Auction Theory

Class 4 – Some applications of revenue equivalence

1

TodayThe machinery in Myerson’s work is useful in many

settings.

Today, we will see two applications: interesting results that are derived almost “for free” from these tools.

1. Equilibrium in 1st-price auctions.2. “Auctions vs. negotiations” – should we really run

the optimal auction?

2

Equilibrium in 1st-price auctionsOld debt:

I promised to prove what is the equilibrium behavior in 1st-price auctions.

This will be an easy conclusion from the results we know.

3

Equilibrium in 1st-price auctionsIn a second price auction:

– The usual notation: bidder i with value vi wins with probability Qi(vi).

– i’s expected payment when he wins: the expected value of the highest bid of the other n-1 bidders given that their value is < vi.

– E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ]

ui(vi) = Qi(vi) ( vi - E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ] )

In 1st-price auction: a winning bidder pays her bid. ui(vi) = Qi(vi) ( vi - bi(vi) )

Revenue equivalence: expected utility in 1st and 2nd price must be equal in equilibrium.

Equilibrium bid in 1st-price auctions: bi(vi) = E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ]

4

Payment when winning with value vi

Payment when winning with value vi

Equilibrium in 1st-price auctionsExample: the uniform distribution on [0,1].

This is the expected highest order statistic of n-1 draws from the uniform distribution on the interval [0,v]

Conclusion:

5

bi(vi) = E[ max{v1,…,vi-1,vi+1,…,vn} | vk < vi for every k ]

0 v 1

ivn

1ivn

2ivn

n 1ivn

n 2….

iii vn

nvb

1)(

TodayThe machinery in Myerson’s work is useful in many

settings.

Today, we will see two applications: interesting results that are derived almost “for free” from these tools.

1. Equilibrium in 1st-price auctions.2. “Auctions vs. negotiations” – should we really run

the optimal auction?

6

Optimal auctions• We saw that Vickrey auctions are efficient, but do

not maximize revenue.

• Myerson auctions maximize revenue, but inefficient.– With i.i.d. distributions, Myerson = Vickrey + reserve price.

• Should one really run Myerson auctions?– We will see: not really…

7

Marketing• Two approaches for improving your revenue:

– Optimize your mechanism. Make sure you make all the revenue theoretically possible.

– Increase the market size: invest in marketing.

• Maybe, instead of optimizing a reserve price, we can just expand the market?

8

Bulow-Klemperer’s result

9

• The efficient auction with one additional bidder earns more revenue than the optimal auction!

• Finding an additional bidder is better than optimizing the reserve price.

Revenue in the optimal auction with n players.

Revenue from the Vickrey auction with n+1 players.

Theorem [Bulow & Klemperer 1996]:

≤

Bulow-Klemperer’s result: discussion

10

• Holds for every n:– Even n=1.

• The optimal (Myerson) auction requires knowledge on the distribution, Vickrey does not.

• Auctions with no reserve price may be more popular.

Additional revenue from optimal auctions is minor.

Revenue in the optimal auction with n players.

Revenue from the Vickrey auction with n+1 players.

Theorem [Bulow & Klemperer 1996]:

≤

Bulow & Klemperer: setting

• We consider the basic auction setting:– Values are drawn i.i.d from some distribution F.– Risk Neutrality– F is Myerson-regular (non-decreasing virtual valuation)

• We will define the “must-sell” optimal auction: the auction with the highest expected revenue among all auctions where the item is always sold.

11

Bulow & Klemperer: proof

• The proof will follow easily from two simple claims.– Taken from “A short proof of the Bulow-Klemperer auctions vs.

negotiations result” by Rene Kirkegaard (2006)

– Still, a bit tricky.

• (the original proof was not so easy…)

12

Bulow & Klemperer: proofClaim 1:

13

The revenue in the “must-sell” optimal auction with n+1 bidders

The optimal revenue with n bidders.≥

Bulow & Klemperer: proof

14

Proof of claim 1:• The following “must sell” auction with n+1 bidders

achieves the same revenue as the optimal revenue:“Run the optimal auction with n players; if item is unsold, give it to bidder n+1 bidder for free.”

But this “must-sell” auction achieves the same revenue as the optimal auction with n bidders….

• The “must-sell” optimal auction can only do better. The claim follows.

Claim 1: the revenue in the “must-sell” optimal auction with n+1 bidders

The optimal revenue with n bidders.≥

Bulow & Klemperer: proof

15

What is the “must-sell” optimal auction?

Claim 2: the “must-sell” optimal auction is the Vickrey auction.

Proof: • Recall: E[revenue] = E[virtual surplus]• When you must sell the item, you would still aim to maximize

expected virtual surplus.• The bidder with the highest value has the highest virtual value.

– When values are distributed i.i.d. from F and F is Myerson regular.

Vickrey auction maximizes the expected virtual surplus when item must be sold.

Bulow & Klemperer: proof

16

Conclusion:

the revenue in the “must-sell” optimal auction with n+1 bidders

The revenue in the “must-sell” optimal auction with n+1 bidders

The optimal revenue with n bidders.

The revenue in the Vickrey auction with n bidders.

≥

The optimal revenue with n bidders.

The revenue in the Vickrey auction with n+1 bidders.

≥=

Claim 1Claim 2

SummaryThe tools developed in the literature on optimal

auctions are useful in many environements.

We saw two applications:1. Characterization of the equilibrium behavior in 1st

price auctions.2. Bulow-Klemperer result: running Vickrey with n+1

bidders achieves more revenue than the optimal auction with n bidders.

17