Announcements Ø HW2 is out, due 10/15 before class 1
CS6501: Topics in Learning and Game Theory (Fall 2019) Optimal Auction Design for Single-Item Allocation (Part I) Instructor: Haifeng Xu
Outline Ø Mechanism Design for Single-Item Allocation Ø Revelation Principle and Incentive Compatibility Ø The Revenue-Optimal Auction 3
Single-Item Allocation Ø A single and indivisible item, 𝑜 buyers 1, ⋯ , 𝑜 = [𝑜] Ø Buyer 𝑗 has a (private) value 𝑤 * ∈ 𝑊 * about the item Ø Outcome: choice of the winner of the item, and payment 𝑞 * from each buyer 𝑗 Ø Objectives: maximize revenue • Last lecture: VCG auction maximizes welfare even for multiple items 4
The Mechanism Design Problem Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where: Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø 𝑊 = 𝑊 6 × ⋯× 𝑊 / is the set of all possible value profiles Ø 𝑌 = {𝑓 9 , 𝑓 6 , ⋯ , 𝑓 / } is the set of all possible allocation outcomes Ø 𝑣 = (𝑣 6 , ⋯ , 𝑣 / ) where 𝑣 * = 𝑤 * 𝑦 * − 𝑞 * is the utility function of 𝑗 for any outcome 𝑦 ∈ 𝑌 and payment 𝑞 * required from 𝑗 Objective: maximize revenue ∑ *∈[/] 𝑞 * 5
The Mechanism Design Problem Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where: Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø 𝑊 = 𝑊 6 × ⋯× 𝑊 / is the set of all possible value profiles Ø 𝑌 = {𝑓 9 , 𝑓 6 , ⋯ , 𝑓 / } is the set of all possible allocation outcomes Ø 𝑣 = (𝑣 6 , ⋯ , 𝑣 / ) where 𝑣 * = 𝑤 * 𝑦 * − 𝑞 * is the utility function of 𝑗 for any outcome 𝑦 ∈ 𝑌 and payment 𝑞 * required from 𝑗 Objective: maximize revenue ∑ *∈[/] 𝑞 * Ø Cannot have any guarantee without additional assumptions Ø Will assume public prior knowledge on buyer values. For convenience, think of 𝑤 * ∼ 𝑔 * independently • Most results of this lecture hold for correlated 𝑤 * ’s, but easier to think for independent cases 6
The Mechanism Design Problem Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where: Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø 𝑊 = 𝑊 6 × ⋯× 𝑊 / is the set of all possible value profiles Ø 𝑌 = {𝑓 9 , 𝑓 6 , ⋯ , 𝑓 / } is the set of all possible allocation outcomes Ø 𝑣 = (𝑣 6 , ⋯ , 𝑣 / ) where 𝑣 * = 𝑤 * 𝑦 * − 𝑞 * is the utility function of 𝑗 for any outcome 𝑦 ∈ 𝑌 and payment 𝑞 * required from 𝑗 Remarks: Ø General mechanism design problem can be defined similarly Ø 𝑣 * = 𝑤 * 𝑦 * − 𝑞 * is called quasi-linear utility function • Not the only form of utility functions, but widely adopted Ø Typically, 𝑊 6 = ℝ A , but can also be intervals like [𝑏, 𝑐] 7
The Mechanism Design Problem Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where: Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø 𝑊 = 𝑊 6 × ⋯× 𝑊 / is the set of all possible value profiles Ø 𝑌 = {𝑓 9 , 𝑓 6 , ⋯ , 𝑓 / } is the set of all possible allocation outcomes Ø 𝑣 = (𝑣 6 , ⋯ , 𝑣 / ) where 𝑣 * = 𝑤 * 𝑦 * − 𝑞 * is the utility function of 𝑗 for any outcome 𝑦 ∈ 𝑌 and payment 𝑞 * required from 𝑗 Remarks: Ø Assume risk neural players – i.e., all players maximize expected utilities Ø Will guarantee 𝔽[𝑣 * ] ≥ 0 (a.k.a., individually rational or IR) • Otherwise, players would not even bother coming to your auction 8
The Design Space – Mechanisms A mechanism (i.e., the game) is specified by ⟨𝐵, ⟩ where: Ø A = 𝐵 6 × ⋯× 𝐵 / where 𝐵 * is allowable actions for buyer 𝑗 Ø : 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome 𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏 6 , ⋯ , 𝑏 / ) ∈ 𝐵 Ø That is, we will design ⟨𝐵, ⟩ 9
The Design Space – Mechanisms A mechanism (i.e., the game) is specified by ⟨𝐵, ⟩ where: Ø A = 𝐵 6 × ⋯× 𝐵 / where 𝐵 * is allowable actions for buyer 𝑗 Ø : 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome 𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏 6 , ⋯ , 𝑏 / ) ∈ 𝐵 Ø That is, we will design ⟨𝐵, ⟩ Ø Players’ utility function will be fully determined by ⟨𝐵, ⟩ Ø This is a game with incomplete information – 𝑤 * is privately known to player 𝑗 ; all other players only know its prior distribution 10
