Post on 18-May-2018
transcript
Market DesignPeter Cramton
Professor of EconomicsUniversity of Maryland
European University InstituteUniversity of Cologne
www.cramton.umd.edu
3 October 2016
With contributions fromAxel OckenfelsUniversity of Cologne
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http://www.cramton.umd.edu/courses/market-design/http://www.cramton.umd.edu/papers/spectrum/
Market design
Establishes rules of market interaction Economic engineering
Economics Computer science Engineering, operations research
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Market design accomplishments
Improve allocations Improve price information Mitigate market failures
Reduce risk Enhance competition
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Applications
Matching (marriage, jobs, schools, kidneys, homes) Auctions
Electricity markets Communication markets Financial securities Transportation markets Natural resource auctions
(timber, oil, diamonds, pollution emissions) Procurement
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Innovation in a few words
Electricity Open access Pay for performance
Communications Auction spectrum Open access
Transportation Price congestion
Climate policy Price carbon
Financial securities Make time discrete
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Resistance to market reforms is the norm: Peter to regulator, "I can save you $10 billion."
Wall Street to regulator, "Hey, thats my $10 billion."Distribution trumps efficiency
Milton Friedman
There is enormous inertiaa tyranny of the status quoin private and especially governmental arrangements. Only a crisisactual or perceivedproduces real change. When that crisis occurs, the actions that are taken depend on the ideas that are lying around. That, I believe, is our basic function [as economists]: to develop alternatives to existing policies, to keep them alive and available until the politically impossible becomes politically inevitable.
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Lectures
9-10:30am, Monday-Friday, 3-7 October 10-1pm, Thursday, 13 October
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Assessment
Select a market design application of your choice Draft an essay of less than ten pages in pdf that
develops some aspect of the market design application
Focus on a narrow question or provide a broader summary of issues
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What is economic engineering?
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Economic engineering
science of designing institutions and mechanisms that align individual incentives
and behavior with the underlying goals
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The Economist as Engineer(Roth 2002)
Understand how markets work in enough detail so we can fix them when theyre broken
Build an engineering-oriented economics literature Otherwise, the practical problems of design will be
relegated to the arena of "just consulting" and we will fail to benefit from the accumulation of knowledge which is so evident in other kinds of engineering
See web.stanford.edu/~alroth marketdesigner.blogspot.com
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http://web.stanford.edu/%7Ealroth/http://marketdesigner.blogspot.com/
Useful to distinguish two kinds of complexities
Institutional detail
Behavior
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Economic engineering
Identifies behavioral and institutional complexities that are relevant for a broader range of real-world contexts
Develops mechanisms, models and methods that can be proven to be robust against such complexities
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Roth (2010) on why experiments are useful in market design Experiments are a natural part of market design, not
least because to run an experiment, an experimenter must specify how transactions are carried out, and so experimenters are of necessity engaged in market design in the laboratory
Experiments can serve multiple purposes in the design of markets outside of the lab
Scientific use of experiments to test hypotheses Testbeds to get a first look at market designs that may not yet
exist outside of the laboratory (Plott 1987) Demonstrations and proofs of concept
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Game theory
Market designBehavioral and Experimental
Economics
2009 ELINOR OSTROM for her analysis of economic governance, especially the commons
2012 ALVIN E. ROTH and LLOYD SHAPLEY for the theory of stable allocations and the practice of market design
2007 LEONID HURWICZ, ERIC S. MASKIN, and ROGER B. MYERSON for having laid the foundations of mechanism design theory
2005ROBERT J.AUMANN andTHOMAS C. SCHELLING for having enhanced our understanding of conflict and cooperation through game-theory analysis
2002VERNON L. SMITH for having established laboratory experiments as a tool in empirical economic analysis, especially in the study of alternative market mechanisms, and
DANIEL KAHNEMAN for having integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty
2001GEORGE A. AKERLOF, A. MICHAEL SPENCE, and JOSEPH E. STIGLITZ, for their analyses of markets with asymmetric information
1996JAMES A. MIRRLEESand WILLIAM VICKREY for their fundamental contributions to the economic theory of incentives under asymmetric information
1994JOHN C. HARSANYI ,JOHN F. NASH andREINHARD SELTEN for their pioneering analysis of equilibria in the theory of non-cooperative games
1978HERBERT A. SIMON for his pioneering research into the decision-making process within economic organizations
Nobel awards
2014 JEAN TIROLE for his analysis of market power and regulation
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ObjectivesEfficiencySimplicityTransparencyFairness
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It is good to keep these goals in mind whenever designing a market
Matching
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One-sided matching
Each of n agents owns house and has strict preferences over all houses
Can the agents benefit from swapping?
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Top trading cycles
Each owner i points to her most preferred house (possibly is own
Each house points back to its owner Creates directed graph; identify cycles
Finite: cycles exist Strict preference: each owner is in at most one cycle
Give each owner in a cycle the house she points to and remove her from the market
If unmatched owners/houses remain, iterate
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Properties of top trading cycles
Top-trading-cycle outcome is a core allocation No subset of owners can make its members better off by
an alternative assignment within the subset
Top-trading-cycle outcome is unique Top-trading-cycle algorithm is strategy proof
No agent can benefit from lying about her preferences
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Example
Owners ranking from best to worst1: (h3, h2, h4, h1)2: (h4, h1, h2, h2)3: (h1, h4, h3, h3)4: (h3, h2, h1, h4)
1
2
3
4
h1
h2
h3
h4
3
1
2
4
h2
h4
4
2
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Two-sided matching
Consider a set of n women and n men Each person has an ordered list of some members of
the opposite sex as his or her preference list Let be a matching between women and men A pair (m, w) is a blocking pair if both m and w prefer
being together to their assignments under . Also, (x, x)is a blocking pair, if x prefers being single to his/her assignment under
A matching is stable if it does not have any blocking pair
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Example
Lucy Peppermint Marcie Sally
Charlie Linus Schroeder Franklin
Schroeder Charlie
Charlie Franklin Linus
Schroeder Linus
Franklin
Lucy Peppermint
Marcie
Peppermint Sally
Marcie
Marcie Sally
Marcie Lucy
Stable!
Charlie Linus
Franklin
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Deferred Acceptance Algorithms
In each iteration, an unmarried man proposes to the first woman on his list that he hasnt proposed to yet
A woman who receives a proposal that she prefers to her current assignment accepts it and rejects her current assignment
This is called the men-proposing algorithm
(Gale and Shapley, 1962)
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Example
Lucy Peppermint Marcie Sally
Charlie Linus Schroeder Franklin
Schroeder Charlie
Charlie Franklin Linus
Schroeder Linus
Franklin
Lucy Peppermint
Marcie
Peppermint Sally
Marcie
Marcie Sally
Marcie Lucy
Stable!
Charlie Linus
Franklin Schroeder Charlie
Charlie Franklin Linus
Schroeder
Charlie Linus
FranklinSchroederCharlie
CharlieLinus
Franklin Linus
Franklin
LucyPeppermint
Marcie
MarcieSally
MarcieLucy
PeppermintSally
Marcie
MarcieLucy
LucyPeppermint
Marcie
PeppermintSally
Marcie 35
Theorem 1. The order of proposals does not affect the stable matching produced by the men-proposing algorithm
Theorem 2. The matching produced by the men-proposing algorithm is the best stable matching for men and the worststable matching for women This matching is called the men-optimal matching
Theorem 3. In all stable matchings, the set of people who remain single is the same
Classical Results
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Applications of stable matching
Stable marriage algorithm has applications in the design of centralized two-sided markets
National Residency Matching Program (NRMP) since 1950s Dental residencies and medical specialties in the US, Canada, and parts
of the UK National university entrance exam in Iran Placement of Canadian lawyers in Ontario and Alberta Sorority rush Matching of new reform rabbis to their first congregation Assignment of students to high-schools in NYC
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Question: Do participants have an incentive to announce their real preference lists?
Answer: No! In the men-proposing algorithm, sometimes women have an incentive to be dishonest about their preferences
Incentive Compatibility
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Example
Lucy Peppermint Marcie Sally
Charlie Linus Schroeder Franklin
Schroeder Charlie
Charlie Franklin Linus
Schroeder Linus
Franklin
Lucy Peppermint
Marcie
Peppermint Sally
Marcie
Marcie Sally
Marcie Lucy
Stable!
Charlie Linus
Franklin Schroeder Charlie
Lucy Peppermint
Marcie
MarcieSally
MarcieLucy
PeppermintSally
Marcie
Charlie Linus
Franklin
MarcieSally
LinusFranklin
MarcieLucy
SchroederCharlie
Lucy Peppermint
Marcie
CharlieLinus
Franklin
PeppermintSally
Marcie
PeppermintSally
Marcie
Charlie FranklinLinus
Schroeder
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Next Question: Is there any truthful mechanism for the stable matching problem?
Answer: No!Roth (1982) proved that there is no mechanism for the stable marriage problem in which truth-telling is the dominant strategy for both men and women
Incentive Compatibility
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Auctions
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Why auctions?
Auctions solve complex problems in infrastructure industries
(communications, energy, transport) in B2B markets in internet markets in financial markets
in an economic way high revenues high efficiency revelation of hidden information
A basic framework with one buyer
c vp
Payoff of sellerp - c
Payoff of buyerv - p
Trade is efficientGains from trade = v c > 0
Money
Price determines the distribution of the surplus
and with many buyers
c v2
Money
v1v3v5 v4
Why auctions?
