mechanism design tutorial - ernetdrona.csa.iisc.ernet.in/.../parkes-mechanismdesign.pdf · ·...

Mechanism Design TutorialDavid C. Parkes, Harvard UniversityIndo-US Lectures Week in Machine Learning, Game Theory and Optimization

1

Outline

� Classical mechanism design� Preliminaries (DRMs, revelation principle)

� Positive results – Groves, Single-parameter (Myerson)▪ min makespan task assignment

� Negative results – Gibbard-Satterthwaite

� Algorithmic mechanism design� Knapsack auction

� Price-of-anarchy analysis

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Mechanism design� � alternatives; � ∈ � agents, value ��: � ↦ , �� ∈ � Utility �� , � � �� Design a game Γ ��, … , �� ∈ � � �, attain

desiderata in equilibrium

3

��

��

1Agent 1

Agent n

… � ∈ �

��, … , �� ∈ �

Mechanism design� � alternatives; � ∈ � agents, value ��: � ↦ , �� ∈ � Utility �� , � � �� Design a game Γ ��, … , �� ∈ � � �, attain

desiderata in equilibrium

4

��

��

1Agent 1

Agent n

… � ∈ �

��, … , �� ∈ �

Examples

� Auction; e.g., servers, bandwidth, ad space

� Coordination; e.g., meetings, tasks

� Public choice; e.g., build a new school

� Matching; e.g., residents to hospitals

� Desiderata: efficiency, maxmin fairness, envy-free,

participation, revenue, budget-balance …

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Game theory for MD� Incomplete information game; Valuation �� ∼ �� Behavior �� ; Strategy �� ∈ �

� Dominant strategy equilibrium

� �� , ��

� �� , �� , all �,all ��, all ��

� Bayes-Nash equilibrium

� ! "#$%��

� ��∗ �� , ��

∗ �� ' � ! "#$%��

� �� , ��∗ �� ',

all �,all ��

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� �� , ��

� �� , �� , all �,all ��, all ��


� ! "#$%��

� ��∗ �� , ��

∗ �� ' � ! "#$%��

� �� , ��∗ �� ',

all �,all ��

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� �� , ��

� �� , �� , all �,all ��, all ��


� ! "#$%��

� ��∗ �� , ��

∗ �� ' � ! "#$%��

� �� , ��∗ �� ',

all �,all ��

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Implementation

� Mechanism Γ implements a social choice

function (: � ↦ � if Γ� �∗ � � ()�* for all

� � ��, … , �� ,in equilibrium �∗.

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��

��

1

… � ∈ �

��, … , �� ∈ �

��∗)��*

��∗)��*

Direct Revelation Mechanisms

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, �- ∈ �

. �- ∈ �

�′�

�′�

1

… ��

∗)��*

��∗)��*

� DRM Γis (Dom/Bayes) incentive compatible if truthful reporting is a (DSE/BNE).

� Choice rule ,; Payment rule .

(“Strategyproof, “Truthful.”)

Direct Revelation Mechanisms

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, �- ∈ �

. �- ∈ �

�′�

�′�

1

… ��

∗)��*

��∗)��*

� DRM Γis (Dom/Bayes) incentive compatible if truthful reporting is a (DSE/BNE).

� Choice rule ,; Payment rule .

(“Strategyproof, “Truthful.”)

Revelation Principle

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��1 ��∗)��*

��∗)��*

… �

��, … , ��

… �′�

�′�

� Theorem: Any scf ( implemented by Γ can be implemented by an incentive compatible DRM.

*Positive results **Negative results

Outline


� Positive results – Groves, Single-parameter (Myerson)▪ Min makespan




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Groves mechanism� ( � ∈ arg max

5∑ 7�� 8 9)�*� ; 7� ; 0

� ,= �- ∈ arg max5

∑ ��-)�*�

� .=,� �- � >� ��- � ∑ �?

-)�*?@� , for � � ,)�-*

� Utility: �� , �- 8 ∑ �?- , �- � >�)��

- *?@�� ⇒ truthful! (and efficient!)

