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Competitive Analysis of Incentive Compatible On-Line Auctions Ron Lavi and Noam Nisan SISL/IST, Cal-Tech Hebrew University. Motivation for On-Line Auctions. Example: auctions are useful for allocating bandwidth. Transmissions arrive over time, each transmission lasts all day. - PowerPoint PPT Presentation
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Competitive Analysis ofIncentive Compatible On-Line
Auctions
Ron Lavi and Noam NisanSISL/IST, Cal-Tech Hebrew University
Motivation for On-Line Auctions
• Transmissions arrive over time, each transmission lasts all day.
• Problem: Need to know all users (and their demands) before the first transmission (auctions are performed off-line).
Transmission 1
Time: 07:00
Transmission 2 Time: 07:30
Transmission 3 Time: 08:00
• Example: auctions are useful for allocating bandwidth
The Model (1)• Goods and players’ utilities
– K indivisible goods (or one divisible good with quantity Q=1) to be allocated among many players.
– Each player has a valuation for each number of goods.– denotes the marginal valuation of player i.
We assume that all marginal valuations are downward sloping. – Thus, player i’s valuation of a quantity q* is :
)(qvi
*0 )(q dqqv
i
p
Vi(q)
q* q
The Model (2)• A non-cooperative game with private values:
– The valuation of each player is known only to him.
– The goal of each player is to maximize his own utility:
(where is the price paid).
• An on-line setting:
– Players arrive one at a time and submit a bid ( )when they arrive (we will relax this in the sequel)
– Auctioneer answers immediately to each player, specifying allocated quantity and total price charged.
– No knowledge of the future: allocation can depend only on previous bids.
)(qbi
ii Pq
dqqvq iu *
)(*)(0 iP
Incentive compatible Auctions
• We want performance guarantees with respect to the true inputs of the players.– But players are “selfish”, and might manipulate us.
• One solution is to design an incentive-compatible auction:– Declaring the true input always maximizes the
utility of the player (a dominant strategy).
Question 1
What on-line auctions are incentive compatible ?
Online Auction based on Supply Curves
An on-line auction is “based on supply curves” if, before receiving the i’th bid, it fixes a function (supply curve), such that:
• The total price for a quantity q* is:
• The quantity q* sold is the quantity that maximizes the player’s utility.
(when the supply curve isnon-decreasing, q* solves: pi(q*) = bi(q*) )
ppi(q)
bi(q)
q* q
*
0)(
qdqqpi
)(qpi
Lemma 1Lemma 1: An on-line auction that is based on
supply curves is incentive compatible
Proof: (for the case of a non-decreasing supply curve)
(1) The price paid depends only on the quantity received. Lying can help only if it changes the quantity received.
(2) The quantity q* maximizes the player’s utility. Lying can’t increase utility.
ppi(q)
vi(q)
q2 q* q1 q
A
B
C
Theorem 1
An on-line auction is incentive compatible
if and only if it is based on supply curves.
A Global Supply Curve
• In general, there is no specific relation between the supply curves.
• Definition: An auction is based on a global supply curve if the i’th supply curve is a left shift of the (i-1)’th supply curve by the quantity sold to the (i-1)’th bidder.
p
q
p1(q)
b1(q)
q*1
A Global Supply Curve
• In general, there is no specific relation between the supply curves.
• Definition: An auction is based on a global supply curve if the i’th supply curve is a left shift of the (i-1)’th supply curve by the quantity sold to the (i-1)’th bidder.
p
q
p1(q)
q*1
A Global Supply Curve
• In general, there is no specific relation between the supply curves.
• Definition: An auction is based on a global supply curve if the i’th supply curve is a left shift of the (i-1)’th supply curve by the quantity sold to the (i-1)’th bidder.
p
q
p2(q)
q*1
A Global Supply Curve
• In general, there is no specific relation between the supply curves.
• Definition: An auction is based on a global supply curve if the i’th supply curve is a left shift of the (i-1)’th supply curve by the quantity sold to the (i-1)’th bidder.
p
q
p2(q)
q*1
Question 2: which supply curve is best?
• Economics approach: average case (Bayesian) analysis. Assuming some fixed distribution.
