Transcript Course Overview - Technion – Israel Institute of Technology

Part 1
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•
•
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Course administration
Overview of course syllabus
An example of a real-life auction: The sale of Bezeq.
Definitions: normal-form games, dominant strategies, Nash
equilibrium.
• One-item auctions: analysis of the 2nd price auction.
Course administration
• Grade is based on two components of equal weight:
– Giving a 20-minutes talk. This assignment can be done in
couples (10 minutes each).
AND
– Multiple choice exam (15 questions, 1.5 hours)
OR solving a final exercise at home. The exercise contains
difficult mathematical questions, and is intended only for
students who want a mathematical challenge.
• Course website on moodle. All material is there.
Presentations
• Possible topics for presentations are detailed in moodle.
– Book chapters, research papers, webpages on research
– Main focus should be the connection of theory to practice, so maybe add
real-life examples that you search over the Internet searches.
• Email me which chapter would you like to do. I will update in
the moodle site with the already occupied subjects and slots.
• Each week, three people will give a talk (a group of two can be
split between two consecutive weeks).
• Next week the presentations should be about “stable matching”. I
will give a 15-points bonus for this topic. The first three to email
will have the slots.
What is Auction Theory?
• Auctions are becoming more and more popular
– The Internet only helps.
• We see different auction formats, different rules.
– Can we say anything about the difference, what will be the
result of one format vs. another.
– Examples: Ebay has proxies, different sites has different end
rules, etc.
• How to analyze this? We use Game Theory, that models
strategic decisions of players.
• The course will describe the mathematical theory that evolved
(but with very little mathematical proofs)
‫הסיפור מאחורי הפרטת בזק‬
‫•‬
‫•‬
‫•‬
‫•‬
‫•‬
‫מקור‪" :‬הארץ"‪ 29 ,‬למאי ‪2005‬‬
‫שתי קבוצות התמודדו במכרז‪ :‬קבוצת סבן‪ ,‬וקבוצת אלג'ם‪.‬‬
‫ההצעות ההתחלתיות ‪ -‬אלג'ם‪ 3.2 :‬מיליארד ‪ ,‬סבן‪ 3.6 :‬מיליארד‬
‫התכנון המקורי של משרד האוצר היה לחשוף את ההצעות‪ ,‬ולקבל הצעות חדשות‬
‫(כלומר לערוך מכרז אנגלי עולה)‪ .‬אבל לנוכח הפער הגדול בין שתי ההצעות‬
‫(‪ 400‬מיליון שקל)‪ ,‬באוצר חששו להמשיך לפי התכנון המקורי‪ ,‬שמא אלג'ם‬
‫תתייאש ותפרוש‪.‬‬
‫היועץ המקצועי‪ ,‬פרופ' מוטי פרי מירושלים‪ ,‬התעקש להמשיך לפי התכנון‪" :‬אין‬
‫ביטחון מלא‪ ,‬אבל בדרך כלל השקיפות מביאה לתוצאות הטובות ביותר‪ .‬יש למסור‬
‫לכל אחד מהמשתתפים את תוצאות הסבב הראשון‪ ...‬ההצעה העיקרית שלי במכרז‬
‫היתה ליצור מערכת שקופה לחלוטין‪ ,‬שבה כל אחד מהצדדים יודע בדיוק מה קורה‬
‫ומקבל את כל המידע‪ .‬ככל שלמתחרים יש יותר ידע‪ ,‬רמת הסיכון שלהם יורדת‪,‬‬
‫מרכיב ההימור בעסקה יורד‪ ,‬והם מוכנים להסתכן בהצעות גבוהות יותר"‪.‬‬
‫הסיפור מאחורי הפרטת בזק‬
‫• ההצעות ההתחלתיות ‪ -‬אלג'ם‪ 3.2 :‬מיליארד ‪ ,‬סבן‪ 3.6 :‬מיליארד‪.‬‬
‫• המשך המכרז‪ :‬אלג'ם העלו ל‪ 3.8-‬מיליארד‪ ,‬וסבן העלה ל‪ 4.11-‬מיליארד‪ ,‬וניצח‪.‬‬
‫• מכרז בזק היה המכרז הראשון בתולדות ההפרטה שנוהל על פי ייעוץ של פרופ'‬
‫לתורת המשחקים‪ .‬תוצאות המכרז הוכיחו שגישה זו הייתה נכונה‪.‬‬
Course Overview
1.
Basic game-theoretic notions, The basic auction model,
description and analysis of classic one-item auctions.
2.
Revenue-maximizing auctions for one item.
3.
Bidders with interdependent values.
4.
Mechanism Design and VCG.
5.
Auctions for multiple goods.
Four “classic” auctions
• First price: bidders submit bids. Winner is the highest bidder.
Pays her bid.
• Second price: bidders submit bids. Winner is the highest
bidder. Pays her bid.
• Dutch auction: price starts from a very high number, and
gradually descends. First one to raise her hand wins. Pays that
price.
• English auction: price starts from zero, and gradually ascends.
Last one to keep her hand up wins. Pays that price.
Simulation
• Each student gets a different amount of virtual dollars: two last
digits of ID number.
• I will conduct two auctions:submit a note that contains your
name and ID number, and your bid for the first and second
price auctions.
• Each winner pays the price she owes me with her virtual
dollars. Each remaining virtual dollar will give one bonus
point in the final grade (and vice versa - if she cannot pay the
entire price with her virtual dollars, I will deduct the difference
from the grade!).
A game (in complete information)
Player II’s actions
Action “L”
Action “T”
Player I’s
actions
Action “B”
Action “R”
Here’re the utilities
if player I plays T
and II plays L
5 , -3
If player I plays B and II plays R then player I gets 5 and II gets -3
Example: Prisoner’s dilemma
• The story: Business partners decide to break down, and each one
needs to hire a lawyer, to reach a settlement on how to split their
property. Total property is worth 10.
– Two types of lawyers: cheap costs 1, expensive costs 5.
– If both choose same type, property is split equally.
– Otherwise the one with the expensive lawyer gets everything.
