Changes
Jump to navigation
Jump to search
Konrad (2007) - Strategy In Contests-An Introduction (view source)
Revision as of 22:14, 11 April 2010
, 22:14, 11 April 2010no edit summary
Keywords: Survey of contests, tournaments, conflict, strategic aspects
==Summary==
The paper provides a series of descriptions of setting in which contests might be applied, and then introduces three types of contests, followed by a detailed review of a number of other specific contest models with advanced assumptions.
The three key models (detailed below) are:
*First price all-pay auctions
*Additive noise contests
*The Tullock contest
The advanced assumption models (not detailed below - see the paper) include:
*Timing and Participation Models:
**Endogeneous Timing
**Voluntary Particpation
**Exclusion
*Cost and Prize Structure
**Choice of Cost
**Multiple Prizes
**Endogenous Prizes
*Delegation
*Externalities
**Joint Ownership
**Sabotage
**Information Externalities
**Public Goods and Free Riding
*Grand Contests
**Nested Contests
***Exogenous Sharing Rules
***The Choice of Sharing Rules
***Intra-group conflict
**Alliances
**Repeated Battles
==Three Key Models==
The following section provides the basic three models.
The general set up is as follows:
*There are <math>N\,</math> players: <math>N=\{1,\ldots,n\}\,</math>
*Players can exert effort <math>x_i\,</math>, <math>X=\{x_1,\ldots,x_n\}\,</math>. Implicitly <math>x_i \in \mathbb{R}_+ \forall i\,</math>
*There is, unless otherwise stated, a single prize <math>B\,</math>
*Players have values <math>v_i(B)\,</math> and costs <math>C_i(x_i)\,</math>
The "contest success function" maps effort into probabilities:
:<math>p_i = p_i(x), p_i: \mathbb{R}_+ \rarrow [0,1]\,</math>
Players have profit functions (i.e. utilities):
:<math>\pi_i(x)=p_i(x)\cdot v_i(B) - C_i(x_i)\,</math>
===All-pay auctions===
To make things interesting but tractable we assume:
*2 Players
*<math>C_i = C \forall i\,</math>
*<math>v_1 \ge v_2 \ge 0\,</math>
The contest function for player 1 (for player 2 it is <math>1-p_1\,</math>) is:
<math>
p_1(x) =
\begin{cases}
1 & x_i > x_2 \\
\frac{1}{2} & x_1 = x_2 \\
0 & x_1 < x_2
\end{cases}
</math>
There is no equilibrium in pure strategies. This can be proved by contradiction. In mixed strategies there is an equilibrium described by the CDF of the distributions of effort as follows:
<math>
F_1(x_1) =
\begin{cases}
\frac{x_1}{v_2} & x_1 \in [0,v_2] \\
1 & x_1 > x_2
\end{cases}
</math>
<math>
F_2(x_2) =
\begin{cases}
\left(1-\frac{v_2}{v_1}\right)+\frac{x_2}{v_1} & x_2 \in [0,v_2] \\
1 & x_1 > x_2
\end{cases}
</math>
