Reserve Pricing in Repeated Second-Price Auctions with Strategic - PowerPoint PPT Presentation

Reserve Pricing in Repeated Second-Price Auctions with Strategic Bidders Alexey Drutsa

Setup

Second-Price (SP) Auction with Reserve Prices ▌ Setting › A good (e.g., an ad space) is offered for sale by a seller to 𝑁 buyers › Each buyer 𝑛 holds a private valuation 𝑤 $ ∈ [0,1] for this good ( 𝑤 $ is unknown to the seller) ▌ Actions › The seller selects a reserve price 𝑞 $ for each buyer 𝑛 › Each buyer 𝑛 submits a bid 𝑐 $ ▌ Allocation and payments › Determine actual buyer-participants: 𝕅 = {𝑛 ∣ 𝑐 $ ≥ 𝑞 $ } › The good is received by the buyer 𝑛 4 = argmax $∈𝕅 𝑐 $ (that has the highest bid) › This buyer pays 𝑞 $ 4 = max {𝑞 $ 4 , max $∈𝕅∖{$ 4} 𝑐 $ }

Repeated Second-Price Auctions with Reserve Equal goods (e.g., ad spaces) are repeatedly offered for sale › by a seller (e.g., RTB platform) to 𝑁 buyers (e.g., advertisers) › over 𝑈 rounds (one good per round). Each buyer 𝑛 › holds a private fixed valuation 𝑤 $ ∈ [0,1] for each of those goods, › 𝑤 $ is unknown to the seller. At each round 𝑢 = 1, … , 𝑈 , the seller conducts SP auction with reserves: › the seller selects a reserve price 𝑞 > $ for each buyer 𝑛 › and a bid 𝑐 > $ is submitted by each buyer 𝑛 .

Seller’s pricing algorithm › The seller applies a pricing algorithm 𝐵 that sets reserve prices {𝑞 > B,C $ } >@A,$@A B,C $ } >@A,$@A in response to bids 𝐜 = {𝑐 > of buyers 𝑛 = 1, … , 𝑁 › A price 𝑞 > $ can depend only on past bids {𝑐 E >GA,C F } E@A,F@A and the horizon 𝑈 .

Strategic buyers ▌ The seller announces her pricing algorithm 𝐵 in advance In each round 𝑢 , each buyer 𝑛 › observes a history of previous rounds (available to this buyer) and › chooses his bid 𝑐 > $ s.t. it maximizes his future 𝛿 $ -discounted surplus: B 4 O (𝑤 $ − 𝑞 E $ ) Sur > 𝐵, 𝑤 $ , 𝛿 $ , {𝑐 E $ } : = 𝔽 M EGA 𝕁 $@$ 𝛿 $ , 𝛿 $ ∈ 0,1 , E@> where 𝕁 $@$ 4 O is the indicator of the event when buyer 𝑛 is the winner in round 𝑡 $ is the payment of the buyer 𝑛 in this case 𝑞 E

Seller’s goal The seller’s strategic regret: $ 𝑤 $ − 𝕁 𝕅 W X∅ 𝑞 > 4 W ) $ SReg 𝑈, 𝐵, 𝑤 $ $ , 𝛿 $ $ : = ∑ B (max >@A She seeks for a no-regret pricing for worst-case valuation: sup \ ] ,…,\ ^ ∈ _,A SReg 𝑈, 𝐵, 𝑤 $ $ , 𝛿 $ $ = 𝑝 𝑈 Optimality : the lowest possible upper bound for the regret of the form 𝑃 𝑔(𝑈) .

Background, Research question & Main contribution

Background: 1-buyer case (posted-price auctions) If one buyer ( 𝑁 = 1 ), a SP auction reduces to a posted-price auction: › the buyer either accepts or rejects a currently offered price 𝑞 > A › the seller either gets payment equal to 𝑞 > A or nothing [Kleinberg et al., FOCS’2003] Optimal algorithm against myopic buyer with truthful regret Θ(log log 𝑈) . [Amin et al., NIPS’2013] The strategic setting is introduced. ∄ no-regret pricing for non-discount case 𝛿 = 1 . [Drutsa, WWW’2017] Optimal algorithm against strategic buyer with regret Θ(log log 𝑈) for 𝛿 < 1 .

