Auction Theory

Connor Chapin

Motivation

Auctions are common
Auctions can model other transactions
Auctions are simple

Objective

Learn the basic assumptions of auctions
Understand the four standard auctions
Learn the "Revenue Equivalence Theorem"
Other fun auction things

Basic Assumptions

Bidders \(i = 1\dots n\)
One object to be sold
The values bidders have are independent and identically distributed random variables \(V_i\)
The support of \(V_i\) is \([\underline v,\overline v]\)
Bidders are risk neutral

Four Standard Auction Types

Open ascending-bid auction (English/Japanese)
Open descending-bid auction (Dutch)
First-price sealed bid auction ("Real Estate")
Second-price descending-bid auction (Vickrey)

Open Ascending-Bid Auction

(English/Japanese)

Price initialized to \(\underline v\)
All bidders are considered "in" to begin
Price raised continuously
Bidders drop out if price is too high
Last bidder remaining wins
Pays the price final drop out price

Thoughts on a bidder's best strategy?

Drop out at \(v_i\)

Thoughts on a bidder's best strategy?

Drop out at \(v_i\)
If you win for less, you are happy
If you end up paying more, you are sad
You are indifferent paying \(v_i\)

Thoughts on a bidder's best strategy?

Let's consider the next auction type...

Second-Price Sealed Auction

(Vickrey)

Bidders submit sealed bids to auctioneer
Highest bid wins
Pays second-highest bid

Thoughts on a bidder's best strategy?

How does this strategy compare to the best strategy in a Japanese auction?

Thoughts on a bidder's best strategy?

They're the same!

Bid \(v_i\)
Winner pays the second-highest value

Thoughts on a bidder's best strategy?

First-Price Sealed-Bid Auction

"Real Estate"

Bidders submit sealed bids to auctioneer
Highest bid wins
Pays highest bid

How does a bidder's strategy here compare their strategy in a Vickrey auction?

Not the same!

How does a bidder's strategy here compare their strategy in a Vickrey auction?

Not the same!

Bidding \(v_i\) results in zero profit

Thoughts on a bidder's best strategy?

Difficult to pin down
Depends on expectations of others
Tradeoff:
- \(\uparrow P(\text{win}) \implies \downarrow \text{profit}\)

Thoughts on a bidder's best strategy?

Let's consider the next auction type...

Thoughts on a bidder's best strategy?

Open Descending-Bid Auction

(Dutch)

Price is initialized to \(\overline v\)
Price lowered continuously
First bidder to accept the price wins
Pays that price

Thoughts on a bidder's best strategy?

Again, difficult to pin down

How does a bidder's strategy here compare to that of the previous auction?

Same tradeoffs

How does a bidder's strategy here compare to that of the previous auction?

Same tradeoffs
The auctions are strategically equivalent
- Choosing any given bid yields the same payoffs in both games as a function of the other players' bids

There seem to be many parallels between auction types

Let's develop a more powerful tool for analyzing them

Revenue Equivalence Theorem

Recall our assumptions:

Bidders \(i = 1\dots n\)
One object to be sold
The values bidders have are independent and identically distributed random variables \(V_i\)
The support of \(V_i\) is \([\underline v,\overline v]\)
Bidders are risk neutral

Revenue Equivalence Theorem

Consider a mechanism for allocating the unit among the \(n\) bidders that satisfies the following:

The bidder with highest value wins in equilibrium
A bidder with valuation \(\underline v\) makes zero profits

Revenue Equivalence Theorem

Consider a mechanism for allocating the unit among the \(n\) bidders that satisfies the following:

The bidder with highest value wins in equilibrium
A bidder with valuation \(\underline v\) makes zero profits

This mechanism will

yield the same expected payment by each bidder
and the same expected revenue for the auctioneer

Implications

All the auctions we've covered satisfy the assumptions
\(E[\text{revenue}] = E[v_{(n-1)}]\)
More specifics on expected payment for a bidder:
- \(E[\text{payment}\mid v ]\) is constant across auction types
- Thus, \(E[\text{payment}]\) is constant across auction types
- Note: \(E[\text{payment}\mid v]\) \(\neq\) \(\text{bid}(v)\)

Solving for bidding strategy of the first price auctions

Strategy: use the RET and the second-price auctions

Solving for bidding strategy of the first price auctions

Let \(P(v)=P(v=v_{(n)})\) be the equilibrium probability of winning as a function of \(v\)
Let \(b_A(v)\) be the equilibrium bid in an auction of type A as a function of \(v\)
In a FPSB auction, the winner pays the price they bid

