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Mechanism Design. Milan Vojnović Lab tutorial, March 2010. Mechanism design is about designing a game so as to achieve a desired goal. b 1. b 2. b n. . Input:. Output: ( x , p ). other input. allocation:. payment:. Ex 1: sponsored search. Ex 1: sponsored search (cont’d). a.

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Mechanism Design


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mechanism design

Mechanism Design

Milan Vojnović

Lab tutorial, March 2010

slide4

b1

b2

bn

...

Input:

Output: (x, p)

other input

allocation:

payment:

ex 1 sponsored search cont d
Ex 1: sponsored search (cont’d)

a

Position 1

Position 2

Position 3

Position 4

advertisers

Position 5

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

x2

x2

x2

C2

C

C

C3

C/w

P

P

P

C

x1

C1

x1

C/w

C

x1

C2

x1

1

x1

x1

w

x2

w

C1

C2

C3

x2

x2

C

1

slide11

... 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...

some developments
Some developments

...

1961

Vickery’s auction

...

1981

Myerson’s optimal auction design

...

1997

Overture’s auction; network resource allocation (Kelly)

1999

Algorithmic mechanism design (Nisan & Ronen)

2001

Competitive auctions and digital goods (Goldberg et al)

2002

Generalized Second Price Auction

...

2007

Algorithmic game theory (Nisan et al)

active research area
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
the importance of irrelevant alternatives
The importance of irrelevant alternatives

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Source: Ariely D. (2008)

this tutorial agenda
This tutorial agenda
  • Design objectives
  • Vickery & Myerson auctions
  • Prior-free auctions
  • Auctions for resource allocation
standard goals
Standard goals

Max seller’s profit

“optimal auction design”

Max social welfare

“efficient”

examples of other goals
Examples of other goals

min makespan, max flow, max weighted flow

machines

...

v2

vn

v1

processing speed

jobs

standard constraints
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-negativeAlso known as voluntary participation

two kinds of games
Two kinds of games
  • Incomplete information
  • Complete information
  • Types are private information
  • Types are public information
  • Types drawn from a distribution F
  • F is public information
vickery auction for allocation of a single item
Vickery auctionfor allocation of a single item
  • Allocation to the buyer with highest bid
  • Payment equal to the second highest bid
incentive compatibility
Incentive compatibility

equal profit

equal profit

win only by overbidding

dominated by truthful

win only if truthful

equal profit = 0

lose in either case

slide22

Vickery auction is a truthful efficient auction

But how do I maximize my profit?

myerson s optimal auction design
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
  • C2)payment:
incentive compatibility25
Incentive compatibility
  • Buyers’ profit:

B

B

A

A

slide27

Under independent buyer’s valuations, every optimal allocation is a solution ofthe virtual surplus maximization

Virtual valuation:

virtual valuation
Virtual valuation
  • Ex. 2 Fi(v) = 1 - exp(-li)
  • Ex. 1 Fi(v) uniform on [0, hi]
optimality of vickery auction with reserve price
Optimality of Vickery auction with reserve price
  • Single-item auction
  • Independent and identical buyers
  • Strictly increasing virtual valuations

The optimal is Vickery auction

with the reserve price r:

competitive framework for auctions
Competitive framework for auctions
  • Competitiveness to a profit benchmark B(v)

Ex. 1 sum valuation

Ex. 3 uniform pricing with at least two winners

Ex. 2 max valuation

Competitive ratio for an auction A =

random reserve price auction lu at al 2006
Random reserve price auction (Lu at al 2006)

Run the second-price auction

1- d

d

Sample reserve price r from

Ifb1 ≥ r thenallocate the item to a buyer with highest bid

random r eserve price cont d
Random reserve price (cont’d)

E[profit] =

E[social welfare] =

h = max valuation

  • A tighter expected revenue can be obtained using a successive composition of log(x+1)
  • Can’t do a better expected revenue !
why incentive compatibility as a requirement
Why incentive compatibility as a requirement?
  • Pros
    • Simplifies buyer’s strategy – just report the type
    • Simplifies the problem for the designer
  • Cons
    • Computational complexity
this tutorial agenda35
This tutorial agenda
  • Design objectives
  • Vickery & Myerson auctions
  • Prior-free auctions
  • Auctions for resource allocation
kelly s resource allocation
Kelly’s resource allocation

b1

bi

bn

C

allocation to buyer i:

payment by buyer i = bi

kelly s resource allocation cont d
Kelly’s resource allocation (cont’d)
  • Extensions to networks of links: the mechanism applied by each link
  • Two user models

scalar bids (TCP like)

vector bids

kelly s resource allocation cont d39
Kelly’s resource allocation (cont’d)
  • Price-taking users:
  • Underprice-taking users with concave, utilityfunctions, efficiency is 100%.
johari tsitsiklis price anticipating users
Johari & Tsitsiklis’ price-anticipating users

User:

  • Underprice-anticipating users with concave, non-negative utility functions, and vector bids, the worst-case efficiency is 75%.
full efficiency loss under scalar bids
Full efficiency loss under scalar bids
  • (Hajek & Yang 2004) Underprice-anticipating users with concave, non-negative utility functions, and scalar bids, theworst-case efficiency is 0.
  • A worst-case: serial network of unit capacity links
the weighted proportional allocation mechanism
The weighted proportional allocation mechanism
  • Guarantees on social welfare and seller’s profit - Thanh-V. 2009
  • Allocation to buyer i:
  • Payment by buyer i = bi
some important aspects not discussed in this tutorial
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
some references
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.
some references cont d
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.