Announcements No class next Tuesday CS Department Research - PowerPoint PPT Presentation

Announcements Ø No class next Tuesday Ø CS Department Research Symposium (10/08, next Tuesday) 1

CS6501: T opics in Learning and Game Theory (Fall 2019) Optimal Auction Design for Single-Item Allocation (Part II) Instructor: Haifeng Xu

Outline Ø Recap: Mechanism Design Basics Ø Optimal Auction Design for Independent Bidders 3

Single-Item Allocation 4

Single-Item Allocation Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣, 𝑔⟩ where: Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø 𝑊 = 𝑊 7 × ⋯× 𝑊 ( is the set of all possible value profiles Ø 𝑌 = {0,1, ⋯ , 𝑜} is the set of winners Ø 𝑣 = (𝑣 7 , ⋯ , 𝑣 ( ) where 𝑣 " = 𝑤 " 𝑦 " − 𝑞 " is the utility function of 𝑗 for any (randomized) allocation 𝑦 ∈ Δ (@7 and payment 𝑞 " Ø 𝑔 is the public prior on buyer values 𝑤 ∈ 𝑊 Ø For convenience, think of 𝑤 " ∼ 𝑔 " independently Ø Objective: maximize revenue ∑ "∈[(] 𝑞 " 5

The Design Space – Mechanisms A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where: Ø A = 𝐵 7 × ⋯× 𝐵 ( where 𝐵 " is allowable actions for buyer 𝑗 Ø 𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to outcome = [an allocation 𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏 7 , ⋯ , 𝑏 ( ) ∈ 𝐵 Ø That is, we will design ⟨𝐵, 𝑕⟩ Ø Players’ utility function will be fully determined by ⟨𝐵, 𝑕⟩ Ø We want to maximize revenue at the Bayes Nash equilibrium of this resulting game 6

The Design Space – Mechanisms A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where: Ø A = 𝐵 7 × ⋯× 𝐵 ( where 𝐵 " is allowable actions for buyer 𝑗 Ø 𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to outcome = [an allocation 𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏 7 , ⋯ , 𝑏 ( ) ∈ 𝐵 Example: second-price auction Ø 𝐵 " = ℝ @ for all 𝑗 Ø 𝑕 𝑏 allocates the item to the buyer 𝑗 ∗ = arg max 𝑏 " and asks " 𝑗 ∗ to pay max2 " 𝑏 " , and all other buyers pay 0 Ø Truthful bidding is a dominant-strategy equilibrium, thus also a BNE Ø Thus expect truthful bidding (i.e., 𝑏 " = 𝑤 " ); Revenue will be 𝔽 P∼Q max2 " 𝑤 " 7

Incentive Compatible Mechanisms Definition . A mechanism ⟨𝐵, 𝑕⟩ is a direct revelation mechanism if 𝐵 " = 𝑊 " for all 𝑗 . In this case, the mechanism is described by 𝑕 . Ø In DR mechanism, we only need to design 𝑕 Definition . A direct revelation mechanism 𝑕 is Bayesian incentive-compatible (a.k.a., truthful or BIC) if truthful bidding forms a Bayes Nash equilibrium in the resulting game Ø A stronger notion of IC is dominant-strategy IC (DIC) Ø A DIC mechanism is also BIC Ø Example: second-price auction is DIC • First price auction can be “modified” to be BIC 8

The Revelation Principle Theorem . If there is a mechanism that achieves revenue 𝑆 at a Bayes Nash equilibrium [resp. dominant-strategy equilibrium], then there is a direct revelation, Bayesian incentive-compatible [resp. DIC] mechanism achieving revenue 𝑆 . Ø Proof idea: let the auctioneer to simulate the strategic behaviors on behalf of bidders, so they only need to react honestly 9

The Revelation Principle Theorem . If there is a mechanism that achieves revenue 𝑆 at a Bayes Nash equilibrium [resp. dominant-strategy equilibrium], then there is a direct revelation, Bayesian incentive-compatible [resp. DIC] mechanism achieving revenue 𝑆 . Ø Proof idea: let the auctioneer to simulate the strategic behaviors on behalf of bidders, so they only need to react honestly Optimal Mechanism Design for Single-Item Allocation Given instance ⟨𝑜, 𝑊, 𝑌, 𝑣, 𝑔⟩ , design the allocation function 𝑦: 𝑊 → 𝑌 and payment 𝑞: 𝑊 → ℝ ( such that truthful bidding is a BNE in the following Bayesian game: Solicit bid 𝑐 7 ∈ 𝑊 7 , ⋯ , 𝑐 ( ∈ 𝑊 1. ( Select allocation 𝑦 𝑐 7 , ⋯ , 𝑐 ( ∈ 𝑌 and payment 𝑞(𝑐 7 , ⋯ , 𝑐 ( ) 2. 10

Optimal Bayesian Mechanism Design Ø Previous formulation and simplification leads to the following optimization problem ( 𝔽 P∼Q ∑ "V7 max 𝑞 " (𝑤 7 , ⋯ , 𝑤 ( ) T,U s. t. 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑤 " , 𝑤 \" − 𝑞 " 𝑤 " , 𝑤 \" ≥ 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑐 " , 𝑤 \" − 𝑞 " 𝑐 " , 𝑤 \" , ∀𝑗 ∈ 𝑜 , 𝑤 " , 𝑐 " ∈ 𝑊 " 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑤 " , 𝑤 \" − 𝑞 " 𝑤 " , 𝑤 \" ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤 " ∈ 𝑊 " ( ∑ "V_ 𝑦 " (𝑤) = 1, ∀𝑤 ∈ 𝑊 𝑦 " 𝑤 ≥ 0, ∀𝑤 ∈ 𝑊, ∀𝑗 = 0,1 ⋯ , 𝑜 11

