters of participants.J: justice warriors.P: pragmatic helpers.R: rational moralists.The highest values of   ,  and   respectively occurred at justice warriors (J), pragmatic helpers (P), and rational moralists (R).For each box plot, the bottom, middle, and top lines of the box respectively indicate the first quartile, the median, and the third quartile; the whiskers represent 1.5 times the IQR, which is the distance between the Q3 and the Q1.i-l, The intervention probability p(yes) in the 100 different conditions for all participants (i) and for each cluster (j-l), with data (top panels) contrasted with the prediction of the full motive cocktail model (bottom panels).Similar to Fig. 2b, each heatmap contains four submaps, horizontally divided by scenario (left for punishment, right for helping) and vertically by impact ratio (bottom for 1.5, top for 3.0).The x-axis of each submap denotes the severity of inequality, running from near equality (left: 50:50) to extreme inequality (right: 90:10).The y-axis of each submap denotes the intervention cost, from low cost (bottom: 10) to high cost (top: 50).Colors code p(yes), where darker colors correspond to higher probabilities to choose intervention.m-r, How the three parameters (  ,  and   ) may contribute to the behavioral differences across the three clusters of participants.Each panel is for one main or interaction effect that was detected across individuals (as in Fig. 1), with the bar height denoting the corresponding effect size in each cluster.The arrow accompanying a parameter in a specific panel indicates the significant correlation between parameters and behavioral measures, and how the parameter modulates the plotted behavioral measure (the orientation of the arrow) in the group level (Fig. S4).For example, panel m shows that higher  was associated with higher overall p(yes) and higher   with lower overall p(yes), which coincide with the observed high p(yes) in pragmatic helpers (k) and low p(yes) in rational moralists (l).The sensitivity for a specific variable was calculated as the normalized intervention probability difference between the high and low manipulated conditions.Specifically, "low inequality" refers to trials with inequality levels of 60:40 and 50:50, whereas "high inequality" refers to inequality levels of 90:10, 80:20, and 70:30."Low cost" and "high cost" respectively refer to trials with cost ≤ 20 and cost > 20."Low ratio" and "high ratio" respectively refer to trials with impact ratio of 1.5 and 3.

Replication in a pre-registered, large-scale online experiment
To test whether our findings can be generalized to a large population with different cultural backgrounds, we performed a pre-registered, large-scale online experiment using the same experimental procedures, with 1258 participants (sample size pre-determined based on a model-based power analysis, Fig. S5) from over 60 countries (or regions, see Table S2).As summarized below, all major findings of Experiment 1 were replicated in Experiment 2.
As in Experiment 1, the full motive cocktail model outperformed the other models and accurately captured the behavioral patterns in Experiment 2 (Fig. 5a-b, see Fig. S7 for model recovery analysis).The behavioral patterns of the 1258 participants were best captured by six clusters (Fig. S6j), in which the first three clusters agreed with those in Experiment 1-justice warriors (16.60%,Fig. 5c), pragmatic helpers (17.30%,Fig. 5d), and rational moralists (27.00%,Fig. 5e).As in Experiment 1, each of these three clusters were best fit by the full motive cocktail model (or its derivatives; Fig. S8).The remaining 3 clusters of participants (39.10%,Fig. S6n-q) were best described by a heuristic model that linearly combines different independent variables (see Methods and Fig. S8b), accounting for phenomena, for example, always helping but seldom punishment.These participants probably have less engaging participation, which is more common in online settings.
Upon completion of the experiment, participants were asked to fill out personality questionnaires that assessed their prosocial inclinations in everyday life, including the social value orientation scale (SVO, Murphy et al., 2011) to measure selfishness, the interpersonal reactivity index (IRI) (Davis, 1983) for empathy concern.We computed Pearson's correlation coefficients (r) between each participant's model parameters (from the motive cocktail model) and the participant's personality measures (Fig. S9 and Fig. S10).In both Experiments 1 and 2, we found that stronger self-centered disadvantageous inequality aversion () or inaction inequality inattention (  ) was associated with more selfishness.When one of these two parameters was controlled, the correlations between   and selfishness (Experiment 1: ρ = -0.22,p = 0.006; Experiment 2: ρ = -0.16,p < 0.001) was still significant, but the correlation between  and selfishness was significant only in Experiment 2 (Experiment 1: ρ = -0.11,p = 0.16; Experiment 2: ρ = -0.12,p < 0.001).We also found that both inaction inequality inattention (  ) and action inequality inattention (  ) was associated with empathy in the opposite direction.When one of these two parameters was controlled, the correlation between   and empathy was still significant in both experiments (Experiment 1: ρ = -0.25,p = 0.002; Experiment 2: ρ = -0.12,p < 0.001), the correlation between   and empathy was significant only in Experiment 2 (Experiment 1: ρ = 0.12, p = 0.13; Experiment 2: ρ = 0.12, p < 0.001).

