Sponsored Search Auctions G. Amanatidis Based on slides by A. - PowerPoint PPT Presentation

Single-parameter Mechanism design: Sponsored Search Auctions G. Amanatidis Based on slides by A. Voudouris

Single-item auctions • A seller with one item for sale • 𝑜 agents • Each agent 𝑗 has a private value 𝑤 𝑗 for the item – This value represents the willingness-to-pay of the agent; that is, 𝑤 𝑗 is the maximum amount of money that agent 𝑗 is willing to pay in order to buy the item • The utility of each agent is quasilinear in money : – If agent 𝑗 loses the item, then her utility is 0 – If agent 𝑗 wins the item at price 𝑞 , then her utility is 𝑤 𝑗 − 𝑞

Single-item auctions • General structure of an auction: – Input: every agent 𝑗 submits a bid 𝑐 𝑗 ( agents = bidders ) – Allocation rule: decide the winner – Payment rule: decide a selling price • Deciding the winner is easy: the highest bidder • Deciding the selling price is more complicated – A selling price of 0 , creates a competition among the bidders as to who can think of the highest number • We are interested in payment rules that incentivize the bidders to bid their true values – Truthful auctions that maximize the social welfare

First-price auction • Allocation rule: the winner is the highest bidder • Payment rule: the winner pays her bid • Is this a truthful auction? 𝑤 1 = 100 𝑤 2 = 50

First-price auction • Allocation rule: the winner is the highest bidder • Payment rule: the winner pays her bid • Is this a truthful auction? 𝑤 1 = 100 𝑐 1 = 50.1 𝑤 2 = 50 𝑐 2 = 50

Second-price auction • Allocation rule: the winner is the highest bidder • Payment rule: the winner pays the second highest bid Theorem [Vickrey, 1961 ] In a second-price auction (a) it is a dominant strategy for every bidder 𝑗 to bid 𝑐 𝑗 = 𝑤 𝑗 , and (b) every truthtelling bidder gets non-negative utility • (b) is obvious: – the selling price is at most the winner’s bid, and the bid of a truthtelling bidder is equal to her true value

Second-price auction • For (a), our goal is to show that the utility of bidder 𝑗 is maximized by bidding 𝑤 𝑗 , no matter what 𝑤 𝑗 and the bids of the other bidders are • Second highest bid: 𝐶 = max 𝑘≠𝑗 𝑐 𝑘 • The utility of bidder 𝑗 is either 0 if 𝑐 𝑗 < 𝐶 , or 𝑤 𝑗 − 𝐶 otherwise Case I: 𝒘 𝒋 < 𝑪 • Maximum possible utility = 0 • Achieved by setting 𝑐 𝑗 = 𝑤 𝑗 Case II: 𝒘 𝒋 ≥ 𝑪 • Maximum possible utility = 𝑤 𝑗 − 𝐶 • Bidder 𝑗 wins the item by setting 𝑐 𝑗 = 𝑤 𝑗 ▢

Sponsored search auctions

Sponsored search auctions • In 2011 Google’s revenue was almost 40.000.000.000 usd • 96% of this was generated by sponsored search auctions

Sponsored search auctions • 𝑙 advertising slots • 𝑜 bidders (advertisers) who aim to occupy a slot • Slot 𝑘 has a click-through-rate (CTR) 𝑏 𝑘 – The CTR of a slot represents the probability that the ad placed at this slot will be clicked on – Assumption: the CTRs are independent of the ads that occupy the slots • The slots are ranked so that 𝑏 1 ≥ ⋯ ≥ 𝑏 𝑙 • Each bidder 𝑗 has a private value 𝑤 𝑗 per click – Bidder 𝑗 derives utility 𝑏 𝑘 ⋅ 𝑤 𝑗 from slot 𝑘

Sponsored search auctions: goals • Truthfulness: It is a dominant strategy for each bidder to bid her true value • Social welfare maximization: σ 𝑗 𝑤 𝑗 ⋅ 𝑦 𝑗 – 𝑦 𝑗 is the CTR of the slot that bidder 𝑗 is assigned to, or 0 otherwise • Poly-time execution: running the auction should be quick

