Auction Theoryתכנון מכרזים ומכירות פומביות

Topic 7 – VCG mechanisms

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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

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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

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Previously …

• We saw that in single-item auctions we can maximize efficiency with dominant strategies.

• Can this be achieved in other models?

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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.

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Outline1. Some examples

2. VCG idea – intuition

3. Formal part:1. Mechanism design model2. The VCG mechanism3. Proof: VCG is truthful

4. Roommates example

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Auctions scheme

v1

v2

v3

v4

b1

b2

b3

b4

values bids

winner

payments$$$

Mechanism Design scheme

t1

t2

t3

t4

b1

b2

b3

b4

types Bids/reports

outcome

payments

p1,p2,p3,p4

Social planner

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?

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I only watch sports

!רק אירוויזיון

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

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.

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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 =

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Welfare of the other players from the chosen outcome

Optimal welfare (for the other players) if

player i was not participating.

Vickrey-Clarke-Groves (VCG) mechanisms

• Let’s see how this payment rule works on our examples:

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Pi =Welfare of the other

players from the chosen outcome

Optimal welfare (for the other players) if

player i was not participating.

VCG idea in single item auctions

• Pi=

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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

VCG in 5-item auctions

• pi=

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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?

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)

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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

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Mechanism Design scheme

t1

t2

t3

t4

b1

b2

b3

b4

types Bids/reports

outcome

payments

p1,p2,p3,p4

Social planner

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

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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.

Formal model

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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

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.

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• 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

TruthfulnessTheorem (Vickrey-Clarke-Groves):

In the VCG mechanism, truth-telling is a dominant strategy for all players.

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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.

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Now, proof.

We will show: no matter what the others are doing,

lying about my type will not help me.

Truthfulness of VCG - Proof

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• 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.

Truthfulness of VCG - Proof

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• 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 ),()( *

Truthfulness of VCG - Proof

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• 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!

Truthfulness of VCG - intuition

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• 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.

The VCG family

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• 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 ),()( *

The VCG family

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• 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 ),()( *

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).

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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.• 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.

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Outline1. Some examples

2. VCG idea – intuition

3. Formal part:1. Mechanism design model2. The VCG mechanism3. Proof: VCG is truthful

4. Roommates example

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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)

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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

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Payment of agent 1

Payment of agent 2

70

80

0 1000

100

v1

v2

Needed to cover the cost

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”

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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.

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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.

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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

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