What selling mechanism results in the highest expected price (of a single item)?
(Seemingly) Trivial answer in case of complete information: take-it-or-leave-it offer
With incomplete information, an ascending auction typically does better than take-it-or-leave-it offer
Why auctions?
Instead of guessing how much buyers are willing to pay, an auction lets buyers name their own price, revealing how much the item is worth
E.g., price = value of strongest competitor
If there are many buyers, this is almost as good as with complete information ideal
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Single-item auctions(assuming that the pre-auction decisions, such as product design, have been made)
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Auction formats English (clock) auction: the price rises
continuously until there is only one bidder left willing to pay
Dutch (clock) auction: the price drops continuously until the first bidder accepts
First price auction: all bidders submit an offer unaware of what the other bidders are offering. The highest bid wins. The price equals the highest bid
Second price auction: As first price auction, but the price equals the second highest bid
Bidding behavior: English auction Which auction format is the best one?
To find out the optimal auction, one has to analyze bidding behavior under the different formats
Winner i gets vi p, and all others get 0
In the English auction, it is optimal to stay in the auction as long as the price is below ones own value
The bidder with the highest value will win at a price equal to the second highest value
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The second price auction looks weird. Why should a seller be satisfied with the second highest bid?
But, in fact, the second-price auction is similar to (the sealed-bid equivalent of) the English auction
Each bidder will bid her value
As a consequence, the bidder with the highest value will win at a price equal to the second highest value
Second-price auction
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0 1vibi max b-i
(Zero-) payoff unchanged
0 1vibimax b-i
Payoff unchanged
0 1vibi max b-i
loss
1) bi < vi < max b-i
2) max b-i < bi < vi
3) bi < max b-i < vi
Should I bid less than my value?
Second-price auction
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0 1vi bi max b-i
(Zero-) Payoff unchanged1) vi < bi < max b-i
2) max b-i < vi < bi
3) vi < max b-i < bi
0 1vi bimax b-i
Payoff unchanged
0 1vi bimax b-i
loss
Should I bid more than my value?
Second-price auction
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Second-price auction
Bidding ones value never hurts and is sometimes rewarded
So it is a dominant strategy to bid value
The underlying idea is that bidders are price takersa winner cannot influence ones price, but only the probability of winning
By bidding her value, she makes sure that shell win if and only if the price is below her value
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First-price auction (pay-as-bid) Because the winners bid determines what she has to pay, the
only way to make positive surplus is to bid less than ones value
Thus, bidders will shade their bids. But by how much?
The winner must think about how much he needs to bid to win, which is difficult because of the risk-return trade-off she faces
She wants to just outbid the bidder with the second-highest value
So, under risk neutrality (an ex ante symmetry), she will bid the expected value of the second-highest value, conditional on winning (only then this is relevant)
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First-price auction (pay-as-bid)
Because the bidder with the highest value wins (in any equilibrium), the revenue equals the expected second-highest value
Thus, expected revenue is the same in the first-price, second-price and English auction
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Dutch auction
The Dutch auction is strategically equivalent to the first price auction
You must decide how much to bid at a time when you do not know how much the others bid
So, all these auctions lead to the same expected revenue
The winner pays the (expected) value of the strongest competitor
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The revenue equivalence theorem
All auction protocols with the properties that the bidder with the highest value wins, and bidders with the lowest-possible value make zerolead to the same expected revenue to the seller!
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But strong assumptions are required
Single item: one item is auctioned Risk neutrality: bidders are risk-neutral Independence: bidders values are private
information and statistically independent Symmetry: values are drawn from the same
probability distribution
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These assumptions never hold in practice, yet the revenue equivalence theorem is incredibly useful!
William Vickrey
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... received the Nobel prize 1996 for the revenue equivalence theorem!
Illustration with complete information
An item is worth 15 to bidder A and 10 to bidder B
In the English auction, B drops out at a price 10
In the second price auction, both bidders bid their value, so that the price is 10
In the first price auction, A bids just enough to win (10 + epsilon), and B bids 10
The Dutch auction is strategically equivalent to the first price auction
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Remarks Standard single unit auction protocols fall under the
revenue equivalence theorem Bidders adjust their behavior in a way that does not allow the seller
to get more revenue by some sophisticated auction procedures
However, when the underlying assumptions do not hold, (e.g. risk aversion, asymmetries, etc.) the choice of mechanism matters!
Still, the RET is useful
Similarities to matching: if you ask me for my preferences, the critical question is what are you going to do with this information
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Remarks:The optimal auction and reserve prices
The optimal auction has a reserve price that is equal to the optimal take-it-or-leave-it price in case of only one bidder
All four auction formats with this reserve price are optimal
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Remarks: Interdependent values
Here is a jar with cents Please write down
Your name: Peter Your estimate of number of cents: E = nnn Your bid in 1st price auction (in cents): 1st = nnn Your bid in 2nd price auction (in cents): 2nd = nnn
The highest bid wins in each auction
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0
5
10
15
20
25
30
35
40
45
50
1 2 3 4 5 6 7 8 9 10 11
Remarks: The winners curse
Real value: 3,00
Distribution of bids
Average bid: 2,70
Highest bid: 11,50
Diagramm1
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34
27
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24
13
15
11
4
7
5
2
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1
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1
1
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2
11
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Hufigkeit
Nagel
SpieltheoretikerZeitungsleserTemplate
Hufigkeit in %Hufigkeit in %
020%5%
17%7%
10%3%
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Nagel
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Zahlen von Spieltheoretikern(Hufigkeiten in %)
Nagel (2)
SpieltheoretikerZeitungsleserTemplate
Hufigkeit in %Hufigkeit in %
0020%5%
17%8%
20%3%
36%2%
42%2%
53%2%
63%2%
77%3%
81%2%
92%3%
100%2%
11110%4%
120%2%
132%3%
142%2%
152%4%
162%3%
173%2%
182%1%
190%1%
200%1%
210%2%
22225%7%
230%2%
245%2%
251%1%
261%1%
270%1%
280%2%
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310%1%
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33334%7%
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354%1%
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Template
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Template
Zahlen der Zeitungsumfrage(Hufigkeiten in %)
Hufigkeit in %
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Nagel (2)
Hufigkeit in %
Aktienindizes
Historical
NEMAX 50 PRC AUC
Type:Stock IndexFamily:NemaxISIN:DE0009665141Epic or local ID:Symbol:Country: Germany
YearFirst trading day.HighLowLast trading day.Average
2003355.74596,24306,31560,240
20021150.131295,15303,93355,99
20012834.262958,76638,481145,03
20005112.949665,812666,492859,28
19994341.065230,583272,435074,93
Source:onvista
Historical
NEMAX 50 PRF AUC
Type:Stock IndexFamily:NemaxISIN:DE0009665133Epic or local ID:Symbol:NMDXCountry: Germany
YearFirst trading day.HighLowLast trading day.
2003358.54604.37308.73567.88
20021,155.231,300.89306.32358.79
20012,843.902,968.82641.311,150.10
20005,127.899,694.072,675.562,869.01
19994,352.155,245.873,281.125,089.76
istorical
DAX (PERFORMANCEINDEX)
Type:Stock IndexFamily:DaxISIN:DE0008469008Epic or local ID:5738566Symbol:Country: Germany
YearFirst trading day.HighLowLast trading day.
