Mechanism DesignMilan Vojnović

Lab tutorial, March 2010

3

• Mechanism design is about designing a game so as to achieve a desired goal

1v

2v

nv

...

0v

b1

b2

bn

Input: ),,,( 21 nbbbb Output: (x, p)

),,,( 21 nxxxx ),,,( 21 npppp allocation: payment:

other input

5

Ex 1: sponsored search

6

a

Ex 1: sponsored search (cont’d)

Position 1

Position 2

Position 3

Position 4

Position 5

advertisers

7

Ex 2: online contests

8

Ex 2 (cont’d)

9

Ex 3: resource allocation... communication networks, data centres, distributed systems

x1

x2

C/w

C/w

C

w

P

w

1

1

x1

x2

C

C2

C2

C3

C1

x2

x1

P

C2C1

x1

C3x2

P

C

C

x2

x1

C

x1x2

10

Part of Computer Science

11

... this mechanism is strategy proof ... however, it is not ex-post individually

rational ... there is a high efficiency loss ...

U(x) – px ...

... maximizes virtual surplus...

12

Some developments

1999 Algorithmic mechanism design (Nisan & Ronen)

2001 Competitive auctions and digital goods (Goldberg et al)

1961 Vickery’s auction

Algorithmic game theory (Nisan et al)2007

1997 Overture’s auction; network resource allocation (Kelly)

2002 Generalized Second Price Auction

......

...

1981 Myerson’s optimal auction design...

13

Active research area

• Algorithmic problems– Efficient and user-friendly mechanisms– Prior-free and online learning– Alternative solutions concepts– Computational / communication complexity

• The use of models to better understand and inform design

• Realistic models of rational agents

14

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Source: Ariely D. (2008)

15

This tutorial agenda

• Design objectives

• Vickery & Myerson auctions

• Prior-free auctions

• Auctions for resource allocation

16

Standard goals

Max social welfare“efficient”

)()(max xcxUi

ii

)(max xcpi

i Max seller’s profit“optimal auction design”

17

Examples of other goals

min makespan, max flow, max weighted flow

jobs

v1 v2 vn

...

processing speed

machines

18

Standard constraints• Incentive-compatibility

= it is to the agents’ best interests to report true types

Also known as implementation theory, the theory of incentives, or strategy-proof mechanisms

• Individual rationality

= ensure the agents’ profits are non-negative

Also known as voluntary participation

19

Two kinds of games

• Incomplete information • Complete information

– Types are private information

– Types drawn from a distribution F

– F is public information

– Types are public information

20

Vickery auctionfor allocation of a single item

• Allocation to the buyer with highest bid• Payment equal to the second highest bid

21

Incentive compatibility

iv ibjijb

max

equal profit

iv ibjijb

max

win only by overbiddingdominated by truthful

iv ib jijb

max

equal profit = 0

ib ivjijb

max

equal profit

ib ivjijb

max

win only if truthful

ib iv jijb

max

lose in either case

o.w.0maxif max

profit jijijijii

bbbv

22

• Vickery auction is a truthful efficient auction

But how do I maximize my profit?

23r

)()1( vf

)(1 )1( rF

v

... setting a reserve price should work

][max)(1 )1( vvPvF ii

)](1[ )1( vFv

v

)](1[max)](1[ )1(0)1( vFvrFrv

BA A B

24

Myerson’s optimal auction design• A mechanism is truthful if and only if for every

buyer i and bids of other agents b-i fixed:

C1) allocation xi(b-i, bi) is non-decreasing with bi

ib

iiiiiii dzzbxbbxbbp0

),(),()(C2) payment:

),( zbx ii

z

),( zbx ii

zib

25

Incentive compatibility

• Buyers’ profit: )()()( bpbxvb iiii

),( zbx ii

ziv

A Ai

ziv

B

BAi

ib

ziv

A BAi

ib

B

26

But how do I determine an optimal allocation?

27

• Under independent buyer’s valuations, every optimal allocation is a solution of

the virtual surplus maximization

)()(1

)(ii

iiiii vf

vFvv

i

iii xcxv )()( maximize

Virtual valuation:

28

Virtual valuation• Ex. 1 Fi(v) uniform on [0, hi]

)( ii v

iv

ih

ih

ii hv 2

• Ex. 2 Fi(v) = 1 - exp(-li)

)( ii v

iv

i1

iiv

1

29

Optimality of Vickery auction with reserve price

• Single-item auction• Independent and identical buyers • Strictly increasing virtual valuations

0)( riThe optimal is Vickery auction with the reserve price r:

30

Optimality of Vickery auction with reserve price (cont’d)

• Ex. F uniform [0, h], 2/hr

r

)(vfi

)(1 rFi

v

BA A B

31

Competitive framework for auctions

• Competitiveness to a profit benchmark B(v)

ii vxB max)(2

)()(

supvAvB

vCompetitive ratio for an auction A =

i

ivvB )(1Ex. 1 sum valuation

Ex. 2 max valuation

)(23 max)( iiivvB

Ex. 3 uniform pricing with at least two winners

32

Random reserve price auction (Lu at al 2006)

1- d

d

Run the second-price auction

Sample reserve price r from

212

for,)1/log(

)( bxbxx

xf

If b1 ≥ r then allocate the item to a buyer with highest bid

1b2b

33

Random reserve price (cont’d)

E[profit] =

E[social welfare] =

1)1log(hh

hd)1(

• A tighter expected revenue can be obtained using a successive composition of log(x+1)

• Can’t do a better expected revenue !

h = max valuation

34

Why incentive compatibility as a requirement?

