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Unformatted text preview: Real Estate Auctions
HKU Real Estate Finance K. S. Maurice Tse
School of Economics and Finance The University of Hong Kong email@example.com Content
Types of Auction Optimal Bidding Strategy Optimal Auctions from the Seller’s Point of View Numerical Examples on Hong Kong Land Auctions The Winner’s Curse Introduction
Source: “Oriental Daily 東方日報” one day before July 1, 1997 Types of Auctions
Four types of auctions commonly used:
(1) (2) (3) (4) English auction Dutch auction First-price sealed bid auction Second-price sealed bid auction 1. English Auction
Bids are freely made and announced publicly until no purchaser wishes to make any further bid. A rational bidder participates if and only if he can win the auction with a bid that is less than his reservation value for the auctioned property and thus drops out only if the bid on the floor equals or exceeds his reservation value. Therefore English bids only depend on individual reservation values and observed bids on the floor. Stay in the bidding competition as long as the bid on the floor does not exceed the bidder’s reservation value and drop out as soon as it exceeds that. The bidding stops at a price approximately equal to the second highest value among the values that the purchasers place on the property. The winner is the bidder to whom the property has the highest value. Every bidder in an English auction will only bid his/her reservation value. At the end, the bidders know each other’s reservation value English Auction-Open Ascending
Source: “Oriental Daily 東方日報” one day before July 1, 1997 2. Second Price Sealed Bid
An English auction is isomorphic to the second-price sealed bid auction. In a second-price sealed bid auction, the bidder with the highest bid wins but pays only the second highest price. Assume a seller has a property for sale and there are 11 bidders with valuations for the property as follows: $5, 10, 20 ,30, 40, 50, 60, 70, 80, 90, $100 Suppose these valuations are common knowledge. The bidders simultaneously submit bids si ∈ [0, +∞). Suppose the size of each bid is $0.01 And the opening bid is $40. What price does the winner pay in this auction in the secondprice sealed bid auction?? What price does the winner pay if this is an English auction?? Second Price Sealed Bid Contd
The highest bidder wins the property and pays only the second highest bid. The gain to the winner is vi - maxj≠i sj The highest bid of all other bidders Bidders in the second-price sealed bid, as in the English auction, should have no incentive to bid differently from their reservation values; that is, si = vi WHY ???? Analysis of Second-Price
In the second-price auction, does it pay to bid differently from my reservation value?? Analysis
Each bidder has an identity i : Bidder i Each bidder has a reservation value: vi Let the bid of bidder j be sj Let ri be the highest bid of all other bidders on the bidding floor NOT including bidder i himself.
What does this notation mean?? ri = maxj≠i sj Analysis of 2nd Price Contd
Example: Consider 4 bidders with
s1 = $100, s2 = 80, s3 = 170, s4 = 140 Then
r3 = maxj≠3 sj = 140 r1 = maxj≠1 sj = 170 r2 = maxj≠2 sj = 170 r4 = maxj≠4 sj = 170 Analysis of 2nd Price Contd
Only one bid to be submitted. There are three bidding strategies
1. Submit a bid lower than reservation value. 2. Submit a bid higher than reservation value. 3. Submit a bid equal to reservation value. Strategy 1
Should bidder i lie about his reservation value by bidding lower than his reservation value vi? Or should he reveal his reservation value by bidding it?? Suppose si < vi as depicted below.
