Steffen Staab 1WeST Web Science & Technologies University of Koblenz ▪ Landau, Germany Network Theory and Dynamic Systems Auctions Dr. Jérôme Kunegis Acknowledgements to Maria Grineva,

Jérôme Kunegis 2WeST

Jérôme Kunegis 3WeST Pay per click Click is a strong indicator for interest Cost per click (CPC) Costs depend on position and query But how to set prices?

Jérôme Kunegis 4WeST Different Costs Example: “web science master” 0.50$ per click vs “mesothelioma” 50.00$ per click

Jérôme Kunegis 5WeST Setting Prices through an Auction Matching Market Vickrey-Clarke-Groves (VCG) mechanism Generalized Second-Price Auction (GSP)

Jérôme Kunegis 6WeST Matching Markets A set of buyers B A set of sellers S

Jérôme Kunegis 7WeST Matching Markets and Constricted set

Jérôme Kunegis 8WeST Matching Markets and Constricted set

Jérôme Kunegis 9WeST General Considerations towards Determining Prices Valuations Seller wants to sell at highest price Buyers hide their valuations Seller should always indicate minimum price Non-truthful bidding E.g., first-price sealed bid auction, English auction Truthful bidding E.g., second-price sealed bid auction / Vickrey auction, Dutch auction Incomplete information leads to errors in valuations Winner's curse

Jérôme Kunegis 10WeST Matching Markets In the real world valuations are not known Truthful bidding vs untruthful bidding: First-price auctions:  bidders underreport, non-truthful bidding, very complex bidding for everyone Second-price auctions:  truthful bidding is a dominant strategy for all advertisers

Jérôme Kunegis 11WeST Clickthrough Rates and Revenues per Click r i - clickthrough rate of a slot i v j - revenue per click of advertiser j

Jérôme Kunegis 12WeST Clickthrough Rates and Revenues per Click Assumptions: Rates are known Don't depend on ad Depend on position Don't depend on ads in other slots

Jérôme Kunegis 13WeST Matching Markets V ij = r i v j

Jérôme Kunegis 14WeST Approaching the Solution: Step 1 - Matching Markets Announcing of prices p i of slots i pipi

Jérôme Kunegis 15WeST Matching Markets Evaluation of payoffs v ij − p i Preferred-seller Graph 17, 12, 6 payoff 7, 7, 4 −3, 2, 2

Jérôme Kunegis 16WeST Matching Markets Prices are market-clearing if graph has a perfect matching (all items are assigned to distinct buyers) market-clearing maximal payoff (bold graph) 17, 12, 6 payoff 7, 7, 4 −3, 2, 2

Jérôme Kunegis 17WeST Matching Markets Prices are market-clearing if graph has a perfect matching (all items are assigned to distinct buyers) market-clearing maximal payoff (bold graph) 17, 12, 6 payoff 7, 7, 4 -3, 2, 2 If no perfect match is found: Sellers identify a constricted set of buyers and raise the prices until perfect match is reached.

Jérôme Kunegis 18WeST Market Clearing by raising prices If no perfect match is found: Sellers identify a constricted set of buyers and raise the prices until perfect match is reached. Next Step: How to determine prices…

Jérôme Kunegis 19WeST Vickrey-Clarke-Groves (VCG) mechanism Vickrey-Clarke-Groves (VCG) principle: Each individual is charged a price equal to the total amount better off everyone else would be if this individual weren't there. Buyers have individual, private values and don't care about the allocation of remaining goods to buyers. Truthful bidding is a dominant strategy. The bidder who values the item most gets it. Suppose, there are n bidders who value the item as v 1 … v n in decreasing order. If bidder 1 were not present, the item would go to bidder 2, that means bidder 1 pays v 2. Each individual is charged the harm they cause to the rest of the world.

