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Auction Theory
‫תכנון מכרזים ומכירות פומביות‬
Topic 7 – VCG mechanisms
1
Previously…
• We studied single-item auctions
• Bidders have values vi for an item
• A winning bidder gets a utility of ui=vi-pi
– A losing bidder pays nothing and get ui=0
2
Previously…
• Seller possible goals:
– Maximize social welfare (efficiency)
• 2nd-price (Vickrey) auction
– Maximize revenue
• 2nd-price auction with a reserve price (Myerson)
– For example, reserve-price=1/2 for the unifom distribution on
[0,1]
– Reserve price is independent of the number of players.
– Optimality assumes a technical assumption on the
distributions.
• Revenue equivalence
3
Previously …
• We saw that in single-item auctions we can maximize
efficiency with dominant strategies.
• Can this be achieved in other models?
4
Today
• This class:
Moving from a specific example (single-item
auctions) to a more general mechanism design
setting.
• Main goal: in the presence of incomplete
information, design the right incentives such that the
efficient outcome will be chosen.
5
Outline
1. Some examples
2. VCG idea – intuition
3. Formal part:
1. Mechanism design model
2. The VCG mechanism
3. Proof: VCG is truthful
4. Roommates example
6
values
Auctions scheme
bids
v1
b1
v2
b2
v3
b3
v4
b4
winner
payments $$$
Mechanism Design scheme
types
Bids/reports
t1
b1
t2
b2
t3
b3
t4
b4
outcome
Social
planner
payments
p1,p2,p3,p4
Example 1: Roommates buy TV
• Consider two roommates who would like to
buy a TV for their apartment.
• TV costs $100
• They should decide:
– Do they want to buy a TV together?
– If so, how should they share the costs?
I only watch
sports
!‫רק אירוויזיון‬
9
Example 2: Selling multiple items
• Each bidder has a value of vi for an item.
• But now we have 5 items!
– Each bidder want only one item.
• An efficient outcome: sell the items to the 5
bidders with the highest values.
$70
$30
$27
$25
$12
$5
$2
10
Vickrey-Clarke-Groves (VCG) mechanisms
• Goal:
implement the efficient outcome in dominant
strategies.
• A general method to do this: VCG
– 2nd-price auction is a special case
• Solution (intuitively):
players should pay the “damage” they impose
on society.
11
VCG basic idea (cont.)
In more details:
• You can maximize efficiency by:
– Choosing the efficient outcome (given the bids)
– Each player pays his “social cost” (how much his
existence hurts the others).
pi =
Optimal welfare (for
the other players) if
player i was not
participating.
Welfare of the other
players from the
chosen outcome
12
Vickrey-Clarke-Groves (VCG) mechanisms
• Let’s see how this payment rule works on our
examples:
Pi =
Optimal welfare (for
the other players) if
player i was not
participating.
Welfare of the other
players from the
chosen outcome
13
VCG idea in single item auctions
• Pi=
Optimal welfare (for
the other players) if
player i was not
participating.
= 2nd-highest value.
When i is not playing,
the welfare will be the
second highest.
Welfare of the other
players from the
chosen outcome
= 0.
When i wins, the total
value of the other is 0.
 By VCG payments, winners pay the 2nd-highest bid
14
VCG in 5-item auctions
• pi=
Optimal welfare (for
the other players) if
player i was not
participating.
Welfare of the other
players from the
chosen outcome
=30+27+25+12+5
=30+27+25+12.
The five winners when i
is not playing.
The other four winners.
What is my
VCG payment?
pays
5
pays ??
$70
$30
$27
$25
$12
$5
$2
15
VCG in k-item auctions
• VCG rules for k-item auctions:
– Highest k bids win.
– Everyone pay the (k+1)st bid.
And truthfulness is a dominant strategy
here too. (we will prove it later)
16
Outline
1. Some examples
2. VCG idea – intuition
 Formal part:
1. Mechanism design model
2. The VCG mechanism
3. Proof: VCG is truthful
1. VCG: the negative side
17
Mechanism Design scheme
types
Bids/reports
t1
b1
t2
b2
t3
b3
t4
b4
outcome
Social
planner
payments
p1,p2,p3,p4
Formal model
• n players
• possible outcome w1,w2,…,wm
• Each player has private info ti
• Each player has a value per
each outcome (depends on ti)
– vi(ti,w)
w is from {w1,…,wm}
• Goal of social planner:
choose w that
maximizes
n
 v (t , w)
i 1
i
i
• Single-item
auction example:
• 2 players
• w1 = “1 wins”,
w2 = “2 wins”
• ti=vi (willingness to
pay)
•
v1(v1, w1) = v1
v1(v1, w2) = 0
• Goal:
choose a winner with
the highest vi.
19
Formal model
w*=w5
w1
w2
w3
w4
w5
Player 1
V1(t1,w1)
V1(t1,w2)
V1(t1,w3)
V1(t1,w4)
V(t1,w5)
Player 2
V2(t2,w1)
V2(t2,w2)
V2(t2,w3)
V2(t2,w4)
V2(t2,w5)
Player 3
V3(t3,w1)
V3(t3,w2)
V3(t3,w3)
V3(t3,w4)
V3(t3,w5)
Player 4
V4(t4,w1)
V4(t4,w2)
V4(t4,w3)
V4(t4,w4)
V4(t4,w5)
n
n
 v (t , w )
 v (t , w )
i 1
i
i
1
i 1
n
 v (t , w )
i 1
i
i
2
i
i
n
3
Assume:w5
maximizes
efficiency
 v (t , w )
n
i 1
i
i
5
 v (t , w )
i 1
i
i
4
20
VCG – formal definition
• Bidders are asked to report their private values ti
• Terminology: (given the reported ti’s)
– w* outcome that maximizes the efficiency.
– Let w*-i be the efficient outcome when i is not playing.
• The VCG mechanism:
– Outcome w* is chosen.
*
*
– Each bidder pays:  v j (t j , wi )   v j (t j , w )
j i
The total value for the
other when player i is not
participating
j i
The total value for the
others when i participates
21
Truthfulness
Theorem (Vickrey-Clarke-Groves):
In the VCG mechanism, truth-telling is a dominant
strategy for all players.
Conclusion: welfare maximization can always be
achieved in dominant strategies.
• No Bayesian distributional assumptions.
• No real multiple-equilibria problem as in Nash.
• Very simple strategy for the bidders.
22
Now, proof.
We will show:
no matter what the others are doing,
lying about my type will not help me.
23
Truthfulness of VCG - Proof
• The VCG mechanism:
– Outcome w* is chosen.
*
v
(
t
,
w
– Each bidder pays:
j
j
i ) 

