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All-Pay Auctions with Different Forfeits

Benjamin Kang, James Unwin

Abstract

In an auction each party bids a certain amount and the one which bids the highest is the winner. Interestingly, auctions can also be used as models for other real-world systems. In an all pay auction all parties must pay a forfeit for bidding. In the most commonly studied all pay auction, parties forfeit their entire bid, and this has been considered as a model for expenditure on political campaigns. Here we consider a number of alternative forfeits which might be used as models for different real-world competitions, such as preparing bids for defense or infrastructure contracts.

All-Pay Auctions with Different Forfeits

Abstract

In an auction each party bids a certain amount and the one which bids the highest is the winner. Interestingly, auctions can also be used as models for other real-world systems. In an all pay auction all parties must pay a forfeit for bidding. In the most commonly studied all pay auction, parties forfeit their entire bid, and this has been considered as a model for expenditure on political campaigns. Here we consider a number of alternative forfeits which might be used as models for different real-world competitions, such as preparing bids for defense or infrastructure contracts.

Paper Structure

This paper contains 10 sections, 6 theorems, 48 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Background: All-Pay Auctions
  3. 3. Auctions with Constant Entrance Fees
  4. 4. Auctions with Fractional Forfeits
  5. 5. Auctions with Exponential Forfeits
  6. 6. Concluding Remarks
  7. Appendix A: Reproduction of the proof of Lemma 1
  8. Appendix B: Proof of Theorem 3
  9. Appendix C: Proof of Theorem 4
  10. Appendix D: Derivation of $\alpha'(x)$ in exponential case

Key Result

Theorem 1

If $v(x,y)f_{Y_1}(y|x)$ is increasing in x, then the formula for symmetric equilibrium function is given by

Theorems & Definitions (12)

  • proof
  • Theorem 1: KM
  • Theorem 2: MW
  • Theorem 3
  • proof : Proof sketch
  • Theorem 4
  • proof : Proof sketch
  • proof
  • Theorem 3
  • proof
  • ...and 2 more