The Design Space – Mechanisms A mechanism (i.e., the game) is specified by ⟨𝐵, ⟩ where: Ø A = 𝐵 6 × ⋯× 𝐵 / where 𝐵 * is allowable actions for buyer 𝑗 Ø : 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome 𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏 6 , ⋯ , 𝑏 / ) ∈ 𝐵 Example 1: first-price auction Ø 𝐵 * = ℝ A for all 𝑗 Ø 𝑏 allocates the item to the buyer 𝑗 ∗ = arg max *∈[/] 𝑏 * and asks 𝑗 ∗ to pay 𝑏 * ∗ , and all other buyers pay 0 11
The Design Space – Mechanisms A mechanism (i.e., the game) is specified by ⟨𝐵, ⟩ where: Ø A = 𝐵 6 × ⋯× 𝐵 / where 𝐵 * is allowable actions for buyer 𝑗 Ø : 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome 𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏 6 , ⋯ , 𝑏 / ) ∈ 𝐵 Example 2: second-price auction Ø 𝐵 * = ℝ A 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 12
The Design Space – Mechanisms A mechanism (i.e., the game) is specified by ⟨𝐵, ⟩ where: Ø A = 𝐵 6 × ⋯× 𝐵 / where 𝐵 * is allowable actions for buyer 𝑗 Ø : 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome 𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏 6 , ⋯ , 𝑏 / ) ∈ 𝐵 Ø In general, 𝐵, can be really arbitrary, up to your design Ø E.g, the following is a valid – though bad – mechanism Ø 𝐵 * = {𝑘𝑣𝑛𝑞 𝑢𝑥𝑗𝑑𝑓 (𝐾), 𝑚𝑝𝑝𝑙 45 ° 𝑣𝑞 (𝑀)} 𝑦(𝑏) gives the item to anyone of 𝑀 uniformly at random Ø Ø 𝑞(𝑏) asks everyone to pay $0 13
Revenue at Equilibrium Ø How to predict/estimate how much revenue we achieve? Ø Revenue = expected revenue at (Bayesian) Nash equilibrium Ø Due to incomplete information, player 𝑗 ’s strategy is 𝑡 * : 𝑊 * → Δ(𝐵 * ) where 𝑡 * (𝑤 * ) is the mixed strategy of 𝑗 with private value 𝑤 * Ø Expected utility of 𝑗 with value 𝑤 * in mechanism ⟨𝐵, ⟩ is 𝔽 (c d ,c ed )∼(f d g d ,f ed g ed ) 𝑤 * 𝑦 * 𝑏 * , 𝑏 h* − 𝑞 * (𝑏 * , 𝑏 h* ) 14
Revenue at Equilibrium Ø How to predict/estimate how much revenue we achieve? Ø Revenue = expected revenue at (Bayesian) Nash equilibrium Ø Due to incomplete information, player 𝑗 ’s strategy is 𝑡 * : 𝑊 * → Δ(𝐵 * ) where 𝑡 * (𝑤 * ) is the mixed strategy of 𝑗 with private value 𝑤 * Ø Expected utility of 𝑗 with value 𝑤 * in mechanism ⟨𝐵, ⟩ is 𝔽 g ed ∼i ed 𝔽 (c d ,c ed )∼(f d g d ,f ed g ed ) 𝑤 * 𝑦 * 𝑏 * , 𝑏 h* − 𝑞 * (𝑏 * , 𝑏 h* ) = 𝑉 * 𝑡 * 𝑤 * 𝑤 * , 𝑡 h* 15
Revenue at Equilibrium Ø How to predict/estimate how much revenue we achieve? Ø Revenue = expected revenue at (Bayesian) Nash equilibrium Ø Due to incomplete information, player 𝑗 ’s strategy is 𝑡 * : 𝑊 * → Δ(𝐵 * ) where 𝑡 * (𝑤 * ) is the mixed strategy of 𝑗 with private value 𝑤 * Ø Expected utility of 𝑗 with value 𝑤 * in mechanism ⟨𝐵, ⟩ is 𝔽 g ed ∼i ed 𝔽 (c d ,c ed )∼(f d g d ,f ed g ed ) 𝑤 * 𝑦 * 𝑏 * , 𝑏 h* − 𝑞 * (𝑏 * , 𝑏 h* ) = 𝑉 * 𝑡 * 𝑤 * 𝑤 * , 𝑡 h* Strategy profile 𝑡 ∗ = (𝑡 6 ∗ , ⋯ , 𝑡 / ∗ ) is a Bayes Nash Equilibrium (BNE) for mechanism ⟨𝐵, ⟩ if for any player 𝑗 and value 𝑤 * ∗ 𝑤 * ∗ ∗ 𝑉 * 𝑡 * 𝑤 * , 𝑡 h* ≥ 𝑉 * 𝑏 * 𝑤 * , 𝑡 h* , ∀ 𝑏 * ∈ 𝐵 * ∗ for any 𝑗 and 𝑤 * . ∗ (𝑤 * ) is a best response to 𝑡 h* That is, 𝑡 * 16
Revenue at Equilibrium Strategy profile 𝑡 ∗ = (𝑡 6 ∗ , ⋯ , 𝑡 / ∗ ) is a Bayes Nash Equilibrium (BNE) for mechanism ⟨𝐵, ⟩ if for any player 𝑗 and value 𝑤 * ∗ 𝑤 * ∗ ∗ 𝑉 * 𝑡 * 𝑤 * , 𝑡 h* ≥ 𝑉 * 𝑏 * 𝑤 * , 𝑡 h* , ∀ 𝑏 * ∈ 𝐵 * ∗ for any 𝑗 and 𝑤 * . ∗ (𝑤 * ) is a best response to 𝑡 h* That is, 𝑡 * Theorem . Any finite Bayesian game admits a mixed BNE. Ø Can be proved by Nash’s theorem Ø It so happens that in many natural Bayesian games we look at, there will be a pure BNE 17
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