1: ri < si si 3: si < ri < vi vi 2: vi < ri Let the highest bid of all other bidders be ri. There are three possible outcomes. 1: ri < si si 3: si < ri < vi vi 2: vi < ri Analysis of Strategy 1
The results by comparing ri with bidder i’s bid si are as follows: Outcome
1: ri < si 2: vi < ri 3: si < ri < vi Lie: si < vi
WIN Gain = vi - ri > 0 Lose Payoff = 0 Lose Payoff = 0 Truth: si = vi
WIN Gain = vi - ri > 0 Lose Payoff = 0 WIN Gain = vi - ri > 0 Telling truth “dominates” telling lie. Submitting a bid lower than reservation value is a bad strategy relative to telling truth Strategy 2
If submitting a bid lower than reservation value is a bad strategy, how about “submitting a bid higher than reservation value” ?? 1: ri < vi vi 3: vi < ri < si si 2: si < ri Analysis of Strategy 2
If bidder i should not bid lower than his reservation value, how about bid higher than reservation value?? Compare ri with bidder i’s bid si. Outcome 1: ri < vi 2: si < ri 3: vi < ri < si Lie: si > vi
WIN Gain = vi - ri > 0 Lose Payoff = 0 WIN Gain = vi - ri < 0 Truth: si = vi
WIN Gain = vi - ri > 0 Lose Payoff = 0 Lose Payoff = 0 Telling truth “dominates” telling lie. Submitting a bid higher than reservation value is a bad strategy relative to telling truth Second Price Sealed Bid Contd
What can we conclude about the equilibrium outcome of 2nd Price auction?? What about 1st price sealed bid auction?? 3. First Price Sealed Bid
In the first price sealed bid auction, the bidder with the highest bid wins and pay his own bid (the highest bid). Analysis
Each bidder has an identity i: Bidder i Each bidder has a reservation value: vi Let the bid of bidder j be sj. Let ri be the highest bid of all other bidders on the bidding floor NOT including bidder i himself. Only one bid to be submitted. There are three bidding strategies
1. Submit a bid higher than reservation value. 2. Submit a bid lower than reservation value. 3. Submit a bid equal to reservation value. ri = maxj≠i sj Strategy 1 of 1st Price Auction
Should bidder i lie about his reservation value by bidding higher than his reservation value vi? Or should he reveal the truth by bidding his reservation value?? Suppose vi < si as depicted below.
1: ri < vi vi 3: vi < ri < si si 2: si < ri Let the highest bid of all other bidders be ri. Then there are three possible outcomes. 1: ri < vi vi 3: vi < ri < si si 2: si < ri Analysis of Strategy 1
The results by comparing ri with bidder i’s bid si are as follows: Outcome
1: ri < vi 2: si < ri 3: vi < ri < si Lie: si > vi
WIN Gain = vi - si < 0 Lose Payoff = 0 WIN Gain = vi - si < 0 Truth: si = vi
WIN Gain = vi - vi = 0 Lose Payoff = 0 Lose Payoff = 0 Submitting a bid higher than reservation value is a bad strategy relative to telling truth 1: ri < si si 3: si < ri < vi vi 2: vi < ri Analysis of Strategy 2
If bidder i should not bid higher than his reservation value, how about bid LOWER than reservation value?? Outcome 1: ri < si 2: vi < ri 3: si < ri < vi Lie: si < vi
WIN Gain = vi - si > 0 Lose Payoff = 0 Lose Payoff = 0 Truth: si = vi
WIN Gain = vi - vi = 0 Lose Payoff = 0 WIN Gain = vi - vi = 0 Telling lie “dominates” telling truth. Submitting a bid lower than reservation value is a better strategy than telling truth First Price Sealed Bid Contd
The equilibrium outcome is to submit a bid less than our own reservation value. If we should submit a bid lower than our reservation value in the first price sealed bid auction, how much below reservation value should we bid? How much should I shade my bid below reservation value? The answer can be found by analyzing the Dutch Auction. 4. Dutch Auction
Offer price starts at an amount believed to be higher than any bidder is willing to pay and is lowered by an auctioneer or a clock device until one of the bidders accepts the last price offer. The first and only bid is the sales price in the Dutch auction. In this case, each bidder must consider the probable bids that might be made by others in order to give a bid soon enough that would give bidder the greatest expected gain. To bid as soon as the price has come down to the reservation value of the bidder maximizes the probability of obtaining the object, but the gain is zero. As the price comes down further, the possibility of a gain to the bidder increases but the chance to get the object also decreases as others might put in a bid before the bidder does. Each bidder must balance between these two factors in terms of whatever he knows regarding other bidders’ probable bids. Dutch Auction
Let vi be Bidder i’s reservation value for a property All other bidders are denoted by j (j ≠ i)
$ Price 0 Should respond in this region vi Opening Bid Should not respond in this region Bidding Strategy ?? The longer bidder i waits, the more likely bidder i can make a profit if he wins. The longer bidder i waits, the more likely bidder i will lose to other bidders in the auction. “Probability of Winning” “Size of Profit” Question: How long should I wait before I bid?? Optimal Strategy ?? Analysis of Dutch Auction
There are n bidders in the auction: bidder i with i = 1, 2 ,3 4 …. Bidder i knows exactly what his/her own reservation value is, but bidder i does not know the reservation values of all other bidders. Bidder i believes the reservation values of all other bidders fall into a range between a low price ($m) and a high price ($M) The reservation values of all other bidders are uniformly distributed over the interval (m, M)
Probability U(m, M) Question: Let X denote the reservation value of other bidders. What is Prob[X ≤ x] ?? x−m Pr ob[X ≤ x ] = M −m
0 m x M $ Price Pr ob[ X ≤ M ] =
Pr ob[X ≤ m] = M −m =1 M −m m−m =0 M −m Analysis Contd.
