I have been working on this problem for a few days but I am completely lost on how to start. Any suggestions, comments, hints are greatly appreciated. Here is a scenario:
Participants are competing in a competition where only the best 3 products will be chosen and awarded in the amount of $R$.
To produce a product, each participant picks an effort level $e$. The quality of the product q is a function of $e$. Each participant can produce one and only one product.
Each participant has a cost function $c$ which in a convex, increasing function of effort level $e$ and increasing in a cost parameter $\theta$. This parameter $\theta$ is uniformly distributed between 0 and 1. Participants know their own $\theta$ but do not know others. Although, they know the distribution of $\theta$.
The questions at hand are:
1) Assuming that participants can observe others' effort, what is the probability that a participant who spend an effort $e$ wins a competition (his product quality is among the top 3)
2) Given the reward amount $R$, what is the equilibrium effort level for a participant with a cost parameter $\theta$.
For me, this question is quite similar to the price setting problem: If there are N sellers in the market with different cost parameters, and buyers are looking for the cheapest product, how are they going to price their product? However, I could not find a reference about the solution to this problem also.
Again, any suggestions, comments, hints are greatly appreciated. Thank you very much!