• Our approach (of Computer Science): worst-case analysis.. The valuations’ distribution is not known. The auction is evaluated for the worst-case scenario.
• We apply competitive analysis: comparing to an off-line auction, in the worst case.
Definitions• Assumption: all valuations are in , and p is also the
auctioneer’s reservation price.
• Definitions (for an auction A and a sequence of bids ) :
– Revenue:
(where is the total payment of player j)
This is equal to the auctioneer’s resulting utility.
– Social Welfare:
(where is the total value of player j of the quantity he received).
This is equal to the sum of all players’ resulting utilities.
],[ pp
quantity) unsold()( pPR j jA
jP
quantity) unsold()( pVW j jA
jV
Off-line benchmark: The Vickrey Auction• The Vickrey auction:
– Allocation: the goods are allocated to maximize social welfare (according to the players’ declarations).
– Payment: Each player pays the total additional valuation of other players when dividing his allocation optimally among them.
• Why do we use this auction as the off-line benchmark?
– Welfare: optimal.
– Revenue:
• Popular and standard (equivalent to the English auction).
• All optimal-efficiency auctions has same revenue.
• But, in general it is not optimal.
Competitiveness• An on-line auction A is -competitive with respect
to the social welfare if for every bid sequence , (where vic is the Vickrey auction)
• Similarly, A is -competitive with respect to the revenue if for every bid sequence ,
)(
)( vicA
RR
)(
)( vicA
WW
A supply curve for a divisible good
• p(q) will have the property that for any point (q*, p*) on p(q), the shaded area (marked A+B) will be exactly equal to p*/c (the constant c will be determined later).
Following the “threat-based”
approach ofEl-yaniv, Fiat,
Karp, and Turpin [FOCS’92]
Intuition: Fixed Marginal Valuations• An example of the case of fixed marginal valuations ( ):
(1) The Vickrey auction allocates the entire quantity to player 3 fora price v2, so:
(2) The on-line revenue (the shaded area) is
P1+P2+P3+A = v3 / c
(3) So on-line welfare and revenue
are higher than v3 / c, and thus:
ii vqv )(
23 , vRvW vicvic
c
RR
c
WW vic
onvic
on ;
The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ
cqpqdxxpqq
/)()1()(:100
The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ
cqpqdxxpqq
/)()1()(:100
cqpqp /)('1)( Taking the derivative
of both sides
The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ
cqpqdxxpqq
/)()1()(:100
cqpqp /)('1)( Taking the derivative
of both sidesqceqp 1)(
The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ
cqpqdxxpqq
/)()1()(:100
cqpqp /)('1)( Taking the derivative
of both sidesqceqp 1)(
11)0( cpc
The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ
cqpqdxxpqq
/)()1()(:100
cqpqp /)('1)( Taking the derivative
of both sidesqceqp 1)(
11)0( cpc ln)1(1)1( ccecp
The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ
cqpqdxxpqq
/)()1()(:100
cqpqp /)('1)( Taking the derivative
of both sidesqceqp 1)(
11)0( cpc ln)1(1)1( ccecp
)(ln
ln
c
c
Results
Definition: The “Competitive On-Line Auction” has the global supply curve:
where c solves
Theorem: The “Competitive On-Line Auction” is c-competitive with respect to both the revenue and the social welfare.
Theorem: No incentive compatible on-line auction can have a competitive ratio less than c.
))1(1()( e qccpqp )ln(
1
1ln
cc
c
Model Variant: time dependent bidding• Consider the following model extensions:
– Delayed bidding
– Split bidding
– Players’ valuations may be time dependent (in a non-increasing way)
• When the supply curves are non-decreasing (even over time), there is no gain from delaying/splitting the bids.
• Since a global supply curve is non-decreasing over time, all the upper bounds still hold for these extensions (the lower bounds trivially remain true).
The case of k indivisible goods
• A randomized auction ( c - competitive ).
• Deterministic auction ( - competitive).
• A lower bound of for deterministic auctions.
)1(1 kk
)1(1 k
Summary
• A demonstration for an integration of algorithmic and game-theoretic considerations.
• Main issue here: design prices to simultaneously achieve– Incentive compatibility
– Good approximation
• Many times, these two (provably) coincide. This opens many interesting questions…