Solving for bidding strategy of the first price auctions

Let \(P(v)=P(v=v_{(n)})\) be the equilibrium probability of winning as a function of \(v\)
Let \(b_A(v)\) be the equilibrium bid in an auction of type A as a function of \(v\)
In a FPSB auction, the winner pays the price they bid
Thus, \(E[\text{payment} \mid v]_\text{FPSB} = P(v)\cdot b_\text{FPSB}(v)\)

Solving for bidding strategy of the first price auctions

Now, consider the ascending (second price) auction:

Solving for bidding strategy of the first price auctions

Now, consider the ascending (second price) auction:

E[\text{payment} \mid v]_{\text{ascending}} = P(v) \cdot E[v_{(n-1)} \mid v=v_{(n)}]

Solving for bidding strategy of the first price auctions

Using the Revenue Equivalence Theorem:

E[\text{payment} \mid v ]_\text{FPSB} = E[\text{payment} \mid v ]_\text{ascending} \\ P(v)\cdot b_\text{FPSB}(v) = P(v) \cdot E[v_{(n-1)} \mid v=v_{(n)}] \\ b_\text{FPSB}(v) = E[v_{(n-1)} \mid v=v_{(n)}]

Solving for bidding strategy of the first price auctions

Let's solve for \(E[v_{(n-1)} \mid v=v_{(n)}]\)

Solving for bidding strategy of the first price auctions

Let's solve for \(E[v_{(n-1)} \mid v=v_{(n)}]\)
The density of the highest of \(n-1\) values is:

g(v) = (n-1)f(v)(F(v))^{n-2}

Solving for bidding strategy of the first price auctions

Let's solve for \(E[v_{(n-1)} \mid v=v_{(n)}]\)
The density of the highest of \(n-1\) values is:

g(v) = (n-1)f(v)(F(v))^{n-2}

After integrating the density function times \(v\), we can come to our final answer

Solving for bidding strategy of the first price auctions

After integrating the density function times \(v\), we can come to our final answer:

b_\text{FPSB}(v) = E[v_{(n-1)} \mid v=v_{(n)}] = v-\frac{\int_{x=\underline v}^v (F(x))^{n-1} dx}{(F(v))^{n-1}}

Solving for bidding strategy of the first price auctions

b_\text{FPSB}(v) = E[v_{(n-1)} \mid v=v_{(n)}] = v-\frac{\int_{x=\underline v}^v (F(x))^{n-1} dx}{(F(v))^{n-1}}

If the valuations are distributed standard uniform

b_\text{FPSB}(v) = v-\frac{\int_{0}^v x^{n-1} dx}{x^{n-1}}=v-\frac vn = (\frac{n-1}{n})v

Solving for bidding strategy of the first price auctions

Let's consider the \(n=2\) case

\(E[\text{revenue}]= \frac 13\) by symmetry

Solving for bidding strategy of the first price auctions

Let's consider the \(n=2\) case

\(E[\text{revenue}]= \frac 13\) by symmetry
\(b_\text{FPSB}(v) = \frac{2-1}{2} v = \frac v2\)
\(P(v) = v\)

Solving for bidding strategy of the first price auctions

\(E[\text{revenue}]= \frac 13\) by symmetry
\(b_\text{FPSB}(v) =\frac v2\)
\(P(v) = v\)
\(E[\text{payment}]_i = \int_{0}^1 v\cdot\frac v2 dx = \frac 16 \)

Solving for bidding strategy of the first price auctions

\(E[\text{revenue}]= \frac 13\) by symmetry
\(b_\text{FPSB}(v) = \frac v2\)
\(P(v) = v\)
\(E[\text{payment}]_i = \int_{0}^1 v\cdot\frac v2 dx = \frac 16 \)
\(E[\text{revenue}]=E[\text{payment}]_1 +E[\text{payment}]_2 = \frac 13\)

There's many more fun types of auctions

If we have time, we can talk about and play some

The end!

Works Cited

Klemperer, Paul. "Auction theory: A guide to the literature." Journal of economic surveys 13, no. 3 (1999): 227-286.
Levin, Jonathan. "Auction theory." Manuscript available at www. stanford. edu/jdlevin/Econ 20286 (2004).