An Introduction to Auctions

In order to study a class of auctions, we must specify both the economic setting in which the auction takes place and the type(s) of auction procedure under investigation.

The economic setting:

• single item vs. multiple items
• private valuations vs. common values (vs. mixed)
• independent valuations vs. dependent valuations
• symmetric vs. asymmetric
• risk-neutral bidders vs. risk-averse (vs. other)
• pure consumption vs. resale
• costless participation vs. costly (entry fees)
• the item will surely be sold to a bidder vs. the seller might retain the item (e.g., the seller submits a bid or sets a reserve price)

Notation:

• n bidders
• Each bidder i has a private valuation X_i .
• X₁, X₂,… , X_n are independent draws from a "taste" distribution F(x).
• Each bidder seeks to maximize his expected payoff.
• If bidder i receives the item and must pay b, his payoff is X_i - b .
• If bidder i doesn't receive the item and must pay b, his payoff is - b .
  
  
• X_[1] ≥ X_[2] ≥…≥ X_[n]

Some procedures:

• First-price
• Second-price (Vickrey)
• English (ascending)
  • Japanese clock auction
  • eBay vs. Yahoo
• Dutch (descending)
  • Aalsmeer cut-flower auction
  • The Aalsmeer auction market (1.2GB video - download to play)
  • Sydney shrimp auction (7MB video - download to play)
  
  
• All-pay
• War of Attrition (e.g., potlatch ceremonies)
  
  
• Multi-stage (e.g., preemptive right)

Mode of analysis

• symmetric equilibrium
  • first-price, Dutch

    b(x) = E[X_[2] | X_[1] = x]

  • second-price, English

    b(x) = x

  • all-pay

    b(x) = E[X_[2] | X_[1] = x] Pr(X_[1] ≤ x)

Amazing result

• The allocation function, i.e., the likelihood of a bidder being awarded the item, given the valuations of all the bidders.
• The expected profit of a bidder with the lowest of all possible valuations.

The Revenue Equivalence Theorem

In the symmetric independent private values setting (with risk-neutral bidders), if

• the bidders all act sensibly (i.e., their strategies together are in equilibrium)
• the bidder with the highest valuation surely wins the item, and
• any bidder with the lowest of all possible valuations has an expected profit of 0,