The motive cocktail model reproduces a broader range of puzzling phenomena
The motive cocktail estimated in participants' Intervene-or-Watch decisions is not necessarily unique to the binary-choice task or third-party intervention, but may underly human responses to inequality in general.With slight adaptations, the motive cocktail model can predict behavioral patterns in second-party punishment (2PP) as well as 3PP and 3PH (Fig. 6), including phenomena that were puzzling to previous theoretical models (Fehr & Fischbacher, 2004;Stallen et al., 2018).
One such phenomenon is that the intervener would spend a greater amount to penalize the transgressor when they themselves are the victim instead of the unaffected third party (i.e., 2PP > 3PP).We assume that the second-party intervener has all the motives a third-party intervener would have except for efficiency concern (see Methods for details), that is, the victim cares less about the welfare of the transgressor than the third party does.We found that the motive cocktail model, with parameters estimated from participants of our Experiment 1, can reproduce the 2PP>3PP phenomenon as well as the increase of punishment with increasing inequality observed in previous laboratory experiments (Fehr & Fischbacher, 2004;Stallen et al., 2018).For both experiments, simulations with the justice warriors' parameters best matched the data.
The task scenario of Stallen et al. (2018) differed from that of Fehr & Fischbacher (2004) and ours in that the first party robs from the second party, which violates social norms in a more aggressive way.In such case we assume that even the unaffected third party would have no efficiency concern.With this modification, our model reproduces the 3PP>3PH phenomenon in Stallen et al. (2018), in contrast to the helping-over-punishment preference found in most previous studies and our two experiments.and the three clusters of our Experiment 1. a, Reproduction of the 2PP and 3PP behaviors in Figure 5 of Fehr & Fischbacher (2004).The amount participants would use to punish the allocator in a dictator game is plotted as a function of the level of inequality favoring the allocator.In simulating 2PP behaviors, participants-as the second party (the receiver)-were treated as a third party who had all the motives of third parties except for efficiency concern.Our model simulation (with no free parameters) reproduced two effects in the data: (1) the amount participants use for punishment decreases almost linearly with the decrease of inequality when the inequality favors the allocator and is nearly zero when the inequality favors the receiver, and (2) 2PP is larger than 3PP.The simulation based on justice warriors' parameters best matched the data.b, Reproduction of the 2PP, 3PP, and 3PH behaviors in Stallen et al. (2018).The amount participants would use to intervene is plotted as a function of the level of inequality.The task scenario of Stallen et al. (2018) differed from that of Fehr & Fischbacher (2004) in that the first party robs from the second party, causing a more severe violation of social norms.In this case we assume the efficiency concern is excluded from the motive cocktail for all intervention behaviors, which leads to larger amounts for punishment than helping.As in a, the simulation based on justice warriors' parameters best matches the data.

Discussion
It has been widely known that people have an innate aversion for inequality, willing to change others' payoffs at the expense of their own, even when the action does not augment their reputation or encourage future cooperation (Dawes et al., 2007).What we have demonstrated here, through 100 different experimental conditions and a total of 1415 participants, is that such altruistic behaviors may result from a trade-off among as many as seven socioeconomic motives.Moreover, we have shown the plausibility to disentangle the effects of these different motives via computational modeling.The motive cocktail model we constructed here can explain a wide range of phenomena observed in the literature as well as in our two experiments.
Part of the phenomena found in our experiments and the literature, such as the increased third-party intervention with the increased inequality between the transgressor and the victim (Bernhard et al., 2020;Ciaramidaro et al., 2018;Civai et al., 2019;Cui et al., 2019;Egas & Riedl, 2008;Gummerum & Chu, 2014;Guo et al., 2020;Jordan et al., 2014;Mothes et al., 2016;Ouyang et al., 2021;Stallen et al., 2018;Yang et al., 2021;Zhong et al., 2016) and with the increased impact-to-cost ratio of the intervention (Bone & Raihani, 2015;Egas & Riedl, 2008), can be readily explained by previous models of inequality aversion as well as by the three different inequality aversion terms in our model.