Optimal Bayesian Mechanism Design Ø Previous formulation and simplification leads to the following optimization problem ( 𝔽 P∼Q ∑ "V7 max 𝑞 " (𝑤 7 , ⋯ , 𝑤 ( ) BIC constraints T,U s. t. 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑤 " , 𝑤 \" − 𝑞 " 𝑤 " , 𝑤 \" ≥ 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑐 " , 𝑤 \" − 𝑞 " 𝑐 " , 𝑤 \" , ∀𝑗 ∈ 𝑜 , 𝑤 " , 𝑐 " ∈ 𝑊 " 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑤 " , 𝑤 \" − 𝑞 " 𝑤 " , 𝑤 \" ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤 " ∈ 𝑊 " Individually rational (IR) ( ∑ "V_ 𝑦 " (𝑤) = 1, ∀𝑤 ∈ 𝑊 constraints 𝑦 " 𝑤 ≥ 0, ∀𝑤 ∈ 𝑊, ∀𝑗 = 0,1 ⋯ , 𝑜 12

Optimal Bayesian Mechanism Design Ø Previous formulation and simplification leads to the following optimization problem Ø If 𝑊 has finite support, this is an LP with variables 𝑦 " 𝑤 , 𝑞 " 𝑤 ",P ( 𝔽 P∼Q ∑ "V7 max 𝑞 " (𝑤 7 , ⋯ , 𝑤 ( ) BIC constraints T,U s. t. 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑤 " , 𝑤 \" − 𝑞 " 𝑤 " , 𝑤 \" ≥ 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑐 " , 𝑤 \" − 𝑞 " 𝑐 " , 𝑤 \" , ∀𝑗 ∈ 𝑜 , 𝑤 " , 𝑐 " ∈ 𝑊 " 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑤 " , 𝑤 \" − 𝑞 " 𝑤 " , 𝑤 \" ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤 " ∈ 𝑊 " Individually rational (IR) ( ∑ "V_ 𝑦 " (𝑤) = 1, ∀𝑤 ∈ 𝑊 constraints 𝑦 " 𝑤 ≥ 0, ∀𝑤 ∈ 𝑊, ∀𝑗 = 0,1 ⋯ , 𝑜 13

Optimal Bayesian Mechanism Design Ø Previous formulation and simplification leads to the following optimization problem Ø If 𝑊 has finite support, this is an LP with variables 𝑦 " 𝑤 , 𝑞 " 𝑤 ",P Ø Drawbacks of this algorithmic approach: (1) Support of 𝑊 may be extremely large in which case LP is large (2) Do not reveal any structure about the optimal auction – do not know what it is like except that it is a solution to an LP Ø Next, will look at continuous 𝑊 and solve out for the optimal function 𝑦 𝑤 , 𝑞 𝑤 • This will also lead to an elegant form of the optimal auction 14

Outline Ø Recap: Mechanism Design Basics Ø Optimal Auction Design for Independent Bidders • That is, will assume 𝑤 " ∼ 𝑔 " independently 15

The Optimal Auction (Myerson’1981) Theorem (informal). For single-item allocation with prior distribution 𝑤 " ∼ 𝑔 " independently, the following auction is BIC and optimal: Solicit buyer values 𝑤 7 , ⋯ , 𝑤 ( 1. 7\a [ (P [ ) Transform 𝑤 " to “virtual value” 𝜚 " (𝑤 " ) where 𝜚 " 𝑤 " = 𝑤 " − 2. Q [ (P [ ) If 𝜚 " 𝑤 " < 0 for all 𝑗 , keep the item and no payments 3. Otherwise, allocate item to 𝑗 ∗ = arg max "∈[(] 𝜚 " (𝑤 " ) 4. and charge him \7 max max the minimum bid needed to win, i.e., 𝜚 " cd" ∗ 𝜚 c (𝑤 c ) , 0 ; Other bidders pay 0 . 16

Stages of a Bayesian Game Ø Stages of a Bayesian game of mechanism design: • Ex-ante: Before players learn their types • Interim: A player learns his own type, but not the types of others • Ex-post: All players types are revealed Ø Interim stage is when players make decisions • The interim allocation for buyer 𝑗 tells us what 𝑗 ’s probability of winning is as a function of his bid 𝑐 " , in expectation over others’ truthful report 𝑦 " 𝑐 " = 𝔽 P Z[ ∼Q Z[ 𝑦 " 𝑐 " , 𝑤 \" e • Similarly, the interim payment is 𝑞 " 𝑐 " = 𝔽 P Z[ ∼Q Z[ 𝑞 " 𝑐 " , 𝑤 \" e • Expected bidder utility of bidding 𝑐 " 𝔽 P Z[ ∼Q Z[ 𝑤 " 𝑦 " 𝑐 " , 𝑤 \" − 𝑞 " 𝑐 " , 𝑤 \" = 𝑤 " e 𝑦 " 𝑐 " − e 𝑞 " 𝑐 " • If BIC, expected revenue ( ( ( 𝔽 P∼Q ∑ "V7 𝑞 " (𝑤 7 , ⋯ , 𝑤 ( ) = ∑ "V7 𝔽 P∼Q 𝑞 " (𝑤 7 , ⋯ , 𝑤 ( ) = ∑ "V7 𝔽 P [ ∼Q [ e 𝑞 " (𝑤 " ) 17