CSC304 Lecture 12 Mechanism Design w/ Money: Revenue maximization - PowerPoint PPT Presentation

CSC304 Lecture 12 Mechanism Design w/ Money: Revenue maximization Myerson’s Auction CSC304 - Nisarg Shah 1

Revenue Maximization CSC304 - Nisarg Shah 2

Welfare vs Revenue • In welfare maximization, we want to maximize σ 𝑗 𝑤 𝑗 𝑏 ➢ VCG = strategyproof + maximizes welfare on every single instance ➢ Beautiful! • In revenue maximization, we want to maximize σ 𝑗 𝑞 𝑗 ➢ We can still use strategyproof mechanisms (revelation principle). ➢ BUT… CSC304 - Nisarg Shah 3

Welfare vs Revenue • Different strategyproof mechanisms are better for different instances. • Example: 1 item, 1 bidder, unknown value 𝑤 ➢ strategyproof = fix a price 𝑠 , let the agent decide to “take it” ( 𝑤 ≥ 𝑠 ) or “leave it” ( 𝑤 < 𝑠 ) ➢ Maximize welfare → set 𝑠 = 0 o Must allocate item as long as the agent has a positive value ➢ Maximize revenue → 𝑠 = ? o Different 𝑠 are better for different 𝑤 CSC304 - Nisarg Shah 4

Welfare vs Revenue • We cannot optimize revenue on every instance ➢ Need to optimize the expected revenue in the Bayesian framework • If we want to achieve higher expected revenue than VCG, we cannot always allocate the item ➢ Revenue equivalence principle! CSC304 - Nisarg Shah 5

Single Item + Single Bidder • Value 𝑤 is drawn from distribution with CDF 𝐺 • Goal: post the optimal price 𝑠 on the item • Revenue from price 𝑠 = 𝑠 ⋅ 1 − 𝐺 𝑠 (Why?) • Optimal 𝑠 ∗ = argmax 𝑠 𝑠 ⋅ 1 − 𝐺 𝑠 ➢ “Monopoly price” ➢ Note: 𝑠 ∗ depends on 𝐺 , but not on 𝑤 , so the mechanism is strategyproof. CSC304 - Nisarg Shah 6

Example • Suppose 𝐺 is the CDF of the uniform distribution over [0,1] (denote by 𝑉 0,1 ). ➢ CDF is given by 𝐺 𝑦 = 𝑦 for all 𝑦 ∈ [0,1] . • Recall: E[Revenue] from price 𝑠 is 𝑠 ⋅ 1 − 𝐺 𝑠 ➢ Q: What is the optimal posted price? ➢ Q: What is the corresponding optimal revenue? • Compare this to the revenue of VCG, which is 0 ➢ This is because if the value is less than 𝑠 ∗ , we are willing to risk not selling the item. CSC304 - Nisarg Shah 7

Single Item + Two Bidders • 𝑤 1 , 𝑤 2 ∼ 𝑉[0,1] • VCG revenue = 2 nd highest bid = min(𝑤 1 , 𝑤 2 ) ➢ 𝐹 min 𝑤 1 , 𝑤 2 = 1/3 (Exercise!) • Improvement: “VCG with reserve price” ➢ Reserve price 𝑠 ➢ Highest bidder gets the item if bid more than 𝑠 ➢ Pays max(𝑠, 2 nd highest bid) o “Critical payment” : Pay the least value you could have bid and still won the item CSC304 - Nisarg Shah 8

Single Item + Two Bidders • Reserve prices are ubiquitous ➢ E.g., opening bids in eBay auctions ➢ Guarantee a certain revenue to auctioneer if item is sold • 𝐹 revenue = 𝐹 max 𝑠, min 𝑤 1 , 𝑤 2 ➢ Maximize over 𝑠 ? Hard to think about. • What about a strategyproof mechanism that is not VCG + reserve price? ➢ What about just BNIC mechanisms? CSC304 - Nisarg Shah 9

Single-Parameter Environments • Roger B. Myerson solved revenue optimal auctions in “single -parameter environments” • Proposed a simple auction that maximizes expected revenue CSC304 - Nisarg Shah 10

Single-Parameter Environments • Each agent 𝑗 … ➢ has a private value 𝑤 𝑗 drawn from a distribution with CDF 𝐺 𝑗 and PDF 𝑔 𝑗 ➢ is “satisfied” at some level 𝑦 𝑗 ∈ [0,1] , which gives the agent value 𝑦 𝑗 ⋅ 𝑤 𝑗 ➢ is asked to pay 𝑞 𝑗 • Examples ➢ Single divisible item ➢ Single indivisible item ( 𝑦 𝑗 ∈ {0,1} – this is okay too!) ➢ Many items, single-minded bidders (again 𝑦 𝑗 ∈ {0,1} ) CSC304 - Nisarg Shah 11

