Auctions Johan Stennek - Auctions (2020)
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Auctions
Johan Stennek
1• Examples
– Antiques, fine arts
– Houses, apartments, land
– Government bonds, bankrupt assets
– Government contracts (roads)
– Radio frequencies
2Why use auctions?
• Seller’s goal
– Maximize revenues (You selling your apartment)
– Efficient use (Government selling radio spectrum)
– Both cases: Sell to person with highest valuation
• Problem
– Seller doesn’t know what people are willing to pay
• What is the highest valuation?
• Who has it?
• Solution
– Buyer claiming highest valuation gets the good
– And will pay accordingly
• Auction = Mechanism to extract information
3But are auctions a good solution?
• Efficiency
– IF: people really “tell the truth” = bid their valuations
– THEN: good will be allocated correctly (highest WTP)
• Revenues
– IF: people really “tell the truth” = bid their valuations
– THEN: price will be high (efficiency & extract WTP)
• Question: Do people “tell the truth”?
– Question: Bid = WTP?
– Need to study bidding behavior
4• Bidding behavior turns out to depend on:
– Exact rules of the auction (Auction design)
– How buyer’s valuations are related (Type of uncertainty)
54 Designs
• Sealed bid, second price • English
(“Vickrey”) (“open cry”)
• Simultaneous
We • Sequential + perf. info
• Winner paysw second bid
ill • Ascending bids
on
• New Zeeland spectrum
auction 1990 ly • Art, antiques
stu
dy
• Sealed bid, first price mo• Dutch
st
• Simultaneous c•o Sequential + perf. info
• Winner pays own bid
mm
on
• Descending offers
• Government contracts
• Flowers, Tobacco
6Types of uncertainty
• Private value
– Different buyers have different values
• Common value e va lue
rivatdy p
– Same value n lyto
stuall buyers
w ill o
– But e
W different buyers have different information
7English Auction
8• Assume
– One indivisible unit of the good
– Two bidders
• Information
– Bidders get to know own valuations, v1 and v2
– Then the bidding game starts
• Bidding rules: a simple model
– Players take turns bidding
– Auctioneer asks if that player is willing to raise the price by
€1
– Whenever one player says no, the good is sold to the
current bid
9• Outcome
– Winner = Highest bidder
– Price = Highest bid
10• Utility
⎧ vi − bi if winning
⎪
ui = ⎨
⎪ 0 otherwise
⎩
11• Define: “Bid up to WTP strategy” for i
– IF: bjt-1 + 1 ≤ vi, THEN: Yes
– IF: bjt-1 + 1 > vi, THEN: No
• Claim
– This strategy is optimal (actually, dominant)
12• Sketch of proof
– If b2t-1 +1 ≤ v1
• Yes: Positive utility with (weakly) positive probability
• No: u1 = 0 for sure
– If b2t-1 +1 > v1
• No: u1 = 0 for sure
• Yes: Negative utility with (weakly) positive probability
• Note - dominance
– Above strategy optimal
– no matter how b2t-1 selected
13• Outcome
– Q: “Truth telling”?
• Sort of…
• Most people keep raising the bid until it is equal to
their valuation
• Person with highest WTP keeps raising the bid until
nobody else continues
– Q: Who gets the good?
14• Outcome
– “Truth telling”
– Efficiency
• Bidder with highest valuation wins the good
– Q: Who gets the surplus?
15• Outcome
– “Truth telling”
– Efficiency
• Bidder with highest valuation wins the good
– Surplus-sharing
• p = SHV (sometimes p = SHV + 1)
16First-Price Sealed-Bids Auction
35• Rules
– Each bidder informed about own valuation only
– Simultaneous bids (= sealed bids)
– Winner pays his bid (= first price)
36• Simplification 1
– Two bidders: v1, v2
37• Important fact
– How much a person bids for a good depends on
that person’s valuation of the good
• Game theory
– A strategy is a complete plan of action
– Here: a strategy for player i prescribes a bid, bi, for
every possible valuation, vi
38• Simplification 2
– Assume linear strategies: bi = zi·vi
39• Simplification 3
– In FPSB-auctions, the distribution matters
– vi uniformly distributed over [0, 1]
g(vi)
vi
0 1
40• Q: Probability that vi < x?