20032,898.683,996.282,188.753,965.16
20025,155.265,467.312,519.302,892.63
20016,431.146,795.143,539.185,160.10
20006,961.728,136.166,110.266,433.61
19994,978.386,992.924,601.076,958.14
19984,270.696,224.523,822.595,006.57
19972,836.824,477.702,815.504,224.30
19962,284.862,916.162,284.862,880.07
19952,079.452,317.011,910.962,253.88
19942,267.982,271.111,960.592,106.58
19931,531.332,266.681,516.502,266.68
19921,601.881,811.571,420.301,545.05
19911,366.101,715.801,322.681,577.98
19901,814.381,968.551,334.891,398.23
19891,335.011,790.371,271.701,790.37
1988943.881,340.41931.181,327.87
19871,000.001,000.001,000.001,000.00
Hufigkeit in %
Template
Template
dax 95-04
DateOpenHighLowCloseclose .00sVolumeAdj. Close*DateClose
6/1/043924.303929.883856.0438641803864.1819951/2/952021
5/3/043972.884029.323710.0239214103921.412/1/952102
4/1/043858.344156.893857.0439852103985.213/1/951922
3/1/044026.194163.193692.4038567003856.704/3/952015
2/2/044062.794150.563960.4140181604018.165/2/952092
1/2/043969.044175.483969.0440586004058.606/1/952083
12/1/033752.723996.283752.7239651603965.167/3/952218
11/3/033657.613814.213576.5237459503745.958/1/952238
10/1/033255.763675.783217.4036559903655.999/1/952187
9/1/033493.343676.883202.8732567803256.7810/2/952167
8/1/033481.813588.533299.7734845803484.5811/1/952242
7/1/033217.213487.863119.3534878603487.8612/1/952253
6/2/032990.933324.442990.9332205803220.5819961/2/962470
5/2/032938.723068.082769.4529826802982.682/1/962473
4/1/032426.243004.792395.722942402942.043/1/962485
3/3/032553.742731.572188.7524238702423.874/1/962505
2/3/032750.402802.932433.152547502547.055/2/962542
1/2/032898.683157.252563.9227478302747.836/3/962561
12/2/023332.733476.832836.0128926302892.637/1/962473
11/1/023152.193443.492987.8533203203320.328/1/962543
10/1/022772.963299.012519.3031528503152.859/2/962651
9/2/023698.693698.692719.492769302769.0310/1/962659
8/1/023699.313930.963235.3837129403712.9411/1/962845
7/1/024377.104483.033265.9637001403700.14199712/2/962888
6/3/024818.074850.383946.7043825604382.561/2/973035
5/2/025043.015126.334741.9548183004818.302/3/973259
4/2/025379.645379.644929.3550412005041.203/3/973429
3/1/025025.965467.315003.0653972905397.294/1/973438
2/1/025109.485165.974706.015039805039.085/2/973562
1/2/025155.265352.164974.5851076105107.616/2/973766
12/1/014986.585341.864872.2551601005160.107/1/974405
11/1/014556.715217.474481.5549899104989.918/1/973919
10/1/014315.064874.314157.6045591304559.139/1/974154
9/3/015195.675220.103539.1843081504308.1510/1/973753
8/1/015859.685930.455124.6951881705188.1711/3/973972
7/2/016053.816131.975551.1458611905861.19199812/1/974224
6/1/016122.966278.045758.6960583806058.381/2/984442
5/2/016272.036337.475960.4661232606123.262/2/984693
4/2/015843.376271.205383.9962645106264.513/2/985097
3/1/016212.666342.545351.4858299505829.954/1/985107
2/1/016788.576788.576071.3562082406208.245/4/985569
1/2/016431.146795.146172.4467951406795.146/2/985897
12/1/006379.266813.306110.266433616842106433.617/1/985873
11/1/007093.847185.666361.4363723310454546372.338/3/984833
10/2/006800.077087.926297.497077443095237077.449/1/984474
9/1/007221.417456.716468.4667981210476196798.1210/1/984671
8/1/007194.037395.816954.6072447910434787244.7911/2/985022
7/3/006912.977503.326842.617190373333337190.3712/1/985002
6/1/007117.577486.406851.8468824406882.4419991/4/995159
5/2/007407.537571.726794.0871096707109.672/1/994911
4/3/007599.787641.536890.9674146807414.683/1/994884
3/1/007645.308136.167411.7675993907599.394/1/995393
2/1/006841.127813.206841.1276445507644.555/3/995069
1/3/006961.727306.486388.9168356006835.606/1/995378
12/1/995891.276992.925836.7369581406958.147/1/995101
11/1/995518.746032.675474.125896405896.048/2/995270
10/1/995166.575557.915078.5955254005525.409/1/995149
9/1/995286.585530.765069.8751498305149.8310/1/995525
8/2/995088.655455.844948.0852707705270.7711/1/995896
7/1/995428.345686.555035.2051018705101.8712/1/996958
6/1/995083.625500.934980.6353785205378.5220001/3/006835
5/3/995372.685449.225003.9750698305069.832/1/007644
4/1/994852.395397.644779.0453931105393.113/1/007599
3/1/994923.315151.014605.2748842004884.204/3/007414
2/1/995193.035287.964749.4549118104911.815/2/007109
1/4/994991.955509.344768.4151599605159.966/1/006882
12/1/984986.815085.474451.2050023905002.397/3/007190
11/2/984722.845176.274617.4150227005022.708/1/007244
10/1/984389.214722.093833.7146711204671.129/1/006798
9/1/984727.355161.404357.4244745104474.5110/2/007077
8/3/985811.425818.154752.4048338904833.8911/1/006372
7/1/985863.256199.585810.9558739205873.9212/1/006433
6/2/985569.895921.405512.5858974405897.4420011/2/016795
5/4/985251.415664.845172.955569805569.082/1/016208
4/1/985084.105436.804992.1051074405107.443/1/015829
3/2/984696.505136.804612.1050973005097.304/2/016264
2/2/984481.004736.104474.3046939004693.905/2/016123
1/2/984270.704458.304053.9044425004442.506/1/016058
12/1/974000.004320.004000.0042243004224.307/2/015861
11/3/973802.703986.403645.8039721003972.108/1/015188
10/1/974154.604363.703366.0037537003753.709/3/014308
9/1/973942.504196.803783.1041549004154.9010/1/014559
8/1/974422.204460.203849.9039198003919.8011/1/014989
7/1/973782.704477.703782.6044055004405.5012/1/015160
6/2/973551.003836.903544.5037669003766.9020021/2/025107
5/2/973448.703695.303448.7035627003562.702/1/025039
4/1/973336.103452.603192.3034381003438.103/1/025397
3/3/973264.503475.003260.2034291003429.104/2/025041
2/3/973034.303287.803032.3032596003259.605/2/024818
1/2/972855.203041.602833.8030352003035.206/3/024382
12/2/962854.802914.602760.8028887002888.707/1/023700
11/1/962675.402845.502665.9028455002845.508/1/023712
10/1/962647.502739.302645.9026593002659.309/2/022769
9/2/962540.802672.602510.3026519002651.9010/1/023152
8/1/962492.202571.702491.0025438002543.8011/1/023320
7/1/962563.902583.702446.2024734002473.4012/2/022892
6/3/962526.802578.702524.4025614002561.4020031/2/032747
5/2/962490.702571.802455.2025428002542.802/3/032547
4/1/962495.402554.802487.1025053002505.303/3/032423
3/1/962494.702526.302400.5024859002485.904/1/032942
2/1/962472.002484.502375.1024736002473.605/2/032982
1/2/962271.402471.702271.4024701002470.106/2/033220
12/1/952259.402289.902232.5022539002253.907/1/033487
11/1/952157.902256.002154.3022428002242.808/1/033484
10/2/952205.402219.302096.1021679002167.909/1/033256
9/1/952233.402320.202163.202187002187.0010/1/033655
8/1/952222.302277.402208.2022383002238.3011/3/033745
7/3/952083.402243.602081.8022187002218.7012/1/033965
6/1/952118.302155.802081.7020839002083.9020041/2/044058
5/2/952029.602110.702015.9020922002092.202/2/044018
4/3/951915.102030.701914.1020159002015.903/1/043856
3/1/952114.502130.401893.6019226001922.604/1/043985
2/1/952037.702145.702037.7021022002102.205/3/043921
1/2/952084.102092.202017.8020213002021.306/1/043864
dax 95-04
Close
Dax-Verlauf 1995 - 2004
winners curse
Hufigkeit
0,00-0,497
0,50-0,9920
1,00-1,4934
1,50-1,9927
2,00-2,4945
2,50-2,9924
3,00-3,4913
3,50-3,9915
4,00-4,4911
4,50-4,994
5,00-5,497
5,50-5,995
6,00-6,492
6,50-6,99
7,00-7,491
7,50-7,99
8,00-8,491
8,50-8,991
9,00-9,491
9,50-9,99
10,00-10,492
10,50-10,99
11,00-11,492
11,50-11,99
12,00-12,49
12,50-12,99
13,00-13,49-
13,50-13,99
14,00-14,49
14,50-14,99
15,00-15,492
15,50-15,99
>15,993
Close
Dax-Verlauf 1997 - 2004
Close
Dax-Verlauf 1995 - 2004
winners curse
Hufigkeit
Hufigkeiten der Gebote
winners curse (2)
Hufigkeit
7
Remarks: Interdependent valuesYou want to bid on this used car ...
but you dont know the true value
65
Remarks: Interdependent values
You estimate the value somewhere between 0 and 12.000 (every value has the same probability)
The seller knows the true value, but he needs money and is willing to sell at a price of 2/3 of the value (otherwise he rejects your bid)
How much do you bid?
66
0 6.000 12.000
The expected value is 6.000
4.000
Assume you bid 4.000
If you win at this price, is this a bargain?
Remarks: Interdependent values
67
0 6.000 12.000
Assume seller accepts your bid
4.000
The (expected) value is thus only 3.000 That is, independent of your bid, if you win you must always
expect to make a loss!
3.000
Then you know that the value of the car cannot exceed 6.000, because otherwise your bid would be rejected (2/3 6.000 = 4.000)
68
Remarks: Interdependent values
Remarks: "The winners curse"UMTS-auctions
MobilCom gave its license back for which it paid 8,5 billion Euros
Road construction and tender ...Book manuscriptsOil fieldsBaseball players and ManagerseBay....
69
Remarks: "The winners curse"Bidders typically have different pieces of information
about the true value
Winning is bad news, because it means that others (other bidders or the seller) have more negative information about the value
Thus, bids need to be substantially shaded: "what is the value of the item conditional on all opponents estimating this value to be lower?"
This bid shading reduces revenues
70
Remarks: "The winners curse" What does this imply for auction design if bidders are
rational? Reduce uncertainty!
Each bidder would revise his value if he also had the information about other bidders
This information might be inferred by the competitors bids in open (but not in sealed-bid) auctions
Better information and less uncertainty allows bidders to bid more aggressively
Thus, the English auction, theoretically, yields larger revenues
71
Multiple Item Auctions
72
Outline Equilibrium of games with incomplete information Mechanism design Auctioning many similar items Revenue equivalence and optimal auctions Applications
Electricity Spectrum Mobile communications Transportation Financial securities
73
First some mathCalculating equilibrium behavior in games with
incomplete information
Warning: A little knowledge is a dangerous thing!
