Optimal Mechanism Design Finance 510: Microeconomic Analysis

Optimal Mechanism Design

Finance 510: Microeconomic Analysis

Optimal mechanism design deals with institutional rules chosen to serve some explicit optimization goal.

Example

Suppose that your learn of a long lost uncle that has died and has left you and your sister $3M. You and your sister need to decide how to split the $3M. However, the lawyers fees are $1M per negotiating round.

You and your sister agree to the following:

Coin flip decides who will make the first offer

Offers are made in $100,000 increments

Once an offer is made, the other has the right of refusal

No communication allowed during settlement

You

Sister

Offer

Accept Reject

Sister

Offer

You

Accept Reject

You

Offer

Sister

Accept Reject

($0,$0)

Round 1

Round 2

Round 3

With $1M left to split, you offer your sister $100,000 (Which is strictly preferred to $0)

You

Sister

Offer

Accept Reject

Sister

Offer

You

Accept Reject

You

Offer

Sister

Accept Reject

($0,$0)

Round 1

Round 2

Round 3

With $2M left to split, your sister offers $1,000,000 (Which is strictly preferred by you to $900,000)

You: $900,000

Sister: $100,000

You

Sister

Offer

Accept Reject

Sister

Offer

You

Accept Reject

You

Offer

Sister

Accept Reject

($0,$0)

Round 1

Round 2

Round 3

With $3M left to split, you offer your sister $1,100,000 (Which is strictly preferred to $1,000,000)

You: $900,000

Sister: $100,000

You: $1,000,000

Sister: $1,000,000

You: $1,900,000

Sister: $1,100,000


We initially had the following rules:

Coin flip decides who will make the first offer

Offers are made in $100,000 increments

Once an offer is made, the other has the right of refusal

No communication allowed during settlement

Suppose that we drop the last rule (no communication) and as a result, you sister is able to convince you that she only cares about what she gets relative to you!

i.e. ($0, $0) is preferred to ($600,000, $400,000)

You

Sister

Offer

Accept Reject

Sister

Offer

You

Accept Reject

You

Offer

Sister

Accept Reject

($0,$0)

Round 1

Round 2

Round 3

With $1M left to split, you offer

You: $400,000

Sister: $600,000

You

Sister

Offer

Accept Reject

Sister

Offer

You

Accept Reject

You

Offer

Sister

Accept Reject

($0,$0)

Round 1

Round 2

Round 3

With $2M left to split, your sister offers $500,000 (Which is strictly preferred by you to $400,000)

You: $400,000

Sister: $600,000

You

Sister

Offer

Accept Reject

Sister

Offer

You

Accept Reject

You

Offer

Sister

Accept Reject

($0,$0)

Round 1

Round 2

Round 3You: $400,000

Sister: $600,000

You: $500,000

Sister: $1,500,000

With $3M left to split, you offer

You: $700,000

Sister: $2,300,0003.2 to one

3 to one


No Communication Communication

You: $1,900,000

Sister: $1,100,000

You: $700,000

Sister: $2,300,000

If you were designing the rules of the negotiation process, which would you choose?

It is customary for the goods or services to be handed out on a first come first serve basis. Therefore, if a line forms, the newest arrival goes to the end of the line.

Could this mechanism be improved on?

With Last Come First Serve

Lines disappear

Goods/services are distributed to those with the highest value (no lines)

Individuals need not alter their schedules

With First Come First Serve

Lines are unnecessarily long

Goods/services aren’t necessarily distributed to those with the highest value

Individuals inefficiently alter their schedules to avoid the line

Auction Design

In 2000, revenues from online auctions was $6.5 Billion. In 2003, that number grew to $30 Billion!! Experts expect revenues in 2006 to exceed $50 Billion!

Auctions have been used for:

•The Babylonians used auctions to arrange marriages

•The Greeks used auctions to award mineral rights

•The French utilized a “candle auction”. Bids were accepted until the candle burned out (similar to EBay's timed auctions)

•The Dutch used auctions to sell tulips (creating the Dutch auction)

•T-Bills are sold by the US Treasury via auction

•The NYSE is an auction market

Auctions are distinguished by their rules

Sequential: There are always re-bid opportunities

Simultaneous: Each player gets one bid

Minimum Improvement: There exists a minimum “unit” for bidding

Continuous: No minimum “unit”

Minimum Improvement: There exists a minimum “unit” for bidding

Continuous: No minimum “unit”

Bids can be sealed (private), open outcry, or posted anonymously

Some auctions have a minimum allowable bid (reserve price)

Who Pays and How Much?