• Pros– Simplifies buyer’s strategy – just report the type– Simplifies the problem for the designer

• Cons– Computational complexity

35

This tutorial agenda

• Design objectives

• Vickery & Myerson auctions

• Prior-free auctions

• Auctions for resource allocation

36

• Every time you send an email you participate in an auction

37

Kelly’s resource allocation

C

b1

bn

bi

payment by buyer i = bi

Cb

bx

jj

ii allocation to buyer i:

38

Kelly’s resource allocation (cont’d)• Extensions to networks of links:

the mechanism applied by each link

ib ib ib ib

scalar bids (TCP like)

1ib

2ib

3ib 4

ibvector bids

• Two user models

39

Kelly’s resource allocation (cont’d)

ipb

ibbU

i

i

i

)(max0

Cbpj

ji /• Price-taking users:

Under price-taking users with concave, utility functions, efficiency is 100%.

40

Johari & Tsitsiklis’ price-anticipating users

Under price-anticipating users with concave, non-negative utility functions, and vector bids, the worst-case efficiency is 75%.

ib

bib

bCUj

j

i

i

)(max

0User:

41

Full efficiency loss under scalar bids(Hajek & Yang 2004) Under price-anticipating users with concave, non-negative utility functions, and scalar bids, the worst-case efficiency is 0.

xxU )(1 xxU )(2xxUn )(

axxU )(0

anna

an

for ,)1(

efficiency2

an

11

• A worst-case: serial network of unit capacity links

42

But how do I maximize my profit?

43

Profit maximization by price discrimination

i

iii xxU )(' max

Px

over

ip price i)(' iii xUp

44

I’d like to run an auction!

45

The weighted proportional allocation mechanism

iC weighti ibi

i

jj

ii C

bb

x

• Allocation to buyer i:

• Payment by buyer i = bi

• Guarantees on social welfare and seller’s profit - Thanh-V. 2009

46

Some important aspects not discussed in this tutorial

• When truthfulness requires side-payments

• Frugality, envy-freeness

• Competitive guarantees of some auctions, ex. digital-goods auctions

• Computational complexity under incentive compatibility

47

Thank you for your attention!

48

Some references• Aggarwal G., Fiat A., Goldberg A. V., Hartline J. D., Immorlica N., Sudan Madhu, Derandomization of

auctions, STOC 2005. • Archer A. and Tardos E, Truthful Mechanisms for one-parameter agents, FOCS 2001. • Balcan M.-F., Blum A., Harline J. D., Mansour Y., Mechanism Design via Machine Learning, FOCS

2005. • Bulow J. and Klemperer P., Auctions versus negotiations, The American Economic Review, Vol 86, No

1, 1996.• DiPalantino D. and Vojnovic M., Crowdsourcing and all-pay auctions, ACM EC ‘09.• Edelman B., Ostrovsky M., Schwartz M., Internet Advertising and the Generalized Second Price

Auction: Selling Billion of Dollars Worth of Keywords, Working Paper, 2005. • Fiat A., Goldberg A. V., Hartline J. D., and Karlin A. R., Competitive Generalized Auctions, STOC 2002.• Goldberg A. V., Hartline J. D., Karlin A. R., Saks M., A lower bound on the competitive ratio of truthful

auctions, FOCS 2004. • Goldberg A. V, Hartline J. D., Wright A., Competitive Auctions and Digital Goods, SODA 2001. • Hajek B. and Yang S., Strategic buyers in a sum bid game for flat networks, IMA Workshop, 2004.• Hartline J. D., The Lectures on Optimal Mechanism Design, 2006.• Hartline J. D., Roughgarden T., Simple versus Optimal Mechanisms, ACM EC ’09.

49

Some references (cont’d)• Johari R. And Tsitsiklis J. N., Efficiency Loss in a Network Resource Allocation Game, Mathematics of

Operations Research, Vol 29, No 3, 2004.• Kelly F., Charing and rate control for elastic traffic, European Trans. on Telecommunications, Vol 8,

1997.• Levin D., LaCurts K., Spring N., Bhattacharjee B., Bittorrent is an auction: analyzing and improving

Bittorrent’s incentives, ACM Sigcomm 2008.• Lu P., Teng S.-H., Yu C., Truthful Auctions with Optimal Profit, WINE 2006• Lucier B. And Borodin A., Price of Anarchy for Greedy Auctions, SODA 2009. • Migrom P. R. And Weber R. J., A Theory of Auctions and Competitive Bidding, Econometrica, Vol 50,

No 5, 1982.• Myerson R. B., Optimal Auction Design, Mathematics of Operations Research, Vol 6, No 1, 1981.• The Prize Committee of the Royal Swedish Academy of Sciences, Mechanism Design Theory, 2007.• Papadimitriou C., Schapira M., Singer Y., On the hardness of being truthful, FOCS 2008.• Ronen A., On approximating optimal auctions, ACM EC ‘01. • Ronen A. And Saberi A., Optimal auctions are hard. • Thanh N. and Vojnovic M., The Weighted Proportional Allocation Mechanism, MSR Technical

Report, MSR-TR-2009-123, 2009.• Varian H. R., Position auctions, Int’l Journal of Industrial Organization, Vol 25, 2007.• Vickery W., Counterspeculation, auctions, and competitive sealed tenders, The Journal of Finance,

1961.