g(vi)
vi
0 1
• A: Prob(vi < x) = x
g(vi)
vi
x 1
41• Payoff = expected utility
– Eπ1(b1) = (v1 – b1) Pr(win) + 0 Pr(loose)
– Eπ1(b1) = (v1 – b1) Pr(b1 > b2)
We need to compute
probability that b2 < b1
42• Payoff = expected utility
– Eπ1(b1) = (v1 – b1) Pr(win) + 0 Pr(loose)
– Eπ1(b1) = (v1 – b1) Pr(b1 > b2)
Depends on
b1 = own choice
b2 = random variable
43• Payoff = expected utility
– Eπ1(b1) = (v1 – b1) Pr(win) + 0 Pr(loose)
– Eπ1(b1) = (v1 – b1) Pr(b1 > b2)
Simplifying assumption:
b2 = z2 · v2
44• Payoff = expected utility
– Eπ1(b1) = (v1 – b1) Pr(win) + 0 Pr(loose)
– Eπ1(b1) = (v1 – b1) Pr(b1 > b2)
prob(v2 < b1/z2) = b1/z2
– Eπ1(b1) = (v1 – b1) Pr(b1 > z2 · v2) g(v2)
– Eπ1(b1) = (v1 – b1) Pr(v2 < b1/z2) b1/z2 v2
– Eπ1(b1) = (v1 – b1) (b1/z2)
45• Conclusion
– IF: b2 = z2 · v2
– THEN: Eπ1(b1) = (v1 – b1) (b1/z2)
46• The normal form game
– Players
• bidder 1 and bidder 2
– Strategies
• A bid for every valuation
• bi = zi · vi ⇒ players need to choose zi
– Payoffs
• Eπi(bi) = (vi – bi) (bi/zj)
47• Payoff
– Eπ1(b1) = (v1 – b1) (b1/z2)
• Trade-off
– Higher bid à Higher probability of winning
– Higher bid à Higher price
• Q: What is player 1’s best reply?
48• What is 1’s best reply?
– Eπ1(b1) = (v1 – b1) (b1/z2)
– FOC: (-1) (b1/z2) + (v1 – b1) (1/z2) = 0
Utility if winning * Increased probability of winning
49• What is 1’s best reply?
– Eπ1(b1) = (v1 – b1) (b1/z2)
– FOC: (-1) (b1/z2) + (v1 – b1) (1/z2) = 0
Decreased utility * probability of winning
50• Assume
– B2(v2) = z2 v2
• What is 1’s best reply?
– Eπ1(b1) = (v1 – b1) (b1/z2)
– FOC: - (b1/z2) + (v1 – b1)/z2 = 0
– Solve: b1 = ½ · v1
51• Conclusion
– IF: Bidder 2 uses a linear strategy: B2(v2) = z2 · v2
– THEN: Best reply for bidder 1: B1(v1) = ½ · v1
• Note
– Since ½ · v1 is linear
– Since players are symmetric
– Both bidding bi = ½ · vi is a Nash equilibrium of a
game where the strategy for each player is to
choose some function Bi(vi).
52• Interpretation
– Why bid ½ v ?
• Answer 1
– Optimal balance between
• probability of winning
• price in case of winning
53g(v2)
• Interpretation 1
– But why exactly ½ ? v2
v1/2 v1 1
• Answer 2
– Assume you have highest valuation
– Q: What is the expected second highest
valuation?
– Winner bids expected wtp of competitor
=> competitor no incentive to bid more
54• Remark
– With more bidders, expected second highest wtp
is closer to highest wtp
– Bid larger share of wtp
– As n è ∞ b è wtp
55• Outcome
– Q: Who gets the good?
56• Outcome
– Efficiency
• Bidder with highest valuation wins the good
– Q: Who gets the surplus?
57• Outcome
– Efficiency
• Bidder with highest valuation wins the good
– Surplus-sharing
• p = ½ HV
– Truth-telling?
58• Outcome
– Efficiency
• Bidder with highest valuation wins the good
– Surplus-sharing
• p = ½ HV
– “Sort of truth-telling”
• Players actually reveal their valuation
59Game Theoretic “Details”
Auction = Game of Incomplete Information
60Game of Incomplete Information
• Game with incomplete information
– Buyers don’t know each others’ valuations
• Ada is not able to predict Ben’s bid exactly
• It depends on Ben’s valuation of the object
– How should Ada and Ben analyze the situation?
