Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)

Main Tests
For each of our main outcomes we will have one main empirical test. Power is based on the first outcome (``measure of influence''), but sample sizes are then adjusted to take into account our ability to test for certain dimensions of heterogeneity (see below) as well as the second main outcome (`` measure of the extent to which people act on their influence'').
\textbf{Measure of Influence}: We will measure influence using only the robot treatments, where we will compare situations where the robot chose ``Go In'' in period 1 with situations where they chose ``Stay Out''. Our empirical measure of influence of a particular position in a particular network is the difference between the share of participants who choose to ``Go In'' following the robot's choice to ``Go In'' in period 1 and the share of participants who choose to ``Go In'' following the robot's choice to ``Stay Out'' in period 1.
Our main specification will be a logistic regression where we regress the binary variable indicating ``Go In'' on a dummy variable indicating whether the robot chose to ``Go In''. The regression will be run separately for each network position. (Positions that are the same up to relabeling are treated as the same position). We will report odds ratios. Standard errors will be clustered at the silo level. We will report several specifications including demographic control variables, round fixed effects and controls for position in the sequence, but our main pre-registered specification will be the one without any controls. We will also conduct linear probability models with the same regressors and report those results in the Appendix.
Power analysis is done based on t-tests comparing the proportion of participants choosing to ``Go In'' following the robot's choice to ``Go In'' or ``Stay Out'' in period 1. The t-tests are conducted on an initial set of sessions conducted with Network 2B (see attached graphs). We conducted these sessions to get a sense of the magnitude of possible effect sizes in order to enable us to do more meaningful power analysis. In these sessions we found the following effect sizes for network positions A/E, B, C and D: 12$\%$, 22$\%$, 37$\%$ and 4$\%$. Power analysis revealed that to detect these effect sizes with 80 percent power at the 5$\%$ level we would need 14, 5, 4 and 37 silos of 10 participants each, respectively.
\textbf{Sample Size} Based on these numbers we decided to use 10 silos of 10 participants each for the 5-player networks and 5 silos of 20 players each for the 10-player networks. (Note that in the robot treatments one participant per network is played by a robot). This should allow us to reliably detect influence at least for the more influential network positions.
\textbf{Awareness of Influence}: We will use the treatments without robots and regress the binary variable indicating whether individual i in network position j chose to ``Go In'' in period 1 on the measure of empirical influence of network position j obtained from the sessions with robot. We will also include a fixed effect for the rate of entry in the network as a whole. Otherwise the empirical specifications will be the same as those above.
\paragraph{Other Tests} We will explore heterogeneity based on answers in a post -experimental questionnaire and based on the sequence of decisions. We will also use decision-time as a secondary measure of awareness of influence. Last we will contrast empirical influence with theoretical measures of influence of a node (e.g. centrality measures).