All Bidders Pay: Anyone with an “acceptable” bid pays and gets the product

First Price Auction: Highest Bid wins and pays his/her bid

Nth Price Auction: Highest Bid wins and pays the amount of the Nth highest bid

English Auctions: Open outcry auction. Last bidder (with the highest offer) wins (ascending auction)

Dutch Auctions: The first bidder to accept wins as the auctioneer reads off descending prices (descending auction)

Does Auction Type Matter?

Sequential

Minimum Bid Improvement

Posted Prices

Multiple Rounds

Open Bidding

Reserve Price

First Price

English Ascending Price

Seller is Known

Simultaneous

Continuous

Posted Prices (Reverse Auction)

One Time (If Seller “Hits”)

Credit Card Immediately Authorized

No Reserve

All Acceptable Bids Pay

Dutch Auction

Seller is Anonymous

VS

Suppose that you are bidding on an object of unknown value to you (but known to the seller). You know its worth between $0 and $100 to the seller and you also know that your value is 50% above the seller’s.

What should your bidding strategy be?

Consider an example with three possible values: $100, $55, and $0

BID

$0

$55

$100

All Offers Refused

V = $100

V = $55V = $0

V = $0

V = $55

V = $100

A ( $-55, $55)

A ($27.50, $0)

A ( $95, -$45)

A ( -$100, $100)

A (-$17.50, $45)

A ($50, $0)

R ( $0, $0)

R ( $0, $0)

R ( $0, $0)

R ( $0, $0)

R ( $0, $0)

R ( $0, $0)

The Winner’s Curse

BID = $0

All offers rejected

Expected Gain = $0

BID = $55

Accepted only if V = $0

Expected Gain = -$18

BID = $100

Accepted if V = $100 or V = $55

Expected Gain = -$39

The Best Strategy is to bid $0!! (the expected value is $51)

The Winner’s curse states that in an Auction with asymmetric information, if you win the auction, you have definitely overpaid!

Bidders are aware of the winner’s curse. Therefore, there is an incentive to underbid (or not bid at all)

The Winner’s Curse

Bids for Offshore Oil Contracts (in Millions of 1969 Dollars)

Santa Barbara Channel

$43.5 $32.1 $18.1 $10.2 $6.3

Alaska North Slope

$10.5 $5.2 $2.1 $1.4 $.5

Bids for FCC Spectrum Rights (in Millions of 1995 Dollars)

Miami Metro Area

$131.7 $126.0 $125.0 $119.4 $119.3

Dallas Metro Area

$84.2 $72.0 $68.7 --- ---

Source: R. Weber, “Making More For Less”, Journal of Economics and Management Strategy, Fall 1997

Open bidding allows bidders to react to information revealed in prior rounds. The FCC used open bidding when they recently auctioned broadband PCS

Market Population Winner Second Bid Price/Pop

New York 26.4M Wireless Alaacr $442.7 $16.76

San Francisco 11.9M PacTel AmerPort $202.2 $17.00

Charlotte 9.8M BellSouth CCI $70.9 $7.27

Dallas 9.7M Wireless GTE $84.2 $8.68

Houston 5.2M PrimeCo Wireless $82.7 $15.93

New Orleans 4.9M PrimeCo Powertel $89.5 $18.17

Louisville 3.6M Wireless PrimeCo $46.6 $13.10

Salt Lake City 2.6M Wireless GTE $46.2 $17.95

Jacksonville 2.3M PrimeCo GTE $44.5 $19.56

Source: P. Crampton, “The FCC Spectrum Auctions”, Journal of Economics and Management Strategy, Fall 1997

Suppose that the value of the Louisville, Kentucky market is a random variable with 6 equally likely possibilities: $10, $20, $30, $40, $50, $60

(Expected Value = $35)

You are competing with one other bidder with the same priors (beliefs about the market value). - common value, common information

Oral English Auction

Your Bid: <$35

Competitor’s Bid: <$35

Sealed Bid Auction

Your Bid: <$35


The open auction yields no benefits over the sealed bid auction because there is no information to reveal.

