Competitive Analysis of Incentive Compatible On-Line Auctions Ron Lavi and Noam Nisan SISL/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). 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 :

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 is non-decreasing, q * solves: p i (q*) = b i (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 q p 1 (q) b 1 (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 p 1 (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 p 2 (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 p 2 (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 of El-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 for a price v 2, so: (2) The on-line revenue (the shaded area) is P 1 +P 2 +P 3 +A = v 3 / c (3) So on-line welfare and revenue are higher than v 3 / 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

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

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

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

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…