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(arbitrary fcn) 0

Affine maximization (Simple case, 7� � 1, 9 � � 0 ) )


5∑ 7�� 8 9)�*� ; 7� ; 0

� ,= �- ∈ arg max5

∑ ��-)�*�

� .=,� �- � >� ��- � ∑ �?

-)�*?@� , for � � ,)�-*

� Utility: �� , �- 8 ∑ �?- , �- � >�)��

- *?@�� ⇒ truthful! (and efficient!)

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(arbitrary fcn)



5∑ 7�� 8 9)�*� ; 7� ; 0

� ,= �- ∈ arg max5

∑ ��-)�*�

� .=,� �- � >� ��- � ∑ �?

-)�*?@� , for � � ,)�-*

� Utility: �� , �- 8 ∑ �?- , �- � >�)��

- *?@�

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(arbitrary fcn) 0


⇒ truthful! (and efficient!)

VCG mechanism

� Special case of Groves.

� Payment rule: Negative externality

� ."BC,� �- � ∑ �?-)��*?@� � ∑ �?

-)�*?@� ,

for � � ,=)�-*, �� ,=)��- *.

� Truthful, efficient, participation, no-deficit*

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(Negative result (Roberts): if � � 3, � EF , then only truthful

mechanisms are Groves mechanisms.)

VCG mechanism

� Special case of Groves.

� Payment rule: Negative externality

� ."BC,� �- � ∑ �?-)��*?@� � ∑ �?

-)�*?@� ,

for � � ,=)�-*, �� ,=)��- *.

� Truthful, efficient, participation, no-deficit*

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(Negative result (Roberts): if � � 3, � EF , then only truthful

mechanisms are Groves mechanisms.)

VCG Example 1

� Single-item Auction

� Values $10, $4, $2

� ,)�*: assign to A1

� .� � �

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VCG Example 1

� Single-item Auction

� Values $10, $4, $2

� ,)�*: assign to A1

� .� � � 4 � 0 � 4;zero to others� … a second-price auction

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VCG Example 2

� Combinatorial Auction

� Items {A,B,C}

� ,)�*: )∅, �, �*� tJ � �

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agent A B AB

1 0 0 10

2 6 0 6

3 0 8 8

VCG Example 2

� Combinatorial Auction� Items {A,B,C}

� ,)�*: )∅, �, �*� tJ � � 10 � 8 � 2� .M v � 10 � 6 � 4� Agent 1 pays zero.

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agent A B AB

1 0 0 10

2 6 0 6

3 0 8 8

(revenue low, and NP-hard winner determination.)

VCG Example 3

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� Agents= Edges; Value = -Cost

� Externality: )- total cost without) – (- total cost

with); e.g., for edge 17 this is -90-(-40)=-50.

VCG Example 4

� Double Auction

� A1: buyer, value $10

� A2: seller, value $8

� ,)�*: trade

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VCG Example 4

� Double Auction

� A1: buyer, value $10

� A2: seller, value $8

� ,)�*: trade� Payments

� A1: 8 � 0 � 8 (pays $8)

� A2: 0 � 10 � �10 (paid $10!)

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Single-parameter domains

� Private infoS� ∈ ; induces �� S� , � ∈ � E.g., Min makespan task assignment

� Agents A1, A2. Tasks T1,T2,T3 (sizes 1, 2 and 4)

� Private :: Unit processing time (S� ��

J, SJ � 1*

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Min make-span =

max(2.5,2)=2.5

What would VCG do?

Single-parameter domains

� Private infoS� ∈ ; induces �� S� , � ∈ � E.g., Min makespan task assignment

� Agents A1, A2. Tasks T1,T2,T3 (sizes 1, 2 and 4)

� Private :: Unit processing time (S� ��

J, SJ � 1*

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Min make-span =

max(2.5,2)=2.5

What would VCG do?

Single-parameter domains�� S� , � � S� � T� �

� Allocation rule ,: U, ∞ � ↦ �� Fix S��

- , monotonic ,

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private S� ∈ %U, ∞*known T�: � ↦ Esummarization fcn

(1) Auction: S�is value, T� � is (prob) agent allocated?