Research question The known optimal algorithms (PRRFES & prePRRFES) from posted-price auctions cannot be directly applied to set reserve prices in second-price auctions › buyers in SP auctions have incomplete information due to presence of rivals › the proofs of optimality of [pre]PRRFES strongly rely on complete information ▌ In this study, I try to find an optimal algorithm for the multi-buyer setup

Main contribution A novel algorithm for our strategic buyers with regret upper bound of Θ(log log 𝑈) for 𝛿 < 1 A novel transformation that maps any pricing algorithm designed for posted-price auctions to a multi-buyer setup

Main ideas

Two learning processes $ 𝑤 $ − 𝕁 𝕅 W X∅ 𝑞 > 4 W ) $ B SReg 𝑈, 𝐵, 𝑤 $ $ , 𝛿 $ $ : = ∑ (max >@A Find which buyer has Find the buyers’ valuations the maximal valuation Learning process #1 Learning process #2

Learning proc.#1: an idea to localize a valuation PRRFES is an optimal learner of a valuation in posted-price auctions. However, its core localization technique relies on: ▌ The buyer completely knows the outcomes of current and all future rounds ▌ given their bids (due to absence of rivals) Can we use PRRFES in the second-price scenario where each buyer does not know perfectly the outcomes of rounds?

Barrage pricing › Reserve prices are personal (individual) in our setup › Thus, we are able to “eliminate” particular buyers from particular rounds › Namely, a buyer 𝑛 will not bid above 1/(1 − 𝛿 $ ) › We call this price as “barrage” one and denote it by ∞ Let “eliminate” all buyers except some buyer 𝑛 in a round 𝑢 Then the buyer 𝑛 will have com round 𝑢 complete i ete inf nfor ormati tion on abo about outcome of this s ro

Learning proc.#2: an idea to find max valuation The search algorithm works by maintaining a feasible interval [𝑣 $ , 𝑥 $ ] that › is aimed to localize the valuation 𝑤 $ , i.e. 𝑤 $ ∈ [𝑣 $ , 𝑥 $ ] › shrinks as 𝑢 → ∞ [𝑣 o , 𝑥 o ] [𝑣 A , 𝑥 A ] [𝑣 p , 𝑥 p ] 𝑤 A 𝑤 p 𝑤 o 0 1 round 𝑢 A round 𝑢 p round 𝑢 o ▌ If, in a round 𝑢 , it becomes that 𝑥 $ < 𝑣 m for some buyers 𝑛 and 𝑜 , ▌ then buyer 𝑛 has non-maximal valuation which should not be searched anymore

Dividing algorithms

Key instrument that implements the ideas transformation di div

Transformation di div : cyclic elimination Let 𝐵 be an algorithm designed for repeated posted-price auctions ▌ Its transformation 𝐞𝐣𝐰 𝐵 is an algorithm for repeated SP auctions as follows Buyers: Reserve prices are set by: Reserve Prices (only one non-barrage in a round): A A A 𝑞 A ∞ ∞ Algorithm 𝐵 𝑞 t ∞ ∞ 𝑞 u ∞ . . . p p p ∞ ∞ Algorithm 𝐵 𝑞 p ∞ ∞ ∞ . . . 𝑞 v 𝑞 w o o ∞ ∞ Algorithm 𝐵 𝑞 o ∞ ∞ ∞ ∞ . . . 𝑞 x Rounds, 𝑢 = 3 1 2 4 5 6 7 8 1 2 3 Periods, 𝑗 =

