Session statistics
The experiment was conducted at a computer laboratory in Germany in February 2017. It was programmed in z-Tree (Fischbacher 2007) and the participants were recruited using ORSEE (Greiner 2015).
Altogether, 10 experimental sessions were concluded with a total of 180 participants (37% males). The average age was 26 years. In terms of educational background, 48% of the participants were undergraduate students and 16% studied Economics or Business Administration. The average earnings were €10.3 (including a show-up fee of €2.5) for approximately 40 minutes spent in the laboratory. All sessions involved three groups of 6 participants, randomly assigned to either of the three treatments.
Group level results
The general findings are summarized in Table 1. Only 60% of the groups were successful in reaching the threshold to decrease the risk of ”climate change” in the baseline, whereas an impressive 90% of them (all but one group) did so in RR2. With 80% of the groups reaching the threshold, RR1 was in-between. Despite the theoretical arguments presented above, the observed differences in the contributions suggest that the introduction of residual risk actually increases the average contributions. The fact that the contributions in RR1 were higher than those in the baseline imply that this effect was even stronger than the effect of the probability gain reported by Milinski et al. (2008).
Table 1 Group-level descriptive statistics. The values of p and π denote the levels of a priori and residual risk, respectively In our view, however, exploring the treatment differences at the group level and as static phenomena is inefficient since 6 subjects were interacting per group over multiple rounds. To take into account variation at the individual level and also to study the time dynamics of the effect, we present a mixed effects models in the next section. For completeness, however, we compared both RR1 and RR2 with the baseline as far as the proportion of the groups reaching the threshold (χ2 = 0.24, one-sided p = 0.313 and χ2 = 1.07, one-sided p = 0.151 for RR1 and RR2 relative to the baseline, respectively) and as far as the total amount contributed to the climate account (W = 39.5, one-sided p = 0.220 and W = 35.0, one-sided p = 0.129 for RR1 and RR2 relative to the baseline, respectively).
Individual level results
Figure 1 presents the average individual contributions over time. In all treatments, the initial contributions fell short of the reference value of 2 ECU per participant/period that would warrant reaching the threshold by the end of the game. Over time, the contributions tended to increase and especially so towards the end of the game, with the notable exception of the baseline.
Figure 2 presents the average difference between the group total amount contributed to the climate account and the amount that would be on the climate account were everyone to contribute 2 ECU each round. One can see varying degrees of success across the treatments in keeping this difference under control. In the baseline, the participants did not appear to react to the fact that their contributions were too low until almost the very end, whereas in RR2 they managed to reverse the initial trend of declining contributions by about the middle of the game. The RR1 groups performed worse than the baseline in the first half of the game but better in the second half.
In order to address the nested structure of the data properly, we estimated a mixed effects model. Besides the binary variables for the fixed effects of the treatments, the model contained random effects at both the group and individual levels. Since Fig. 1 suggested that most of the between-treatment variation manifested itself in the last period, the corresponding binary variable was included as an additional fixed effect along with its interactions with the treatment variables. Finally, we controlled for risk aversion (Holt and Laury 2002) and environmental attitude (Dunlap et al. 2000) of the participants by adding the corresponding metrics as individual fixed effects to the model.
The coefficient estimates are summarized in Table 2. Since defining the number of degrees of freedom of the fixed effects for such model is somewhat problematic (Bates et al. 2015), all p values were obtained using bootstrap (10K samples). We also considered a number of more complex specifications — e.g., including lagged individual and group contributions — but none of them could improve the model fit according to the AIC criterion.
Table 2 Fixed factor estimates for the mixed effects model of individual contribution over time, p values obtained via bootstrap (10K samples) According to the model estimation results, the last period contributions in treatments RR1 and RR2 were significantly higher than in the baseline (either p value < 0.01), which corroborates the trends observed at the group level. No other significant effects were found.
An in-depth investigation of the group dynamics suggested that one group in the baseline condition could be considered an outlier as its members contributed exceptionally little during the game (Fig. 3). However, excluding that group did not qualitatively affect our resultsFootnote 4.
Distinguishing between a priori and residual risk
Since our experimental design is compatible with the one in Milinski et al. (2008), we expanded the analysis by including the data from their treatment with p = 0.9 and π = 0.0. Together with our three treatments, this allowed us to employ the full factorial design — i.e., 2 × 2 levels of p ∈{0.7,0.9}× π ∈{0.0,0.2} — to distinguish between the effects of a priori and residual risk.
Comparing the average group contribution across the four treatments resulted in the following ranking: RR2 (121.6) ≻ RR1 (119.4) ≻ Milinski (118.2) ≻ Baseline (117). This suggests that higher levels of both a priori and residual risk result in higher contributions. In order to test for the statistical significance of this finding, we estimated a new mixed effects model using the merged dataset. Save for absence of the individual level controls and addition of a binary variable to control for the source of the the data, the specification was identical to the one used previously.
As shown in Table 3, there is no significant difference between the two datasets (p = 0.87) and the estimation results support our previous findings that both a priori and residual risk result in significantly higher contributions in the last period, both effects being of similar magnitude.
Table 3 Fixed factor estimates for the mixed effects model of individual contribution over time that distinguishes between the effects of a priori and residual risk, p-values obtained via bootstrap (10K samples)