(2) Task assignment: S� is (−processing time), T� � total load

T�), S� *

S� S�

Single-parameter domains�� S� , � � S� � T� �

� Allocation rule ,: U, ∞ � ↦ �� Fix S��

- , monotonic ,

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private S� ∈ %U, ∞*known T�: � ↦ Esummarization fcn

(1) Auction: S�is value, T� � is (prob) agent allocated?

(2) Task assignment: S� is (−processing time), T� � total load

T�), S� *

S� S�

Myerson mechanism (s.p. domain)

� Given monotonic ,, then mechanism truthful if:

� .� S- � S�-T� , S- � W T� , X, S��

- YX � >��)S��- *

Z$[

\]^

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0

106

1

=critical

value in 0-1

domains

Myerson mechanism (s.p. domain)

� Given monotonic ,, then mechanism truthful if:

� .� S- � S�-T� , S- � W T� , X, S��

- YX � >��)S��- *

Z$[

\]^

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0

10

(basically

necessary)

6

1

=critical

value in 0-1

domains

value

payment

_-value

_-payment

� Unit processing time (S� ��

J, SJ � 1*

� Thm. VCG is an n-approx, and truthful.

� Proof. UB: `a

`bcde ∑ fg��,?? /)1/i* ∑ fg��,?? � i

� LB: i machines, n tasks (size 1). � Machine 1 unit cost 1. Machine 2..n unit cost 1 8 j, j ; 0� Min makespan 1 8 j. VCG make-span n.

� limm→oi/)1 8 j* � i

Min makespan scheduling

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`a Z

`bcd)Z*p�cc-approx:

(Archer and Tardos’01)


J, SJ � 1*


� Proof. UB: `a

`bcde ∑ fg��,?? /)1/i* ∑ fg��,?? � i

� LB: i machines, n tasks (size 1). � Machine 1 unit cost 1. Machine 2..n unit cost 1 8 j, j ; 0� Min makespan 1 8 j. VCG make-span n.

� limm→oi/)1 8 j* � i


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`a Z




J, SJ � 1*


� Proof. UB: `a

`bcde ∑ fg��,?? /)1/i* ∑ fg��,?? � i

� LB: i machines, n tasks (size 1). � Machine 1 unit cost 1. Machine 2..n unit cost 1 8 j, j ; 0� Min makespan 1 8 j. VCG makespan n.

� limm→oi/)1 8 j* � i


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`a Z



What else can we do?

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LexOpt mechanism

� Adopt ,)�*to min makespan, particular tie-breaking rule.

� Thm. LexOpt is monotonic (S’� ; S�, load�’� � ��)

� Suppose �’ r �.

� (Case 1) ��iZ$[ �’ � ��iZ$

�’ . ��iZ$[ � e

��iZ$� e ��iZ$

)�’* � ��iZ$[)�’*. Contradiction.

� (Case 2)��iZ$[ �’ p ��iZ$

�’ .

�S�T� � e ��iZ$� e ��iZ$

)�’* � �S�T�)�’*, since i is

bottleneck in �’ at S�. Monotone.

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LexOpt mechanism




� (Case 1) ��iZ$[ �’ � ��iZ$

�’ . ��iZ$[ � e

��iZ$� e ��iZ$


� (Case 2)��iZ$[ �’ p ��iZ$

�’ .

�S�T� � e ��iZ$� e ��iZ$

)�’* � �S�T�)�’*, since i is


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LexOpt mechanism




� (Case 1) ��iZ$[ �’ � ��iZ$

�’ . ��iZ$[ � e

��iZ$� e ��iZ$


� (Case 2)��iZ$[ �’ p ��iZ$

�’ .

�S�T� � e ��iZ$� e ��iZ$

)�’* � �S�T�)�’*, since i is


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Aside: Computation

� min- makespan is NP-hard

� Standard PTAS optimizes over a restricted range of

candidate assignments, set construction violates

monotonicity.

� Exists a monotone PTAS, both randomized and

deterministic.

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(Dhangwatnotai et al. 11, Christodoulou and Kovacs 10)

Outline






40

Gibbard-Satterthwaite

� No money. ≡ all strict preferences (e.g.,

� ≻ � ≻ f)

� � � 3, ,)�* onto. � Dictatorial: Same agent always gets top choice

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Gibbard-Satterthwaite

� No money. ≡ all strict preferences (e.g.,

� ≻ � ≻ f)

� � � 3, ,)�* onto. � Dictatorial: Same agent always gets top choice

� Theorem. The only truthful mechanisms are

dictatorial with all strict prefs, |A|>=3 onto.