However, inequality aversion alone cannot explain why participants in our experiments
were more likely to help the victim than to punish the transgressor, a finding in line with many previous studies (Batistoni et al., 2022;FeldmanHall et al., 2014FeldmanHall et al., , 2018;;Lotz et al., 2011;Singh & Garfield, 2022;van Doorn et al., 2018;Wiessner, 2020).Helping and punishment are equally efficient in reducing victim-centered inequality but differ in their influences on self-centered inequality, which would lead to a preference of punishment over helping, unless one is more uneasy about their advantage over others than the reverse.What enables our motive cocktail model to explain the preference for helping over punishment is the inclusion of efficiency concern as a utility term, that is, people also care about the overall payoff of the transgressor and the victim.
With an additional assumption that the motive of efficiency concern is weakened when oneself is the victim or when the transgressor violates social norms in a more aggressive way such as robbing or stealing from the victim (Stallen et al., 2018;van Prooijen, 2009), we can also explain why people spend more resources for 2PP than for 3PP (Fehr & Fischbacher, 2004;Stallen et al., 2018) and why a preference for punishment than helping is found in some studies (Stallen et al., 2018;van Prooijen, 2009), as our simulation shows (Fig. 6).The simulation also reveals that the proportional increase of punishment with inequality can be the consequence of the joint effect of the motive cocktail, without any "willingness to punish" to be assumed.Our motive cocktail model thus provides a unified account for 2PP, 3PP and 3PH behaviors.One motive documented in previous studies, which seems to go against inequality aversion, is rank reversal aversion (Li et al., 2022;Xie et al., 2017).Our motive cocktail model includes a generalized form of this motive and found that participants in our experiment prefer to reverse the initial inequality so that the victim has an advantage over the transgressor, similar to what happens in Shakespeare's The Merchant of Venice.This reversal preference motive is opposite to rank reversal aversion and our finding suggests that the latter may only apply to situations where the inequality is not caused by the benefited party (Li et al., 2022;Xie et al., 2017).
The sixth and seventh motive terms-the inequality inattention terms-distinguished our model from all previous models in that they assume interactions between cost and inequality aversion.In other words, different motives are not necessarily independent; the perceived inequality between others may be modulated by one's self-interest.That is, when the intervention cost is high, people act as if the inequality between others is minimal.In line with the joint functioning of multiple motives identified in our modeling analysis, we found a three-way interaction between cost, impact ratio, and the inequality between the transgressor and the victim.Such interaction was not found in previous studies, probably because most studies used cost as a dependent variable, measuring the amount of money participants were willing to spend on the intervention, rather than as an independent variable to manipulate.In contrast, the cost is manipulated by the experimenter in our task, resembling real-world scenarios where individuals are confronted with limited options when it comes to addressing others' inequalities.
In both the laboratory experiment (N=157) and the large-scale online replication (N=1258), we found diverse behavior patterns across individuals.Three types of interveners-justice warriors, pragmatic helpers, and rational moralists-differed in their probability of intervention, their sensitivity to variables such as cost and inequality, and their preference for helping over punishment.The motive parameters estimated from the motive cocktail model provides a multi-facet measure of such individual differences, opening questions such as how individuals' motives may depend on their personal experiences, cultural background, or genetic makeup.
The motive cocktail model we developed here extends the economic modeling of altruistic behaviors and may be applied to a broader range of socioeconomic tasks than second-party and third-party interventions.The new motives and individual differences we identified may also shed light on future research that aims to uncover the neural basis of human morality and its disorders, one of the most important scientific questions (Sanders, 557 2021).
intervene.On each trial, the transgressor allocated the 100 game tokens between himself/herself and the victim, while the victim had to accept the offer without any other options.Participants were told that all offers between a transgressor and a victim were made by other real participants, and that their decisions would affect their own payoffs as well as those of the victims and the transgressors.In reality, the offers between the transgressors and the victims were generated by a custom code and were designed to disentangle different hypotheses.To give the participants a more realistic experience and to familiarize them with the roles in the game, they were instructed to play two trials of the dictator game, in which they played the role of transgressor and victim respectively.In the Intervene-or-Watch task, participants had 50 game tokens in each trial which could be used to reduce the payoff of the transgressor in the punishment scenario or increase the payoff of the victim in the helping scenario.To avoid serial or accumulative effects, participants were instructed that their payoff was independent across trials and would not be accumulated through the task.They were also informed that 10% of the trials will be randomly selected and implemented at the end of the study to determine the payoffs of all players (or roles).Additionally, participants were explicitly informed that the roles of the transgressor and the victim were played by different participants in each trial, hence encouraging them to make decisions based solely on the current situation.Since all players in the task were anonymous, no reputation concern was involved in this task.The players also had no opportunities for interaction, thus reciprocity could be excluded.Therefore, participants' decisions to help and to punish in the Intervene-or-Watch task were altruistic.
Each trial (Fig. 1c & 1d) began with a fixation cross (600-800 ms), followed by an inequality window (1500 ms) displaying the allocation between the transgressor and the victim, and an intervention offer window (1500 ms) showing the intervention cost for the participant and the consequence of the intervention (impact ratio × intervention cost) to the transgressor or victim.Subsequently, in the decision window, participants were asked whether they would like to accept the intervention offer: Yes (to intervene) or No (not to intervene).The intervention would only be implemented if participants chose Yes.For example, if the intervention offer window displays an intervention cost of x in a trial, a decision of intervention would result in the transgressor losing (or the victim gaining) 1.5x or 3.0x in the punishment (or helping) scenario.There was no time limit for the decision.A visual feedback window after the decision highlighted the selected choice in red.Four independent variables were varied across trials: scenario (punishment and helping), inequality (transgressor vs victim, 50:50, 60:40, 70:30, 80:20, 90:10, jitter ± 2), cost (10, 20, 30, 40, 50), and impact ratio (1.5 and 3.0).This led to 100 unique conditions, with each condition repeated 3 times for each participant.The scenario variable varied between blocks and the other three variables were randomly interleaved within blocks.Before each block, participants were told whether the following section was the "increase" condition (the helping scenario) or the "reduce" condition (the punishment scenario).In total, each participant completed 300 trials in 6 blocks, with 3 blocks each for the punishment and helping scenarios.

