Motivation for On-Line Auctions

Competitive Analysis ofIncentive Compatible On-Line AuctionsRon Lavi and Noam NisanSISL/IST, Cal-TechHebrew University

Motivation for On-Line Auctions • Example: auctions are useful for allocating bandwidth • 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).

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 :

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.

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*) )

Lemma 1 Lemma 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.

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 p1(q) b1(q) 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 p1(q) 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 p2(q) 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.

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 ,

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) isP1+P2+P3+A = v3 / c(3) So on-line welfare and revenue are higher than v3 / c, and thus:

The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ

The supply curve (for [1,Φ]) p(0)=c ; p(1)=Φ Taking the derivative of both sides

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.

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.

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…

Motivation for On-Line Auctions