“C”
• So, the game is:
“E”
“C”
4,4
-1 , 5
“E”
5 , -1
0,0
What will happen in this game?
DFN: Action A dominates action B (for player i) if for any
combination of actions of the other players, a-i, ui(A,a-i) > ui(B,a-i).
Action A is a dominant action if it dominates all other actions.
• Observation:
“E” is a dominant strategy.
• Assumption: Players indeed
play a dominant strategy if
they have one.
“C”
“E”
“C”
4,4
-1 , 5
“E”
5 , -1
0,0
What will happen in this game?
DFN: Action A dominates action B (for player i) if for any
combination of actions of the other players, a-i, ui(A,a-i) > ui(B,a-i).
Action A is a dominant action if it dominates all other actions.
• Observation:
“E” is a dominant strategy.
“C”
“E”
• Assumption: Players indeed
play a dominant strategy if
they have one.
“C”
4,4
-1 , 5
• Demonstration:
the reality show
“E”
5 , -1
0,0
A one-item auction
The story:
• A seller wishes to sell a good to n players. Player i will obtain
a value of vi > 0 from having the good. vi is known only to i.
• The seller can charge a payment pi from player i.
In this case player i’s utility is: vi – pi
• Like we did in the beginning of the class.
Analysis of second price
• Example: three players. v1 = 10, v2 = 7, v3 = 4.
• If they all bid their true values, player 1 wins, and pays 7.
• Notice that none of the players can improve her utility by
changing her bid.
THM: In the second price auction, bidding the true value
is a dominant strategy.
Drawing the game (for two players)
• The strategy space of each player is composed of all integers.
• For fixed v1, v2, if player 1 plays b1 and player 2 plays b2,
and b2 > b1, then the utility of player 1 is 0, and of 2 is v2 – b1.
b2
.
.
.
b1
0, v2 – b1
Analysis for n=2
Claim: A dominant action for player i=1,2 is to play bi=vi
Proof: Fix any b2
• If v1 > b2 then winning is better than losing for player 1.
Declaring b1=v1 will cause player 1 to win.
• If v1 < b2 then losing is better than winning for player 1.
Declaring b1=v1 will cause player 1 to lose.
• Conclusion: no matter what the other player declared, the
strategy b1=v1 dominates all other strategies of player 1.
Analysis for n>2
• For n > 2 players the game and the claim are similar.
Claim: The dominant action of player i is to play bi=vi
Proof: Fix any b-i , and let x = max j  i bj
• If vi > x then winning is better than losing for player i.
Declaring bi=vi will cause player i to win.
• If vi < x then losing is better than winning for player i.
Declaring bi=vi will cause player i to lose.
Equivalence of second price and English
auctions
• In the English auction, the dominant strategy of every
player is to drop exactly when the price reaches her value.
• If all players do that, then the player with the highest value
wins, and she pays the second highest value (since the
price stops exactly when the second highest player drops).
Next: 1st price auction
• What about the 1st price rule?
• If players continue to bid their true value, then this auction
will clearly have a higher revenue.
• But of-course players adjust their behavior according to the
rules of the auction.
• Observation: There is no dominant strategy in the 1st price
auction (if a player wins with a bid b he would have
preferred to say slightly less than b).
Part 2
• We started by talking about the four classic auctions, and about
analyzing them using game theory.
• We continue by reviewing the theoretical analysis of these
auctions:
– Analyze the equilibrium strategies of these auctions.
– Describe another auction with a better revenue.
– Talk about two insightful lemmas, that are related to the
construction of the auction.
Nash Equilibrium
• In the “coordination game”, players gain a positive utility if and
only if they play the same action.
“A”
“B”
“A”
2,1
DFN
• Action ai is best response to a-i if
0,0
for any other action a’i of player i,
“B”
ui(ai, a-i) > ui(a’i, a-i)
• The actions (a1,…,an) are in Nash equilibrium
ai is best response to a-i for all players i.
• What are the Nash equilibria in the coordination game?
0,0
1,2
Example: the Braess paradox
x
1
S
T
1
x
• Using Nash Equilibrium to study transportation networks.
• The picture shows a road network. All cars need to arrive from
node S to node T.
• There are two types of roads:
– label “1”: wide and long road, travel time is one hour regardless
of number of cars.
– label “x”: narrow and short road, x is the fraction of cars that
take this road, travel time is x times one hour. For example
if ½ of all cars take this road then travel time is half an hour.
Example: the Braess paradox
x
1
100%
S
T
1
x
• If all cars take the upper path, total travel time will be 2.
• Drivers care only about their total travel time, which is influenced
by the choices of all other drivers. Thus this is a game.
• What is a Nash equilibrium of this game?
Example: the Braess paradox
x
1
50%
S
50%
1
T
x
• If all cars take the upper path, total travel time will be 2.
• Drivers care only about their total travel time, which is influenced
by the choices of all other drivers. Thus this is a game.
• What is a Nash equilibrium of this game?
Answer: 50% of the drivers take each path. Total driving time of
each driver is 1.5 hours.
Remark: This is the unique Nash equilibrium.
Example: the Braess paradox
x
1
0
S
1
T
x
• Now the government has decided to expand the network and
build a new fast road. What will happen?
• In other words what will be the new Nash equilibrium, and
what will be the travel times in this equilibrium?
Example: the Braess paradox
x
1
100% 0
S
1
T
x
• Now the government has decided to expand the network and
build a new fast road. What will happen?
• In other words what will be the new Nash equilibrium, and
what will be the travel times in this equilibrium?
Answer: As in the picture. This is again the unique Nash
equilibrium. Why?
Example: the Braess paradox
x
1
100% 0
S
1
T
x
• Now the government has decided to expand the network and
build a new fast road. What will happen?
• In other words what will be the new Nash equilibrium, and
what will be the travel times in this equilibrium?
Answer: As in the picture. This is again the unique Nash
equilibrium.
• Driving time of each driver is 2 hours, worse than before.
• The paradox: instead of helping, the government only made
things worse. The lesson: strategic behavior can cause
unexpected phenomena, and must be taken into account.
