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Consider an $n$-player simultaneous game ($n\geq2$).

Each player $i$ chooses a costly bid, $q_i\in R_+$. There are $p<n$ prizes to be awarded to the $p$ bidders with the highest bids. Bids are not observed by other players. The value of prizes is strictly decreasing $V_1 > V_2 > ...V_p$. The payoff from being ranked below $p$ is zero.

Each player $i$ has an "ability", $a_i\in[0, m]$ which is distributed randomly according to distribution $F_i$. Each distribution has continuous support with density $f_i$ and is twice continuously differentiable. Each distribution $F_i$ is common knowledge, but a player's ability is private information.

The cost of bidding depends on the bid and the player's ability. This is $C(a_i, q_i)=\gamma(a_i)\cdot c(q_i)$ which is continuous and with $\gamma'<0$, $c(0)=0$ and $c'>0$.

The problem is to characterize the equilibrium bidding strategy for the players.

For the solution of the original symmetric case (where $F_i = F$ for every $i$) see: http://www.econ2.uni-bonn.de/pdf/papers/pearson22.pdf

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  • $\begingroup$ The answer to this question would be a entire new research paper. Do you realize that? $\endgroup$ Aug 8, 2014 at 13:47

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The answer to this question would be a paper worth publishing in the top Microeconomic Theory journals (JET, GEB, TE or ET) so no one is ever going to answer it, however, there are several points that may help the op to start atacking the question:

  1. Without any loss of generality you can take $c(\cdot)$ to be the identity. Players instead of choosing $q_i$ choose $c(q_i)$.
  2. Also without any loss of generality you can also have $\gamma_i(\cdot)$ to be the indentity if you adjust the $F_i$ accordingly (remember you are assuming the $F_i$ maybe different).
  3. Even with a single prize the bid strategies in the case $n\ge 3$ are not trivial as in the symmetric case because they may be discontinuous (see Parreiras & Rubinchik, GEB, 2010).
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