Transformation di div : stopping rule We stop considering a buyer 𝑛 in periods when 𝑥 $ < 𝑣 m for some buyer 𝑜. ▌ The number of periods with buyer 𝑛 is referred to as subhorizon, 𝐽 $ . We stopped learning of 𝑤 A and 𝐽 A = 𝑙 , when 𝑥 A < 𝑣 p Buyers: Reserve prices are set by: Reserve Prices : A 𝑞 E ∞ ∞ ∞ Algorithm 𝐵 ∞ ∞ ∞ ∞ . . . p p p p ∞ ∞ Algorithm 𝐵 𝑞 E|A ∞ ∞ 𝑞 E|u . . . 𝑞 E|o 𝑞 E|v o o o ∞ ∞ Algorithm 𝐵 𝑞 E|p ∞ ∞ ∞ . . . 𝑞 E|t 𝑞 E|x 𝑡 + 2 𝑡 𝑡 + 1 𝑡 + 3 𝑡 + 4 𝑡 + 5 𝑡 + 6 𝑡 + 7 Rounds, 𝑢 = 𝑙 + 3 𝑙 𝑙 + 1 𝑙 + 2 Periods, 𝑗 =

� � Transformation div: regret decomposition Lemma 1. For the described transformation, strategic regret has decomposition: SReg 𝑈, 𝐞𝐣𝐰 𝐵 , 𝑤 $ $ , 𝛿 $ $ = 𝑤 m − 𝑤 $ ) = M Reg $ (𝑈, 𝐵, 𝑤 $ , 𝛿 $ ) 𝐽 $ (max + M m $ $ Individual regrets Deviation regret Measure how the algorithm 𝐵 learns Measures how fast we stop learning of non-maximal valuations the valuation of each buyer

Key challenge against strategic buyer Strategic buyer may lie and mislead algorithms, thus a good algorithm must Extract correct information about a buyer’s valuation from his actions (bids) ▌ Dividing structure in a round allows to construct a tool to locate valuations: ▌ it is enough to make complete information situation in a round

Upper bound on valuation of strategic buyer Let buyer 𝑛 is the non-”eliminated ” one in a round 𝑢 . ▌ If the buyer accepts (bids above) the current reserve price 𝑞 > $ B 4 W (𝑤 $ − 𝑞 > 4 O (𝑤 $ − 𝑞 E >GA 𝕁 $@$ $ ) + 𝔽 M EGA 𝕁 $@$ $ ) Surplus > = 𝔽 𝛿 $ 𝛿 $ E@>|A = ≤ Ž (𝑤 $ − 𝑞 > >GA (𝑤 $ − 𝑞 > >GA 𝕁 • W $ ) = 𝛿 $ $ ) 0 𝛿 $ Ž ‘’ W ▌ If the buyer rejects (bids below) the current reserve price 𝑞 > $ >|†GA B ≤ 𝛿 $ 4 O (𝑤 $ − 𝑞 E (𝑤 $ − [lowest_price]) $ ) EGA 𝕁 $@$ Surplus > = 𝔽 M 𝛿 $ 1 − 𝛿 $ E@>|† If we observe that a buyer rejects non-”barrage” reserve price, then: • 𝑤 $ − 𝑞 > $ < $ − [lowest_price]) • Ž • (𝑞 > AG• Ž G• Ž

Optimal algorithm

Pricing algorithm divPRRFES Apply the transformation div div to PRRFES algorithm

divPRRFES: individual and deviation regrets ▌ Individual regrets Our tool to locate valuations provides the upper bound (as in 1-buyer case): Reg $ 𝑈, 𝐵, 𝑤 $ , 𝛿 $ = 𝑃 log p log p 𝑈 ∀𝑛 ▌ Deviation regrets › For each buyer 𝑛 with non-maximal valuation (i.e., 𝑤 $ < max 𝑤 m ) m › We can upper bound its subhorizon 𝐽 $ : 𝐷 𝐽 $ ≤ 𝑤 m − 𝑤 $ max m