Example: the Braess paradox
• This has happened in reality:
• In NYC, the 42nd street was closed on Earth Day in 1990. Lucius
J. Riccio, the transportation ommissioner at the time, said that
“many predicted it would be doomsday“. However, to everyone’s
surprise, traffic flow improved when the street closed.
• In Seoul, South Korea, traffic improved when a motorway was
removed as part of the Cheonggyecheon restoration project -- an
elevated highway was removed to uncover a historic waterway.
• In Stuttgart, Germany, initial investments into the road network in
1969 did not improve traffic. However, once a section of this
newly-built road was closed, the traffic situation improved.
• In 2008, an academic research specified roads that, if closed, will
reduce travel time in NYC, Boston, and London.
Back to auctions
Reminder: Four “classic” auctions
• First price: bidders submit bids. Winner is the highest bidder.
Pays her bid.
• Second price: bidders submit bids. Winner is the highest
bidder. Pays her bid.
We saw that it is a dominant strategy to bid the true value.
• Dutch auction: price starts from a very high number, and
gradually descends. First one to raise her hand wins. Pays that
price.
• English auction: price starts from zero, and gradually ascends.
Last one to keep her hand up wins. Pays that price.
Analysis of 1st price
• If players continue to bid their true value, then this auction
will clearly have a higher revenue.
• But of-course players adjust their behavior according to the
rules of the auction.
• Observation: There is no dominant strategy in the 1st price
auction (if a player wins with a bid b he would have
preferred to say slightly less than b). We will therefore use
the notion of a Nash equilibrium.
Equilibrium in incomplete information
• Assumption: vi is drawn from a probability distribution fi, and
these distributions are known to all )“common priors”(.
• Useful example: vi is drawn from the uniform distribution on
[0,1]. In this case:
– Pr(vi < 1/2) = 1/2. In general, for any 0 < x < 1, Pr(vi < x) = x.
– E[vi]=1/2.
• The strategies of the players are in (Bayesian-Nash) equilibrium
if each player maximizes her expected profit by following her
equilibrium strategy, given that the other players follow their
strategies.
Equilibrium in the 1st price auction
THM: Assume each value is drawn from the uniform distribution
on [0,1]. Then a Bayesian-Nash equilibrium of the 1st price
auction is when every player i bids bi(vi) =[(n-1)/n] · vi
• Clearly, in the 1st price auction a bidder needs to “shade down”
his value. This tells the players by how much.
– As the number of players grow, shading down becomes
smaller.
Example
• n=2. Therefore, in equilibrium, every player bids half his value.
Let’s see if this is indeed best for player 1, if his value is 2/3:
• u1(1/4) = (2/3 - 1/4) · Pr[ v2 / 2 < 1/4]
profit if wins
probability to win
Example
• n=2. Therefore, in equilibrium, every player bids half his value.
Let’s see if this is indeed best for player 1, if his value is 2/3:
• u1(1/4) = (2/3 - 1/4) · Pr[ v2 / 2 < 1/4] = (5/12)(1/2)  0.208
 Pr[ v2 < 1/2] = 1/2
Example
• n=2. Therefore, in equilibrium, every player bids half his value.
Let’s see if this is indeed best for player 1, if his value is 2/3:
• u1(1/4) = (2/3 - 1/4) · Pr[ v2 / 2 < 1/4] = (5/12)(1/2)  0.208
 Pr[ v2 < 1/2] = 1/2
• u1(1/3) = (2/3 - 1/3) · Pr[ v2 / 2 < 1/3] = (1/3)(2/3)  0.222
 Pr[ v2 < 2/3] = 2/3
• u1(1/2) = (2/3 - 1/2) · Pr[ v2 / 2 < 1/2] = 1/6  0.167
 Pr[ v2 < 1] = 1
Remarks
• 1st price auction is equivalent to a descending )“Dutch”( auction:
the auctioneer gradually lowers the price, the first to accept wins,
for this price.
• A comparison to second price:
– No dominant strategies, so less obvious how to play.
– In both auctions, in equilibrium, the bidder with the highest
value gets the item )the “efficient” outcome(.
– 2nd price may look bad - the winner’s price may be much lower
than his bid. An extreme once happened in a New-Zealand governmentauction: One firm bid NZ$100,000 for a license, and paid the second-highest
price of only NZ$6 (http://www.economicprincipals.com/issues/06.05.21.html).
– Compare these strategies to what you chose to do last class.
Revenue considerations
• Which of the two auctions raises more revenue?
• Is there an auction with a higher revenue than both these
auctions?
Numerical example
VAL 1
VAL 2
96
52
36
94
58
12
29
47
88
2
87
96
44
11
38
5
52
50
51
64
18
78
39
23
38
24
33
19
78
64
84
79
27
72
82
46
16
14
1st price revenue
48
26
39
47
29
19
14.5
23.5
44
39
43.5
48
39.5
13.5
36
41
26
25
25.5
33
2nd price revenue
64
18
36
39
23
12
24
33
19
2
64
84
44
11
38
5
46
16
14
31.15789
optimal revenue
64
50
50
50
50
0
0
0
50
50
64
84
50
0
50
50
50
50
50
42.73684
The optimal auction for symmetric bidders
•
•
The symmetric case: players’ values are drawn from the same
distribution, F(v). Choose v* that solves v = (1-F(v))/f(v).
Then, English auction when the price starts from v* is the
revenue maximizing auction:
– If the highest value is below v*, no one wins
– If the highest value is above v* and the second highest is
below v*, the highest player wins and pays v*.
– If the second highest is above v*, the highest wins and
pays the second highest.
Numerical example
VAL 1
VAL 2
96
52
36
94
58
12
29
47
88
2
87
96
44
11
38
5
52
50
51
64
18
78
39
23
38
24
33
19
78
64
84
79
27
72
82
46
16
14
1st price revenue
48
26
39
47
29
19
14.5
23.5
44
39
43.5
48
39.5
13.5
36
41
26
25
25.5
33
2nd price revenue
64
18
36
39
23
12
24
33
19
2
64
84
44
11
38
5
46
16
14
31.15789
optimal revenue
64
50
50
50
50
0
0
0
50
50
64
84
50
0
50
50
50
50
50
42.73684
Practical Conclusion: Devote a large effort to
figuring out the correct reservation price.
