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 AcceptReject Sister Offer You AcceptReject You Offer Sister AcceptReject ($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 AcceptReject Sister Offer You AcceptReject You Offer Sister AcceptReject ($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 AcceptReject Sister Offer You AcceptReject You Offer Sister AcceptReject ($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 AcceptReject Sister Offer You AcceptReject You Offer Sister AcceptReject ($0,$0) Round 1 Round 2 Round 3 With $1M left to split, you offer You: $400,000 Sister: $600,000

You Sister Offer AcceptReject Sister Offer You AcceptReject You Offer Sister AcceptReject ($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 AcceptReject Sister Offer You AcceptReject You Offer Sister AcceptReject ($0,$0) Round 1 Round 2 Round 3 You: $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, to one 3 to one

Optimal mechanism design deals with institutional rules chosen to serve some explicit optimization goal. No CommunicationCommunication 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 = $55 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)

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$72.0$ 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 MarketPopulationWinnerSecondBidPrice/Pop New York26.4MWirelessAlaacr$442.7$16.76 San Francisco11.9MPacTelAmerPort$202.2$17.00 Charlotte9.8MBellSouthCCI$70.9$7.27 Dallas9.7MWirelessGTE$84.2$8.68 Houston5.2MPrimeCoWireless$82.7$15.93 New Orleans4.9MPrimeCoPowertel$89.5$18.17 Louisville3.6MWirelessPrimeCo$46.6$13.10 Salt Lake City2.6MWirelessGTE$46.2$17.95 Jacksonville2.3MPrimeCoGTE$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 valueon the object Bidder A places valueon the object Both are following strategies of bidding an amount equal to some fraction of their true value

Bidder A places valueon the object Bidder A places valueon the object Both are following strategies of bidding an amount equal to some fraction of their true value Bidder A wins if

10M 1

Optimal Bidding by Player A First Order Necessary Conditions

Bidder A places valueon the object Bidder A places valueon 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. With to bidders, optimal strategy is to underbid by 50%!!!

-50% -20% -10% 2510 Number if Bidders It can be shown that with N bidders, the optimal strategy is 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

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 = 1 st 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 = 2 nd 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. 1 st Price Auctions (sealed bid) vs. 2 nd Price Auctions (sealed bid)?? In first price auctions, the high bid is paid, but everybody has the strategy of underbidding.

Revenue Equivalence Private ValuesCommon Values Risk Neutral 1 st Price = 2 nd Price1 st Price < 2 nd Price Risk Averse 1 st Price > 2 nd Price1 st Price ?? 2 nd 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 1 st Price = 2 nd Price1 st Price < 2 nd Price Risk Averse1 st Price > 2 nd Price1 st Price ?? 2 nd 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!!