Announcements Ø HW 3 and proposal due today 1
CS6501: T opics in Learning and Game Theory (Fall 2019) Selling Information Instructor: Haifeng Xu
Outline Ø Bayesian Persuasion and Information Selling Ø Sell to a Single Decision Maker Ø Sell to Multiple Decision Makers 3
Recap: Bayesian Persuasion Persuasion is the act of exploiting an informational advantage in order to influence the decisions of others Ø One of the two primarily ways to influence agents’ behaviors • Another way is through designing incentives Ø Accounts for a significant share in economic activities • Advertising, marketing, security, investment, financial regulation,… 4
The Bayesian Persuasion Model Ø Two players: a sender (she) and a receiver (he) • Sender has information, receiver is a decision maker Ø Receiver takes an action 𝑗 ∈ 𝑜 = {1,2, ⋯ , 𝑜} • Receiver utility 𝑠(𝑗, 𝜄) and sender utility 𝑡(𝑗, 𝜄) • 𝜄 ∼ 𝑞𝑠𝑗𝑝𝑠 𝑒𝑗𝑡𝑢. 𝑞 is a random state of nature Ø Both players know prior 𝑞 , but sender additionally observes 𝜄 5
The Bayesian Persuasion Model Ø Two players: a sender (she) and a receiver (he) • Sender has information, receiver is a decision maker Ø Receiver takes an action 𝑗 ∈ 𝑜 = {1,2, ⋯ , 𝑜} • Receiver utility 𝑠(𝑗, 𝜄) and sender utility 𝑡(𝑗, 𝜄) • 𝜄 ∼ 𝑞𝑠𝑗𝑝𝑠 𝑒𝑗𝑡𝑢. 𝑞 is a random state of nature Ø Both players know prior 𝑞 , but sender additionally observes 𝜄 Ø Sender reveals partial information via a signaling scheme to influence receiver’s decision and maximize her utility Definition : A signaling scheme is a mapping 𝜌: Θ → Δ ; where Σ is the set of all possible signals. 𝜌 is fully described by 𝜌 𝜏, 𝜄 >∈?,@∈; where 𝜌 𝜏, 𝜄 = prob. of sending 𝜏 when observing 𝜄 (so ∑ @∈; 𝜌 𝜏, 𝜄 = 1 for any 𝜄 ) 6
Example: Recommendation Letters Ø Sender = advisor, receiver = recruiter Ø Θ = {𝑓𝑦𝑑𝑓𝑚𝑚𝑓𝑜𝑢, 𝑏𝑤𝑓𝑠𝑏𝑓} , 𝜈 𝑓𝑦𝑑𝑓𝑚𝑚𝑓𝑜𝑢 = 1/3 Ø Receiver decides Hire or NotHire • Results in utilities for receiver and sender Ø Optimal strategy is a signaling scheme 7
Optimal Signaling via Linear Program Revelation Principle . There always exists an optimal signaling scheme that uses at most 𝑜(= # receiver actions) signals, where signal 𝜏 L induce optimal receiver action 𝑗 Ø Optimal signaling scheme is computed by an LP • Variables: 𝜌 𝜏 L , 𝜄 = prob of sending 𝜏 L conditioned on 𝜄 • Send 𝜏 L = recommend action 𝑗 8
Many Other Examples and Extensions Ø Prosecutor persuades judge 9
Many Other Examples and Extensions Ø Prosecutor persuades judge Ø Lobbyists persuade politicians 10
Many Other Examples and Extensions Ø Prosecutor persuades judge Ø Lobbyists persuade politicians Ø Election candidates persuade voters 11
Many Other Examples and Extensions Ø Prosecutor persuades judge Ø Lobbyists persuade politicians Ø Election candidates persuade voters Ø Sellers persuade buyers 12
Many Other Examples and Extensions Ø Prosecutor persuades judge Ø Lobbyists persuade politicians Ø Election candidates persuade voters Ø Sellers persuade buyers 13
Many Other Examples and Extensions Ø Prosecutor persuades judge Ø Lobbyists persuade politicians Ø Election candidates persuade voters Ø Sellers persuade buyers Ø Executives persuade stockholders 14
Many Other Examples and Extensions Ø Prosecutor persuades judge Ø Lobbyists persuade politicians Ø Election candidates persuade voters Ø Sellers persuade buyers Ø Executives persuade stockholders Ø . . . Many persuasion models built upon Bayesian persuasion Ø Persuading many receivers, voters, attackers, drivers on road network, buyers in auctions, etc.. Ø Private vs public persuasion Ø Selling information is also a variant 15
Selling Information – the Basic Model Ø Sender = seller, Receiver = buyer who is a decision maker Ø Buyer takes an action 𝑗 ∈ 𝑜 = {1, ⋯ , 𝑜} Ø Buyer has a utility function 𝑣(𝑗, 𝜄; 𝜕) where • 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 is a random state of nature • 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 captures buyer’s (private) utility type 16