Remark
• How does such an auction increases the revenue of the
regular English auction?
• The reservation price introduces two opposite effects:
– Sometimes the optimal auction does not sell the item, so
the revenue is 0, while the English auction always sells
the item (and thus always has a positive revenue).
– Sometimes the optimal auction sells the item but charges
more than the second highest value, if it is below v*,
while the price in the English auction is always the
second highest value.
• The correct choice of v* enables the auctioneer to gain more
from the second point and lose less from the first point.
Two observations that helped to find and
prove this
The Revelation Principle
• Problem: there are infinitely many possible auction formats, so
it is hard to go over all of them…
• Reminder: In a direct-revelation auction, the strategy space of a
player is simply to report her type (value).
THM: Given any auction format with equilibrium strategies s(),
there exists a direct-revelation auction for which truthfulness is
an equilibrium, with the same outcome, and the same prices.
– Remark: This holds for any type of equilibrium (dominant
strategies, Bayesian-Nash,...)
• The implication: we need only search for a direct-revelation
auction.
Example
Input
v1
v2
New Auction
Convert to (1/2) v1
Convert to (1/2) v2
1st price
Auction
Original
Output
• We mimic the equilibrium strategy, hence truthfulness is an
equilibrium.
• For example, suppose the true value of player 1 is 2/3.
– If she declares her true value it is as if she plays 1/3 in the
1st price auction.
– If she instead declares 1/2, it is as if she declares 1/4 in the
1st price auction. But we already know that this is not
better in the 1st price auction.
Proof (sketch)
New Auction
Input v
“Proxy”
Output s(v)
Original
Auction
Original
Output
• A proxy mimics the equilibrium strategy: if others are truthful,
player i would like to play si(vi), so she needs to declare vi.
• Examples:
– In the 1st price auction, the proxy will convert vi to [(n-1)/n] vi
– In the English auction, the proxy will take vi and will keep
“raising the hand” until the price reaches vi.
Conclusion: there is no revenue advantage for
“complicated” indirect auctions )though there may
be a marketing advantage)
The Revenue Equivalence Theorem
THM (The Revenue Equivalence Theorem):
Suppose vi is drawn independently from some Fi(x). Take any
two auctions such that, in both auctions:
- The expected utility of a player with value 0 is the 0.
- The winner (in equilibrium) is always the same.
Then these two auctions raise the same expected revenue (in
equilibrium) .
The Revenue Equivalence Theorem
THM (The Revenue Equivalence Theorem):
Suppose vi is drawn independently from some Fi(x). Take any
two auctions such that, in both auctions:
- The expected utility of a player with value 0 is the 0.
- The winner (in equilibrium) is always the same.
Then these two auctions raise the same expected revenue (in
equilibrium) .
• The 1st and 2nd price auctions satisfy the conditions (in
equilibrium, the item is sold to the bidder with the highest value),
so they raise the same revenue. Another example: an all-pay
auction. Here, too, the winner is the person with the highest
value, so the revenue is the same.
• Why is the optimal auction different?
Conclusion: To design a revenue-maximizing auction, the
only question is who will be the winner (in
equilibrium).
Risk aversion in 1st and 2nd price
auctions
• A crucial assumption in the analysis of the 1st price auction
is that players aim to maximize the expected profit.
– This is termed risk-neutral players.
– Many times this is not true: for example, we might care
about the variance (smaller variance might be better).
• In a 2nd price auction, the dominant strategy is to bid
truthfully, so risk-aversion does not change anything.
– There is no expectation in the considerations of a player,
since dominant strategy maximizes the player’s utility, no
matter what the others are doing.
The picture for risk-averse bidders
Revenue of 1st price with risk-aversion
>
Revenue of 1st price with risk-neutrality
=
Revenue of 2nd price with risk-neutrality
=
Revenue of 2nd price with risk-aversion
• Remark: There are examples where the revenue is strictly
higher in a 1st price auction, and examples where the
revenue is equivalent.
Summary
• We discussed the issue of “private-value” auctions:
– Analyzed equilibrium behavior in the classic auctions.
– Saw the revenue-maximizing auction.
– Studied the “revelation principle” and the “revenue
equivalence” theorem.
– Discussed a crucial assumption to the above: riskneutrality vs. risk-aversion.
• Next week: interdependent values.
Part 3
Next topic: “interdependent values”.
–
–
–
–
Introduction: selling a jar of money, and a newspaper article.
Definition of the model, and the “winner’s curse”.
Equilibrium in the English auctions.
Discussion about revenue.
Interdependent values
• Up to now we have assumed that the player determines and
knows her value (the “private value” model).
• Many times this is not the case:
– oil rights, a sale of a large company,…
– valuable paintings, real-estate,…
• In interdependent values, each player receives a signal Xi, and
her value is a function of all signals:
Vi = Vi(X1,…,Xn)
– In the private value model, Vi(X1,…,Xn)=Xi
– In the “common value” model, Vi(X1,…,Xn)=V(X1,…,Xn)
– Assumption: Vi(0,…,0)=0.
• We assume risk-neutral players.
Example
• Xi is uniformly distributed in [0,100], and Vi = 1/n(X1 + … + Xn).
• Suppose we conduct a second price auction. The problem of a
player: How to bid without knowing the value?
• Solution: estimate
– In real life firms dedicate much effort to this issue.
– Another problem: sometimes we over-estimate.
• For example, we may be tempted to bid E( Vi | Xi ).
• Is this a good idea?
The winner’s curse
• Suppose everyone indeed bid E( Vi | Xi ):
– In the example, bi = (1/n)Xi + (n-1)/n · 50. Therefore all
bids will be “close to” 50 )for example if n=5 then all bids
will be at least 40).
– Now suppose all signals are very low (between 1 and 10).
In this case, the winner will pay more than her value!
The winner’s curse
• Suppose everyone indeed bid E( Vi | Xi ):
– In the example, bi = (1/n)Xi + (n-1)/n · 50. Therefore all
bids will be “close to” 50 (for example if n=5 then all bids
will be at least 40).