Sponsored search auctions: goals • Truthfulness: It is a dominant strategy for each bidder to bid her true value • Social welfare maximization: σ 𝑗 𝑤 𝑗 ⋅ 𝑦 𝑗 – 𝑦 𝑗 is the CTR of the slot that bidder 𝑗 is assigned to, or 0 otherwise • Poly-time execution: running the auction should be quick • If the bidders are truthful, then maximizing the social welfare is easy: sort the bidders in decreasing order of their bids • So, the problem is to incentivize them to be truthful, again • Can we extend the ideas we exploited for single-item auctions?

Generalized second-price auction • Allocation rule: sort the bidders in decreasing order of their bids and rename them so that 𝑐 1 ≥ ⋯ ≥ 𝑐 𝑜 • Payment rule: every bidder 𝑗 ≤ 𝑙 (who is assigned at slot 𝑗 ) pays the next highest bid 𝑐 𝑗+1 per click, and every bidder 𝑗 > 𝑙 pays 0 𝑤 1 = 100 𝑏 1 = 1 𝑤 2 = 50 𝑏 2 = 3 5

Generalized second-price auction • Allocation rule: sort the bidders in decreasing order of their bids and rename them so that 𝑐 1 ≥ ⋯ ≥ 𝑐 𝑜 • Payment rule: every bidder 𝑗 ≤ 𝑙 (who is assigned at slot 𝑗 ) pays the next highest bid 𝑐 𝑗+1 per click, and every bidder 𝑗 > 𝑙 pays 0 𝑤 1 = 100 𝑐 1 = 100 𝑣 1 = 1 ⋅ 100 − 50 𝑏 1 = 1 = 50 𝑤 2 = 50 𝑐 2 = 50 𝑏 2 = 3 𝑣 2 = 3 5 ⋅ 50 = 30 5

Generalized second-price auction • Allocation rule: sort the bidders in decreasing order of their bids and rename them so that 𝑐 1 ≥ ⋯ ≥ 𝑐 𝑜 • Payment rule: every bidder 𝑗 ≤ 𝑙 (who is assigned at slot 𝑗 ) pays the next highest bid 𝑐 𝑗+1 per click, and every bidder 𝑗 > 𝑙 pays 0 𝑤 1 = 100 𝑐 1 = 49 𝑣 2 = 1 ⋅ 50 − 49 𝑏 1 = 1 = 1 𝑤 2 = 50 𝑐 2 = 50 𝑏 2 = 3 𝑣 1 = 3 5 ⋅ 100 = 60 5

Myerson’s Lemma • That didn’t work for sponsored search auctions, so what now? • Let’s try to see how the optimal truthful auction should look like, for any single parameter environment • Input by bidders: 𝒄 = (𝑐 1 , … , 𝑐 𝑜 ) • Allocation rule: 𝒚(𝒄) = (𝑦 1 (𝒄), … , 𝑦 𝑜 (𝒄)) • Payment rule: 𝒒(𝒄) = (𝑞 1 (𝒄), … , 𝑞 𝑜 (𝒄)) • The utility of bidder 𝑗 is 𝑣 𝑗 𝒄 = 𝑤 𝑗 ⋅ 𝑦 𝑗 (𝒄) − 𝑞 𝑗 (𝒄) • Focus on payment rules such that 𝑞 𝑗 𝒄 ∈ 0, 𝑐 𝑗 ⋅ 𝑦 𝑗 𝒄 – 𝑞 𝑗 𝒄 ≥ 0 ensures that the seller does not pay the bidders – 𝑞 𝑗 𝒄 ≤ 𝑐 𝑗 ⋅ 𝑦 𝑗 𝒄 ensures non-negative utility for truthful bidders

Myerson’s Lemma • An allocation rule 𝒚 is implementable if there exists a payment rule 𝒒 such that (𝒚, 𝒒) is a truthful auction • An allocation rule 𝒚 is monotone if for every bidder 𝑗 and bid vector 𝒄 −𝑗 , the allocation 𝑦 𝑗 (𝑨, 𝒄 −𝑗 ) is non-decreasing in the bid 𝑨 of bidder 𝑗 Lemma [Myerson, 1981 ] (a) An allocation rule 𝒚 is implementable if and only if it is monotone (b) For every allocation rule 𝒚 , there exists a unique payment rule 𝒒 such that (𝒚, 𝒒) is a truthful auction