74
Equilibrium in first-price auction
Each of n bidders private value v drawn from distribution F Bidder's expected profit:
(v,b(v)) = (v - b(v))Pr(Win|b(v))
By the envelope theorem,
But then d/dv = Pr(Win|b(v)) = Pr(highest bid)= Pr(highest value) = F(v)n-1
75
ddv b
bv v v
=
+
=
Equilibrium in first-price auction
By the Fundamental Theorem of Calculus,
Substituting into (v,b(v)) = (v - b(v))Pr(Win|b(v)) yields
76
( ) ( ) ,v F(u) du F(u) dun 1v
n 1v
= +XZY
=XZY
00 0
b v v vPr(Win)
v F(v) F(u) du(n 1) n 1v
( ) ( ) .= = XZY
0
Example
v U on [0,1] Then F(v) = v, so
b(v) = v - v/n = v(n-1)/n The optimal bid converges to the value as n, so in
the limit the seller is able to extract the full surplus In equilibrium, the bidder bids the expected value of
the second highest value given that the bidder has the highest value
77
Bargaining: Simultaneous offers(Chatterjee & Samuelson, Operations Research 1983)
A seller and a buyer are engaged in the trade of a single item worth s to the seller and b to the buyer
Valuations are known privately, as summarized below
78
TradersValue Distributed Payoff
PrivateInfo
CommonKnowledge
Strategy(Offer)
Seller s sF on [s, s] u =P s s F, G p(s)
Buyer b bG on [ , ]b b v =b P b F, G q(b)
Traders
Value
Distributed
Payoff
Private
Info
Common Knowledge
Strategy (Offer)
Seller s
s(F on
[
s
,
s
]
u =P s
s
F, G
p(s)
Buyer b
b(G on
[
,
]
b
b
v =b P
b
F, G
q(b)
Simultaneous Offers Independent private value model: s and b are
independent random variables
Ex post efficiency: trade if and only if s < b
Game: Each player simultaneously names a price;if p q then trade occurs at the price P = (p + q)/2;if p > q then no trade (each player gets zero)
79
Simultaneous Offers Payoffs:
Seller
Buyer
where the trading price is P = (p + q)/2
80
u(p,q,s,b) =P - s if p q
0 if p > qRST
v(p,q,s,b) =b - P if p q
0 if p > qRST
Let F and G be independent uniform distributions on [0,1]
Equilibrium conditions:
81
Example
(1) s [s, s],p(s) argmaxE {u(p,q,s,b)|s,q( )}
(2) b [b,b],q(b) argmaxE {v(p,q,s,b)|b,p( )p
b
qs
}
Assume p and q are strictly increasing Let x() = p-1() and y() = q-1() Optimization in (1) can be stated as
First-order condition-y'(p)[p - s] + [1 - y(p)]/2 = 0,
since q(y(p)) = p
82
Sellers Problem
max [(p q(b)) / 2 s]dbp y(p)
1
+ XZY
Optimization in (2) can be stated as
First-order condition
x'(q)[b - q] - x(q)/2 = 0,since p(x(q)) = q
83
Buyers Problem
max [b - (p(s) q) / 2]dsq
x(q)
+XZY0
Equilibrium Equilibrium condition:
s=x(p) and b=y(q)
Equilibrium first-order conditions:
(1') -2y'(p)[p - x(p)] + [1 - y(p)] = 0,(2') 2x'(q)[y(q) - q] - x(q) = 0
84
Solving (2') for y(q) and replacing q with p yields
Substituting into (1') then yields
85
Solution
(2 ) y(p) p 12
x(p)x (p)
, so y (p)= 32
12
x(p)x (p)[x (p)]2
= +
(1 ) [x(p) -p] 3 - x(p)x (p)[x (p)]
1 p 12
x(p)x (p)
=02
LNM
OQP+
LNM
OQP
Linear Solution:x(p) = p +
with = 3/2 and = -3/8
Using (2") yieldsy(q) = 3/2 q - 1/8
Inverting these functions results inp(s) = 2/3 s + 1/4 and q(b) = 2/3 b + 1/12
86
Analytical Solution
Figure 1
87
0.0
1.0
Valuations s, b
0.8
0.6
0.4
0.2
0.0 0.2 0.4 0.6 0.8 1.0
Offers p, q
p(s) q(b)
Trade occurs if and only if
p(s) q(b), or b - s 1/4
The gains from trade must be at least 1/4 or no tradetakes place
The outcome is inefficient
88
Outcome
Mechanism designGeneral understanding of incentives in decision settings
(bargaining, auctions, exchange, public goods, )
89
A General Model(Myerson and Satterthwaite, JET 1983)
Direct Revelation Game: Bilateral Exchange with independent private value s F with positive pdf f on b G with positive pdf g on F and G are common knowledge
In the DRG, the traders report their valuations and then an outcome is selected. Given the reports (s,b), an outcome specifies a probability of trade (p) and the terms of trade (x)
90
[s, s][b,b]
DefinitionDirect MechanismA direct mechanism is a pair of outcome functions p,x, where:
p(s,b) is the probability of trade given the reports (s,b), and x(s,b) is the expected payment from the buyer to the seller
91
PayoffsEx post utilities: Seller's ex post utility:
u(s,b) = x(s,b) - sp(s,b)
Buyer's ex post utility:v(s,b) = bp(s,b) - x(s,b)
Both traders are risk neutral (quasi-linear utility)
92
PayoffsDefine:
X(s) is the seller's expected revenue given sY(b) is the buyer's expected payment given bP(s) is the seller's probability of tradeQ(b) is the buyer's probability of trade
93
X(s) x(s,b)g(b)db Y(b) x(s,b)f(s)ds
P(s) p(s,b)g(b)db Q(b) p(s,b)f(s)ds.
b
b
s
s
b
b
s
s
= XZY =XZY
= XZY =XZY
Payoffs Interim Utilities:
U(s) = X(s) - sP(s) V(b) = bQ(b) - Y(b)
The mechanism p,x is incentive compatible if for all s, b, s, and b:(IC) U(s) X(s') - sP(s') V(b) bQ(b') - Y(b')
The mechanism p,x is individually rational if for alland
(IR) U(s) 0 V(b) 0
94
s [s, s] b [b, b]
Lemma 1(Mirrlees, Myerson)
The mechanism p,x is IC if and only if P() is decreasing, Q() is increasing, and
(IC)
95
U(s) U(s) P(t)dt
V(b) V(b) Q(t)dt
s
s
b
b
= +XZY
= +XZY
Lemma 1: ProofOnly if: By definition, U(s) = X(s) - sP(s) and
U(s') = X(s') - s'P(s'). This and (IC) imply
U(s) X(s') - sP(s') = U(s') + (s' - s)P(s'), andU(s') X(s) - s'P(s) = U(s) + (s - s')P(s)
Putting these inequalities together yields
(s' - s)P(s) U(s) - U(s') (s' - s)P(s')
96
Lemma 1: Proof
Taking s' > s implies that P() is decreasing Dividing by (s' - s) and letting s' s, then yields
dU(s)/ds = -P(s) Integrating produces (IC') The same is true for the buyer
97
Lemma 1: ProofIf: To prove (IC) for the seller, note that it suffices to show
that
s[P(s) - P(s')] + [X(s') - X(s)] 0 for all s, s
Substituting for X(s') and X(s) using (IC') and the definition of U(s) yields
98
[s, s]
X(s) sP(s) U(s) P(t)dts
s
= + +XZY .
Lemma 1: ProofIf: Then it suffices to show for every s,s that
which holds because P() is decreasing
The proof for the buyer is similar
99
[s, s]
0 s[P(s) P(s )] + s P(s ) + P(t)dt sP(s) P(t)dt
(s s)P(s ) P(t)dt [P(t) P(s )]dt ,
s
s
s
s
s
s
s
s
XZY XZY
= +XZY = XZY
Lemma 2
An incentive compatible mechanism p,x is individually rational if and only if
(IR') and V( ) 0
Proof Clearly, (IR') is necessary for p,x to be IR
By Lemma 1, U() is decreasing; hence, (IR') is sufficient aswell
100
U(s) 0 b_
eq \O(b,_)
An incentive-compatible, individually rational mechanism p,x satisfies
101
Theorem:Characterization of IC & IR mechanisms
(*) U(s) + V(b)
b 1 G(b)g(b)
s F(s)f(s)
p(s,b)f(s)g(b)dsdb 0b s
sb
=
LNM
OQP
XZY
z .
Proof Using (IC') and the definition of U(s) yields
102
Theorem:Characterization of IC & IR mechanisms
X(s) sP(s) U(s) P(t)dt.s
s= + + z
Taking the expectation with respect to s (and substituting in the definitions of X(s) and P(s)) shows that
103
b
b
s
s
b
b
s
s
b
b
s
s
x(s,b)f(s)g(b)dsdb
U(s) sp(s,b)f(s)g(b)dsdb
p(s,b)F(s)g(b)dsdb.
XZY
XZY =
+XZYXZY
+XZYXZY
Theorem:Characterization of IC & IR mechanisms
The third term in the right hand side follows, since
104
Theorem:Characterization of IC & IR mechanisms
s
s
s
s
s
s
s
t
s
s
p(t,b)f(s)dtds
p(t,b)f(s)dsdt p(s,b)F(s)ds .
XZY
XZY
= XZYXZY =
XZY
Preceding analogously for the buyer yields
Equating the right-hand sides of the last two equations andapplying (IR') completes the proof
105
Theorem:Characterization of IC & IR mechanisms
b
b
s
s
b
b
s
s
b
b
s
s
x(s,b)f(s)g(b)dsdb
= -V(b) bp(s,b)f(s)g(b)dsdb
- p(s,b)f(s)[1- G(b)]dsdb.
