Proxy optimisation problem Suppose we have a set of participants $p$ who should attend $e$ number of events and everyone of them must declare his presence with signature. Each can however sign for $s$ number of other participants including himself.
Given $e, s$ (let us say $e = 6, s = 3$), how can one find the lowest possible number of attendances $a$ for each participant?
For example, should there be $p = 3$ participants to attend $e = 6$ events and able to sign for $s = 3$ participants including themselves, the answer would be $a = 2$ because $A$ would sign for $A,B,C$ twice, $B$ the same twice and $C$ all the same.
 A: I assume a participant cannot "attend", in actuality or by proxy, a single event more than once.
I assume $s\ge p$.  Otherwise the number of distinct signatures at an event is limited to $p$ and having a larger $s$ is irrelevant.
I assume there are as many distinct events as necessary -- two different participants never need actually attend the same event.  (If there is a small population of attendable events the answer will necessarily be different.)
Put the participants in a ring, and instruct each one to sign for himself and the next $s-1$ participants.  E.g., with 3 participants numbered $0,1,2$ and $s=2$, participant 0 signs 0 and 1, participant 1 signs 1 and 2, and participant 2 signs 2 and 0.  When $s=p$ each participant signs all names.
View this as an iterative process:  In the first step, the $p$ participants go to $p$ distinct events and sign $s$ times.  Now each participant has "attended" 1 event in actuality, and in actuality or by proxy $s$ events.
After $a$ such steps the participants have $as$ attendances in actuality and by proxy.
The minimal number of steps (and actual attendances) is $$a=\lceil e/s \rceil,$$ as this is the smallest integer giving $as\ge e$.
