# Generalized Tournament Model

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

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

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).