Personality questionnaires
Following the Intervene-or-Watch task, participants completed several personality questionnaires that allowed us to access their prosocial tendencies in daily life.Specifically, a Social Value Orientation (SVO, Murphy et al., 2011) was used to measure individual preference about how to allocate financial resources between themselves and others.A higher score on the SVO scale reflects a greater degree of concern for others' payoffs and, therefore, indicates a more prosocial personality.A Machiavellianism Scale (MACH-IV, Rauthmann, 2013) was used to assess an individual's level of Machiavellianism, related to manipulative, exploitative, deceitful, and distrustful attitudes.Individuals with higher scores on the MACH-IV scale are indicative of a more pronounced degree of Machiavellian traits.
An Interpersonal Reactivity Index (IRI, Davis, 1983) was used to measure the multidimensional assessment of empathy, including (1) perspective-taking, assessing an individual's tendency to consider a situation from others' perspective; (2) fantasy, evaluating an individual's inclination to identify with the situation and emotions of characters in books, movies, or theatrical performances; (3) empathy concern, measuring an individual's inclination to care about the feelings and needs of others; (4) personal distress, assessing an individual's tendency to experience distress and discomfort in challenging social situations.

Model-free analysis
All statistical analyses (except for group-level Bayesian model comparisons) were conducted in R 4.2.1 (R Core Team, 2013).Generalized linear mixed models (GLMM) assuming binomial distributed responses were used to model the probability of intervention, given various predictors (e.g., scenario, inequality) and their interactions.The GLMMs were implemented by "lme4" package (Bates et al., 2015), with the fixed-effect coefficients output from the binomial GLMM on the logit scale and the significance of each coefficient determined by the z-statistics.The standard linear mixed-effect models (LMM), which assume the error term is normally distributed, were estimated using the "afex" package to model participants' decision times.For the estimation of marginal effects and the post hoc analysis, the "emmeans" package was used (Lenth et al., 2018).Interaction contrasts were performed for significant interactions and, when higher-order interactions were not significant, pairwise or sequential contrasts were performed for significant main effects.GLMM1: participants' choices of all trials in Experiment 1 are the dependent variable; fixed effects include an intercept, the main effects of the scenario, inequality, cost, ratio, trial number, and all possible interaction effects of the independent variables; random effects include correlated random slopes of scenario, inequality, cost, ratio and trial number within participants and random intercept for participants.The scenario is a category variable.Trial number, inequality, cost, and ratio are continuous variables that were normalized to Z-score prior to model estimation.See Table S1 for the statistical results of GLMM1.
GLMM2: participants' choices of all trials in Experiment 2 are the dependent variable.The fixed and random effects remain the same as GLMM1.See Table S3 for the statistical results of GLMM2.
LMM1: participants' decision times of all trials in Experiment 1 are the dependent variable.
In addition to the fixed and random effects included in GLMM1, participants' intervention decisions (choice) are added as well.See Table S4 and Fig. S11 for the statistical results of LMM1.
We found an inverted U-shaped relationship between the intervention probability (p(yes)) and decision time (Fig. S11j), which implies that participants made decisions with more difficulty when the decision uncertainty (or entropy) was higher.This result is in line with prior research demonstrating an inverted U-shaped relationship between confidence levels and decision times (Lee & Dry, 2006;Ratcliff & Starns, 2013).
LMM2: participant's decision times of all trials in Experiment 2 are the dependent variable.