XZY
XZY
+XZYXZY
XZY
XZY
Corollary:Impossibility of efficient trade
106
If it is not common knowledge that gains exist (the supports of the traders' valuations have non-empty intersection), then no incentive-compatible, individually rational trading mechanism can be ex-post efficient
A mechanism is ex-post efficient if and only if tradeoccurs whenever s b:
107
Proof
p(s,b)1 if s b0 if s b.
=>
RST
Proof To prove that ex-post efficiency cannot be attained, it
suffices to show that the inequality (*) in the Corollary fails when evaluated at this p(s,b). Hence,
108
b s
min{b,s}b
b 1 G(b)g(b)
s F(s)f(s)
f(s)g(b)dsdbz LNM OQPXZY
Proof
109
min{b,s} min{b,s}b b
b bs s
b b
b b
b b
b s
b
[bg(b)+G(b) 1]f(s)dsdb [sf(s)+F(s)]dsg(b)db
[bg(b)+G(b) 1]F(b)db min{bF(b),s}g(b)db (by parts)
[1 G(b)]F(b)db (b s)g(b)db
[1 G(b)]F(b)db
=
=
= +
= b b
s
s
b
[1 G(b)]db (by parts)
[1 G(t)]F(t)dt 0, since b s.
+
= <
Proof
The second term in the second line follows, since by integrating by parts
Since ex-post efficiency is unattainable, we need a weaker efficiency criterion with which to measure a mechanism's performance
110
[sf(s) F(s)]ds xF(x)s
x
+ =XZY .
What IC & IR mechanism maximizes expected gains from trade?
Ex ante efficient mechanism maximizes the expected gains from trade:
subject to IC & IR
111
U(s)f(s)ds V(b)g(b)dbs
s
b
bXZY +
XZY
Optimal trading game
Myerson and Satterthwaite show that the ex ante efficient decision rule (probability of trade) is:
where
and is chosen so that U( ) = V( ) = 0
112
p s,bif c(s, ) d(b, )if c(s, ) d(b, )
( ) =>
RST10
c( ,s) = s + F(s)f(s)
d( ,b) = b 1 G bg b
( )( )
s b
Remarks
If = 0, then p is ex post efficient (all the weight on the objective function)
If = 1, p maximizes the expression in (*); a constrained maximization
The ex ante efficient trading rule has the property that, given the reports, trade either occurs with probability one or not at all
113
Example
Valuations are uniformly distributed on [0,1] Ex ante efficient mechanism: linear equilibrium in
which trade occurs if and only if the gains from trade are at least 1/4 (Chatterjee & Samuelson)
If the traders cannot commit to walking away from gains from trade, then they would be unable to implement this mechanism
So long as it is not common knowledge that gains exist, the traders will, with positive probability, make incompatible demands in situations where gains from trade exist
114
Accomplishments
Characterization of the set of all BE of all bargaining games in which the players' strategies map their private valuations into a probability of trade and a payment from buyer to seller
Proof that ex post efficiency is unattainable if it is uncertain that gains from trade exist
Determination of the set of ex ante efficient mechanisms
Proof that ex ante efficiency is incompatible with sequential rationality
115
Dissolving a Partnership(Cramton, Gibbons, and Klemperer, 1987)
n traders. Each trader i {1,...,n} owns a shareri 0 of the asset, where r1 + ... + rn = 1
As in MS, player i's valuation for the entire good is vi The utility from owning a share ri is rivi Private values, vis are iid F() on A partnership (r,F) is fully described by the vector of
ownership rights r = {r1,...,rn} and the traders' beliefs F about valuations
116
[v, v]
Dissolving a PartnershipMS Case: n = 2 and r = {1,0} There does not exist a BE of the trading game such
that:(1) is (interim) individually rational and(2) is ex post efficient
CGK Case: If the ownership shares are not too unequally
distributed, then it is possible to satisfy both (1) and (2), (satisfying IC, IR, EE and BB)
117
Dissolving a Partnership
A partnership (r,F) can be dissolved efficiently if there exists a Bayesian Equilibrium of a Bayesian trading game such that is interim individually rational and ex post efficient
118
Theorem
The partnership (r,F) can be dissolved efficiently if and only if
(*)
where vi* solves F(vi)n-1 = ri and G(u) = F(u)n-1
119
[1 F(u)]udG(u) F(u)udG(u)v
v
v
v
i 1
n
i*
i*
LNMOQP z z= 0
Examples F(vi) = vi n=2, then (*)
n=3, then
(*)
120
r 3 4i3 2
i 1
3
=
=1
2
2 23 0 1
0.21 0.79
Proposition
For any distribution F, the one-owner partnershipr = {1,0,0,...,0} cannot be dissolved efficiently
The one-owner partnership can be interpreted as an auction Ex post efficiency is unattainable because the seller's value
v1 is private information: the seller finds it in her best interest to set a reserve price above her value v1
An optimal auction maximizes the seller's expected revenue over the set of feasible (ex post inefficient) mechanisms
121
Theorem If a partnership (r,F) can be dissolved efficiently, then
the unique symmetric equilibrium of the followingbidding game is interim individually rational andachieves ex-post efficiency: given an arbitrary mini-mum bid b,
each player receives a side-payment, independent of thebidding,
the players choose bids bi [b,) the good goes to the highest bidder each bidder i pays
122
p (b , ,b ) b 1n 1
bi 1 n i jj i
n =
c (r , , r ) udG(u) udG(u).i 1 nv
v1n
v
v
j 1
ni*
j*
= XZY XZY=
Auctioning manysimilar items
Lawrence M. Ausubel, Peter Cramton,Marek Pycia, Marzena Rostek, and Marek Weretka (2014)
"Demand Reduction and Inefficiency in Multi-Unit Auctions,"Review of Economic Studies, 81:4, 1366-1400
123
http://www.cramton.umd.edu/papers2010-2014/acprw-demand-reduction.pdf
124
Examples of auctioningsimilar items Treasury bills Stock repurchases and IPOs Telecommunications spectrum Electric power Emission allowances
125
Ways to auction many similar items Sealed-bid: bidders submit demand schedules
Pay-as-bid auction (traditional Treasury practice) Uniform-price auction (Milton Friedman 1959) Vickrey auction (William Vickrey 1961)
Bidder 1 Bidder 2
+
AggregateDemand
=
P
Q1 Q2 Q
P P
126
Pay-as-bid Auction:All bids above P0 win and pay bid
Price
Quantity
Supply
Demand(Bids)
QS
P0Clearing price
127
Uniform-Price Auction:All bids above P0 win and pay P0
Price
Quantity
Supply
Demand(Bids)
QS
P0Clearing price
128
Vickrey Auction:All bids above P0 win and pay opportunity cost
Price
Quantity
Residual SupplyQS ji Qj(p)
DemandQi(p)
Qi(p0)
p0
129
Vickrey Auction: m Discrete Items
Allocate m items efficiently: m highest marginal values
Winning bidder pays kth highest losing bid of others on kth item won
Payment = social opportunity cost of items won
3 bidders, 3 itemsmarginal values
A B C
1st 10 8 4
2nd 6 7 2
3rd 3 5 1
5 6
4
130
Payment rule affects behavior
Price
Quantity
Residual SupplyQS ji Qj(p)
DemandQi(p)
Qi(p0)
p0
Pay-as-bid
Uniform-Price
Vickrey
131
Optimal bids by payment rule
Price
Quantity
True demandQi(p)
p0
Uniform-Price
Vickrey
132
More ways to auction many similar items
Ascending-bid: Clock indicates price; bidders submit quantity demanded at each price until no excess demand
Clock auction (single price) Clock auction with Vickrey prices (Ausubel 1997)
133
Ascending clock:All bids at P0 win and pay P0
Price
Quantity
Supply
Demand
QS
P0
ClockExcessDemand
134
Ascending clock with Vickrey pricesAll bids at P0 win and pay price at which clinched
Price
Quantity
Residual SupplyQS ji Qj(p)
DemandQi(p)
Qi(p0)
p0
Clock
Excess Demand
135
More ways to auction many similar items Ascending-bid
Simultaneous ascending auction (FCC spectrum) Sequential
Sequence of English auctions (auction house) Sequence of Dutch auctions (fish, flowers)
Optimal auction Maskin & Riley 1989
136
Research ProgramHow do standard auctions compare? Efficiency
FCC: those with highest values win Revenue maximization
Treasury: sell debt at least cost
137
Efficiency(not pure common value; capacities differ)
Uniform-price and standard ascending-bid Inefficient due to demand reduction
Pay-as-bid Inefficient due to different shading
Vickrey Efficient in private value setting Strategically simple: dominant strategy to bid true demand Inefficient with affiliated information
Dynamic Vickrey (Ausubel 1997) Same as Vickrey with private values Efficient with affiliated information
138
Inefficiency TheoremIn any equilibrium of uniform-price auction, with
positive probability objects are won by bidders other than those with highest values
Winning bidder influences price with positive probability Creates incentive to shade bid Incentive to shade increases with additional units Differential shading implies inefficiency
139
Inefficiency from differential shading
P0
Large Bidder Small Bidder
Q1 Q2
mv1
mv2
Large bidder makes room for smaller rival
D1 D2b1 b2
140
Vickrey inefficient with affiliation Winners Curse in single-item auctions
Winning is bad news about value Winners Curse in multi-unit auctions
Winning more is worse news about value Must bid less for larger quantity Differential shading creates inefficiency in Vickrey
141
What about seller revenues?
