A Computational Model of Non-optimal Suspiciousness in the Minnesota Trust Game

This study modelled spite sensitivity, the worry that others are willing to incur a loss to hurt you, which is thought to undergird suspiciousness and persecutory ideation. Two samples performed a parametric, non-iterative trust game known as the Minnesota Trust Game (MTG). The MTG distinguishes suspicious decision-making from otherwise rational mistrust by incentivizing the player to trust in certain situations but not others. In Sample 1, 243 undergraduates who completed the MTG showed less trust as the amount of money they could lose increased. However, only for choices where partners had a financial disincentive to betray the player was variation in the willingness to trust associated with suspicious beliefs. We modified the Fehr-Schmidt (1999) inequity aversion model, which compares unequal outcomes in social decision-making tasks, to include the possibility for spite sensitivity. An anticipated partner’s dislike of advantageous inequity (i.e., guilt) parameter included negative values, with negative guilt indicating spite. We hypothesized that the anticipated guilt parameter would be strongly related to suspicious beliefs. Our modification of the Fehr-Schmidt model improved estimation of MTG behavior. Furthermore, the estimation of partner’s spite-guilt was highly correlated with choices associated with beliefs in persecution. We replicated our findings in a second sample. This parameter was weakly correlated with a self-reported measure of persecutory ideation in Sample 2. The “Suspiciousness” condition, unique to the MTG, can be modeled to isolate spite sensitivity, suggesting differentiation from inequity aversion or risk aversion. The MTG offers promise for future studies to quantify persecutory beliefs in clinical populations.


Figure S1. Correlations of First Mover and Second Mover Thresholds. Spearman correlations of
Sample 1 and b) Sample 2 showed that participants did not simply assume their partner was like themselves, but instead had a significant negative relationship in Sample 1 and a negative trend in Sample 2. This result suggests that those who were the least trusting in the First Mover Game were the most selfish in the Second Mover Game.p < .05*,p < .01**,p < .001***

Figure S2. Random Simulated Model.
This model simulated responses in the game if they were assumed to be random.We see that, overall, all choices had a mean of .5 (as expected from our random model).Note, risk increases along the x-axis, as indicated by the decreasing Adverse Payoff.RA15 is the low temptation condition against the coin.RA25 is the high temptation condition with the coin.SUS15 is the Suspiciousness condition with the human partner.RMT25 is the Rational Mistrust condition with the human partner.The shading indicates 95% CI.   included two separate risk aversion parameters which related to distinct behavior in the game.We found that the additional estimated spite-guilt parameter was important to model suspicious behavior, thus allowing us to model spite sensitivity for the first time (H4a).However, separating positive and negative guilt into guilt and spite did not improve the model (H4b).Our results suggest that a fear of spite and risk aversion are separate mechanisms in decision-making.

Hypothesis 1
We tested H1 by comparing the Fehr-Schmidt model to the random model (Figure S1).Based on BIC scores, the Fehr-Schmidt model (Figure 2) is better than a random model (Table S1).Just adding a single parameter of envy a1 adds significant value to the performance of the model.However, when looking at the graph of actual to simulated data, we see that the Fehr-Schmidt model does not represent the decisions in the Suspiciousness condition well (Figure 2).

Hypothesis 2
To test H2, we assumed that the addition of the coin will act as a random partner (i.e., probability is equal to .5),whereas the human partner will behave in a more rational and predictable way.To test this, we first separated out the two conditions using the Fehr-Schmidt model, such that each condition had its own envy a and inverse temperature l.However, we also tested models in which there is only one l shared between the two conditions, even though there are two envy parameters (H2 Coin vs. Human in Table S1).Finally, we also tested if envy was only necessary for one of the conditions (either coin or partner), but not the other.
We hypothesize that the coin, as a random entity (50-50), will not require an envy parameter, but the human partner will.In addition, there will only be a shared l across the two conditions.
To compare the two partner conditions, we also tested if multiple envy a and inverse temperature l parameters were necessary.Of these sets of models, we found that the best model included separate envy a parameters for each condition, but a shared l across each softmax equation for the player and the estimated partner's behavior (H2 Shared Inverse Temperature in Table S1).However, modeling multiple envy parameters was not significantly better than a single envy parameter for both in the Fehr-Schmidt model.