The fixed and random effects remain the same as LMM1.See Table S5 for the statistical results of LMM2.The inverted U-shaped relationship between the probability of intervention (p(yes)) and decision time was replicated in Experiment 2 (Fig. S12).
The sensitivity analysis to different variables.We measured participants' intervention sensitivity to different variables, which was defined as the normalized intervention probability difference after the corresponding variable was dichotomized (Fig. 4n-r and Fig. S4).Specifically, participants' sensitivity to the main effects, including scenario, ratio, cost and inequality, was calculated as the intervention probability difference in the helping trials compared with the punishment trials, the high-impact ratio trials (i.e., 3.0) compared with the low-impact ratio trials (i.e., 1.5), the low-cost trials (i.e., cost<=20) compared with high-cost trials (cost>20), and the high-inequality trials (i.e., the inequality level between the transgressor and the victim is 80:20 and 90:10) compared with the low-inequality trials (i.e., 70:30, 60:40 and 50:50), divided by their overall p(yes), respectively.For the interaction effects, the sensitivity (i.e., the normalized intervention probability difference) was calculated in a similar way as the main effect, that is, marginalizing over the other variables (see Fig. S4 for details).

Behavioral modeling
We assumed that participants would make decisions on each trial by calculating the utility of the two options (yes and no) and choosing the option with the higher utility.In the Intervene-or-Watch task, participants were given the context regarding inequality between a transgressor and a victim as well as other related variables (e.g., cost, impact ratio) from the perspective of a third-party and afterwards made a decision between two alternatives, yes (to intervene) and no (not to intervene).In general, participants calculated the utilities of the choices by estimating the reduction in inequality for others through their intervention and considering the associated cost to themselves.Specifically, if they chose "yes" (decide to intervene), they could reduce the inequality between the transgressor and the victim to some extent but at a cost.On the contrary, by choosing "no" (decide not to intervene), they could keep the inequality between the transgressor and the victim without incurring any cost.To investigate how individuals make decisions in the Intervene-or-Watch task, we constructed a series of computational models with different utility calculation hypotheses (i.e., combinations of multiple socioeconomic motives) and compared their goodness of fit.
Participants' choices were then modelled using the Softmax function (Eq. 1, Luce, 1959), with the utilities of not intervention (  ) and intervention (  ) from different models as the inputs: where the inverse temperature, parameter  ∈ [0, 10], controls the stochasticity of participants' choices, with a larger λ corresponding to less noisy choices.
In the following descriptions, we will use  1 ,  2 and  3 to denote the payoffs of the transgressor, the victim, and the third-party (participant) if the third-party does not intervene (choose 'no'), and use  1 ′ ,  2 ′ and  3 ′ to denote the counterpart payoffs if the third-party intervenes (choose 'yes').In particular,  3 ′ is equal to  3 −  in both scenarios.In the punishment scenario,  1 ′ =  1 −   ×  and  2 ′ =  2 .While in the helping scenario,  1 ′ =  1 and  2 ′ =  2 −   × .  =  3 (4) where  3 denotes the payoff of the third-party when choosing "no" (without intervention), which is always 50 tokens in each trial.′ 3 denotes the payoff of the third-party after choosing "yes" (with intervention), equaling to 50 − .
Building upon the SI model, the following hypothetical socioeconomic components were progressively introduced into the utility calculation and participants' choices were modeled using the Softmax function.The necessity of each component to explain participants' decisions was determined through model comparisons.

