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 𝑤𝑗 for the item

– This value represents the willingness-to-pay of the agent; that is, 𝑤𝑗 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 𝑤𝑗 − 𝑞

Single-item auctions

• General structure of an auction:

– Input: every agent 𝑗 submits a bid 𝑐𝑗 (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?

𝑤1 = 100 𝑤2 = 50

First-price auction

• Allocation rule: the winner is the highest bidder
• Payment rule: the winner pays her bid
• Is this a truthful auction?

𝑤1 = 100 𝑤2 = 50 𝑐1 = 50.1 𝑐2 = 50

Second-price auction

• Allocation rule: the winner is the highest bidder
• Payment rule: the winner pays the second highest bid
• (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 Theorem [Vickrey, 1961] In a second-price auction (a) it is a dominant strategy for every bidder 𝑗 to bid 𝑐𝑗 = 𝑤𝑗, and (b) every truthtelling bidder gets non-negative utility

Second-price auction

• For (a), our goal is to show that the utility of bidder 𝑗 is maximized by

bidding 𝑤𝑗, no matter what 𝑤𝑗 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 𝑐𝑗 < 𝐶, or 𝑤𝑗 − 𝐶 otherwise

Case I: 𝒘𝒋 < 𝑪

• Maximum possible utility = 0
• Achieved by setting 𝑐𝑗 = 𝑤𝑗

Case II: 𝒘𝒋 ≥ 𝑪

• Maximum possible utility = 𝑤𝑗 − 𝐶
• Bidder 𝑗 wins the item by setting 𝑐𝑗 = 𝑤𝑗

▢

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) 𝑏𝑘

– 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 𝑏1 ≥ ⋯ ≥ 𝑏𝑙
• Each bidder 𝑗 has a private value 𝑤𝑗 per click

– Bidder 𝑗 derives utility 𝑏𝑘 ⋅ 𝑤𝑗 from slot 𝑘

Sponsored search auctions: goals

• Truthfulness: It is a dominant strategy for each bidder to bid her true

value

• Social welfare maximization: σ𝑗 𝑤𝑗 ⋅ 𝑦𝑗

– 𝑦𝑗 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: σ𝑗 𝑤𝑗 ⋅ 𝑦𝑗

– 𝑦𝑗 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 𝑐1 ≥ ⋯ ≥ 𝑐𝑜

• Payment rule: every bidder 𝑗 ≤ 𝑙 (who is assigned at slot 𝑗) pays the

next highest bid 𝑐𝑗+1 per click, and every bidder 𝑗 > 𝑙 pays 0

𝑤1 = 100 𝑤2 = 50 𝑏1 = 1 𝑏2 = 3 5

Generalized second-price auction

• Allocation rule: sort the bidders in decreasing order of their bids and

rename them so that 𝑐1 ≥ ⋯ ≥ 𝑐𝑜

• Payment rule: every bidder 𝑗 ≤ 𝑙 (who is assigned at slot 𝑗) pays the

next highest bid 𝑐𝑗+1 per click, and every bidder 𝑗 > 𝑙 pays 0

𝑤1 = 100 𝑤2 = 50 𝑏1 = 1 𝑏2 = 3 5 𝑐1 = 100 𝑐2 = 50 𝑣1 = 1 ⋅ 100 − 50 = 50 𝑣2 = 3 5 ⋅ 50 = 30

Generalized second-price auction

• Allocation rule: sort the bidders in decreasing order of their bids and

rename them so that 𝑐1 ≥ ⋯ ≥ 𝑐𝑜

• Payment rule: every bidder 𝑗 ≤ 𝑙 (who is assigned at slot 𝑗) pays the

next highest bid 𝑐𝑗+1 per click, and every bidder 𝑗 > 𝑙 pays 0

𝑤1 = 100 𝑤2 = 50 𝑏1 = 1 𝑏2 = 3 5 𝑐1 = 49 𝑐2 = 50 𝑣2 = 1 ⋅ 50 − 49 = 1 𝑣1 = 3 5 ⋅ 100 = 60

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 = (𝑐1, … , 𝑐𝑜)
• Allocation rule: 𝒚(𝒄) = (𝑦1(𝒄), … , 𝑦𝑜(𝒄))
• Payment rule: 𝒒(𝒄) = (𝑞1(𝒄), … , 𝑞𝑜(𝒄))
• The utility of bidder 𝑗 is 𝑣𝑗 𝒄 = 𝑤𝑗 ⋅ 𝑦𝑗(𝒄) − 𝑞𝑗(𝒄)
• Focus on payment rules such that 𝑞𝑗 𝒄 ∈ 0, 𝑐𝑗 ⋅ 𝑦𝑗 𝒄

– 𝑞𝑗 𝒄 ≥ 0 ensures that the seller does not pay the bidders – 𝑞𝑗 𝒄 ≤ 𝑐𝑗 ⋅ 𝑦𝑗 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 𝑦𝑗(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) = 𝑦𝑗(𝑨, 𝒄−𝑗) – 𝑞 𝑨 = 𝑞𝑗 𝑨, 𝒄−𝑗 – 𝑣 𝑨 = 𝑣𝑗 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:
• True value = 𝑧, deviating bid = 𝑨:

𝑣 𝑨 ≥ 𝑣 𝑧 ⟺ 𝑨 ⋅ 𝑦 𝑨 − 𝑞 𝑨 ≥ 𝑨 ⋅ 𝑦 𝑧 − 𝑞 𝑧 ⟺ 𝑞 𝑧 − 𝑞 𝑨 ≥ 𝑨 ⋅ 𝑦 𝑧 − 𝑦 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:
• This also implies that
• 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 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≤ 𝑞 𝑧 − 𝑞 𝑨 ≤ 𝑧 ⋅ 𝑦 𝑧 − 𝑦 𝑨 𝑧 − 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≥ 0

Proof of Myerson’s Lemma

• We can now assume that 𝒚 is monotone
• Assume 𝒚 is piecewise constant, like in sponsored search auctions
• The break points are defined by the highest bids of the other bidders

𝑦(𝑨) 𝑨

Proof of Myerson’s Lemma

𝑦(𝑨) 𝑨

Proof of Myerson’s Lemma

• By fixing 𝑨 and taking the limit as 𝑧 tends to 𝑨, we have that

𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≤ 𝑞 𝑧 − 𝑞 𝑨 ≤ 𝑧 ⋅ 𝑦 𝑧 − 𝑦 𝑨 jump of 𝑞 at 𝑨 = 𝑨 ⋅ (jump of 𝑦 at 𝑨)

Proof of Myerson’s Lemma

• By fixing 𝑨 and taking the limit as 𝑧 tends to 𝑨, we have that
• Therefore, we can define the payment of the bidder as

where 𝑧 enumerates all break points of 𝑦 in [0, 𝑐] 𝑨 ⋅ 𝑦 𝑧 − 𝑦 𝑨 ≤ 𝑞 𝑧 − 𝑞 𝑨 ≤ 𝑧 ⋅ 𝑦 𝑧 − 𝑦 𝑨 jump of 𝑞 at 𝑨 = 𝑨 ⋅ (jump of 𝑦 at 𝑨) 𝑞 𝑐 = σ𝑧∈[0,𝑐] 𝑧 ⋅ (jump of 𝑦 at 𝑧)

Proof of Myerson’s Lemma

• Example:

𝑦(𝑨) 𝑨 𝑐 𝑧1 𝑧2

𝑞 𝑐 = σ𝑧∈[0,𝑐] 𝑧 ⋅ (jump of 𝑦 at 𝑧) = 𝑧1 ⋅ 𝑦1 + 𝑧2 ⋅ (𝑦2 − 𝑦1)

𝑦1 𝑦2

Proof of Myerson’s Lemma

• Example:

𝑦(𝑨) 𝑨 𝑐 𝑧1 𝑧2

𝑞 𝑐 = σ𝑧∈[0,𝑐] 𝑧 ⋅ (jump of 𝑦 at 𝑧) = 𝑧1 ⋅ 𝑦1 + 𝑧2 ⋅ (𝑦2 − 𝑦1)

𝑦1 𝑦2

Proof of Myerson’s Lemma

• Example:

𝑦(𝑨) 𝑨 𝑐 𝑧1 𝑧2

𝑞 𝑐 = σ𝑧∈[0,𝑐] 𝑧 ⋅ (jump of 𝑦 at 𝑧) = 𝑧1 ⋅ 𝑦1 + 𝑧2 ⋅ (𝑦2 − 𝑦1)

𝑦1 𝑦2

Proof of Myerson’s Lemma

𝑦(𝑨) 𝑨 𝑤 𝑐 𝑦(𝑨) 𝑨 𝑤 𝑐 𝑦(𝑨) 𝑨 𝑐 = 𝑤

Proof of Myerson’s Lemma

𝑦(𝑨) 𝑨 𝑤 𝑐 𝑦(𝑨) 𝑨 𝑤 𝑐 𝑦(𝑨) 𝑨 𝑐 = 𝑤

Proof of Myerson’s Lemma

𝑦(𝑨) 𝑨 𝑤 𝑐 𝑦(𝑨) 𝑨 𝑤 𝑐 𝑦(𝑨) 𝑨 𝑐 = 𝑤

Sponsored search auctions

• 𝑧 enumerates the break points: the bids that are smaller than 𝑐

– In other words, 𝑧 enumerates the slots from worst to best

• jump of 𝑦 at 𝑧: the difference in CTR between two consecutive slots
• The total payment of the 𝑗-th highest bidder is:

𝑞 𝑐 = σ𝑧∈[0,𝑐] 𝑧 ⋅ (jump of 𝑦 at 𝑧) 𝑞𝑗 𝑐𝑗, 𝒄−𝑗 = ෍

𝑘=𝑗 𝑙

𝑐

𝑘+1(𝑏𝑘 − 𝑏𝑘+1)

Summary

• Auctions: allocation rule + payment rule
• An allocation rule is implementable is there exists a payment rule,

so that together they define a truthful auction

• An allocation rule is monotone, if larger bids give more stuff
• Single-item auctions: first-price is not truthful, second-price is

truthful and maximizes the social welfare (sells to the bidder with the highest value)

• Sponsored search auctions: generalized second-price auction is

not truthful

• Myerson’s Lemma: a characterization of truthful mechanisms in

single-parameter environments

• Using Myerson’s Lemma we can design a truthful sponsored search

auction