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Simple proof (T ⇒ u) for 2-agent case

� abcdef → badcfd

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�� ′�

If v wx, w�x- � y,then

v wx-, w�x

- � y.

Proof. If report ..cd… and

get d, then report ..dc…

and get g, if g > {c,d} then

“cd” type deviates; else,

“dc” type deviates.

Monotonicity (M)

� If every agent z ≻ {then don’t pick b.

Proof. Suppose pick b. Still

pick b if all a>b > … (M)

Onto, so exists v with

x(v)=a. Still pick a if all

a>b>… (M).

Contradiction.

Consistency (C)

� “1 is a dictator on a”: if 1

reports a top, a picked

� P1: a>b>c; b>a>c.

� Can’t pick c (C). Consider a.

� P2: a > b > c; b > c > a

� Can’t pick c (C). Can’t pick b

(T). Select a.

� Consider any P3, top(1)=a

Pick a (M, consider P2 -> P3)

� Argue 1 also dictator on b, c.

Impossibility



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�� ′�


v wx-, w�x

- � y.






Monotonicity (M)

� If every agent .. z ≻ { .. then don’t pick b.





a>b>… (M).

Contradiction.

Consistency (C)



� P1: a>b>c; b>a>c.


� P2: a > b > c; b > c > a


(T). Select a.




Impossibility



45

�� ′�


v wx-, w�x

- � y.

Monotonicity (M) Consistency (C)



� P1: a>b>c; b>a>c.


� P2: a > b > c; b > c > a


(T). Select a.




Impossibility










a>b>… (M).

Contradiction.




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�� ′�


v wx-, w�x

- � y.

Monotonicity (M) Consistency (C)



Impossibility

P1: a>b>c; b>a>c.

Can’t pick c (C). Consider a.

P2: a > b > c; b > c > a

Can’t pick c (C). Can’t pick b (T).

Select a.

Consider any P3, top(1)=a


Argue 1 also dictator on b, c.










a>b>… (M).

Contradiction.


(Svensson’99)

Outline






47

Algorithmic Mechanism Design

� New concern is to obtain computational tractability

as well as incentive compatibility

� Emphasis also placed on bidding languages,

preference elicitation.

48

Knapsack auction

� | items, agent � value S� for }� units (known)� Goal: maximize total value. 0-1 knaspack problem. NP-hard

� Can’t use VCG. 2-approx: `bcd Z

`a Ze 2,all S

� x: order by decreasing S’�/}�. � If ∑ S’�~ � max S ’�sell �1 … �� else sell to >� Charge critical value (Myerson)

� Example: $5@2, $6@1, $6@3, $12@5; supply 5 units � Compare (6+5,12) -> allocated to A4. Pay $11. � Suppose A2 reports 8? Now {1,2} allocated. A2 pays $7.

49

(Mu’alem and Nisan’08)

Knapsack auction



`a Ze 2,all S



50


Knapsack auction



`a Ze 2,all S



51


Knapsack auction - Analysis

� Theorem. Truthful and 2-approx.

� Monotone: (Case 1) Allocated and in {1..k}. Still in. (Case 2)

Allocated and highest. May cause {1..k} to win but still in.

� 2-approx: suppose � p i.

� �� e �� ∑ S?

~?]� 8 �S~E� e ∑ S?

~E�?]� e ∑ S?

~?]� 8

max?

S? � ��..~ 8 �� e 2max)��..~ , ��* � 2�g

52







� �� e �� ∑ S?

~?]� 8 �S~E� e ∑ S?

~E�?]� e ∑ S?

~?]� 8

max?

S? � ��..~ 8 �� e 2max)��..~ , ��* � 2�g

53







� �� e �� ∑ S?

~?]� 8 �S~E� e ∑ S?

~E�?]� e ∑ S?

~?]� 8

max?