Myerson’s Lemma • Myerson’s Lemma: For a single-parameter environment, a mechanism is strategyproof if and only if for all 𝑗 1. 𝑦 𝑗 is monotone non-decreasing in 𝑤 𝑗 𝑤 𝑗 𝑦 𝑗 𝑨 𝑒𝑨 + 𝑞 𝑗 (0) 2. 𝑞 𝑗 = 𝑤 𝑗 ⋅ 𝑦 𝑗 𝑤 𝑗 − ׬ 0 (typically, 𝑞 𝑗 0 = 0 ) • Generalizes critical payments ➢ For every “ 𝜀 ” allocation, pay the lowest value that would have won it CSC304 - Nisarg Shah 12

Myerson’s Lemma • Note: allocation determines unique payments 𝑤 𝑗 𝑞 𝑗 = 𝑤 𝑗 ⋅ 𝑦 𝑗 𝑤 𝑗 − න 𝑦 𝑗 𝑨 𝑒𝑨 + 𝑞 𝑗 (0) 0 • A corollary: revenue equivalence ➢ If two mechanisms use the same allocation 𝑦 𝑗 , they “essentially” have the same expected revenue • Another corollary: optimal revenue auctions ➢ Optimizing revenue = optimizing some function of allocation (easier to analyze) CSC304 - Nisarg Shah 13

Myerson’s Theorem • “Expected Revenue = Expected Virtual Welfare” 𝑤 𝑗 𝑦 𝑗 𝑨 𝑒𝑨 + 𝑞 𝑗 (0) ➢ Recall: 𝑞 𝑗 = 𝑤 𝑗 ⋅ 𝑦 𝑗 𝑤 𝑗 − ׬ 0 ➢ Take expectation over draw of valuations + lots of calculus 𝐹 {𝑤 𝑗 ∼𝐺 𝑗 } Σ 𝑗 𝑞 𝑗 = 𝐹 {𝑤 𝑗 ∼𝐺 𝑗 } Σ 𝑗 𝜒 𝑗 ⋅ 𝑦 𝑗 1−𝐺 𝑗 (𝑤 𝑗 ) • 𝜒 𝑗 = 𝑤 𝑗 − = virtual value of bidder 𝑗 𝑔 𝑗 (𝑤 𝑗 ) • σ 𝑗 𝜒 𝑗 ⋅ 𝑦 𝑗 = virtual welfare CSC304 - Nisarg Shah 14

Myerson’s Theorem • Myerson’s auction : ➢ A strategyproof auction maximizes the (expected) revenue if its allocation rule maximizes the virtual welfare subject to monotonicity and it charges critical payments. • Charging critical payments is easy. • But maximizing virtual welfare subject to monotonicity is tricky. ➢ Let’s get rid of the monotonicity requirement! CSC304 - Nisarg Shah 15

Myerson’s Theorem Simplified • Regular Distributions ➢ A distribution 𝐺 is regular if its virtual value function 𝜒 𝑤 = 𝑤 − (1 − 𝐺 𝑤 )/𝑔 𝑤 is non-decreasing in 𝑤 . ➢ Many important distributions are regular, e.g., uniform, exponential, Gaussian, power- law, … • Lemma ➢ If all 𝐺 𝑗 ’s are regular, the allocation rule maximizing virtual welfare is already monotone. • Myerson’s Corollary: ➢ When all 𝐺 𝑗 ’s are regular, the strategyproof auction maximizes virtual welfare and charges critical payments. CSC304 - Nisarg Shah 16

Single Item + Single Bidder • Setup: ➢ Single indivisible item, single bidder, value 𝑤 drawn from a regular distribution with CDF 𝐺 and PDF 𝑔 • Goal: 1−𝐺 𝑤 ➢ Maximize 𝜒 ⋅ 𝑦 , where 𝜒 = 𝑤 − and 𝑦 ∈ {0,1} 𝑔 𝑤 • Optimal auction: 1−𝐺 𝑤 ∗ 1−𝐺 𝑤 ⇔ 𝑤 ≥ 𝑤 ∗ where 𝑤 ∗ = ➢ 𝑦 = 1 iff 𝜒 ≥ 0 ⇔ 𝑤 ≥ 𝑔 𝑤 ∗ 𝑔 𝑤 ➢ Critical payment: 𝑤 ∗ ➢ This is VCG with a reserve price of 𝜒 −1 (0) ! CSC304 - Nisarg Shah 17