Suppose bidder i has reservation value equal to v Bidder i is going to submit a bid equal to s. If bidder i wins, his profit is Profit = (v – s) What is the probability that s is the winning bid (i.e. the highest bid)?? Prob[s is the highest bid] = Prob[All other bids are lower than s] Probability that a particular bid is lower than s = (s-m)/(M-m) The probability that ALL OTHER bidders’ bids are lower than s is ⎛ s − m ⎞⎛ s − m ⎞⎛ s − m ⎞ ⎛ s − m ⎞ ⎛ s − m ⎞ ⎟ ⎟⎜ ⎟.....⎜ ⎟=⎜ ⎜ ⎟⎜ ⎝ M − m ⎠⎝ M − m ⎠⎝ M − m ⎠ ⎝ M − m ⎠ ⎝ M − m ⎠
All others = (n-1) n −1 What is the expected profit to bidder i ?? Analysis Contd.
If bidder i bids $s, his expected profit is ⎛ s−m ⎞ π = (v − s)⎜ ⎟ ⎝M −m⎠
Gain if win n −1 Chance of winning Bidder i wants to choose an optimal bid s* that maximizes his
expected profit. Differentiation: Differentiate π with respect to s.
∂π ⎛ s−m ⎞ = (−1)⎜ ⎟ ∂s ⎝M −m⎠
n −1 ( s − m) n − 2 + (v − s )(n − 1) =0 n −1 ( M − m) Solving for the optimal bid s*, we have s* = m + (n − 1)v n Graphical Analysis
Plot the expected profit π against the size of bid s.
Exp Profit π π = (v − s )⎜
max π ⎛ s−m ⎞ ⎟ ⎝M −m⎠ n −1 0 m s* v M Bid (s) As s↑, expected profit (π) ↓ if win As s↑, Prob of winning ↑ Analysis Contd.
Implication of the results: Let us rearrange the result slightly.
* ⎛v−m⎞ s = v −⎜ ⎟ ⎝n⎠ m + (n − 1)v n This result tells us exactly how much we should shade below our reservation value when bidding in a Dutch Auction. The winner in the Dutch (1st Price) auction is the one who has the highest reservation value. Bidders will always submit a bid less than their reservation values. As the number of competing bidders in the auction increases, that is, n BIG & LARGE, then s* v Analysis Contd ⎛v−m⎞ s = v−⎜ ⎟ ⎝n⎠
* Example Suppose there are 5 bidders including myself bidding for a property in a Dutch/1st price Auction. My own reservation value for a property is $1 million. All bidders’ reservation values are uniformly distributed in [$0.7m, $1.3m]. What should be my optimal bid??? My optimal bid is $0.94m [=1 – (1-0.7)/5]. I will also submit a bid less than my reservation value. What have we achieved at this point
We have shown, from the bidders’ (buyers’) perspective the optimal Bidding Strategy in
English Auction 2nd Price Auction 1st Price Auction Dutch Auction Question What about from Seller’s perspective? Theoretically, should the seller prefer one to others???? Optimal auctions from the seller’s point of view
Under the assumptions that bidders are risk neutral, values are independent, and reservation values are identically distributed, what type of auction maximizes the seller’s expected revenue? Does it matter what auction to use in terms of expected revenue? Question: Is there an optimal base price to announce? (Riley & Samuelson, “Optimal Auctions” American Economic Review, 80, ) Theoretical Analysis
Assume a simple auction with only 2 bidders A & B. They do not know each other’s reservation value. But they believe that the reservation value of the other falls between the interval ($0, $1) (simple standardization); i.e. m = 0 and M = 1.