j i
*
v
(
t
,
w
 j j )
j i
• Method of proof: we will assume that there is a
profitable lie for some player I, and this will result
in a contradiction.
24
Truthfulness of VCG - Proof
• Buyer’s utility (when w* is chosen):
vi (ti , w )  pi
*


*
*
 vi (ti , w )    v j (t j , wi )   v j (t j , w ) 
j i
 j i



*
*
*
 vi (ti , w )   v j (t j , w )    v j (t j , wi ) 
j i
 j i

n


*
*
  vi (ti , w )    v j (t j , wi ) 
i 1
 j i



*
*
 SocialWelfare( w )    v j (t j , wi ) 
 j i

*
• Assume: bidder i reports a lie t’  outcome x is chosen.
• Buyer’s utility (when x is chosen):


*
SocialWelfare( x)    v j (t j , wi ) 
 j i

25
Truthfulness of VCG - Proof
• Buyer’s utility from truth (w* is chosen):


*
SocialWelfare( w )    v j (t j , wi ) 
 j i

*
• Buyer’s utility from lying (x is chosen):


*
SocialWelfare( x)    v j (t j , wi ) 
 j i

• Lying is good when:


*
SocialWelfare( x)    v j (t j , wi ) 
 j i

>


*
SocialWelfare( w )    v j (t j , wi ) 
 j i

*
• Impossible since w* maximizes social welfare!
26
Truthfulness of VCG - intuition
• The trick is actually quite simple:
—By lying, players may be able to change the
outcome.
—But their utility depends not only on the
outcome, but also on their payments.
—With VCG payments, the utility of each player is
the total efficiency.
 Therefore, players want the efficient outcome
to be chosen. Lying my ruin this.
27
The VCG family
• From the proof, we can see that the VCG
mechanism is actually a family of mechanisms.
• The VCG mechanism:
– Outcome w* is chosen.
– Each bidder pays:
*
*
*
v
h
(
(
t
t
)
,
w

