1v
2v
nv
...
0v
b1
b2
bn
Input: ),,,( 21 nbbbb Output: (x, p)
),,,( 21 nxxxx ),,,( 21 npppp allocation: payment:
other input
6
a
Ex 1: sponsored search (cont’d)
Position 1
Position 2
Position 3
Position 4
Position 5
advertisers
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
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
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
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
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
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
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.