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)
- n bidders
- Each bidder i has a private valuation Xi .
- X1, X2,… , Xn 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 Xi - b .
- If bidder i doesn't receive the item and must pay b, his payoff is - b .
- X ≥ X ≥…≥ X[n]
- Second-price (Vickrey)
- English (ascending)
- Japanese clock auction
- eBay vs. Yahoo
- Dutch (descending)
- War of Attrition (e.g., potlatch ceremonies)
- Multi-stage (e.g., preemptive right)
Mode of analysis
The seller's expected profit (at equilibrium, in the independent private values setting, with risk-neutral bidders) depends on only two things:
- 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
then the seller's expected revenue from the auction is E[X], irrespective of the choice of auction procedure. In particular, all the procedures listed above yield the same expected revenue.
Here's a simulation spreadsheet illustrating the Revenue Equivalence Theorem.
- 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,