CSC304 Lecture 11 Mechanism Design w/ Money: Revenue maximization; - PowerPoint PPT Presentation

CSC304 Lecture 11 Mechanism Design w/ Money: Revenue maximization; Myerson’s auction CSC304 - Nisarg Shah 1

Announcements • Returning graded midterm ➢ Was only able to keep my promise due to wonderful TAs • Delighted by your performance! ➢ Given that the midterm was relatively hard • Coming up: 4-5 questions of homework 2 CSC304 - Nisarg Shah 2

Welfare vs Revenue • In the auction setting… ➢ We choose an outcome 𝑏 based on agent valuations {𝑤 𝑗 } ➢ And charge payments 𝑞 𝑗 to each agent 𝑗 • In welfare maximization, we want to maximize σ 𝑗 𝑤 𝑗 𝑏 ➢ VCG = DSIC + maximizes welfare on every single instance ➢ Beautiful! • In revenue maximization, we want to maximize σ 𝑗 𝑞 𝑗 ➢ We can still use DSIC mechanisms (revelation principle). BUT… CSC304 - Nisarg Shah 3

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

Single-Item Auction Framework • 𝑜 bidders • Value 𝑤 𝑗 of bidder 𝑗 is drawn from distribution 𝐺 𝑗 with density 𝑔 𝑗 and support 0, 𝑤 𝑛𝑏𝑦 𝑗 , and wants to maximize 𝐹 σ 𝑗 𝑞 𝑗 • Principal knows 𝐺 ➢ Expectation over each 𝑤 𝑗 drawn i.i.d. from 𝐺 𝑗 ➢ Principal wants to use a DSIC mechanism o IC part is without loss of generality (revelation principle) o Will see that can’t do better using BNIC mechanisms CSC304 - Nisarg Shah 5

Single Item, Single Bidder • Revisiting 1 item, 1 bidder • Value 𝑤 ∼ 𝐺 • Want to post a price 𝑠 on the item • Revenue from price 𝑠 → 𝑠 ⋅ 1 − 𝐺 𝑠 (Why?) • Awesome! Select 𝑠 ∗ = argmax 𝑠 𝑠 ⋅ 1 − 𝐺 𝑠 ➢ “Monopoly price” ➢ Note: 𝑠 ∗ depends on 𝐺 , but not on 𝑤 ⇒ DSIC CSC304 - Nisarg Shah 6

Single Item, Single Bidder • Suppose the bidder’s value is drawn from the uniform distribution 𝑉 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 CSC304 - Nisarg Shah 7

An Aside • In welfare maximization, we are bound to always selling the item • In revenue maximization, we are willing to risk leaving the item unsold ➢ If the item is not sold, you get 0 revenue ➢ But if sold, you can get more than 2 nd highest bid • Subject to always selling the item, VCG actually has the highest revenue ➢ Revenue equivalence: “same allocation ⇒ same payment” CSC304 - Nisarg Shah 8

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 • A possible 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) CSC304 - Nisarg Shah 9

Single Item, Two Bidders • Reserve prices are ubiquitous ➢ E.g., opening bids in eBay auctions ➢ Guarantee a certain revenue to the auctioneer if item is sold • 𝐹 revenue = 𝐹 max 𝑠, min 𝑤 1 , 𝑤 2 ➢ Maximize over 𝑠 ? • What about other DSIC mechanisms? What if there are more bidders? Other distributions? CSC304 - Nisarg Shah 10

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 11

Single-Parameter Environments • Each agent 𝑗 has a private value 𝑤 𝑗 ∼ 𝐺 𝑗 , ➢ Value if the agent is “served” ➢ Example: single-item auction → win the item ➢ Example: combinatorial auction + single-minded bidder → get the desired set ➢ Can potentially allow agents to be “fractionally” served • Fixing bids of other agents… ➢ Let 𝑦 𝑗 𝑤 𝑗 = fraction served when reporting 𝑤 𝑗 o When fractional serving not allowed, this is in {0,1} ➢ Let 𝑞 𝑗 𝑤 𝑗 = payment charged when reporting 𝑤 𝑗 CSC304 - Nisarg Shah 12

Myerson’s Lemma • Myerson’s Lemma: For a single-parameter environment, a strategy profile is in BNE under a mechanism if and only if 1. 𝑦 𝑗 (𝑤 𝑗 ) is monotone non-decreasing 𝑤 𝑗 𝑦 𝑗 𝑨 𝑒𝑨 + 𝑞 𝑗 (0) 2. 𝑞 𝑗 𝑤 𝑗 = 𝑤 𝑗 ⋅ 𝑦 𝑗 𝑤 𝑗 − ׬ 0 (typically, 𝑞 𝑗 0 = 0 ) • Intuition similar to 2 nd price auction ➢ For every “ 𝜀 ” allocation, pay the lowest value that would have won it CSC304 - Nisarg Shah 13

Myerson’s Lemma • Note: allocation determines unique payments 𝑤 𝑗 𝑞 𝑗 𝑤 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 14

