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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!

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    $\begingroup$ Some info seems to be missing here. Shouldn't the cost be related to effort in some way? If $c(\theta)$ is only the cost for entering the competition, and that effort is costless, then all participants who find it worthwhile entering the competition will exert the maximal amount of effort (assuming $q(e)$ strictly increasing). If this is the case, there will be a threshold value for $\bar\theta(R)$; a participant will enter if and only if $\theta\ge \bar\theta(R)$. Those who enter will choose the highest $e$ and each has probability $3/N$ of getting $R$ ($N$ is the number of people entering). $\endgroup$ – Herr K. May 26 '16 at 17:10
  • $\begingroup$ You are absolutely right. I am sorry I missed it. The cost function has two parameters $c(e,\theta)$. It is increasing and convex in $e$ and increasing in $\theta$. $\endgroup$ – gdsquare May 26 '16 at 21:14
  • $\begingroup$ In that case, question 1 doesn't make a lot of sense (or maybe it's trivial). If others' effort can be observed, say $(e_2,\dots,e_n)$ (and wlog assume $e_2\ge e_3\ge \cdots\ge e_n$), then the probability of player 1 with effort $e_1$ getting the reward would simply be $$\begin{cases}1&\text{if }e_1> e_4\\0&\text{if }e_1<e_4\\3/N&\text{if } e_1=e_2=\cdots=e_N\end{cases}$$ where $N>3$. I suppose there are other cases, but you get my point here. $\endgroup$ – Herr K. May 26 '16 at 21:58
  • $\begingroup$ Perhaps I screwed up the description again. But with your explanation, I still need the probability that $e_1 > e_4$. Also, I can assume that $e$ is monotone in $\theta$ to eliminate the last case. $\endgroup$ – gdsquare May 26 '16 at 22:27
  • $\begingroup$ I think you should start out by solving the problem for the case that only the best product wins. This is a well known problem in the literature. $\endgroup$ – HRSE May 27 '16 at 2:03
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This problem is similar to first-price (sealed-bid) auctions with independent private values for the buyers. You can read about those for exaxmple here.

Your first question was already ansewered in the comments: given others' efforts the probability of getting the reward is or 1. Let us assume that $e_1 \geq e_2 \geq \dots$ are the other players' efforts and $e$ is own effort. Then $$ \Pr[\text{getting }R | e, e_e, e_2, \dots] = \begin{cases} 1 & \text{if } e > e_3 \\ 0 & \text{if } e < e_3 \\ \dots & \text{otherwise} \end{cases} $$ What happens in case of ties is not exactly specified in the question, but as $\theta$ has a continuous distribution, if $e(\theta)$ is strictly monotone edge cases happen with 0 probability. Based on this, you can also calculate what happens if you do not condition on other players' strategies (you should probably look into order statistics).

Now the player want to maximize the following: $$ EU(e) = \Pr[\text{getting }R | e]\times R - c(\theta, e) $$

  1. You can start by assuming that players follow pure strategies, so $e_i = e(\theta_i)$.
  2. Also assume that effort is strictly monotone decreasing in $\theta$.
  3. And last, assume that players have symmetric strategies.

Now you have to write down the expected utility defined above in terms of integrals. As of assumption 2. (and the convexity of $c$) you can differentiate it with respect to $e$ to get a (sufficient) first order condition. Then using assumption 3. and luck will get you a solvable differential equation, and hopefully you can find a closed form for $e(\theta)$. That is going to be your equilibrium strategy.

Or to be precise, one of the equilibrium strategies, as we have only looked at a subset (pure symmetric strategies). But finding others (or proving that none exist) is much more challenging.

I hope this is enough to start. If you get stuck, I will gladly expand this answer with some concrete formulas.

Disclaimer: I have not solved this problem, but as it is remarkably similar to the previously mentioned auction problems, this method should work.

There might be another way using the revelation principle. You could come up with a (second price auction-like) game where reporting the truthful $e$ is the dominant strategy, and has the same expected payoffs. Then you could relatively simply calculate the expected payoffs, and based on that, getting the strategies might be quite easy. I am not sure this method works either, but if you are interested I can look into it.

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