Price
Quantity
Residual SupplyQS ji Qj(p)
DemandQi(p)
Qi(p0)
p0
Pay-as-bid
Uniform-Price
Vickrey
Exercise 2 bidders (L and R), 2 identical items L has a value of $100 for 1 and $200 for both R has a value of $90 for 1 and $180 for both Uniform-price auction
Submit bid for each item Highest 2 bids get items 3rd highest bid determines price paid
Ascending clock auction Price starts at 0 and increases in small increments Bidders express how many they want at current price Bidders can only lower quantity as price rises Auction ends when no excess demand (i.e. just two
demanded); winners pay clock price
142
143
Uniform price may perform poorly Independent private values uniform on [0,1] 2 bidders, 2 units; L wants 2; S wants 1 Uniform-price: unique equilibrium
S bids value L bids value for first and 0 for second Zero revenue; poor efficiency
Vickrey price = v(2) on one unit, zero on other
144
Standard ascending-bid may be worse 2 bidders, 2 units; L wants 2; S wants 2 Uniform-price: two equilibria
Poor equilibrium: both L and S bid value for 1 Zero revenue; poor efficiency
Good equilibrium: both L and S bid value for 2 Get v(2) for each (max revenue) and efficient
Standard ascending-bid: unique equilibrium Both L and S bid value for 1
Ss demand reduction forces L to reduce demand Zero revenue; poor efficiency
145
Efficient auctions tend to yield high revenues
Theorem. With flat demands drawn independently from the same regular distribution, sellers revenue is maximized by awarding good to those with highest values
Generalizes to non-private-value model with independent signals:vi = u(si,s-i)
Award good to those with highest signals if downward sloping MR and symmetry
146
Downward-sloping demand:pi(qi) = vi gi(qi)
Theorem. If intercept drawn independently from the same distribution, sellers revenue is maximized by
awarding good to those with highest values if constant hazard rate
shifting quantity toward high demanders if increasing hazard rate
Note: uniform-price shifts quantity toward lowdemanders
147
But uniform price has advantages
Participation Encourages participation by small bidders
(since quantity is shifted toward them and less strategy) May stimulate competition
Post-bid competition More diverse set of winners may stimulate competition
in post-auction market
148
Bidding behavior in electricity markets Marginal cost bidding is a useful benchmark, but
not a norm of behavior Profit maximization is an appropriate norm of
behavior in markets Profit maximization should be expected and
encouraged Market rules should be based on this norm
149
Uniform-price auction:All bids below p0 win and get paid p0
Price
Quantity
Supply(as bid)
Demand
q0
p0(clearing price)
150
Residual demand removessupply of other bidders
Demand Supplyof others
Residual demand
=
q qi qi
p p
q0
Supplyfirm i
Supply
p
p0
q q
151
Price
Quantity
Di(p) = D(p) ji Sj(p)
As-bid supply Si(p)
Residual demand curve
qi
p0
Residual demand
152
Price
Quantity
Di
As-bid supplySi = MCi
Bidding strategywith perfect competition
qi
p0 Residual demand
Loss
153
Incentive to bid above marginal cost:tradeoff higher price with reduced quantity
p
q
Residual demandDi
As-bid supplySi
qi
p0
MCi
Loss
Gain
154
Optimal bid balances marginal gain and loss
p
q
Residual demandDi
As-bid supplySi
qi
p0 MCi
Loss
Gain
155
p
0
Still bid above marginal cost when others bid marginal cost
q-i
p-i
10GW
Otherbidders
p
0 qi
p0
1GW
Firm i
D
S-i=MC-i
Di
MCi
Si
p0
156
Residual demand response reduces incentive to inflate bids
p
q
Residual demandDi
As-bid supplySi
qi
p0
MCi
LossGain
157
p
0
Residual demand is steeper for large bidders
ql
p0
10GW
Large bidder p
0 qs
p0
1GW
Small bidder
Dl
Sl
Ds
Ss
158
p
0
Large bidder makes roomfor its smaller rivals
ql
p0
10GW
Large bidder p
0 qs
p0
1GW
Small bidder
Dl
Sl
Ds
Ss
MCl
MCs
159
Economic vs. Physical Withholding
p
q
Di
Si
qiqi
p0MCiMCi
qe
p
q
Di
SiSi
qiqi
p0MCiMCi
qe
160
Forward contracts mitigate incentive to bid above marginal cost
p
q
Residual demandDi
Si no forward
qi
p MCi
Si with forward
p0
qiqF
qS
Forward sale
Revenue equivalence and optimal auction in general model
Lawrence M. Ausubel and Peter Cramton (2001) The Optimality of Being Efficient, Working Paper,
University of Maryland.
161
http://www.cramton.umd.edu/papers1995-1999/98wp-optimality-of-being-efficient.pdf
162
Identical items model Seller has quantity 1 of divisible good (value = 0) n bidders; i can consume qi [0,i]
q = (q1,,qn) Q = {q | qi [0,i] & iqi 1} ti is is type; t = (t1,,tn); ti ~ Fi w/ pos. density fi Types are independent Marginal value vi(t,qi) is payoff if gets qi and pays xi:
v t y dy xi iqi ( , ) z0
163
Identical items model (cont.)
Marginal value vi(t,qi) satisfies: Value monotonicity
non-negative increasing in ti weakly increasing in tj weakly decreasing in qi
Value regularity: for all i, j, qi, qj, ti, ti > ti,vi(ti,ti,qi) > vj(ti,ti,qj) vi(ti,ti,qi) > vj(ti,ti,qj)
164
Identical items model (cont.)
Bidder is marginal revenue:marginal revenue seller gets from awarding additional quantity to bidder i
MR t q v t q F tf t
v t qti i i i
i i
i i
i i
i( , ) ( , ) ( )
( )( , )
=
1
165
Revenue Equivalence
Theorem 1. In any equilibrium of any auction game in which the lowest-type bidders receive an expected payoff of zero, the sellers expected revenue equals
E MR t y dyt iq t
i
ni ( , )( )
01z
=
LNM
OQP
166
Optimal Auction MR monotonicity
increasing in ti weakly increasing in tj weakly decreasing in qi
MR regularity: for all i, j, qi, qj, ti, ti > ti,MRi(t,qi) > MRj(t,qj) MRi(ti,ti,qi) > MRj(ti,ti,qj)
Theorem 2. Suppose MR is monotone and regular. Sellers revenue is maximized by awarding the good to those with the highest marginal revenues, until the good is exhausted or marginal revenue becomes negative.
167
Optimal Auction is Inefficient
Assign goods to wrong parties High MR does not mean high value
Assign too little of the good MR turns negative before values do
168
Three Seller Programs
1. Unconstrained optimal auction(standard auction literature)Select assignment rule and pricing rule to
max E[Seller Revenue]s.t. Incentive Compatibility
Individual Rationality
169
Three Seller Programs
2. Resale-constrained optimal auction(Coase Theorem critique)Select assignment rule and pricing rule to
max E[Seller Revenue]s.t. Incentive Compatibility
Individual RationalityEfficient resale among bidders
170
Three Seller Programs
3. Efficiency-constrained optimal auction(Coase Conjecture critique)Select assignment rule and pricing rule to
max E[Seller Revenue]s.t. Incentive Compatibility
Individual RationalityEfficient resale among bidders
and seller
171
1. Unconstrained optimal auction
Select assignment rule to
All feasible assignment rules.}
q t
E MR t y dy
Qq t Q t i
q t
i
ni
( )
max ( , )
{( )
( )
=zLNM
OQP
=
01
172
1. Unconstrained optimal auction(two bidders)
quantity
price
0
d2 d1 MR1 MR2
D
q
S
1
p2
MR
173
3. Efficiency-constrained optimal auction
Select assignment rule to
Ex post efficient assignment rules.}
q t
E MR t y dy
Q
R
q t Qt i
q t
i
n
R
R
i
( )
max ( , )
{
( )
( )
=zLNM
OQP
=
01
174
3. Efficiency-constrained optimal auction (two bidders)
quantity
price
0
d2 d1 MR1 MR2
D
q=1
S p1
175
2. Resale-constrained optimal auction
Select assignment rule to
Resale - efficient assignment rules.}
q t
E MR t y dy
Q
R
q t Qt i
q t
i
n
R
R
i
( )
max ( , )
{
( )
( )
=zLNM
OQP
=
01
176
2. Resale-constrained optimal auction(two bidders)
quantity
price
0
d2 d1 MR1 MR2
D MRR
qR
S
1
p1
177
Theorem. In the two-stage game (auction followed by perfect resale), the seller can do no better than the resale-constrained optimal auction.
Proof. Let a(t) denote the probability measure on allocations at end of resale round, given reports t. Observe that, viewed as a static mechanism, a(t) must satisfy IC & IR. In addition, a(t) must be resale-efficient.
178
Can we obtain the upper bound on revenue?resale process is coalitionally-rational against
individual bidders if bidder i obtains no more surplus si than i brings to the table: si v(N | q,t) v(N ~ i | q,t).
That is, each bidder receives no more than 100% of the gains from trade it brings to the table.
179
Vickrey auction with reserve pricingSeller sets monotonic aggregate quantity that will
be assigned to the bidders, an efficient assignment q*(t) of this aggregate quantity, and the payments x*(t) to be made to the seller as a function of the reports t where
Bidders simultaneously and independently report their types t to the seller.