– Now suppose all signals are very low (between 1 and 10).
In this case, the winner will pay more than her value!
• More generally, if player i wins. This means that he has the
highest value, and so
E( Vi | Xi = x ) > E( Vi | Xi = x and Y1 < x)
( Y1 = maxi≠1Xi )
• So winning is an indication that the value is not as high as you
first thought…
The English auction
• Suppose we have 3 players, Vi = 1/3(X1 + X2 + X3).
• Consider the following strategies:
– In the beginning, each player will drop at her signal.
– After the first player drops, the other two can infer her
signal, and so they can update their value. As a result,
each i of the remaining two will drop at 1/3(X1 + Xi + Xi).
• With signals 10, 20, and 30, we will have: the first player will
drop at price=10, the second player will drop at price=50/3,
and the first player will win and will pay this. Notice that her
actual value is 20 > 50/3.
Equilibrium in the English Auction
• Step 1:
– Each player i estimate her value by bN(Xi) = V(Xi,…, Xi) and
drops when the price reaches this estimate.
– The first player that drops is the player with the lowest signal.
Call her player N. All other players see the price at which that
player dropped. Then they can infer her signal.
• Step 2:
– All remaining players update the estimated value, by plugging in
the signal of player N. I.e. bN-1(Xi) = V(Xi,…,Xi,XN). Each player
remains until the price reaches her estimate. When the next
player drops, all other players infer her signal.
• And so on and so forth, until one player remains.
THM The strategy s* = [b1(),…,bN()] forms a symmetric equilibrium.
Remarks
• When a player drops he knows that his value is higher than the
price! (but still he has no way to win the auction with a profit)
• No equivalence between second-price and English auction.
• This equilibrium is in fact stronger than Bayesian-Nash:
DFN: The strategies s1,…, sn are in ex-post equilibrium if for
any i, v-i, vi, ai :
ui(si(vi),s-i(v-i) > ui(ai,s-i(v-i)
– Implies a Bayesian-Nash equilibrium for any possible
distribution.
– Has the “no regret” property: a player does not regret her
action even after knowing the signals and actions of the
other players.
Remarks (2)
• The English auction does not have dominant-strategies: if the
player with the lowest signal stays after he is supposed to
retire, the rest will get a false picture of his signal, and can pay
more then their value.
• For example, suppose two players with signals X1=10, X2=2,
and the value for both is the average of the signals. If player 2
decides to drop when the price reaches 8, the first player will
still win, but he will pay 8 which is larger than his value (which
is 6).
Revenue comparison and Information
Revelation
• Second-price auctions have lower revenue, on average, and firstprice auctions have even lower than that. Thus among the classic
auctions, the English auction is best.
• The reason is that the English auction manages to reveal all
signals of losing players to the winner.
• Thus, the practical conclusion is that in case of interdependent
values, the auctioneer should design the auction as to reveal as
much information as possible on the other players: information
revelation increases the revenue for interdependent values.
Social Efficiency
• Efficiency: the item goes to the player with highest value.
Measures the society’s welfare, not for the auctioneer’s own utility.
• Why is this good? For example,
– In the FCC auction, the US law requires the government to
maximize the efficiency, and not the revenue.
– Super-huge firms sometime have “dummy money” to make
inside decisions more efficient )e.g. IBM has “blue-money”(.
– Ideological reasons: economists should know how to improve
global social welfare.
• With private values, we know that first-price, second-price, and
English auction are all efficient. This is true with symmetric
interdependent values as well.
• With general interdependent values, this is not the case.
Part 4
• Mechanism design and VCG -- how to “implement” a social
choice function.
• Truthful cost-sharing and auctions for unlimited number of
identical items.
Two Examples
• “Public Project”: The government considers building a bridge.
Each citizen, i, will increase his productivity by some value vi if
the bridge will be built. The cost of building the bridge is C. The
government wants to build the bridge iff i vi > C, but does not
know the values. What to do?
• “Efficient allocation of a resource”: A manager has a unique
machine it can give only one of his workers. Each worker, i, will
increase his productivity by some value vi. The manager wants to
give the machine to the worker with the highest productivity
increase. What to do?
The setting
• A social designer has a set of alternatives.
• Each player has a value for every alternative.
• The social planner wants to choose the alternative that
maximizes the sum of values.
• Example:
a
I
1
II
7
III
9
sum 17
b
5
6
8
19
c
10
4
7
21
The setting
• A social designer has a set of alternatives.
• Each player has a value for every alternative.
• The social planner wants to choose the alternative that
maximizes the sum of values.
• Example:
a
I
1
II
7
III
9
sum 17
b
5
6
8
19
c
10
4
7
21
The planner wants to
choose alternative c.
Vickrey-Clarke-Groves (VCG)
• Problem: the designer does not know the values of the players.
• A solution (VCG mechanism):
– Request players to reveal values.
– Choose the alternative according to players' declarations.
– Charge a payment from player i that is equal to the “damage”
she causes the other players: the aggregate value of the best
alternative, if player i was absent, minus the sum of values of
all players besides i to the chosen alternative.
THM: 1) The VCG mechanism is truthful.
2) Payments are always non-negative (players always pay).
3( A player’s utility is non-negative.
4) If the same outcome will be chosen whether the player
participates or not, then his price is zero.
Example
I
II
III
sum
a
1
7
9
17
b
5
6
8
19
c
10
4
7
21
price
5
0
0
utility
10-5=5
4
7
If player I is absent, alternative a is chosen, with value 16. The
chosen alternative when player I is present is c, and its value to
the other players is 11. Thus the price of player I is 16 – 11 = 5.
Example
I
II
III
sum
a
1
7
9
17
b
5
6
8
19
c
10
4
7
21
price
5
0
0
utility
10-5=5
4
7
What if player I lies and says that her value for c is 7?
I
II
III
sum
a
1
7
9
17
b
5
6
8
19
c
10 7
4
7
18
price utility
2
5-2=3
Example
I
II
III
sum
a
1
7
9
17
b
5
6
8
19
c
10
4
7
21
price
5
0
0
utility
10-5=5
4
7
What if player I lies and says that her value for c is 7?