v 0 1 We are going to consider two cases. Case 1: 1st price sealed bid auction Case 2: 2nd price sealed bid auction Evaluate the expected revenue and the risk of revenue in terms of variance for each auction. Case 1: 1st price sealed bid
Let v be the reservation value of the two bidders. Recall the optimal bid in the 1st price sealed bid auction is
vv ⎛v−m⎞ b = v−⎜ ⎟=v− = 22 ⎝n⎠ The winning (highest) bid comes from the highest reservation value between the two bidders. The probability that this bid is the winning (highest bid) is simply equal to v.
v 0 1 Case 1: 1st Price Contd.
Since there are 2 bidders, there are two possible outcomes because the highest bid could come from either bidder A or bidder B. Revenue from a particular bidder with a given reservation value (v) = v/2 Revenue expected from 2 different bidders with given reservation value (v) = 2(v/2)*v = v2. The expected revenue from the 2 different bidders with reservation value v between (0, 1) is equal to the SUM of all possible revenues weighted by the probability of occurrence. v 2 E ( R ) = ∫ v dv = 3 0 1 31 0 13 03 1 =−= 333 This is what the seller expects to get on average. It’s NOT actual revenue!! Risk !! Case 1: 1st Price Contd
What is the revenue risk of 1st price sealed bid as measured by variance?? Variance = E(Rev2) – [E(Rev)]2 v ⎛v⎞ ⎛1⎞ = ∫ ⎜ ⎟ ( 2 v ) dv − ⎜ ⎟ = 2⎠ 8 ⎝3⎠ 0⎝
1 2 2 41 0 1 1 −= 9 72 Squared Revenue Prob of v being the highest reservation value What about 2nd price sealed bid?? Case 2: 2nd Price Auction
In the 2nd price auction, the highest bidder wins and pays only the second highest price. Also the optimal bid by either bidder A or bidder B given his/her reservation value is simply their reservation value (v). What is the probability that this given bid (v) is the lower of the 2 bids ?
1-v 0 v 1 Revenue expected from bidders A & B given their reservation value (v) = v(1-v) * 2 Case 2: 2nd Price Contd
What’s the expected revenue?? Expected revenue of this 2-bidder 2nd price auction is the sum of all possible revenues expected over ($0, $1). How does this result compare with the expected revenue of 1st price auction?? What is the revenue risk of this 2nd price auction ?? ⎡ 2 2v ⎤ 21 ∫ v(1 − v)2dv = ⎢v − 3 ⎥ = 1 − 3 = 3 ⎣ ⎦0 0
1 3 1 Case 2: 2nd Price Contd
What is the revenue risk of 2nd price sealed bid as measured by variance?? Variance = E(Rev2) – [E(Rev)]2
11 ⎛ 1 ⎞ ⎡ 2v 2v ⎤ − = ∫ v (1 − v)(2)dv − ⎜ ⎟ = ⎢ ⎥− = 4 ⎦ 0 9 18 ⎝3⎠ ⎣ 3 0
1 2 3 4 2 1 Squared Revenue Prob of v being the lower reservation value Summary of Analysis
2 bidders Reservation value v: U(0, 1) Results Expected Revenue Revenue Variance 1st Price 1/3 1/72 2nd Price 1/3 4/72 Expected Revenues are the same for both 1st and 2nd price auction Variance of 2nd price > Variance of 1st price Which auction should the seller use ??? Summary Contd
Which auction to use depends on the risk attitude of the seller. If the seller is risk neutral, all 4 types of auctions are just equally desirable. If the seller is risk-averse, he/she will prefer 1st price/Dutch auction. If the seller is risk-taking, he/she will prefer 2nd price/English. Optimal Reserve Price
Question: Can the seller increase the expected sales revenue by announcing some minimum reserve price below which the seller will not accept bids ??? The minimum reserve price is to be appropriately chosen. Riley and Samuelson (1981) provides the answer to this question. They show that for any of these auctions, the expected revenue of the seller can be determined if one can determine the lowest reservation value for which it is worthwhile bidding, the expected seller revenue is the same in all four types of auctions, and all four types of auctions are optimal for the seller if the seller chooses the minimum reserve price well whatever the number of bidders. Riley, J. G. and W. F. Samuelson. (1981). “Optimal Auctions.” American Economic Review, 71, 381-92. Intuition on Choice of Reserve Price
Suppose there are 3 bidders (A, B, C) in an English auction with the following reservation values:
A= $40, B= $45, C= $60 Suppose the bid size is $1. What should be the base price to be set by the seller??
