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## Mechanism Design

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

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

... 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

...

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

- 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

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Welcome to

The Economist Subscription Centre

Pick the type of subscription you want to buy

or renew

Economist.com subscription – US $59.00

One-year subscription to Economist.com

Includes online access to all articles from

The Economist since 1997.

Print & web subscription – US $125.00

One-year subscription to the print edition

of The Economist and online access to all

articles from The Economist since 1997.

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Source: Ariely D. (2008)

This tutorial agenda

- Design objectives
- Vickery & Myerson auctions
- Prior-free auctions
- Auctions for resource allocation

Examples of other goals

min makespan, max flow, max weighted flow

machines

...

v2

vn

v1

processing speed

jobs

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

- Incomplete information

- Complete information

- Types are private information

- Types are public information

- Types drawn from a distribution F

- F is public information

Vickery auctionfor allocation of a single item

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

Incentive compatibility

equal profit

equal profit

win only by overbidding

dominated by truthful

win only if truthful

equal profit = 0

lose in either case

Vickery auction is a truthful efficient auction

But how do I maximize my profit?

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:

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

Virtual valuation:

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:

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

- Ex.F uniform [0, h],

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)

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 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?

- Pros
- Simplifies buyer’s strategy – just report the type
- Simplifies the problem for the designer
- Cons
- Computational complexity

This tutorial agenda

- Design objectives
- Vickery & Myerson auctions
- Prior-free auctions
- Auctions for resource allocation

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’d)

- Price-taking users:

- Underprice-taking users with concave, utilityfunctions, efficiency is 100%.

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

- (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

- 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

- When truthfulness requires side-payments
- Frugality, envy-freeness
- Competitive guarantees of some auctions, ex. digital-goods auctions
- Computational complexity under incentive compatibility

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)

- 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.

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