Proof of Myerson’s Lemma • Fix a bidder 𝑗 , and the bids 𝒄 −𝑗 of the other bidders • Given that these quantities are now fixed, we simplify our notation: – 𝑦(𝑨) = 𝑦 𝑗 (𝑨, 𝒄 −𝑗 ) – 𝑞 𝑨 = 𝑞 𝑗 𝑨, 𝒄 −𝑗 – 𝑣 𝑨 = 𝑣 𝑗 𝑨, 𝒄 −𝑗 • The idea: – assuming (𝒚, 𝒒) is a truthful auction, the bidder has no incentive to unilaterally deviate to any other bid – This will give us a relation between 𝒚 and 𝒒 , which we can use to derive an explicit formula for 𝒒 as a function of 𝒚

Proof of Myerson’s Lemma • Consider two bids 0 ≤ 𝑨 < 𝑧 and assume 𝒚 is implementable by 𝒒 • True value = 𝑨 , deviating bid = 𝑧 : 𝑣 𝑨 ≥ 𝑣 𝑧 ⟺ 𝑨 ⋅ 𝑦 𝑨 − 𝑞 𝑨 ≥ 𝑨 ⋅ 𝑦 𝑧 − 𝑞 𝑧 ⟺ 𝑞 𝑧 − 𝑞 𝑨 ≥ 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 • True value = 𝑧 , deviating bid = 𝑨 : 𝑣 𝑧 ≥ 𝑣 𝑨 ⟺ 𝑧 ⋅ 𝑦 𝑧 − 𝑞 𝑧 ≥ 𝑧 ⋅ 𝑦 𝑨 − 𝑞 𝑨 ⟺ 𝑞 𝑧 − 𝑞 𝑨 ≤ 𝑧 ⋅ 𝑦 𝑧 − 𝑦 𝑨

Proof of Myerson’s Lemma • Combining these two, we get: 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≤ 𝑞 𝑧 − 𝑞 𝑨 ≤ 𝑧 ⋅ 𝑦 𝑧 − 𝑦 𝑨 • This also implies that 𝑧 − 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≥ 0 • Since 0 ≤ 𝑨 < 𝑧 , this is possible if and only if 𝒚 is monotone so that 𝑧 − 𝑨 > 0 and 𝑦 𝑧 − 𝑦 𝑨 > 0 ⇨ (a) is now proved

Proof of Myerson’s Lemma • We can now assume that 𝒚 is monotone • Assume 𝒚 is piecewise constant, like in sponsored search auctions 𝑦(𝑨) 0 𝑨 • The break points are defined by the highest bids of the other bidders

Proof of Myerson’s Lemma 𝑦(𝑨) 0 𝑨

Proof of Myerson’s Lemma 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≤ 𝑞 𝑧 − 𝑞 𝑨 ≤ 𝑧 ⋅ 𝑦 𝑧 − 𝑦 𝑨 • By fixing 𝑨 and taking the limit as 𝑧 tends to 𝑨 , we have that jump of 𝑞 at 𝑨 = 𝑨 ⋅ ( jump of 𝑦 at 𝑨)

Proof of Myerson’s Lemma 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≤ 𝑞 𝑧 − 𝑞 𝑨 ≤ 𝑧 ⋅ 𝑦 𝑧 − 𝑦 𝑨 • By fixing 𝑨 and taking the limit as 𝑧 tends to 𝑨 , we have that jump of 𝑞 at 𝑨 = 𝑨 ⋅ ( jump of 𝑦 at 𝑨) • Therefore, we can define the payment of the bidder as 𝑞 𝑐 = σ 𝑧∈[0,𝑐] 𝑧 ⋅ ( jump of 𝑦 at 𝑧) where 𝑧 enumerates all break points of 𝑦 in [0, 𝑐]