המכרזים של הערוצים הייעודיים סמינר- מכרזים ורשיונות בתקשורת תל-אביב, 30 ביולי 2000
Auction Theory תכנון מכרזים ומכירות פומביות
description
Transcript of Auction Theory תכנון מכרזים ומכירות פומביות
![Page 1: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/1.jpg)
Auction Theoryתכנון מכרזים ומכירות פומביות
Topic 7 – VCG mechanisms
1
![Page 2: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/2.jpg)
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
![Page 3: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/3.jpg)
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
![Page 4: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/4.jpg)
Previously …
• We saw that in single-item auctions we can maximize efficiency with dominant strategies.
• Can this be achieved in other models?
4
![Page 5: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/5.jpg)
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
![Page 6: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/6.jpg)
Outline1. Some examples
2. VCG idea – intuition
3. Formal part:1. Mechanism design model2. The VCG mechanism3. Proof: VCG is truthful
4. Roommates example
6
![Page 7: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/7.jpg)
Auctions scheme
v1
v2
v3
v4
b1
b2
b3
b4
values bids
winner
payments$$$
![Page 8: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/8.jpg)
Mechanism Design scheme
t1
t2
t3
t4
b1
b2
b3
b4
types Bids/reports
outcome
payments
p1,p2,p3,p4
Social planner
![Page 9: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/9.jpg)
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?
9
I only watch sports
!רק אירוויזיון
![Page 10: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/10.jpg)
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.
10$70 $30 $27 $25 $12 $5 $2
![Page 11: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/11.jpg)
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
![Page 12: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/12.jpg)
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 =
12
Welfare of the other players from the chosen outcome
Optimal welfare (for the other players) if
player i was not participating.
![Page 13: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/13.jpg)
Vickrey-Clarke-Groves (VCG) mechanisms
• Let’s see how this payment rule works on our examples:
13
Pi =Welfare of the other
players from the chosen outcome
Optimal welfare (for the other players) if
player i was not participating.
![Page 14: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/14.jpg)
VCG idea in single item auctions
• Pi=
14
Welfare of the other players from the chosen outcome
Optimal welfare (for the other players) if
player i was not participating.
= 0.
When i wins, the total value of the other is 0.
= 2nd-highest value.
When i is not playing, the welfare will be the
second highest.
By VCG payments, winners pay the 2nd-highest bid
![Page 15: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/15.jpg)
VCG in 5-item auctions
• pi=
15
Welfare of the other players from the chosen outcome
Optimal welfare (for the other players) if
player i was not participating.
$70 $30 $27 $25 $12 $5 $2
=30+27+25+12.
The other four winners.
=30+27+25+12+5
The five winners when i is not playing.
pays 5
pays ??
What is my VCG payment?
![Page 16: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/16.jpg)
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
![Page 17: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/17.jpg)
Outline1. Some examples
2. VCG idea – intuition
Formal part:1. Mechanism design model2. The VCG mechanism3. Proof: VCG is truthful
3. VCG: the negative side
17
![Page 18: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/18.jpg)
Mechanism Design scheme
t1
t2
t3
t4
b1
b2
b3
b4
types Bids/reports
outcome
payments
p1,p2,p3,p4
Social planner
![Page 19: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/19.jpg)
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
19
n
iii wtv
1
),(
• 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.
![Page 20: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/20.jpg)
Formal model
20
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
iii wtv
11),(
n
iii wtv
12 ),(
n
iii wtv
13),(
n
iii wtv
14 ),(
n
iii wtv
15 ),(
Assume:w5 maximizes efficiency
w*=w5
![Page 21: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/21.jpg)
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.
21
• The VCG mechanism:– Outcome w* is chosen.– Each bidder pays:
The total value for the other when player i is not
participating
ij
jjij
ijj wtvwtv ),(),( **
The total value for the others when i participates
![Page 22: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/22.jpg)
TruthfulnessTheorem (Vickrey-Clarke-Groves):
In the VCG mechanism, truth-telling is a dominant strategy for all players.
22
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.
![Page 23: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/23.jpg)
23
Now, proof.
We will show: no matter what the others are doing,
lying about my type will not help me.
![Page 24: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/24.jpg)
Truthfulness of VCG - Proof
24
• The VCG mechanism:– Outcome w* is chosen.– Each bidder pays:
ijjj
ijijj wtvwtv ),(),( **
• Method of proof: we will assume that there is a profitable lie for some player I, and this will result in a contradiction.
![Page 25: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/25.jpg)
Truthfulness of VCG - Proof
25
• Buyer’s utility (when w* is chosen):iii pwtv ),( *
• Assume: bidder i reports a lie t’ outcome x is chosen.
ijjj
ijijjii wtvwtvwtv ),(),(),( ***
ijijj
ijjjii wtvwtvwtv ),(),(),( ***
ijijj
n
iii wtvwtv ),(),( *
1
*
ijijj wtvwareSocialWelf ),()( **
• Buyer’s utility (when x is chosen):
ijijj wtvxareSocialWelf ),()( *
![Page 26: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/26.jpg)
Truthfulness of VCG - Proof
26
• Buyer’s utility from truth (w* is chosen):
ijijj wtvwareSocialWelf ),()( **
• Buyer’s utility from lying (x is chosen):
ijijj wtvxareSocialWelf ),()( *
• Lying is good when:
ijijj wtvxareSocialWelf ),()( *
ijijj wtvwareSocialWelf ),()( **>
• Impossible since w* maximizes social welfare!
![Page 27: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/27.jpg)
Truthfulness of VCG - intuition
27
• 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.
![Page 28: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/28.jpg)
The VCG family
28
• 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:
ijjj
ijijj wtvwtv ),(),( **
This could be any function of the other bids.
ij
jji wtvth ),()( *
![Page 29: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/29.jpg)
The VCG family
29
• 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:
• Choosing 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?).
ij
ijji wtvth ),()( *
ij
jji wtvth ),()( *
![Page 30: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/30.jpg)
Single vs. Multi parameterWe actually proved before how to implement the
efficient outcome:– Max{v1,….,vn} is a monotone function we know how to
construct mechanisms implementing it.• What do VCG mechanisms add?• But, this 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).– 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).
30
![Page 31: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/31.jpg)
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
![Page 32: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/32.jpg)
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.• efficient (Vickrey) auctions, optimal (Myerson) auctions,
characterization of implementable social-choice functions.– Multi-parameter are mostly still an open problem
• 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 multi-dimensional domains.
– But still, most real problems are multi dimensional. We will consider them in the coming classes.
32
![Page 33: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/33.jpg)
Outline1. Some examples
2. VCG idea – intuition
3. Formal part:1. Mechanism design model2. The VCG mechanism3. Proof: VCG is truthful
4. Roommates example
33
![Page 34: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/34.jpg)
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
![Page 35: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/35.jpg)
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
![Page 36: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/36.jpg)
Example 1: Roommates buy TV
36
Payment of agent 1
Payment of agent 2
70
80
0 1000
100
v1
v2
Needed to cover the cost
![Page 37: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/37.jpg)
Example 1: Roommates buy TVConclusion: 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
![Page 38: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/38.jpg)
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
![Page 39: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/39.jpg)
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
![Page 40: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/40.jpg)
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 monotonicity40
![Page 41: Auction Theory תכנון מכרזים ומכירות פומביות](https://reader036.fdocuments.net/reader036/viewer/2022062302/5681640d550346895dd5b4cc/html5/thumbnails/41.jpg)
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