Konrad (2007) - Strategy In Contests-An Introduction

Reference(s)

Konrad, Kai A. (2007), "Strategy in Contests-An Introduction", WZB-Markets and Politics Working Paper No. SP II 2007-01, link pdf

Abstract

Competition in which goods or rents are allocated as a function of the various efforts expended by players in trying to win these goods or rents is a very common phenomenon. A subset of examples comes from marketing, litigation, relative reward schemes or promotion tournaments in internal labor markets, beauty contests, influence activities, education filters, R&D contests, electoral competition in political markets, military conflict and sports. I survey here this type of competition which is sometimes called contest or tournament. I focus on the role of its various design aspects, such as prize structure, sequencing, nesting, repetition, elimination contests and many others. Some key insights about the nature and properties of this type of competition emerge from this analysis.

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 \gt x_2 \\ \frac{1}{2} & x_1 = x_2 \\ 0 & x_1 \lt 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 \gt 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 \gt x_2 \end{cases} [/math]