I
II
III
sum
a
1
7
9
17
b
5
6
8
19
c
10 7
4
7
18
price utility
2
5-2=3
smaller
than her
utility
when
telling
the
truth!
Intuition to proof: Redoing the case of
the 2nd price auction
• Another mechanism:
– Each bidder reports his type.
– The winner is the player with the highest value.
– The mechanism pays all other players the winners’ value.
• In other words, we equate the utility of all players to be the
highest value.
• Example: three players with true values 10, 8, 7.
10 gets the item, pays nothing; 8, 7 (each) get a payment 10.
• Is this truthful?
• Problem: But we pay the players, instead of getting paid??
Solution
• Solution: subtract a “constant” hi(v-i) from the prices, so that the
total will be negative.
• For example: hi(v-i) = max j ≠ i vj . Therefore the payments in the
example will be:
– The 10 player will additionally pay max(8,7)=8.
– The 8 player will get 10 and will pay max(10,7)=10, so his
total payment will be zero.
– The 7 player will get 10 and will pay max(10,8)=10, so his
total payment will also be zero.
– Conclusion: this is exactly the second price auction.
Two main disadvantages of VCG
• Suitable only when our goal is welfare maximization.
Other goals, like revenue maximization, are not answered.
• If some values are negative, the mechanism may end up
paying the players more than the payments it collects.
Solution to the public project
• Reminder: The government considers building a bridge. Each
citizen, i, will increase his productivity by some value vi if the
bridge will be built. The cost of building the bridge is C. The
government wants to build the bridge iff i vi > C, but does not
know the values. What to do?
• We can use VCG. We have two alternatives (YES/NO), and each
player, including the government, has a value for each alternative
(zero for NO, vi or -C for YES). VCG will choose YES iff i vi C > 0, which is what the government wants.
Properties of the payments
• A player will pay nothing if j ≠ i vj > C, otherwise (in a
YES case) he will pay C - j ≠ i vj.
• These payments may not cover the entire cost C.
• The main problem: suppose C=100, we have 102 players,
and each player has value=1. Then the price that each
player will pay is zero!
• If we can exclude some players from using the bridge, we
can use the CostShare method that we will next see.
Truthful cost sharing
• What happens if we must cover the cost?
• The mechanism CostShare(R): collect bids b1,…,bn, and repeat:
– Suppose we have k bids left (initially we have k=n).
– If there exists a bid lower than R/k, delete it, and repeat.
– Otherwise all remaining bidders win, and each one pays R/k.
• Example: bids (5,4,3,2,1), R=9
– In the 1st round the tentative price is 9/5. Bid 5 is deleted.
– In the 2nd round the tentative price is 9/4. Bid 4 is deleted.
– In the 2nd round the tentative price is 9/3. All three remaining
bids win, each one pays 9/3=3. Total revenue is 9.
Properties
• Upon success, we get a revenue R, so for example we will
manage to cover the cost of the bridge, if we decide to build it.
• Sometimes we will fail, although the sum of values is larger than
the cost.
– Example: two bidders with values 9,2. The cost is 10.
– In other words, this method is not efficient. There is a
theorem that if we want efficiency we have to use VCG.
• Another difference from VCG: here we assume we can exclude
losers from using the bridge.
• Most importantly, is this truthful??
Truthfulness of CostShare
• Answer: YES!
• Explanation:
– An auction is “value-monotone” if a winner that increases
his bid (while the rest stays the same) still wins.
– The “threshold-bid” of a player: the smallest bid for which
he still wins (while the rest stays the same).
– Any value-monotone auction, in which every winner pays
his threshold-bid, and every loser pays 0, is truthful.
– Example: 2nd price. (What about 1st price)?
– Check at home that CostShare is value-monotone, and
every winner indeed pays his threshold bid.
An application: Unlimited Supply
• Infinitely many identical items. Each bidder wants one
item.
– Corresponds to a situation were we have no marginal
production cost.
– Very common in “digital goods” – songs sold over the
Internet, software, etc.
• Bidder i has private value vi for the item.
– We assume throughout that bidders are ordered so that
v1>v2>…>vn
Known probability distribution
• If the values of the players are i.i.d from a probability
distribution F, the we can use the optimal auction we
already know:
– Since we have unlimited supply, we need to determine,
for every bidder separately, if she wins or loses.
– So we need an optimal auction for one bidder.
– We already know that this is a take-it-or-leave-it
auction, with a reservation price p such that p=(1F(p))/f(p)
Worst-case analysis
• What if we do not know, and cannot estimate, the underlying
probability distribution?
• We will design an auction with revenue close to the following
benchmark auction:
• Suppose we know the values of the players, but must sell all
items in the same price. The price will then be:
F(v) = maxi i·vi
• Remark: a non-discriminatory monopoly chooses this price.
• A slightly weaker benchmark: F(2)(v) = maxi>2 i·vi
– For “most” cases F)v(= F(2)(v), and we want to avoid the
extreme situations for which this is not true.
• We want to find a small constant c and a truthful auction with
revenue R(v) > F(2)(v)/c for any input v.
The auction
1.
Randomly partition the bids to two sets (A, B) by tossing a
fair coin for each player and associating her to A if the coin
came out “head”, and otherwise to B.
2.
Compute FA=F(A) and FB=F(B).
3.
Run CostShare(A, FB) and CostShare(B, FA) to determine
winners and prices.
Theorem: This auction is truthful, no matter what the coin tosses
are, and its expected revenue, for any v, is at least F(2)(v)/4.
Part 5
• Multi-item auctions with identical items.
• Multi-item auctions with different items.
Identical items
• Three possible bidder types:
– Unit-demand bidders
– Decreasing marginal values
– General valuations
Unit demand bidders
• Each bidder desires one item.
• Two popular “sealed-bid” auction formats:
– Uniform-price auctions: The M highest bidders win, each
pays the M+1 highest bid.
– Discriminatory auctions: The M highest bidders win, each
pays her bid.
• Two equivalent “open-cry” auctions:
– Ascending price (English): The price ascends until M
bidders remain.