Base Price 0 43 50 55 62 Revenue 46 46 50 55 0 What’s the implication
Seller can choose an optimal reserve price (base price) in order to extract additional profits from the high bidders. However, there is a tradeoff.
Initially, the higher the base price, the higher the additional profits. Too high a base price, additional profits may become zero as all bidders will drop out. Result 1 from Riley & Samuelson Optimal Minimum Reserve Price
The optimal minimum reserve price to be announced by the auctioneer is higher than the seller’s valuation and is given by 1 − F (v * ) v* = v0 + F ' (v * )
where F(x) is the probability that a competing bidder draws a reservation value less than x, and v0 is the seller’s personal valuation of the object that is being auctioned. The minimum reserve price to be announced by the seller is b0 = v*. The seller should always announce a reserve price strictly greater than his/her personal valuation. Result 2 from Riley & Samuelson Optimal Bidding Strategy
The optimal bidding strategy of a typical bidder with reservation value v > b0 is b (v ) = v − 1 F n −1 (v) b0 F n −1 ( x)dx ∫ v The bidding function is strictly increasing in v. b(v) is equal to the expected second highest valuation conditional on v being the highest valuation. In equilibrium, each bidder submits a bid less than his/her valuation. The equilibrium bidding strategy highly depends on the number of rival bidders and on the dispersion of the valuation distributions. As the number of rivals increases, the bidder will bid closer to his reservation value. If there are few rivals, the bidder will bid less. As the valuation dispersion among the bidders increases, the optimal bids will decrease. Result 3 from Riley & Samuelson Expected Revenue to Seller
The expected price revenue to the seller is n ∫ (ν F ' (ν ) + F (ν ) − 1) F n −1 (ν ) d ν
n is the number of bidders, b0 is the announced minimum reserve price below which no bids will
b0 v be accepted, v is the estimated maximum price that the bidders would bid. The expected revenue to the seller increases with the number of bidders in the auction and with the announcement of the optimal reserve price b0. Illustration (Uniform Distribution)
Assume that the estimated maximum bidding price from the bidders is equal to M and the minimum bidding price is m. Suppose that the reservation values v of the bidders are uniformly distributed on the unit interval [m, M] such that v−m F (v ) = M −m F ' (v ) = 1 M −m
v0 + M v= 2
* Optimal minimum reserve price to be announced by the seller: 1 − F (v * ) v = v0 + = v0 + M − v * F ' (v * )
* The seller should design the auction so that only those with reservation values larger than (M + v0)/2 will find it worthwhile bidding. The optimal reserve price to be announced by the seller should be b0 = v*. Optimal Bid by the Bidders
To bidder i with reservation value vi > (M + v0)/2, the optimal bid is
vi 1 b(vi ) = vi − n −1 F n −1 ( x )dx ∫ F (vi ) b0
. (vi − m) n − (b0 − m) n = vi − n(vi − m) n −1 As the number of rival bidders increases, b(vi ) approaches to vi . Expected Revenue to the Seller
Setting v = M and noting F(v) and F (v) are given by the expressions before, we obtain the expected revenue to the seller as follows