{ }
* ( )*
0
*
( ) ( ( , ), , ) , where
( , ) inf | ( , ) .
i
i
q t
i i i i i
i i i i i it
x t v t t y t y dy
t t y t q t t y
=
=
180
Can we attain the upper bound on revenue?Theorem 5 (Ausubel and Cramton 1999). Consider
the two-stage game consisting of the Vickrey auction with reserve pricing followed by a resale process that is coalitionally-rational against individual bidders. Given any monotonic aggregate assignment rule , sincere bidding followed by no resale is an ex post equilibrium of the two-stage game.
ApplicationsElectricity market design
Climate policySpectrum auction design
Future of mobile communicationsFuture of transportation
Future of financial markets181
Electricity
182
Goals of electricity markets
Short-run efficiency Least-cost operation of existing resources
Long-run efficiency Right quantity and mix of resources
183
Key idea: open access
Challenges of electricity markets
Must balance supply and demandat every instantat every location
Physical constraints of network Absence of demand response Climate policy
184
Three Markets Short term (5 to 60 minutes)
Spot energy market Medium term (1 month to 3 years)
Bilateral contracts Forward energy market
Long term (4 to 20 years) Capacity market (thermal system) Firm energy market (hydro system)
Address risk, market power, and investment
185
Long-term market:Buy enough in advance
186
Product
What is load buying? Energy during scarcity period (capacity)
Enhance substitution Technology neutral where possible Separate zones only as needed in response to binding
constraints Long-term commitment for new resources to reduce risk
187
Pay for Performance
Strong performance incentives Obligation to supply during scarcity events
Deviations settled at price > $5000/MWh Penalties for underperformance Rewards for overperformance
Tend to be too weak in practice, leading to Contract defaults Unreliable resources
But not in best markets: ISO New England, PJM
188
David MacKay, Peter Cramton, Axel Ockenfels and Steven Stoft), Nature, 526, 315-316, 15 October 2015.
Symposium: International Climate NegotiationsCramton, Ockenfels, Stoft (eds.), Gollier, Stiglitz, Tirole, WeitzmanEconomics of Energy & Environmental Policy, 4:2, September 2015
Price CarbonI Will If You WillMacKay, Cramton, Ockenfels & StoftNature, 15 October 2015
Global Carbon PricingWe Will If You WillCramton, MacKay, Ockenfels & StoftMIT Press, under review, 2016carbon-price.com
190
http://www.cramton.umd.edu/papers2015-2019/eeep-symposium-international-climate-negotiations.pdfhttp://www.cramton.umd.edu/papers2015-2019/mackay-cramton-ockenfels-stoft-price-carbon.pdfhttp://www.cramton.umd.edu/papers2015-2019/global-carbon-pricing-cramton-mackay-okenfels-stoft.pdfhttp://www.cramton.umd.edu/papers2015-2019/mackay-cramton-ockenfels-stoft-price-carbon.pdfhttp://www.cramton.umd.edu/papers2015-2019/global-carbon-pricing-cramton-mackay-okenfels-stoft.pdf
191
Consensus Aspiration: 2C goal
192
IPCC, SYR Figure SPM.10
1.5
How to bridge gulf between goal and intentions?
Paris Agreement
193
Until 2020max political power
no excusesnational policy
Until 2030moderate power
some excusesnational plans
Until 2100no power or blame
many excusesglobal aspiration
Abat
emen
t effo
rt
No down-payment
No payments first 15 years
Non-binding jumbo
payments
Economics:Price carbon
194
Direct Efficient Transparent Promotes international cooperation
195
Remarkably, price never appears in 31 page COP21 Final Agreement
Treaty Design:Promoting cooperation in international negotiations
196
197
Individual commitments(intended nationally determined contributions)
cannot promote cooperation
Individual commitments cannot promote cooperation 10 players; individual endowment = $10 Each $ pledged will be doubled and distributed evenly
to all players Voluntary pledges are enforced Result: Zero cooperation, all pledge $0
Pledge $10$0
Uniqueequilibrium No cooperation
198
Dynamics of individual commitments: Upward spiral of ambition?
History: Japan, Russia, Canada, and New Zealand left the Kyoto agreement
Ostrom (2010), based on hundreds of field studies: insufficient reciprocity leads to a downward cascade
Supported by theory and laboratory experiments
199
Common commitment: I will if you will
200
Trump: I wont cause you wont
201
I will if you will promotes cooperation
10 players; Individual endowment = $10 Each $ pledged will be doubled and distributed evenly
to all players Pledge is commitment to reciprocally match the
minimum pledge of others Voluntary pledges are enforced Result: Full cooperation, all pledge $10
Pledge $10$0
Uniqueequilibrium
Full cooperation
202
Price is focalcommon commitment
203
Direct, efficient, common intensity of effort Consistent with tax or cap & trade
(flexible at country level) Consensus that price should be uniform reduces
dimensionality problem:
PCountry = Pglobal
(No such consensus exists for quantity commitment)
204
Price commitment reduces risk
Countries keep carbon revenues
Eliminates the risk of needing to buy credits
205
But are quantity commitments equivalent?
China, you will be safe, if you accept a Business-as-Usual target for 2008 2012.
Jeffery Frankel, 1998
Business as Usual means what experts think 1999 US Dept. of Energy: 7.5 Gt of CO2 Reality in 2008 2012: 36.6 Gt of CO2
206
Cap targets P = $30, but then P $45
Chinas unexpected costs > $1 trillion $817 B Payments to US, EU, India ??
207
$0
$15
$30
$45
0 10 20 30 40
Carb
on P
rice
EmissionsDOE prediction Actual
Unexpectedabatement cost under Global Cap
$225 B
$817 BUnexpected Trading Cost
underCap & Trade
Gt
Prediction-Error Trading Costs for China, 2008 2012
Carbon Pricing: P = $30
Chinas unexpected costs = $88 B Payments clean up Chinas pollution
208
$0
$15
$30
$45
0 10 20 30 40
Carb
on P
rice
EmissionsDOE prediction Actual
Unexpectedabatement cost under
Global Pricing$88 B
Gt
Prediction-Error Pricing Costs for China, 2008 2012
Sharing the burden
209
Use Green Fund to maximize abatement As before, reduce dimensionality
Carbon price = intensity of cooperation Generosity parameter = intensity of Green Fund
Last resort enforcement with trade sanctions
210
Designing the Green Fund
Excess emissions = deviation from world per capital average (+ for US, - for India)
This addresses differentiated responsibilities Rich, high-emission countries pay into fund Poor, low-emission countries receive from fund
211
Payment into Green Fund =
Generosity parameter
Excess emissions
Global carbon price
Maximizing treaty strength
If G is high, rich countries will want P* low If G is low, poor countries will want P* low Some moderate G maximizes the P* that a super-
majority will accept
212
A mechanism for the willing (G20?) Countries with little stake in the Green Fund (near
average emissions) first determine G G will be determined so that both rich and poor countries
benefit from an effective agreement
Then countries vote for P*; low price wins No country i commits to a P* > Pi, so any country could
protect itself by naming a low Pi if G were unacceptable
Mechanism promotes a strong agreement
213
China USIndia
7.2 440.1
QatarRwanda
1.6 17.2Emissions per capita (tons/year)
Summary
Keys to a strong climate treaty I will if you will (common commitment) Two parameters
Carbon price (common intensity of effort) Green fund intensity (addresses asymmetries)
Further research Equilibrium simulations using standard climate models
to identify best climate club Develop details of treaty (e.g. voting mechanism)
214
215
Price CarbonI will if you will
Spectrum
216
Spectrum auctions
Many items, heterogeneous but similar Competing technologies and business plans Complex structure of substitutes and complements
Government objective: Efficiency Make best use of scarce spectrum Address competition issues in downstream market
217
Key design issues
Establish term to promote investment Enhance substitution
Product design Auction design
Encourage price discovery Dynamic price process to focus valuation efforts
Encourage truthful bidding Pricing rule Activity rule
218
Simultaneous ascending auction
219
Prepare
220
Italy 4G Auction, September 2010470 rounds, 22 days, 3.95B Auction conducted on-site with pen and paper Auction procedures failed in first day No activity rule
221
Thailand 3G Auction, October 2012 3 incumbents bid 3 nearly identical licenses Auction ends at reserve price + 2.8%
222
223
Conflict and cooperation in Germanys spectrum auctionsIn Germany (1999) 10 spectrum blocks were simultaneously sold:
New bid for a particular block must exceed prior bid by at least 10 percent
Auction ends when no bidder is willing to bid higher on any block
What happened?
224
The auction begins 1 2 3 4 5 6 7 8 9 10
1
2
3
225
The offer of Mannesmann (=M)1 2 3 4 5 6 7 8 9 10
1 36.4M
36.4M
36.4M
36.4M
36.4M
40M
40M
40M
40M
56M
2
3
226
T-Mobile (=T) accepted1 2 3 4 5 6 7 8 9 10
1 36.4M
36.4M
36.4M
36.4M
36.4M
40M
40M
40M
40M
56M
2 40T
40T
40T
40T
40T
- - - - -
3
227
Perfect cooperation1 2 3 4 5 6 7 8 9 10
1 36.4M
36.4M
36.4M
36.4M
36.4M
40M
40M
40M
40M
56M
2 40T
40T
40T
40T
40T
- - - - -
3 - - - - - - - - - -
Combinatorial Clock Auction
228
Combinatorial clock auction
Auctioneer names prices; bidder names package
Price increased if excess demand Process repeated until no excess demand
Supplementary bids Improve clock bids Bid on other relevant packages
Optimization to determine assignment/prices
No exposure problem (package auction) Second pricing to encourage truthful bidding Activity rule to promote price discovery
229
CCA SucksLook at Switzerland!