Example • Optimal auction: 1−𝐺 𝑤 ➢ 𝑦 = 1 iff 𝜒 ≥ 0 ⇔ 𝑤 ≥ 𝑔 𝑤 1−𝐺 𝑤 ∗ ➢ Critical payment: 𝑤 ∗ such that 𝑤 ∗ = 𝑔 𝑤 ∗ • Distribution is 𝑉 0,1 : 1−𝑤 1 ➢ 𝑦 = 1 iff 𝑤 ≥ 1 ⇔ 𝑤 ≥ 2 1 ➢ Critical payment = 2 ➢ That is, we post the optimal price of 0.5 CSC304 - Nisarg Shah 18

Single Item + 𝑜 Bidders • Setup: ➢ Single indivisible item, each bidder 𝑗 has value 𝑤 𝑗 drawn from a regular distribution with CDF 𝐺 𝑗 and PDF 𝑔 𝑗 • Goal: 1−𝐺 𝑗 𝑤 𝑗 ➢ Maximize σ 𝑗 𝜒 𝑗 ⋅ 𝑦 𝑗 where 𝜒 𝑗 = 𝑤 𝑗 − and 𝑦 𝑗 ∈ 𝑔 𝑗 𝑤 𝑗 {0,1} such that σ 𝑗 𝑦 𝑗 ≤ 1 CSC304 - Nisarg Shah 19

Single Item + 𝑜 Bidders • Optimal auction: ➢ If all 𝜒 𝑗 < 0 : o Nobody gets the item, nobody pays anything o For all 𝑗 , 𝑦 𝑗 = 𝑞 𝑗 = 0 ➢ If some 𝜒 𝑗 ≥ 0 : o Agent with highest 𝜒 𝑗 wins the item, pays critical payment o 𝑗 ∗ ∈ 𝑏𝑠𝑕𝑛𝑏𝑦 𝑗 𝜒 𝑗 𝑤 𝑗 , 𝑦 𝑗 ∗ = 1, 𝑦 𝑗 = 0 ∀𝑗 ≠ 𝑗 ∗ −1 max 0, max 𝑘≠𝑗 ∗ 𝜒 𝑘 𝑤 𝑘 o 𝑞 𝑗 ∗ = 𝜒 𝑗 ∗ • Note: The item doesn’t necessarily go to the highest value agent! CSC304 - Nisarg Shah 20

Special Case: iid Values • Suppose all distributions are identical (say CDF 𝐺 and PDF 𝑔 ) • Check that the auction simplifies to the following ➢ Allocation: item goes to bidder 𝑗 ∗ with highest value if her value 𝑤 𝑗 ∗ ≥ 𝜒 −1 0 ➢ Payment charged = max 𝜒 −1 (0), max 𝑘≠𝑗 ∗ 𝑤 𝑘 • This is again VCG with a reserve price of 𝜒 −1 (0) CSC304 - Nisarg Shah 21

Example • Two bidders, both drawing iid values from 𝑉[0,1] 1−𝑤 ➢ 𝜒 𝑤 = 𝑤 − 1 = 2𝑤 − 1 ➢ 𝜒 −1 0 = 1/2 • Auction: ➢ Give the item to the highest bidder if their value is at least ½ ➢ Charge them max(½, 2 nd highest bid) CSC304 - Nisarg Shah 22

Example • Two bidders, one with value from 𝑉[0,1] , one with value from 𝑉[3,5] ➢ 𝜒 1 𝑤 1 = 2𝑤 1 − 1 1− 𝑤2−3 1−𝐺 2 𝑤 2 2 ➢ 𝜒 2 𝑤 2 = 𝑤 2 − = 𝑤 2 − = 2𝑤 2 − 5 1 2 Τ 𝑔 2 𝑤 2 • Auction: ➢ If 𝑤 1 < ½ and 𝑤 2 < 5/2 , the item remains unallocated. ➢ Otherwise… o If 2𝑤 1 − 1 > 2𝑤 2 − 5 , agent 1 gets it and pays max ½, 𝑤 2 − 2 5 2 , 𝑤 1 + 2 o If 2𝑤 1 − 1 < 2𝑤 2 − 5 , agent 2 gets it and pays max Τ CSC304 - Nisarg Shah 23