61Game of Incomplete Information
• Solution I: Change definition of strategy
– Strategy = Function prescribing bid for every possible valuation a
player may have
• Example of strategy
– IF wtp = vH THEN bid = bH
– IF wtp = vL THEN bid = bL
• Then, players able to
– Predict rival’s strategy,
even if uncertainty about type and bid remains
– Maximize expected payoff
62Game of Incomplete Information
• But why are strategies functions?
– Ada knows she has high valuation, vH
– Why should she choose strategy with instruction for vL?
• Answer
– Ben doesn’t know Ada’s valuation. Could be vH or vL
– Ben must consider
• What would Ada bid if vH
• What would Ada bid if vL
– To predict Ben’s bid, Ada must also consider what she
herself would have bid in case of vL
63Game of Incomplete Information
• Think of Ada’s choice as two-step procedure
1. Find optimal bid for all possible valuations:
bAda(vH) and bAda(vL)
2. Select the relevant bid: bAda(vH)
64Game of Incomplete Information
• Solution II: Change definition of payoff
– Payoff = expected utility
65Game of Incomplete Information
Bayesian Nash Equilibrium
• Pair of strategies (bAda, bBen) such that
• bAda is a best reply to bBen
– bAda(vH) maximizes Ada’s expected utility
• If Ada’s valuation is vH
• Assuming Ben uses bBen
– bAda(vL) maximizes Ada’s expected utility
• If Ada’s valuation is vL
• Assuming Ben uses bBen
• bBen is a best reply to bAdam
66Comparison of Auction Designs
(Revenues)
67• Which auction gives the highest expected
price?
– FPSB (and Dutch): p = ½ HV
– English (and SPSB): p = SHV Recall: E(SHV) = ½ HV
68• Expected Revenue Equivalence Thm.
(Vickrey, 1961)
– All four auctions give the same expected price
69• Is there any other way to sell the goods which
would give a higher expected profit?
– Lots of different possible ways
• Bargaining
• Other auction formats
• Strange games
70• Generalization of Revenue Equivalence Thm
– No!
– This is example of “mechanism design” and uses
the “revelation principle” (Leonid Hurwicz, Eric
Maskin, Roger Myerson)
71Most fundamental result of
auction theory
72Note 1: No individual knows who has the highest valuation
Note 2: But if people play the auction game
⇒ person with highest valuation walks away with the good
No individual (even a dictator)
could have implemented the efficient allocation,
since nobody has sufficient information
But the market mechanism actually solves the maximization problem
May say the market aggregates information
* must use all the information to solve the max-problem
* despite the fact that it is scattered
73• Empirical test - Laboratory experiments
– Double auctions
– NB: must use laboratory to know people’s
valuations
74• Experimental procedure
– Recruit a group of people
– Divide into “buyers” and “sellers”
– Let them trade an imaginary good
75• Information
– Each seller informed about his cost of selling one unit
– Each buyer informed about his value of buying one unit
– Costs and valuations are private information
76• Payoffs
– Each agent can trade one unit every “trading period”
– Buyer utility: V – P (if trade)
– Seller utility: P – C (if trade)
– Here: no monetary incentives
77Supply curve
Defined by sellers costs
78Demand curve
Defined by buyers values
79Equilibrium price
Defined by values and costs
80Nobody knows
-Demand
-Supply
-Equilibrium price
81• Trading
– Every trading period is 5 – 10 minutes
– Each agent can raise his hand and offer to buy or
sell at some price
– Each other agent can accept
– The trading period continues until nobody is
making additional offers
– Then new trading period
82Contract price as function of
transaction number
83Contract price as function of
transaction number
84Contract price as function of
transaction number
85Increase in demand and supply
86Increase in demand and supply
87• Laboratory experiments
– It works! (Vernon Smith)
– Even with “few” buyers and “few” sellers
market quickly converges to competitive price
88Sure, it is not perfect…
…there is also market failure…
– Coordination failures (mis-pricing; recessions)
– Double coincidence of wants (kidneys, apartments)
– Externalities (global warming; telecom)
– Public goods (R&D; legal system to enforce all contracts)
– Market power (medicines; district heating)
– Incomplete information (cars, insurance, labor, credit)
…and an uneven distribution of wealth
89• But even public policies to correct market
failure use markets to aggregate information
– Cap and trade
– Public procurement
90You can also read