Model 3: SI and self-centered inequality aversion model (SI+SCI).
Based on SI model, we added a self-centered inequality aversion component, which assumes that participants are averse to the inequality between themselves and others in both directions (Fehr & Schmidt, 1999).The self-centered Disadvantageous Inequality (DI) aversion corresponds to that participants are averse to others having more payoffs than themselves, while the self-centered Advantageous Inequality (AI) aversion denotes participants are averse to themselves having more payoffs than others.The contributions of self-centered disadvantageous and advantageous inequality are controlled separately by the parameters  ( ∈ [0, 10]) and  ( ∈ [0, 10]) and are subtracted from the self-interest.Under the assumption of the SI+SCI model, participants are motivated to maximize their self-interest and meanwhile, minimize the inequality between themselves and others and then make a choice between no intervention and intervention based on their respective utilities: where j denotes the index of the transgressor and victim;  1 and  2 represent the payoffs of the transgressor and the victim when participants (the third-party) choose "no";  1 ′ and  2 ′ represent the payoffs of the transgressor and the victim after the intervention of the third party.
Model 4: SI+SCI and victim-centered disadvantageous inequality aversion model (SI+SCI+VCI).Based on the SI + SCI model, we introduced another previously proposed inequality component, the victim-centered disadvantageous inequality aversion (VCI).The VCI assumes that participants are averse to the transgressor having more payoff than the victim (Zhong et al., 2016), with its contribution to the utility calculation determined by a parameter  ( ∈ [0,10]) .Participants with larger  will be more willing to intervene in almost all punishment and helping scenarios.Within this model, participants were motivated to maximize self-interest and simultaneously, minimize the two kinds of inequality aversions (SCI and VCI): Participants with larger ω will more likely intervene in the helping scenario, but not in the punishment scenario: having more money than the transgressor, while a negative value indicates that they are averse to the newly created reverse inequality.Participants with larger  will more likely intervene when the initial victim-centered disadvantageous inequality is small enough or the impact is large enough to guarantee an inequality reversal: (self-centered disadvantageous inequality aversion) and  (efficiency concern),  and   (inequality inaction inattention).To exclude the possibility that the correlation was due to parameter redundancy in the model, we performed redundancy checks as follows.We first randomly shuffled participants' labels for different parameters to eliminate correlations in the shuffled parameters.Based on these shuffled parameters, we generated 157 synthetic datasets and used them to estimate the model parameters.We found little correlations between the parameters estimated from these synthetic datasets, which indicates that the high correlations found in the data reflect the behavioral characteristics of human participants rather than redundancy in the model itself (Fig. S13).

Model fitting and model comparison
For each participant, we fit each model to their intervention decisions across all trials using maximum likelihood estimates.The likelihood function derived from the binomial distribution was used to describe the relationship between participants' choice and the model's prediction.The function fmincon in MATLAB (MathWorks) was used to search for the parameters that minimized negative log-likelihood.To increase the probability of finding the global minimum, we repeated the search process 500 times with different starting points.We compared the goodness of fit of each model based on two metrics: Akaike information criterion with a correction for sample size (AICc) (Hurvich & Tsai, 1989) and the protected exceedance probability of group-level Bayesian model selection (Rigoux et al., 2014).