n ∫ (vF ' (v) + F (v) − 1) F n −1 (v)dv
b0 M v−m ⎛v ⎞⎛ v − m ⎞ n∫⎜ + − 1⎟⎜ ⎟ M − m M − m ⎠⎝ M − m ⎠ b0 ⎝
n n −1 dv Expected revenue to the seller increases with:
Number of bidders in the auction Announced optimal reserve price b0. = = 2m + (n − 1) M ⎛ b0 − m ⎞ ⎛ M (n + 1) − 2(m + nb0 ) ⎞ +⎜ ⎟ ⎟⎜ n +1 n +1 ⎠ ⎝M −m⎠ ⎝ Case Analysis
Consider the market forecasts made the day before the land auction in Hong Kong on February 20, 2001. Pre-Auction Market Forecasts
Ma On Shan Sai Kung Location Use (Residential) Land Area (sq ft) Plot Ration Building Area Market Forecast: Five Forecasts Forecast in $100m Forecast: $ per sq ft Forecast in $100m Forecast: $ per sq ft Case Analysis Contd
The general market sentiment about the range of market value of the two sites is
($420 million, $580 million) for Ma On Shan ($40 million, $70 million) for Sai Kung. The announced base prices for the two sites are
$50 million for Sai Kung $450 million for Ma On Shan. In the auction, there are 8 bidders for Ma On Shan and 12 bidders for Sai Kung. Use the auction theory that we’ve discussed in class to predict the auction outcomes (revenue to government) 1 − F (v * ) v = v0 + = v0 + M − v * F ' (v * )
* v* = v0 + M 2 Analysis
Ma On Shan Sai Kung (vi − m) n − (b0 − m) n bi = vi − n(vi − m) n −1
4.17 0.54 E1
5.20 0.42 E2
5.00 0.50 E4
5.10 0.46 E5
4.70 0.63 E6
4.90 0.58 Land
Ma On Shan Sai Kung
. Avg Max(M) Min (m)
4.85 0.52 5.80 0.70 4.20 0.40 v*
4.50 0.50 [v0]
3.20 0.30 [bi]
4.76 0.51 E(Rev)
5.44 0.65 n
8 12 E1, E2, E3, etc = forecasts made by market Avg = average forecast made by market M = maximum of market sentiment m = minimum of market sentiment v* = opening bid announced by the auctioneer v0 = HK Government’s theoretical reservation value of the land sites. bi = bid that would be made by a bidder with reservation value simply equal to the average market forecast (Avg) with vi is set equal to the average forecast Avg. E(Rev) = expected selling price (expected revenue) of the land site. Notes to analysis: What is the actual outcome???? Case 2: Auction of 2 Land Sites in Hong Kong in 1998
Site 1: Shatin, New Territories, ST318 Site 2: Shatin, New Territories, ST468 Case 2: Contd
Site 1:Shatin, New Territories
ST318 /////// /////// Case 2 Contd
Site 2:Shatin, New Territories
ST468 /////// /////// Case 2 Contd
Pre-Auction Market Opinions Case 2: Analysis Results
Land E1 E2 E3 E4 E5 E6 E7 318 100 59.00 95.00 48.00 68.00 92.0 80.00 468 9.30 7.50 9.00 7.00 7.50 7.50 11.00 Avg 77.4 8.4 M 100 12.00 m 58.00 6.00 v* 38.00 4.50 [bi] 78.06 8.18 E(R) 90.72 11.00 N 8 11 E1, E2, E3, etc = forecasts made by market Avg = average forecast made by market M = maximum of market sentiment m = minimum of market sentiment v* = base price announced by the auctioneer bi = bid that would be made by a bidder with reservation value simply equal to the average market forecast (Avg) with vi is set equal to the average forecast Avg. E(Rev) = expected selling price (expected revenue) of the land site. Notes to analysis: Case 2 Contd
Results Prediction: ST318=90.72m ST468 = 11m End
Please do the problem set end of lecture note. ...
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