I am actually unsure if this game would be considered a minimum effort game or an all-pay auction, so please forgive me if I'm misusing these names for the title.
In a game where 2 players each choose a level of effort and the player with a higher level of effort wins, how would I go about finding the mixed strategy Nash equilibria? Would the form of the answer look like finding which levels of effort would result in payoffs of 0 for both players?
I would like to understand the most general case for this, but for more context on the specific question I am working on, each player at level $s_{i}$ also incurs a cost $\beta_{i}s_{i}$, with $\beta_{1}$ < $\beta_{2}$. This cost is incurred regardless of whether a player wins. The values of winning and losing are normalized to 1 and 0 respectively, and if the players choose the same level of effort, the probability that either one wins is $\frac{1}{2}$.
I think that there are no pure strategy Nash equilibria.