– Descending price (Dutch): The price descends until M
bidders accept.
• Similarly to the single-item case, uniform-price is equivalent
to English, and Discriminatory price is equivalent to Dutch.
Example
• Two items, three bidders, with values 4,7,10:
– The players with values 7,10 win, and each one pays 4.
Example “Real” Applications
• Government securities were sold by the US government
using discriminatory auctions, until 1992.
• From 1992, some securities (e.g. 2-years and 5-years) are
being sold using a uniform-price auction.
• In the UK, electricity generators bid to sell their output on
a daily basis. Until 2000 the auctions were uniform-price,
and after that they switched to discriminatory price
Efficiency and Revenue in Unit-Demand
• In uniform-price: is in fact a VCG mechanism (check at home).
Therefore:
– Truth-telling is a dominant strategy
– The resulting allocation is efficient
• The revenue equivalence theorem and the optimal auction analysis
can be extended to unit-demand bidders:
– Any two auctions with the same outcome in equilibrium raise
the same revenue (e.g. unifrom-price and discriminatory-price).
– The optimal auction is to sell the M items to the M bidders
with the highest virtual valuations.
Decreasing Marginal Valuations
• Each player has a marginal valuation function vi: {1,…,M}-> R
– The value of receiving q items is vi(1(+…+vi(q)
• Marginal decreasing means: vi(q+1) < vi(q) for any 1<q<M
• Implication: Every bidder submits many bids
• Example for the uniform-price auction with two items:
Red is player 1
17
Black is player 2
15
Blue is player 3.
14
Result: the red player wins two items
7
and pays 2·14=28 => utility=4
6
Decreasing Marginal Valuations
• Each player has a marginal valuation function vi: {1,…,M}-> R
– The value of receiving q items is vi(1(+…+vi(q)
• Marginal decreasing means: vi(q+1) < vi(q) for any 1<q<M
• Implication: Every bidder submits many bids
• Example for the uniform-price auction with two items:
Red is player 1
17
Black is player 2
15
Blue is player 3.
14
Result: the red player wins two items
7
and pays 2·14=28 => utility=4
6
Can one of the players improve his utility??
Decreasing Marginal Valuations
• Each player has a marginal valuation function vi: {1,…,M}-> R
– The value of receiving q items is vi(1(+…+vi(q)
• Marginal decreasing means: vi(q+1) < vi(q) for any 1<q<M
• Implication: Every bidder submits many bids
• Example for the uniform-price auction with two items:
Red is player 1
17
Black is player 2
15
Blue is player 3.
14
Result: the red player wins two items
7
and pays 2·14=28 => utility=4
6
Observation: if the red player only bids 17 then he will win one
item and will pay price=7, increasing his utility!
Conclusions and remarks
• It is no longer true that the dominant strategy of a player in the
uniform-price auction is to bid truthfully.
• As we saw, it is beneficial for the players to decrease their
stated values for the items. This phenomena is termed
“demand reduction”.
• There are no dominant strategies. However, the uniform-price
auction is known to have a pure strategy equilibrium, in which:
– “demand reduction” occurs.
– the result is inefficient.
• It is also possible to show that every equilibrium of the
discriminatory auction is inefficient.
VCG
• VCG continues to have dominant strategies and an efficient
outcome.
• The VCG price for this case: suppose player i won q items,
and let x1,…,xq be the q highest non-winning bids of the other
players. Then player i pays x1+…+xq.
• In the previous example (2-item auction),
Result: the red player wins two items
and pays 14+7=21 => utility=11
17
15
14
7
6
The residual supply, and the Ausubel auction
• di(p) = max {q | vi(q) > p }
• s-i(p) = M - Σj≠i dj(p)
• 1 bids: 17, 15. 2 bids: 14, 6. 3 bids: 7.
• While price < 6, the demand of player 1 is 2, the demand of
player 2 is 2, and the demand of player 3 is 1. Therefore the
residual supply of each player < 0.
• At a price=6, 2’s demand decreases to 1. Therefore 1’s residual
supply is 0.
• At a price=7, 3’s demand decreases to 0. Therefore 1’s residual
supply is 1. Therefore he gets one item at price 7.
• At price=14, 2’s demand decreases to 0. Therefore 1’s residual
supply is 1. Therefore he gets a second item for a price 14.
This is equivalent to the VCG prices. As a result, truthfulness is
an ex-post equilibrium in this auction.
bids
bids
di(p)
s-i(p)
quantity
uniform price
di(p)
s-i(p)
quantity
VCG price
General valuations (with complementarities)
• In general, marginal valuations may increase. For example
v(1)=0, v(2)=100 represents a situation where the player
must get two units in order to obtain any value from the
items.
• In this case, the discriminatory-price and the uniform-price
have no real meaning.
• VCG, again, has dominant strategies, and reaches the
efficient outcome. However, no “natural” way of
representing VCG or its price is known. We simply use the
general mechanism.
Multi-item )“combinatorial”( auctions
• We have M different items.
• Players value subsets of items: vi (S) is the value for subset S. It
is usually assumed that:
– Normalization: vi() = 0
– Free disposal: S  T => vi(S) < vi(T)
• This represents different types of interactions among the items:
– Substitutability: item “a” can substitute item “b”. For
example, an e-store on eBay sells different types of cell
phones. In this case we would expect v({a,b}) < v(a) + v(b).
– Complementarities: item “a” helps to extract more value from
item “b”. For example, different computer parts. In this case
we would expect v({a,b}) > v(a) + v(b).
Real world applications
• An early documentation in the beginning of the 20th century: a
company went bankrupt, had to sell two plants (say A and B), and
requested price offers. One offer stated a price x for A, a price y
for B, and: “if we were to receive A and B, then our price is z”
where z > x+y.
• The London bus routes market: 800 routes are auctioned to bus
companies, by the 1984 “Transport Act”.
– First auction in 1985, until 1995 half of the network was
auctioned at least once. Today 15%-20% of the network is
auctioned each year.
• Spectrum rights: hundreds of spectrum intervals, combinations of
wave lengths are needed for technical reasons.
