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Pre- and Post-Auction Discounts in First-Price Auctions

Miguel Alcobendas, Eric Bax

Abstract

One method to offer some bidders a discount in a first-price auction is to augment their bids when selecting a winner but only charge them their original bids should they win. Another method is to use their original bids to select a winner, then charge them a discounted price that is lower than their bid should they win. We show that the two methods have equivalent auction outcomes, for equal additive discounts and for multiplicative ones with appropriate adjustments to discount amounts. As a result, they have corresponding equilibria when equilibria exist. We also show that with the same level of multiplicative adjustments, bidders with discounts should prefer an augmented bid to a discounted price. Then we estimate optimal bid functions for valuation distributions based on data from online advertising auctions, and show how different discount levels affect auction outcomes for those bid functions.

Pre- and Post-Auction Discounts in First-Price Auctions

Abstract

One method to offer some bidders a discount in a first-price auction is to augment their bids when selecting a winner but only charge them their original bids should they win. Another method is to use their original bids to select a winner, then charge them a discounted price that is lower than their bid should they win. We show that the two methods have equivalent auction outcomes, for equal additive discounts and for multiplicative ones with appropriate adjustments to discount amounts. As a result, they have corresponding equilibria when equilibria exist. We also show that with the same level of multiplicative adjustments, bidders with discounts should prefer an augmented bid to a discounted price. Then we estimate optimal bid functions for valuation distributions based on data from online advertising auctions, and show how different discount levels affect auction outcomes for those bid functions.
Paper Structure (11 sections, 3 theorems, 17 equations, 2 figures, 1 table)

This paper contains 11 sections, 3 theorems, 17 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Additive Discounts and Matching Equilibria
  3. Multiplicative Discounts and Corresponding Equilibria
  4. Corresponding Equilibria
  5. Bid Augmentation vs. Price Reduction with Equal Rates
  6. Optimal Bidding for Uniform Value Distributions with Compensating Discounts
  7. Bid Functions and Auctions Under Realistic Valuations
  8. High-Level Process
  9. Estimating Bid Functions
  10. Auction Outcomes
  11. Conclusion

Key Result

theorem 1

For additive discounts, first-price auctions with bid augmentation and those with price reductions have corresponding Nash equilibria (if either has an equilibrium), with each bidder having that same utility under both forms of discounting, if the bid augmentations are equal to the price reductions.

Figures (2)

  • Figure 1: Estimated optimal bids as a function of valuations for a bidder with a price reduction rate $r$ (dashed line) and one of 4 undiscounted bidders (solid line) in auctions with the discounted bidder.
  • Figure 2: Estimated optimal bids as a function of valuations for a bidder with a price reduction rate $r$ (dashed line) and one of 4 undiscounted bidders (solid line) in auctions with the discounted bidder, showing detail for higher values.

Theorems & Definitions (3)

  • theorem 1
  • theorem 2
  • theorem 3