## 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 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]

### Mode of analysis

• symmetric equilibrium

• first-price, Dutch

b(x) = E[X | X = x]

• second-price, English

b(x) = x

• all-pay

b(x) = E[X | X = x] Pr(X ≤ x)

### 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], 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.