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…