– Spectrum auctions are conducted in the US, UK, Europe.
The representation issue
• There are 2M subsets )“bundles”(, so each player “holds” many
numbers. How should she communicate with the auctioneer?
– With 100 items the total size of the valuation is enormous.
• Two approaches:
– “Bidding languages”: the player writes down her values in a
“sophisticated” language that may reduce the total size
– “Iterative queries”: the auctioneer repeatedly asks questions,
the answers keep the process going until we reach a result.
• In single-item auctions we also saw these two approaches but
here things get complicated.
The strategic issue by an example
Temp. price
0
0
Temp. winner
--
--
a
b
Item
Player I: v(a)=v(b)=3, v(a,b)=3
Player II: v(a)=v(b)=5, v(a,b)=5
Player III: v(a)=2, v(b)=0, v(a,b)=2
=1
Example
Temp. price
1
0
=1
Temp. winner
I
--
(phase 1)
a
b
Item
Player I: v(a)=v(b)=3, v(a,b)=3
Player II: v(a)=v(b)=5, v(a,b)=5
Player III: v(a)=2, v(b)=0, v(a,b)=2
Example
Temp. price
1
1
=1
Temp. winner
I
II
(phase 2)
a
b
Item
Player I: v(a)=v(b)=3, v(a,b)=3
Player II: v(a)=v(b)=5, v(a,b)=5
Player III: v(a)=2, v(b)=0, v(a,b)=2
Example
Temp. price
2
1
=1
Temp. winner
III
II
(phase 3)
a
b
Item
Player I: v(a)=v(b)=3, v(a,b)=3
Player II: v(a)=v(b)=5, v(a,b)=5
Player III: v(a)=2, v(b)=0, v(a,b)=2
Example
Temp. price
3
1
=1
Temp. winner
I
II
(phase 4)
a
b
Item
Player I: v(a)=v(b)=3, v(a,b)=3
Player II: v(a)=v(b)=5, v(a,b)=5
Player III: v(a)=2, v(b)=0, v(a,b)=2
Example
Temp. price
3
1
=1
Temp. winner
I
II
(phase 4)
a
b
Item
Player I: v(a)=v(b)=3, v(a,b)=3
Player II: v(a)=v(b)=5, v(a,b)=5
Player III: v(a)=2, v(b)=0, v(a,b)=2
Player I did not bid for
the item with lowest
price.
Example
Temp. price
3
1
Temp. winner
I
II
a
b
Item
=1
Result:
Player I: v(a)=v(b)=3, v(a,b)=3
Player I wins item 1 and pays 3.
Player II: v(a)=v(b)=5, v(a,b)=5
Player II wins item 2 and pays 1.
Player III: v(a)=2, v(b)=0, v(a,b)=2
Properties of the auction
DFN: A player is myopic if he always bids on the item x in
argmaxxS [ v(x) - p(x) ]
THM: Assume all players are unit demand. Then:
• When all players are myopic then the ascending auction
terminates in an allocation with maximal welfare*.
• When all other players are myopic, player i will maximize*
his utility by behaving myopically.
* up to a difference of about .
• In the previous example, suppose  is negligible, then players
I and II will win and will both pay price=2, which is exactly
the VCG price.
What happens with
complementarities?
• Example: Two items a,b are for sale. Bidder 1 needs both,
for a value of 140. Bidder 2 needs only one (either a or b),
for a value of 75.
– Suppose we change the ascending auction by simply
allowing the bidders to place their name on several
items together.
– We would like bidder 1 to win. However, bidder 2
raises the prices of both items to 75, so in order to win
bidder 2 needs to pay 150, which is too much for him.
– This is called the “exposure” problem.
Package bidding
• We can attempt to solve this by introducing prices for bundles,
and not only for items.
– In our example, we will have three prices: for {a}, for {b},
and for {ab}.
– Note that we have exponential number of prices.
• The course of the auction:
– Bidders state all bundles for which price < value.
– The seller composes an allocation with maximal value.
– Prices on all bundles desired by a losing player are increased
• This will solve the exposure problem:
– The prices on all bundles will reach 75, and then stop.
Package bidding (2)
• Package bidding creates another problem:
– Suppose a third bidder that wants item {b} for 40. We
would then like bidders 2+3 to win.
– Now, the prices for {a} and {b} will advance more slowly,
as the winning allocation can contain both {a} and {b}.
– At some point, the prices may be 40 for {a}, 40 for {b},
and {80} for {a,b}. Now, player 2 will not be willing to
increase his bid for {a}, as she’s hoping that player 3 will
do this first for {b}, and so nobody raises the bid for the
singletons, and player 1 wins.
• This is called the “threshold” problem.
• This can be solved by having bundle prices for each bidder
separately. This turns out to be equivalent to VCG.
The disadvantages of VCG
• Exponential complexity (both communication and computation).
• Low and non-monotonic seller’s revenue :
– Example: Bidder 1 wants {a,b} for 2M. Bidders 2,3 value any
singleton {a} or {b} by 2M. In VCG 2 and 3 are the winners.
– The price of bidder 2 is the difference between the value, to
the other bidders, of one item or two items. But this is zero!
– So the revenue is zero. If the two items were bundled and sold
as a whole, we would get a price of 2M.
– Removing the third bidder will increase the revenue.
• Collusion: If bidders 2,3 were to have values 0.5M, they were
losing. But, if together they would have raised their values to 2M,
they would win, and would pay zero, i.e. gaining from the lie.
• Shill bidding: If there were no bidder 3, bidder 2 will profit from
“inventing” him.
So what happens in practice?
• Ascending auctions are usually the choice, despite their problems.
• The FCC runs a “Simultaneous Ascending Auction” )SAA(, where
prices are per item.
• A “clock auction” aims to resolve the threshold problem by
increasing prices in a constant rate )“clock”(.
– Sometimes people add “activity rules”, e.g. the demand of a
player must be consistent among different rounds and prices.
• The “clock-proxy” auction format suggests to perform a clock
auction, followed by a package bidding concluding phase.
• Those are all heuristics, with incomplete theoretical background,
and much more research is still needed.