The setting of my problem is as follows:
I have a large pool of random variables $X_1,...., X_n$ which are i.i.d, which I'll refer to as nodes.
At each round of my random process I select by random $N << n$ nodes $X_{p_1}, ..., X_{p_N} $ which we'll be referred to as participants. I also select randomly one extra node independently from the participants. Let this node be called leader.
At each round, the selected leader performs some selection process where it votes for a single participant. The selection is dependent on the value of the leader node and nothing else (e.g. the leader selects the participant $X_{p_i}$ with value "closest" to its own).
Can I say anything about the expected value $\mathbf{E}[X_{p_i}]$ where $X_{p_i}$ is the selected participant.
The intuition is that since every node has equal chance to become a leader or a participant the expected value will be that of the shared distribution of each node. However, I can't formulate the conditions on the selection process that support that.