Model identifiability and parameter recovery analyses
We further performed a model identifiability analysis to rule out the possibility of model mis-

Clustering analysis
To gain further insight into whether the motive cocktail model (model 7: SI+SCI+VCI+EC+RP+II) could explain the varying behavioral patterns of individuals, we classified participants' intervention decisions using K-means clustering and then investigated the distributions of the estimated parameters across participants as well their unique contributions to behavioral patterns within each cluster.K-means clustering is an unsupervised machine learning algorithm relying on the Euclidean distance to classify each participant into a specific cluster with the nearest mean (Hartigan & Wong, 1979).The clustering evaluation criterion was based on silhouette value which denotes how well each participant was matched to its own cluster compared to other clusters, with a higher silhouette value indicating the clustering solution is more appropriate (Rousseeuw, 1987).
The optimal cluster solution for 157 participants in Experiment 1 is 3 (Fig. 4a).

Correlation analysis between the estimated parameters and behavioral or personality measures
To further validate the psychological significance of the hypothetical socioeconomic motives in the motive cocktail model, we calculated Pearson's correlation between the estimated parameters and the scores on the personality measurements.A similar

Fig. 2 |
Fig. 2 | The seven socioeconomic motives and their hypothetical effects on the third-party's utility gain to intervene relative to not to intervene.a, Five classes of computationally well-defined socioeconomic motives that expand into seven motive terms in utility calculation.The parameters  and  respectively control the motives of disadvantageous (self < other) and advantageous (self > other) inequality aversion.The parameter  controls the motive of victim-centered disadvantageous (victim < transgressor) inequality aversion.The parameter  (could be positive or negative) controls the direction and strength of the reversal preference motive (victim > transgressor after intervention).The parameter  controls the strength of efficiency concern (maximizing the total payoff of others).The parameters   and   respectively control the motives of inaction and action inequality inattention (attenuated perception of inequality under higher intervention cost).b, How the strength of each motive would influence the △U (utility of choosing yes -utility of choosing no) of the third-party intervention decision.

Fig. 3 |
Fig. 3 | The "motive cocktail" model with all seven socioeconomic motives best predicts participants' behavioral patterns.a, Model comparison results.For each participant, the model with the lowest AICc was used as a reference to compute ∆AICc by subtracting it from the AICc of the other models (△AICc = AICc -AICclowest).Lower ∆AIC indicates better fit.Protected exceedance probability (PEP) is a group-level measure of the likelihood of each model's superiority over others.The name of a model (e.g., SI+SCI) conveys the motives included in its utility calculation.SI: self-interest.SCI: self-centered inequality.VCI: victim-centered inequality.EC: efficiency concern.RP: reversal preference.II: inequality inattention.Both ∆AICc and PEP suggest that the full "motive cocktail" model best fits participants' decision behaviors.b, Model recovery analysis.Each model was used to generate 100 synthetic datasets, for each of which model fitting and comparison were performed.Each column is for one generative model.Each row is for one fitting model.The color in each cell codes the probability that the synthetic datasets from the generative model in the column are best fit by the fitting model in the row, with darker color indicating higher probability.c, Data versus the predictions of the full model.The probability of intervention, p(yes),

Fig. 4 |
Fig. 4 | Three types of 3PP and 3PH behaviors: justice warriors, pragmatic helpers, and rational moralists.a, Illustration of the three types of behavioral patterns.b, The clustering performance of behavioral patterns was best for three clusters.To identify the distinct types of behaviors in the 157 participants of Experiment 1, we used k-means to classify participants' behavioral patterns into clusters

Fig. 5 |
Fig. 5 | All major findings were replicated in the pre-registered, large-scale online Experiment 2. a, Model comparison results.As in Experiment 1, the full motive cocktail model best fit participants' decision behaviors, as indicated by the lowest △AICc and a PEP over 99.9%.b, Data versus model prediction.As in Experiment 1, the full model can accurately predict not only participants' average behaviors, but also that of individual clusters.c-e, The median value of the motive parameters for the first three clusters.The three clusters had similar behavioral patterns and parameter combinations to those of the justice warriors, pragmatic helpers, and rational moralists identified in Experiment 1. Conventions follow Fig. 4.