Selling Information – the Basic Model Ø Sender = seller, Receiver = buyer who is a decision maker Ø Buyer takes an action 𝑗 ∈ 𝑜 = {1, ⋯ , 𝑜} Ø Buyer has a utility function 𝑣(𝑗, 𝜄; 𝜕) where • 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 is a random state of nature • 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 captures buyer’s (private) utility type Remarks: Ø 𝑣, 𝑞, 𝑔 are public knowledge Ø Assume 𝜄, 𝜕 are independent Ø In mechanism design, seller also does not know buyer’s value 17
Selling Information – the Basic Model Ø Sender = seller, Receiver = buyer who is a decision maker Ø Buyer takes an action 𝑗 ∈ 𝑜 = {1, ⋯ , 𝑜} Ø Buyer has a utility function 𝑣(𝑗, 𝜄; 𝜕) where • 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 is a random state of nature • 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 captures buyer’s (private) utility type Remarks: Ø 𝑣, 𝑞, 𝑔 are public knowledge Ø Assume 𝜄, 𝜕 are independent Ø In mechanism design, seller also does not know buyer’s value Q : How to price the item if seller knowns buyer’s value of it? 18
Selling Information – the Basic Model Ø Sender = seller, Receiver = buyer who is a decision maker Ø Buyer takes an action 𝑗 ∈ 𝑜 = {1, ⋯ , 𝑜} Ø Buyer has a utility function 𝑣(𝑗, 𝜄; 𝜕) where • 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 is a random state of nature • 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 captures buyer’s (private) utility type Ø Seller observes the state 𝜄 ; Buyer knows his private type 𝜕 Ø Seller would like to sell her information about 𝜄 to maximize revenue Key differences from Bayesian persuasion Ø Seller does not have a utility fnc – instead maximize revenue Ø Buyer here has private info 𝜕 , which is unknown to seller 19
Outline Ø Bayesian Persuasion and Information Selling Ø Sell to a Single Decision Maker Ø Sell to Multiple Decision Makers 20
Warm-up: What if Buyer Has no Private Info Ø 𝑣(𝑗, 𝜄; 𝜕) where sate 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 and buyer type 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 Ø When seller also observes 𝜕 . . . Q : How to sell information optimally? 21
Warm-up: What if Buyer Has no Private Info Ø 𝑣(𝑗, 𝜄; 𝜕) where sate 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 and buyer type 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 Ø When seller also observes 𝜕 . . . Q : How to sell information optimally? Ø Seller knows exactly how much the buyer values “any amount” of her information à should charge him just that amount 22
Warm-up: What if Buyer Has no Private Info Ø 𝑣(𝑗, 𝜄; 𝜕) where sate 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 and buyer type 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 Ø When seller also observes 𝜕 . . . Q : How to sell information optimally? Ø Seller knows exactly how much the buyer values “any amount” of her information à should charge him just that amount Ø How to charge the most? • Reveal full information helps the buyer the most. Why? • So OPT is to charge him following amount and then reveal 𝜄 directly Payment = ∑ >∈? 𝑞 𝜄 ⋅ [max ∑ >∈? 𝑞 𝜄 ⋅ 𝑣(𝑗, 𝜄; 𝜕) 𝑣(𝑗, 𝜄; 𝜕)] − max L L 23
Warm-up: What if Buyer Has no Private Info Ø 𝑣(𝑗, 𝜄; 𝜕) where sate 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 and buyer type 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 Ø When seller also observes 𝜕 . . . Q : How to sell information optimally? Ø Seller knows exactly how much the buyer values “any amount” of her information à should charge him just that amount Ø How to charge the most? • Reveal full information helps the buyer the most. Why? • So OPT is to charge him following amount and then reveal 𝜄 directly Payment = ∑ >∈? 𝑞 𝜄 ⋅ [max ∑ >∈? 𝑞 𝜄 ⋅ 𝑣(𝑗, 𝜄; 𝜕) 𝑣(𝑗, 𝜄; 𝜕)] − max L L Buyer expected utility if learns 𝜄 24
Warm-up: What if Buyer Has no Private Info Ø 𝑣(𝑗, 𝜄; 𝜕) where sate 𝜄 ∼ 𝑒𝑗𝑡𝑢. 𝑞 and buyer type 𝜕 ∼ 𝑒𝑗𝑡𝑢. 𝑔 Ø When seller also observes 𝜕 . . . Q : How to sell information optimally? Ø Seller knows exactly how much the buyer values “any amount” of her information à should charge him just that amount Ø How to charge the most? • Reveal full information helps the buyer the most. Why? • So OPT is to charge him following amount and then reveal 𝜄 directly Payment = ∑ >∈? 𝑞 𝜄 ⋅ [max ∑ >∈? 𝑞 𝜄 ⋅ 𝑣(𝑗, 𝜄; 𝜕) 𝑣(𝑗, 𝜄; 𝜕)] − max L L Buyer expected utility without knowing 𝜄 25
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