)

v
(
t
v
,
w
(
t
)
,
w
)
 j ij 

i
j j j
j
j i
j i
j i
This could be any function of
the other bids.
28
The VCG family
• From the proof, we can see that the VCG
mechanism is actually a family of mechanisms.
• The VCG mechanism:
– Outcome w* is chosen.
– Each bidder pays:
h(ti )   v j (t j , w* )
j i
• Choosing h(ti )   v j (t j , w*i )
j i
ensures individual rationality (when values are positive)
(the utility of each player is never negative, why?)
and no positive transfers
(players are not paid to participate, why?).
29
Single vs. Multi parameter
We actually proved before how to implement the
efficient outcome:
–
•
•
What do VCG mechanisms add?
But, this holds for very specific environments:
players’ values are single parameter
–
–
•
Max{v1,….,vn} is a monotone function  we know how to
construct mechanisms implementing it.
That is, can be represented by a single real number (or
more formally, an ordered space).
We needed the concept of “raising the value of a player”
which implicitly implies an ordered space.
The VCG mechanism is more general: multiparameter domains.
30
Single vs. Multi parameter (cont.)
•
What we learnt in previous classes holds for very
specific environments:
players’ values are single parameter
–
That is, can be represented by a single real number (or
more formally, an ordered space).
•
–
•
Even the interdependent/correlated models.
We needed the concept of “raising the value of a player”
which implicitly implies an ordered space.
The VCG mechanism is more general:
multi-parameter domains.
–
Even if the private value consists of many values (as in
multi-unit auctions).
31
Single vs. Multi parameter (cont.)
•
From a mechanism design point of view, the
difference between single- and multi parameter
domains is huge:
–
The single parameter case is well-understood.
•
–
Multi-parameter are mostly still an open problem
•
•
efficient (Vickrey) auctions, optimal (Myerson) auctions,
characterization of implementable social-choice functions.
For example, no-one knows what is the optimal (revenue maximal)
auctions even for 2 bidders and 2 items.
VCG is one of the few general results known for multidimensional domains.
–
But still, most real problems are multi dimensional. We will consider
them in the coming classes.
32
Outline
1. Some examples
2. VCG idea – intuition
3. Formal part:
1. Mechanism design model
2. The VCG mechanism
3. Proof: VCG is truthful
4. Roommates example
33
Example 1: Roommates buy TV
• TV cost $100
• Bidders are willing to pay v1 and v2
– Private information.
• VCG ensures:
– Efficient outcome (buy if v1+v2>100)
– Truthful revelation.
In our model:
Welfare when buying: v1+v2
Welfare when not buying: 100 (saved the construction cost)
34
Example 1: Roommates buy TV
• Let’s compute VCG payments.
• Consider values v1=70, v2=80.
– With player 1: value for the others is 80.
– Without player 1: welfare is 100.
 p1= 100-80=20
– Similarly: p2 = 100-70 = 30
– Total payment received: 20+30 < 100
• Cost is not covered!
In general, p1=100-v2, p2=100-v1
p1+p2 = 100-v1+100-v2 = 100-(v1+v2-100) < 100
• Whenever we build, cost is not covered.
35
Example 1: Roommates buy TV
100
Payment of agent 1
80
v2
Needed to cover the cost
Payment of agent 2
0
0
v1
70
100
36
Example 1: Roommates buy TV
Conclusion: in some cases, the VCG mechanism is
not budget-balanced.
(spends more than it collects from the players.)
This is a real problem!
There isn’t much we can do:
It can be shown that there is no mechanism that
is both efficient and budget balanced.
– Even in simple settings: one seller and one buyer with
private values.
– “Myerson-Satterthwaite theorem”
37
Roommates (cont.)
Now, assume that the values are v1=110, v2=130.
How much each one pays (in VCG)?
0
Reason: agents do not affect the outcome
Players that affect the outcome: pivots.
Therefore, the VCG mechanism is also known as
the pivot mechanism.
38
Context: Public goods
• The roommate problem is knows as the
“public good” problem.
• Consider a government that wants to build
a bridge.
– When to build? If the total welfare is greater
than the cost.
– How the cost is shared?
– Efficiency vs. Budget Balance (cannot achieve
both).
• Another example: cable infrastructure.
39
More problems with VCG
• We saw one important flaw of VCG
mechanisms: not budget balanced
• Other problems with VCG:
– Auctions with externalities
– Collusions
– False name bids
– Revenue monotonicity
40
Summary: VCG
• Maximizing efficiency is desired in various settings.
• We saw: one can always achieve this with
(dominant-strategy) equilibrium.
– “implementation”
• This is the only general goal that is known to be
“implementable”.
• Pros: No distributional assumptions, strong
equilibrium concept, individually rational.
• Cons: not budget balanced, prone to other
manipulations.
41
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