Fig. 6 |
Fig. 6 | The motive cocktail model successfully reproduces a broader range of puzzling phenomena.We used the full motive cocktail model estimated from the Intervene-or-Watch task (3PP and 3PH) to simulate the second-party punishment (2PP) as well as the 3PP and 3PH behaviors in

Model 1 :
The baseline model.We modeled each participant's choices of intervention in each trial (i.e., whether to choose the yes option) as outcomes from a Bernoulli distribution, where the intervention probability is controlled by a parameter,  ∈ [0, 1] .For each participant, the probabilities of choosing the intervention (p(yes)) and not choosing the intervention (p(no)) are denoted as follows: -interest model (SI).The models based on socioeconomic motives started with SI, where participants only consider self-interest when making decisions and thus, always leading to a reduced utility of the intervention.Participants' choices were then modeled using the Softmax function (Eq.1).

9 )
Model 5: SI+SCI+VCI and efficiency concern model (SI+SCI+VCI+EC).Based on the SI+SCI+VCI model, efficiency concern(Engelmann & Strobel, 2004) component was added into the model.The efficiency concern (EC) assumes that participants are motivated to maximize the total payoff of others, which is weighted by parameter  ( ∈ [0,10]).

)
Model 6: SI+SCI+VCI+EC and reversal preference for victim-centered advantageous inequality model (SI+SCI+VCI+EC+RP).Based on the SI+SCI+VCI+EC model, we introduced another component, the reversal preference for victim-centered advantageous inequality (RP), into the model.RP is mutually exclusive to VCI and assumes that participants prefer to reverse the economic status of the victim.That is, RP motivates participants to make the victim have more payoff than the transgressor by punishing the transgressor or helping the victim.The reversal preference is controlled by the parameter  ( ∈ [−10,10]).A positive value of  indicates that participants are in favor of the victim

)
Model 7: SI+SCI+VCI+EC+RP and inequality inattention model (the motive cocktail model, SI+SCI+VCI+EC+RP+II).Based on the SI+SCI+VCI+EC+RP model, we included a newly proposed inequality inattention (II) component.Thus, the motive cocktail model includes seven socioeconomic motives.II is derived from the rational framework of economic decisions and is implemented to capture the interaction between self-interest and VCI.Specifically, II assumes that people will systematically disregard the victimcentered disadvantageous inequality as costs increase.We proposed two types of inequality inattention: inaction inequality inattention (III, controlled by parameter   ) and action inequality inattention (AII, controlled by   ), which are respectively blind to the initial and residual disadvantageous inequalities between the transgressor and the victim under no intervention and intervention with rising costs, respectively.In the model fitting, the range of parameters   and   is restricted to between 0 and 20.Participants with larger   would have a lower probability to intervene.The effect differs from victim-centered disadvantageous inequality aversion (small γ) in that at large   the tendency to intervene would barely increase with inequality.Conversely, participants with larger   , who subjectively exaggerate the reduction of inequality by intervention, would have a higher probability to intervene.Those with large   would have similarly high probability to intervene regardless of the impact ratio, as if they optimistically believe that the inequality would be minimized by any of their intervention:   =  3 −  max( 1 −  2, 0)   −  ∑ max(  −  3  (cos /50) (17)Redundancy checks on the parameter space for the motive cocktail modelIn the estimated parameters, we observed three highly correlated pairs in the parameter space of the motive cocktail model: the value of parameter  (self-centered advantageous inequality aversion) and  (victim-centered disadvantageous inequality aversion), identification in model comparisons.For each model, the parameters estimated from the data of all participants were used to generate a synthetic dataset of 157 participants.Each synthetic dataset regarding a specific model was then used to fit each of the 7 alternative models and identify the best fitting model by model comparison.We repeated the above procedure 100 times to calculate the percentages that each model was identified as the best model based on all synthetic datasets from a specific generating model.The highest percentage assigned to the same fitting model as the generating model suggests that the model is identifiable.To assess parameter recovery in the motive cocktail model (model 7: SI+SCI+VCI+EC+RP+II), we computed the Pearson correlation between the parameters estimated from the 100 synthetic datasets (recovered parameters) and the parameters used to generate the synthetic datasets.The larger correlation coefficient between the recovered parameter and the estimated parameter indicates a non-redundancy in parameter space.