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
Optimal mechanism design deals with institutional rules chosen to serve some explicit optimization goal.
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
Optimal mechanism design deals with institutional rules chosen to serve some explicit optimization goal.
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
Competitor’s 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
Competitor’s Bid: <$33
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
Competitor’s Bid: <$34
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
Competitor’s 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
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
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
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
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!!