## 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} .
- X
_{1}, X_{2},… , 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)

- All-pay
- War of Attrition (e.g., potlatch ceremonies)

- Multi-stage (e.g., preemptive right)

### Mode of analysis

### Amazing result

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

- 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,

then the seller's expected revenue from the auction is E[X_{[2]}], 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.