S? � ��..~ 8 �� e 2max)��..~ , ��* � 2�g

54


Outline






55

Price of anarchy + MD

� PoA: worst-case ratio of optimal obj to obj in equilibrium

� Extension theorems (Roughgarden, 09, 12; Lucier, Paes Leme 11; Syrgkanis 12,

Syrgkanis Tardos 13)

� For auctions:

� PoA for complete-information auction -> PoA in Bayes Nash

equilibrium

� PoA for complete-information auction -> PoA for composition of

auctions.

56

Price of anarchy + MD

� PoA: worst-case ratio of optimal obj to obj in equilibrium

� Extension theorems (Roughgarden, 09, 12; Lucier, Paes Leme 11; Syrgkanis 12,

Syrgkanis Tardos 13)

� For auctions:

� PoA for complete-information auction under property P -> PoA in

Bayes Nash equilibrium

� PoA for complete-information auction under property P -> PoA for

composition of auctions.

� Comment: now worry about all equilibrium

57

Example: Extension from NE to BNE

� For any b, exists ��- s.t. ∑ �� ’� , �� 8 ��

��.)S* (smoothness)

� Do this under P: �’� only depends on S�

� If b is a NE then ��U � � ∑ �� , �� ∑ �� ’� , ��⇒ ��U � 8 �� .)�*⇒ �� 8 � � 1 �� .)�*⇒ �� 8 � � 1 �� .)�*

⇒ �� e ��.)�*,and �� e �/�� Extends to BNE immediately

58



��.)S* (smoothness)

� Do this under P: �’� only depends on S�

� If b is a NE then ��U � � ∑ �� , �� ∑ �� ’� , ��⇒ ��U � 8 �� .)S*

⇒ �� 8 � � 1 �� .)S*⇒ �� 8 � � 1 �� .)S*

⇒ �� .)S*,and �� e �/�� Extends to BNE immediately

59



��.)S* (smoothness)� Do this under P: �’� only depends on S�

� If b is a NE then ��U � � ∑ �� , �� ∑ �� ’� , ��⇒ ��U � 8 �� .)S*

⇒ �� 8 � � 1 �� .)S*⇒ �� 8 � � 1 �� .)S*

⇒ �� .)S*,and �� e �/�

� Extends to BNE immediately

60

Apply to FPSB auction

� PoA for FPSB is 1! But, want to bound under P.

� Want: ∑ �� ’� , �� 8 �� .)S*

� For any bids, ��Z$

J, �� 8 � � � S� /2

� Either gain Z$

Jfrom deviation, or � � ;

Z$

J

� ⇒ ��Z$

J, �� 8 � � ,�

∗ S �Z$

J,�

∗)S*

� ⇒ ∑ �� ’� , �� 8 ��

J��.)�* ; thus PoA e 2

61



� Want: ∑ �� ’� , �� 8 �� .)S*


J, �� 8 � � � S� /2

� Either gain Z$


Z$

J

� ⇒ ��Z$

J, �� 8 � � ,�

∗ S �Z$

J,�

∗)S*

� ⇒ ∑ �� ’� , �� 8 ��


62



� Want: ∑ �� ’� , �� 8 �� .)S*


J, �� 8 � � � S� /2

� Either gain Z$


Z$

J

� ⇒ ��Z$

J, �� 8 � � ,�

∗ S �Z$

J,�

∗)S*

� ⇒ ∑ �� ’� , �� 8 ��


63



� Want: ∑ �� ’� , �� 8 �� .)S*


J, �� 8 � � � S� /2

� Either gain Z$


Z$

J

� ⇒ ��Z$

J, �� 8 � � ,�

∗ S �Z$

J,�

∗)S*

� ⇒ ∑ �� ’� , �� 8 ��


64

Direction for AMD?

� Lucier and Borodin (2010)

� First price Single-minded Combinatorial auction

� Optimal allocation rule, the PoA is m (m items)

� But, if the allocation rule is approximate (sqrt-m greedy),

then (½,sqrt(m)) – smooth, and O(sqrt(m)) PoA.

� Design mechanisms that are smooth, and provide good

worst-case properties.

65

References

� See “Economics and Computation”, Parkes and

Seuken CUP (forthcoming, 2014)

� Chapters 8 and 10

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