Myerson’s Theorem • “Expected Revenue = Expected Virtual Welfare” 𝑤 𝑗 𝑦 𝑗 𝑨 𝑒𝑨 + 𝑞 𝑗 (0) ➢ Recall: 𝑞 𝑗 𝑤 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 𝑤 𝑗 = 𝐹 {𝑤 𝑗 ∼𝐺 𝑗 } Σ 𝑗 𝜒 𝑗 𝑤 𝑗 ⋅ 𝑦 𝑗 (𝑤 𝑗 ) 1−𝐺 𝑗 (𝑤 𝑗 ) • 𝜒 𝑗 𝑤 𝑗 = 𝑤 𝑗 − is called the virtual value of bidder 𝑗 𝑔 𝑗 (𝑤 𝑗 ) • Virtual welfare = sum of virtual values*allocations CSC304 - Nisarg Shah 15

Myerson’s Auction • Need the allocation 𝑦 𝑗 to be monotonic • E[revenue] = E[virtual welfare] • Myerson’s auction: “The auction that maximizes (expected) revenue is the one whose allocation maximizes the virtual welfare subject to monotonicity” • Let’s apply this to some examples! CSC304 - Nisarg Shah 16

Example • 2 bidders, 1 item, values drawn i.i.d. from 𝑉[0,1] 1−𝐺 𝑤 1−𝑤 ➢ 𝜒 𝑤 = 𝑤 − = 𝑤 − 1 = 2𝑤 − 1 𝑔 𝑤 ➢ Note: virtual value can be negative!! • Given bids 𝑤 1 , 𝑤 2 , … ➢ Maximize 𝑦 1 ⋅ 2𝑤 1 − 1 + 𝑦 2 ⋅ 2𝑤 2 − 1 ➢ Subject to 𝑦 1 , 𝑦 2 ∈ {0,1} and 𝑦 1 + 𝑦 2 ≤ 1 CSC304 - Nisarg Shah 17

Optimal Auction Example • Maximize 𝑦 1 ⋅ 2𝑤 1 − 1 + 𝑦 2 ⋅ 2𝑤 2 − 1 ➢ 𝑦 1 , 𝑦 2 ∈ {0,1} and 𝑦 1 + 𝑦 2 ≤ 1 • Prove on the board: ➢ Allocation: o If ∃ bidder with value ≥ ½ , give to the highest bidder. o If both have value < ½ , neither gets the item. ➢ Payment if item sold = max(½, lesser bid) • Precisely VCG with reserve price ½ CSC304 - Nisarg Shah 18

Optimal Auctions • Theorem: For a single item and 𝑜 bidders whose valuations are drawn i.i.d., the optimal auction is VCG with reserve price 𝜒 −1 (0) . ➢ Note: Reserve price is independent of #bidders! • Wait! We didn’t check for monotonicity of allocation! • It turns out that for “nice” distributions, maximizing virtual welfare already gives a monotonic allocation rule! CSC304 - Nisarg Shah 19

Special Distributions • Regular Distributions: A distribution 𝐺 is regular if its virtual value function 𝑤−(1−𝐺 𝑤 )/𝑔(𝑤) is non-decreasing. • Lemma: If all 𝐺 𝑗 ’s are regular, the virtual welfare maximizing rule is monotone. • Monotone Hazard Rate (MHR): A distribution 𝐺 has monotone hazard rate if (1−𝐺 𝑤 )/𝑔(𝑤) is non-increasing. ➢ Important special case (MHR ⇒ Regular) CSC304 - Nisarg Shah 20

Special Distributions • Not crazy assumptions ➢ Many practical distributions are MHR: e.g., uniform, exponential, Gaussian. ➢ Some important distributions are not MHR, but still regular: e.g., power-law distributions. CSC304 - Nisarg Shah 21

Optimal Single-Item Auction • Allocation: Give the item to agent 𝑗 with highest 𝜒 𝑗 𝑤 𝑗 if that is non-negative • Payment: “lowest bid that still would have won” 𝑤 𝑗 𝑦 𝑗 𝑨 𝑒𝑨 + 𝑞 𝑗 (0) ➢ Follows from 𝑞 𝑗 𝑤 𝑗 = 𝑤 𝑗 ⋅ 𝑦 𝑗 𝑤 𝑗 − ׬ 0 • All 𝐺 𝑗 ’s are equal to 𝐺 and regular: ➢ 𝑠 ∗ = monopoly price of 𝐺 ➢ Item goes to the highest bidder if bid more than 𝑠 ∗ ➢ Payment charged is max(𝑠 ∗ , 2nd highest bid) , ➢ VCG with reserve price 𝑠 ∗ ! CSC304 - Nisarg Shah 22

Extensions • Irregular distributions: ➢ E.g., multi-modal or extremely heavy tail distributions ➢ Need to add the monotonicity constraint ➢ Turns out, we can “iron” irregular distributions to make them regular, and use standard Myerson’s framework • Relaxing DSIC to BNIC ➢ Myerson’s mechanism has optimal revenue among all DSIC mechanisms ➢ Turns out, it also has optimal revenue among the much larger class of BNIC mechanisms! CSC304 - Nisarg Shah 23