Hypothesis 3
H3 examined the impact of risk aversion.The Risk Aversion conditions with the coin partner assisted in measuring risk aversion of the individual, since the coin is essentially a random partner.We model risk aversion (R1) by comparing the monetary amount of the adverse payoff x1 to the amount of the safe option, S.
We modeled risk aversion alone and with the estimated partner's guilt above.We also model risk aversion for both decision agents, human partner alone, and coin alone.
Several models were tested, including separating risk aversion between the two different conditions.The addition of risk aversion was most effective when envy a1 and inverse temperature l were shared across conditions, but risk aversion was considered separately (H3 Coin vs. Human Risk Aversion in Table S1).Despite the improved BIC score, the simulated data did not strongly separate the two human conditions (Rational Mistrust and Suspiciousness).

Hypothesis 4
Estimated Spite-Guilt We tested H4 by modeling the partner's behavior in several different ways.Because it is likely that the second mover is not deciding randomly, we added a parameter in which the first mover estimates the probability of the second mover's choice.We call this parameter estimated spite-guilt, as it is the ability to estimate the partner's guilt.We hypothesize this parameter will improve the model above and beyond the Fehr-Schmidt model of just envy a1.
We model this by estimating the parameter of the partner's guilt ( 1) ).Additionally, a human partner may also include some risk aversion associated with the potential loss of money, as well as require an estimate of the amount of guilt of the partner.Because in our dataset there is never a time in which the first mover has more money than the second mover, we simplify the equation to remove the partner's envy a2.
From this equation, we again use the softmax equation to estimate the probability of the second mover choosing the temptation.The probability is used as p, which influences weighted utilities of the two possible outcomes should the first mover choose to trust.

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We tested the model with three possible inverse temperature l options -a shared l across both individuals, distinct l for each player, and no estimated l for the second mover.As we assume that the second mover will have guilt, but not the coin, we additionally model the estimated partner's guilt 1 ) in three separate ways: both the fair coin and human partner, just the human, and just the coin (as a sanity check).When partner spite-guilt was not estimated, p was assumed to be .5.
We compared several models using estimated spite-guilt, including separating out the behavior between coin and human partners.First, we tested estimated spite-guilt with envy but without risk aversion.Estimated spite-guilt was most effective in the model when it was only used to estimate the human partner's behavior (H4a Estimated Spite-Guilt: Human), compared to estimating both partner conditions separately (H4a Estimated Spite-Guilt: Both) or just for the coin partner (H4a Estimated Spite-Guilt: Coin).The addition of estimated spite-guilt for only the human partner improved the model compared to the Fehr-Schmidt model.
When including risk aversion, we compared two models of a human partner's guilt with either a single shared envy and a shared risk (H4a Estimated Spite-Guilt: Human & shared Risk Aversion) or shared envy with a separated risk (H4a Estimated Spite-Guilt: Human & Split Risk Aversion).While the second model of risk aversion and estimated spite-guilt was a higher BIC, the simulated data using the H4a Estimated Spite-Guilt: Human & Split Risk Aversion model matched the behavior better.We also compared these models to splitting Risk Aversion as a general term (for all conditions and decision agents -General Risk Aversion) and one only for the human partner (Social Risk Aversion).This model had improved BIC scores and simulated data matched the expected pattern of behavior (Spite Sensitivity model in Table S1).