230
No:Rules were poor
Bidding likely poorSetting was difficult
Pricing rule
231
Bidder-optimal core pricing
Minimize payments subject to core constraints Core = assignment and payments
such that Efficient: Value maximizing assignment Unblocked: No subset of bidders offered seller a better
deal
232
Optimization
Core point that minimizes payments readily calculated
Solve Winner Determination Problem Find Vickrey prices Constraint generation method
(Day and Raghavan 2007) Find most violated core constraint and add it Continue until no violation
Tie-breaking rule for prices is important Minimize distance from Vickrey prices
233
5 bidder example with bids on {A,B}
b1{A} = 28
b2{B} = 20
b3{AB} = 32
b4{A} = 14
b5{B} = 12
Winners
Vickrey prices:p1= 14
p2= 12
234
The Core
b4{A} = 14b3{AB} = 32
b5{B} = 12
b1{A} = 28
b2{B} = 20
Bidder 2Payment
Bidder 1Payment
14
12
3228
20
The Core
Efficient outcome
235
The Core
b4{A} = 14b3{AB} = 32
b5{B} = 12
b1{A} = 28
b2{B} = 20
Bidder 2Payment
Bidder 1Payment
Vickrey prices
14
12
3228
20
Vickrey prices: How much can each winners bid be reduced holding others fixed?
Problem: Bidder 3 can offer seller more (32 > 26)!
236
The Core
b4{A} = 14b3{AB} = 32
b5{B} = 12
b1{A} = 28
b2{B} = 20
Bidder 2Payment
Bidder 1Payment
Vickrey prices
14
12
3228
20
Bidder-optimal core prices: Jointly reduce winning bids as much as possible
Problem: bidder-optimal core prices are not unique!
237
Uniquecore prices
b4{A} = 14b3{AB} = 32
b5{B} = 12
b1{A} = 28
b2{B} = 20
Bidder 2Payment
Bidder 1Payment
Vickrey prices
14
12
3228
20
Core point closest to Vickrey prices
17
15
Each pays equal share above Vickrey
238
Activity rule
239
Clock stage performs wellProposition: With revealed preference w.r.t. final round
If clock stage ends with no excess supply,final assignment = clock assignment
Supplementary bids cant change assignment;but can change prices
May destroy incentive for truthful bidding in supplementary round
Supplementary round still needed to determine competitive prices
Possible solutions Do not reveal demand at end of clock stage; possibility of excess
supply motivates more truthful bidding (Canada 700 MHz) Do not impose final price cap (UK 4G)
240
Summary:CCA is an important tool Eliminates exposure Reduces gaming Enhances substitution Allows auction to determine band plan, technology Readily customized to a variety of settings Many other applications
241
242
Broadcast Incentive Auction
29 March 2016
Motivation
Year
Value per MHz
1985 1990 1995 2000 2005 2010 2015
Value of over-the-air broadcast TV
Value of mobile broadband
Gains from trade
243
What is the Incentive Auction?
The worlds most complicated auction!
Goal: clear broadcast spectrum, and repurpose for wireless use Consumer demand for broadcast TV has waned, while
demand for mobile broadband keeps increasing Spectrum in use for broadcast television is very
desirable Excellent propagation characteristics
Convert a set of 6MHz channels into 5+5MHz paired wireless blocks
Ex: 21 channels = 126MHz of spectrum = 10 paired blocks + guard bands
What is the Incentive Auction? Auction has been in the works for a long time
Part of the FCCs National Broadband Plan released in 2010 Initiated by Spectrum Act legislation passed in 2012 Initial design released in 2014; final design released June 2015 Began 29 March 2016; ends late 2016 or early 2017 Stage 1, reverse auction ended 29 June 2016; revenue requirement
$88 billion
Auction involves many novel challenges Combines both a reverse and a forward auction into a single
process Clearing television spectrum is a hard problem Interplay between forward and reverse auctions introduces
additional complexity
Auction summary Oct 16: FCC announces opening price for each station Mar 29: Station decides whether to accept opening price (participate) Apr 29: FCC optimization (min impairment) to determine clearing target
Largest target with impairment < equivalent of 1 nationwide block (10% at 126MHz)
Stage 1: initial clearing target (126MHz, 10 blocks) May 31-Jun 29: Reverse auction to determine clearing cost ($88 billion) Aug 16-30: Forward auction to determine auction proceeds ($23 billion)
Stage 2: reduce clearing target (114MHz, 9 blocks) Sep 10-Oct 11: Reverse continues with lower target (need fewer volunteers) Forward continues with fewer blocks; End if proceeds > cost
Stage End: FCC optimization to determine channel assignments
Auction progress
Quantity(5+5 MHz blocks)
Price (billion $/block)
Stage 1
Stage 2(example)
Stage 3(example)
7 8 9 10
247
88
23
54
303337
Reverse clearing cost
Forward auction proceeds
8.8
2.3Done!
Keys features of good design
Simple expression of supply and demand Station: at this lower price are you still willing to clear? Carrier: at this higher price how many blocks do you
want?
Monotonicity Reverse: prices only go down (excess supply) Forward: prices only go up (excess demand)
Activity rule Reverse: station exit is irreversible Forward: 95% activity requirement
Reverse auction(broadcasters: supply side) Identifies prices and stations to be cleared to achieve
target
There are many different ways for broadcasters to participate
Relinquish their broadcast license entirely Move to a lower band (change from UHF to VHF) Share a channel and facilities with another broadcaster
Decision to participate is voluntary Any broadcaster that does not participate can continue
broadcasting with their current coverage, but channel may change
Forward auction(carriers: demand side) Ascending clock, rather than simultaneous multiple round
Issue: participation in reverse auction determines licenses on offer
Nonparticipating broadcasters must be assigned a channel FCC needs to impair some licenses if too many nonparticipants
Issue: closing requires forward auction to generate enough revenue
Must pay for reverse auction payments + relocation costs + FCC costs
If closing conditions are not met, FCC holds an extended round to increase bidding in top-40 markets, or continues process with a lower clearing target
What is Repurposing broadcast spectrum?CURRENT CHANNEL ALLOCATIONS (sample large market) (29 stations total; 15 above channel 30)
15 Occupied Channels
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV
High V's Ch 7-13 UHF Broadcast UHF Broadcast
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV
High V's Ch 7-13 UHF Broadcast UHF Broadcast
POST AUCTION ALLOCATIONS
TV TV TV TV TV TV TV TV TV
Available for mobile broadband6 stations from Ch. 31 51 moved into Ch. 14 - 30
VHF unchanged
REQUIRED AUCTION VOLUNTEERS: (9 stations from channels 31-51 have no available channel)
Repack Stations
Sheet1
High V's Ch 7-13UHF BroadcastUHF Broadcast
789101112131415161718192021222324252627282930313233343536373839404142434445464748495051
TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV
Sheet2
Sheet3
Sheet1
High V's Ch 7-13UHF BroadcastUHF Broadcast
789101112131415161718192021222324252627282930313233343536373839404142434445464748495051
TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV
High V's Ch 7-13UHF BroadcastUHF Broadcast
789101112131415161718192021222324252627282930313233343536373839404142434445464748495051
TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV
TVTVTVTVTVTVTVTVTV
6 added
9 without home in UHF
Sheet2
Sheet3
Sheet1
High V's Ch 7-13UHF BroadcastUHF Broadcast
789101112131415161718192021222324252627282930313233343536373839404142434445464748495051
TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV
High V's Ch 7-13UHF BroadcastUHF Broadcast
789101112131415161718192021222324252627282930313233343536373839404142434445464748495051
TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV
TVTVTVTVTVTVTVTVTV
6 added
9 without home in UHF
Sheet2
Sheet3
Possible band plans
Notes:1. Numbers in the table represent current broadcast television channels2. Letters in the table represent paired blocks for mobile wireless services3. Grey blocks represent guard bands and their width (in MHz)4. Impaired markets will have a band plan from the table, but not at the nationwide clearing target5. Band plan above stops at UHF channel 21 (does not show UHF channels 14 to 20)
Number of Paired
Blocks
BroadcastSpectrumCleared (MHz)
Current Broadcast Television Band
Proposed Mobile Wireless ServicesRepacked Broadcasters
Why is this complicated? Have a supply side (broadcasters) and demand side (wireless carriers),
so why not just determine where supply meets demand?
Issue: the supply side is extremely complicated Buy stations until # stations = # channels only works at local scale Two stations can only use the same channel if they do not cause interference
Interference between stations creates a complex set of constraints Depends on physical factors: distance, transmitting power, terrain, antenna If stations interfere strongly enough, cannot even be on adjacent channels Large markets can interfere with each other, e.g. New York and Philadelphia
Nationwide interference constraints
Finding channel assignments Finding a nationwide channel assignment is extremely
complicated Hundreds of thousands of interference constraints between
stations Border regions must respect treaties with Canada and Mexico Some channels reserved for emergency communications in many
markets
Checking if a set of stations can be repacked is computationally hard
Equivalent to classic Satisfiability (SAT) problem from computer science
No generally efficient solution known; widely believed not to exist The auction needs to make hundreds of thousands of these checks Fortunately, this can be made (mostly) efficient in practice
Design of the reverse auctionThe reverse auction is a scored descending clock auction: Each television station is assigned a score (starting price) A single clock ticks down from 100% to 0%; at each time
step: Every station is offe