Fig. 5: Results from the additional experiment, manipulating loners’ externality (ρ) on group outcome.

A Number of participants who chose each action (cooperate [C]: orange, defect [D]: blue, or leave [L]: green) in each \({{{{{\rm{\rho }}}}}}\) condition. Note that ρs are placed inversely from right (0) to left (1) for the sake of consistency with Fig. 4. B Effective cooperation rate (\({r}_{{{{{{\rm{C}}}}}}}/\left({r}_{{{{{{\rm{C}}}}}}}+{r}_{{{{{{\rm{D}}}}}}}+\rho {r}_{{{{{{\rm{L}}}}}}}\right)\); left) and group success rate (right) as a function of \({{{{{\rm{\rho }}}}}}\). Error bars indicate bootstrapped 95% CIs (\(n=182\) individuals), with the midpoints showing the exact values computed from the experimental data. See Methods for the exact procedures to calculate the CIs. C The distributions of participants’ beliefs about (effective) cooperation rate (gray bars) and probability of players choosing each action conditioned on the beliefs (color lines; C: orange, D: blue, or L: green). As participants reported their beliefs about the number of other participants (of 30) choosing each action, beliefs about the effective cooperation rate were computed based on Eq. (4).