Jérôme Kunegis 20WeST Vickrey-Clarke-Groves Step 1: Auctioneer collects valuations of buyers: xyz a30156 b20104 c 52

Jérôme Kunegis 21WeST Vickrey-Clarke-Groves Step 2: Auctioneer computes optimal assignment : Maximize total valuation: xyz a30156 b20104 c 52 Set of sellers Set of buyers = 42

Jérôme Kunegis 22WeST Step 3: auctioneer charges buyer j for slot i the VCG price p ij the total harm caused by buyer j to the rest of the buyers is the difference between how they'd do without j present and how they do with j present. Vickrey-Clarke-Groves the best total valuation without seller i and buyer j

Jérôme Kunegis 23WeST Vickrey-Clarke-Groves

Jérôme Kunegis 24WeST Vickrey-Clarke-Groves

Jérôme Kunegis 25WeST Truth-Telling as a Dominant Strategy (2): If buyers report their valuations truthfully, then the assignment of items is designed to maximize the total valuation by definition. (1): Payoff for buyer j : v ij − p ij. we want to show that j has no incentive to deviate from truthful announcement if j lies to get item h instead of i: payoff would be v hj − p hj we want to show that v ij − p ij ≥ v hj − p hj Claim: If items are assigned and prices computed according to the VCG mechanism, then (1) truthfully announcing valuations is a dominant strategy for each buyer, and (2) the resulting assignment maximizes the total valuation of any perfect matching of items and buyers

Jérôme Kunegis 26WeST Analyzing the VCG Procedure Truthful bidding as a dominant strategy

Jérôme Kunegis 27WeST Analyzing the VCG Procedure

Jérôme Kunegis 28WeST Analyzing the VCG Procedure VCG is not used in practice, e.g., Google uses GSP VCG is focused on maximizing total valuation for advertisers Search engines care about maximizing their own revenue

Jérôme Kunegis 29WeST Generalized Second Price Auction GSP is a generalization of the second-price single item auction Procedure: Step 1: Advertiser j announces a bid consisting of a single price b j it is willing to pay per click. Step 2: Each slot i is awarded to the i-th highest bidder at a price of (i+1)-st highest bid Each advertiser shown on the result page is paying a price per click equal to the bid of the advertiser just below them.

Jérôme Kunegis 30WeST Analyzing the Generalized Second-Price Problem: truth-telling may not constitute a Nash equilibrium It is not theoretically proven that GSP gives better revenue to the seller then VCG But: it is computationally simpler

Jérôme Kunegis 31WeST Truth-Telling May Not Be an Equilibrium If each advertiser bids its true valuation, then x gets the top slot at a price per click of 6, pays 6 × 10 = 60. x's valuation of the top slot = 7 × 10 = 70 x's payoff = 70 − 60 = 10

Jérôme Kunegis 32WeST Truth-Telling May Not Be an Equilibrium If x lowers his bid to 5, it gets the second slot at a price of 1 and pays 4 × 1 = 4 for the slot x's valuation for the second slot = 7 × 4 = 28 x's payoff 28 − 4 = 24 > 10 (an improvement over the truthful bid)

Jérôme Kunegis 33WeST Multiple and Nonoptimal Equilibria GSP There can be more than one equilibrium set of bids, and among these equilibria are some that produce a socially nonoptimal assignments of advertisers to slots suppose, x bids 5, y bids 4, z bids 2 – this is an equilibrium, produces a socially optimal allocation suppose, x bids 3, y bids 5, z bids 1 – this an equilibrium too – but not socially optimal (does not maximize the sum of the bidders payoffs)

Jérôme Kunegis 34WeST The Revenue of GSP and VCG For GSP: equilibrium #1: x bids 5, y bids 4, z bids 2 ; search engine revenue = 48 equilibrium #2: x bids 3, y bids 5, z bids 1 ; search engine revenue = 34 For VCG: determining the harm search engines revenue = < 44 < 48

Jérôme Kunegis 35WeST Ad Quality Assumption: Fixed Clickthrough Rate A slot's clickthrough rate does not depend on the ad. In general, this is not true: users look at the ad title and this affects whether they click or not. This user behavior affects search engine revenue, too.

Jérôme Kunegis 36WeST Ad Quality Google uses an estimated quality factor q j for an ad j if advertiser j appears in slot i, then the clickthrough rate is estimated to be not r i but the q j r i so that the clickthrough rate depends on the ad too It is easy to incorporate ad quality q j into GSP model: change the valuation of the advertiser j for a slot i: v ij = r i v j ⇒ v ij = q j r i v j A lot of factors are taken into account: observing clickthrough rates when shown; ad text, “landing page”...