Now, suppose that you and your competitor have the same values, but different information about the distribution - common value, private information

Sealed Bid Auction

Your Bid: <$40


You: $20, $40, $60 (each with the same probability)

Opponent: $10, $40, $60 (each with the same probability)

Expected Value = $40

Expected Value = $33.67

You should win the auction and pay less than $40

Now, suppose that you and your competitor have the same values, but different information about the distribution - common value, private information

You: $20, $40, $60 (each with the same probability)

Opponent: $10, $40, $60 (each with the same probability)

Expected Value = $40

Expected Value = $33.67

Oral English Auction: Round 1

Your Bid: <$40


Both parties learn that $10, $20, $30, and $50 are not possibilities (you eliminated $10, $30, and $50 while your opponent eliminated $20 ,$30, and $50)

Oral English Auction: Round 1

Your Bid: <$50


Both bids in round 2 are more informed!!

Private Value Auctions

In private value settings, each bidder has the same information, but a places a different value on the object (e.g. fine art). In this setting, those with high valuation prefer not to reveal themselves and, hence, would underbid in an open outcry auction

Suppose that there are two bidders for an object. (A and B). Both believe the value of the object to be between $0 and $10M (with a uniform distribution).

Bidder A places value aV

1 kkVBkVB bbaa

on the object

Bidder A places value bV on the object

Both are following strategies of bidding an amount equal to some fraction of their true value


1 kkVBkVB bbaa

on the object



Bidder A wins if ba kVB

000,000,10

1

000,000,10

1Pr

0 k

BdV

k

BV a

bk

Ba

b

a

ba Vk

B

000,000,10

1

000,000,10

1Pr

0 k

BdV

k

BV a

bk

Ba

b

a

bV

)Pr( bV

10M

1

10M

k

Ba

000,000,10

1max

k

BBV aa

Ba

Optimal Bidding by Player A

First Order Necessary Conditions

0000,000,10

1

000,000,10

11

aa

B

kkBV

aa BBV 2a

a

VB


1 kkVBkVB bbaa

on the object



The Nash equilibrium of this game is for both bidders to submit a bid equal to ½ of their private values.

2

1

2

2

k

VB

VB b

ba

a

With to bidders, optimal strategy is to underbid by 50%!!!

-50%

-20%

-10%

2 5 10Number if Bidders

It can be shown that with N bidders, the optimal strategy is

N

VB ii

With Private Value auctions, it pays to have a lot of bidders (as the number if bidders gets arbitrarily large, everyone bids their true value!)

Alternatively, we could deal with the underbidding problem by holding a second price auction

In this setup, the highest bidder wins, but pays the amount equal to the second highest bid

Lets repeat the previous example, but with a second price auction

000,000,10

1max

k

BBV ab

Ba

000,000,10

1max

k

BBV aba

Ba

Is there any incentive to bid higher than your private valuation?

No. By raising your bid, you increase your odds of winning, but you face the possibility of paying more than you private value!

Is there any incentive to bid lower than your private valuation?

No. Lowering your bid has no impact on your purchase price, but lowers you odds of winning.

Second price auctions avoid underbidding as well as the winner’s curse by giving bidders the incentive to reveal their values (incentive compatibility)

Do All Auctions Yield the Same (Expected) Revenues?

Dutch Auctions = 1st Price Auctions (sealed bid)

As the price falls, the individual with the highest value will be the first to speak. He/She will win, and pay an amount equal to his/her bid

English Auctions = 2nd Price Auctions (sealed bid)

As the price rises, the individual with the highest value will be the last to bid and will offer an amount just slightly higher than the previous bidder.

1st Price Auctions (sealed bid) vs. 2nd Price Auctions (sealed bid)??

In first price auctions, the high bid is paid, but everybody has the strategy of underbidding.

Revenue Equivalence

Private Values Common Values

Risk Neutral

1st Price = 2nd Price 1st Price < 2nd Price

Risk Averse

1st Price > 2nd Price 1st Price ?? 2nd Price

It turns out that you can rake the expected returns from different auction rules. The two important questions are

•Are valuations privately or commonly held?

•Are bidders risk neutral or risk averse?

Revenue Equivalence

Private Values

(More Asymmetric Information)

Common Values

(Less Asymmetric Information)

Risk Neutral

1st Price = 2nd Price 1st Price < 2nd Price

Risk Averse 1st Price > 2nd Price 1st Price ?? 2nd Price

Consider the following Products. If you were the seller, which auction type would you prefer?

Treasury Bills?

IPOs?

Artwork?

Logging Rights?

The type of auction you choose depends on the environment you face!!

Documents

Optimal Mechanism Design Finance 510: Microeconomic Analysis