There are $n$ people who can participate in some event in exchange of some participation cost $x$. If $m$ people participate in the event, then the payoff for each participant is $\frac{1}{m}$. This is repeated $k$ times, and everyone has access to the number of participants history for all the past events.
What's the best strategy to determine when to participate in the event?
Without more specification about how choices are made, there's not enough information to determine an optimal strategy.
Here's one possible interpretation . . .
Let $n,x$ be fixed and known, where $n$ is an integer with $n\ge 2$, and $x$ is a real number with $0 < x < 1$.
Suppose the players agree in advance on a probability $p\in [0,1]$ (and are contractually obligated to abide by their agreement) such that each player will independently choose to play with probability $p$.
The question is then: What is the optimal value of $p$, given $n$ and $x$?
With this interpretation, history is irrelevant.
Thus, it suffices to determine the optimal value of $p$ for a single game.
By optimal, assume the goal is to maximize expected value.
For a single game, if $e$ is a player's expected value with all players using the mixed strategy $p$, then $$ e=\sum_{m=1}^n {\small{\binom{n-1}{m-1}}}p^{m}(1-p)^{n-m}\left({\small{\frac{1}{m}}}-x\right) $$ which is maximized when $$p=1-x^{\bigl({\Large{\frac{1}{n-1}}}\bigr)}$$ yielding the expected value $$ \frac{1}{n}-x+\left(\frac{n-1}{n}\right)x^{\left({\Large{\frac{n}{n-1}}}\right)} $$