Estimated Spite & Guilt Separately
H4b was tested by separating the estimated spite-guilt into negative guilt (spite) and positive guilt.Instead of allowing guilt to be negative or positive, we modeled spite separately as a variable adding positive utility to the partner's advantage over the first mover.Each parameter will be constrained to positive values only.This will test if there is a bivalent relationship between guilt and spite, as opposed to a continuum with negative or positive guilt.
Including two envy parameters with estimated guilt and spite for the human partner only (H4b Estimated Spite & Guilt: Split Envy) was better than using one envy parameter with estimated guilt and spite (H4b Estimated Spite & Guilt: Shared Envy).
However, when we additionally included risk aversion, we found that 1 envy and 2 risk parameters (along with partner spite and guilt) were better fit (H4b Estimated Spite & Guilt: Shared Envy & Split Risk Aversion) than 2 envy and 2 risk parameters (H4b Estimated Spite & Guilt: Split Envy & Split Risk Aversion).
Based on BIC and simulated graphs, we identified that a single envy parameter, two risk aversion parameters, and human partner estimated guilt (as a continuous negative or positive value) best fit the data (Spite Sensitivity in Table S1).

Social Risk Aversion
To further examine this relationship between social risk aversion and the human partner thresholds, we compared the difference between the two conditions, RMT25-SUS15, and identified a negative relationship (rs(241) = .288,p < .001).This relationship means that when RMT25 thresholds are higher than SUS15, social risk aversion is also higher such that more risk aversion in the Rational Mistrust condition drove the social risk aversion parameter estimation.
As we had hoped it would represent general mistrust of other partners, it appears to match our expectations; however, the negative relationship with SUS15 thresholds seems to be driven by a disconnect between the SUS15 condition and the RMT25 condition.There was also a moderately strong correlation between the estimated spite-guilt parameter and the social risk aversion parameter (rs(241) = .490,p < .001).Social risk aversion was not correlated with the lower Risk Aversion condition thresholds (RA15: rs(241) = .10,p = .139)and somewhat associated with the higher Risk Aversion condition thresholds (RA25: rs(241) = .13,p = .046).As the social risk aversion parameter was added to the general risk aversion parameter in the human partner conditions, it is necessarily more associated with the human partner conditions.

Variables and Demographic Predictors
We examined the extent to which demographic variables predicted behavioral and computational measures in our study.See Table S2 for details.Overall, sex differences predicted several measures in the study.Men were more alienated and less harm avoidant in the MPQ measures; Previous research has shown that men are more likely to have higher alienation scores but no differences in harm avoidance (Finkel & McGue, 1997).In terms of behavior, men had lower thresholds in the RA15, RA25, and RMT25 conditions, suggesting men were less risk averse than women.Finally, computational variables corroborated this result, in which men had lower general risk aversion and higher inverse temperature.These results are consistent with previous literature on the Trust Game, in which men were more trusting of unknown partners (Lemmers-Jansen et al., 2017).Women have also been shown to be more risk averse in financial decision making (Charness & Gneezy, 2012).Overall, this suggests that women in our sample are more risk averse in their behavior, personality measures, and computational measures than their male counterparts.

Second Mover Analyses
We compared thresholds in the First Mover Game to those from the Second Mover Game and found that they were significantly negatively correlated (SUS15: rs(241) = -.15,p <.001; RMT25: rs(241) = -.29,p <.001), in which higher distrust (and higher thresholds) in the First Mover Game would be associated with Higher selfishness and spite (lower thresholds) in the Second Mover Game and vice versa.This result suggests that players assumed their partners were like themselves (Figure S1).We additionally examined the relationship between Second Mover thresholds and MPQ-Alienation, as previous work has suggested that paranoia may influence generosity (Raihani et al., 2020).However, we did not see a correlation between thresholds in either sample for either condition of the Second Mover Game and MPQ-Alienation (p's >.46).

Figure S6 .
Figure S6.Parameter Correlations for Sample 1. Correlation matrix of the five parameter distributions for Sample 1. Values in the upper left corner of each graph show the spearman correlation.Y-axis labels for all show the range for the parameter estimations.Gray lines indicate least squares regression line.p < .05*,p < .01**,p < .001***