Announcements Ø HW 3 postponed to this Thursday Ø Project proposal due this Thursday as well 1
CS6501: T opics in Learning and Game Theory (Fall 2019) Bayesian Persuasion Instructor: Haifeng Xu
Ø Prediction markets and peer prediction study how to elicit information from others Ø This lecture: when you have information, how to exploit it? • Relevant to mechanism design 3
Outline Ø Introduction and Bayesian Persuasion Ø Algorithms for Bayesian Persuasion 4
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions You only pay the second highest bid! 5
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions • Discounts/coupons 6
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions • Discounts/coupons • Job contract design Bonus depends on performance, and is up to $1M! 7
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions Mechanism Design • Discounts/coupons • Job contract design 8
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions Mechanism Design • Discounts/coupons • Job contract design Ø Influence agents’ beliefs • Deception in wars/battles All warfare is based on deception. Hence, when we are able to attack, we must seem unable; when using our forces, we must appear inactive… -- Sun Tzu, The Art of War 9
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions Mechanism Design • Discounts/coupons • Job contract design Ø Influence agents’ beliefs • Deception in wars/battles Strategic inventory • Strategic information disclosure information disclosure 10
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions Mechanism Design • Discounts/coupons • Job contract design Ø Influence agents’ beliefs • Deception in wars/battles Strategic inventory • Strategic information disclosure information disclosure 11
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions Mechanism Design • Discounts/coupons • Job contract design Ø Influence agents’ beliefs • Deception in wars/battles • Strategic information disclosure • News articles, advertising, tweets, etc. 12
T wo Primary Ways to Influence Agents’ Behaviors Ø Design/provide incentives • Auctions Mechanism Design • Discounts/coupons • Job contract design Ø Influence agents’ beliefs • Deception in wars/battles • Strategic information disclosure Persuasion • News articles, advertising, tweets … • In fact, most information you see is there for a goal 13
Persuasion is the act of exploiting an informational advantage in order to influence the decisions of others Ø Intrinsic in human activities: advertising, negotiation, politics, security, marketing, financial regulation,… Ø A large body of research –– The American Economic Review Vol. 85, No. 2, 1995. 14
Example: Recommendation Letters Ø Advisor vs. recruiter Ø 1/3 of the advisor’s students are excellent; 2/3 are average Ø A fresh graduate is randomly drawn from this population Ø Recruiter Utility 1 + 𝜗 for hiring an excellent student; −1 for an average student • Utility 0 for not hiring • • A-priori, only knows the advisor’s student population (1 + 𝜗)×1/3 − 1×2/3 < 0 hiring Not hiring 15
Example: Recommendation Letters Ø Advisor vs. recruiter Ø 1/3 of the advisor’s students are excellent; 2/3 are average Ø A fresh graduate is randomly drawn from this population Ø Recruiter Utility 1 + 𝜗 for hiring an excellent student; −1 for an average student • Utility 0 for not hiring • • A-priori, only knows the advisor’s student population Ø Advisor • Utility 1 if the student is hired, 0 otherwise • Knows whether the student is excellent or not 16
Example: Recommendation Letters What is the advisor’s optimal “recommendation strategy”? Ø Attempt 1: always say “excellent” (equivalently, no information) • Recruiter ignores the recommendation • Advisor expected utility 0 Remark Advisor commitment: cannot deviate and recruiter knows his strategy 17
Example: Recommendation Letters What is the advisor’s optimal “recommendation strategy”? Ø Attempt 2: honest recommendation (i.e., full information) • Advisor expected utility 1/3 1/3 1/3 1/3 excellent recruiter 2/3 2/3 2/3 average 18
Example: Recommendation Letters What is the advisor’s optimal “recommendation strategy”? Ø Attempt 3: noisy information à advisor expected utility 2/3 1/3 P(excellent | ) = 1/2 2/3 1/3 excellent 1/3 (1 + 𝜗 − 1)/2 > 0 recruiter 2/3 Not hiring Hiring 1/3 1/3 average 19
Model of Bayesian Persuasion Ø Two players: persuader (Sender, she), decision maker (Receiver he) • Previous example: advisor = sender, recruiter = receiver Ø Receiver looks to take an action 𝑗 ∈ 𝑜 = {1, 2, … , 𝑜} Receiver utility 𝑠(𝑗, 𝜄) • 𝜄 ∈ Θ is a random state of nature • Sender utility 𝑡(𝑗, 𝜄) Ø Both players know 𝜄 ∼ 𝑞𝑠𝑗𝑝𝑠 𝑒𝑗𝑡𝑢. 𝜈 , but Sender has an informational advantage – she can observe realization of 𝜄 Ø Sender wants to strategically reveal info about 𝜄 to “persuade” Receiver to take an action she likes • Concealing or revealing all info is not necessarily the best Well…how to reveal partial information? 20
Revealing Information via Signaling Definition : A signaling scheme is a mapping 𝜌: Θ → Δ P where Σ is the set of all possible signals. 𝜌 is fully described by 𝜌 𝜏, 𝜄 R∈S,T∈P where 𝜌 𝜏, 𝜄 = prob. of sending 𝜏 when observing 𝜄 (so ∑ T∈P 𝜌 𝜏, 𝜄 = 1 for any 𝜄 ) Note: scheme 𝜌 is always assumed public knowledge, thus known by Receiver Example Ø Θ = {𝐹𝑦𝑑𝑓𝑚𝑚𝑓𝑜𝑢, 𝐵𝑤𝑓𝑠𝑏𝑓} , Σ = {𝐵, 𝐶} Ø 𝜌 𝐵, 𝐵𝑤𝑓𝑠𝑏𝑓 = 1/2 21
Revealing Information via Signaling Definition : A signaling scheme is a mapping 𝜌: Θ → Δ P where Σ is the set of all possible signals. 𝜌 is fully described by 𝜌 𝜏, 𝜄 R∈S,T∈P where 𝜌 𝜏, 𝜄 = prob. of sending 𝜏 when observing 𝜄 (so ∑ T∈P 𝜌 𝜏, 𝜄 = 1 for any 𝜄 ) What can Receiver infer about 𝜄 after receiving σ ? Bayes updating: Z T,R ⋅\ R Pr 𝑓𝑦𝑑𝑓𝑚𝑚𝑓𝑜𝑢 𝐵 = 1/2 Pr(𝜄|𝜏) = ∑ ]^ Z T,R ^ ⋅\ R ^ 22
Revealing Information via Signaling Definition : A signaling scheme is a mapping 𝜌: Θ → Δ P where Σ is the set of all possible signals. 𝜌 is fully described by 𝜌 𝜏, 𝜄 R∈S,T∈P where 𝜌 𝜏, 𝜄 = prob. of sending 𝜏 when observing 𝜄 (so ∑ T∈P 𝜌 𝜏, 𝜄 = 1 for any 𝜄 ) Would such noisy information benefit Receiver? Ø Expected Receiver utility conditioned on 𝜏 : Z T,R ⋅\ R ∑ R∈S 𝑠 𝑗, 𝜄 ⋅ max b∈[d] [ ] 𝑆 𝜏 = ∑ ]^ Z T,R ^ ⋅\ R ^ Ø Pr(𝜏) = ∑ R ^ 𝜌 𝜏, 𝜄 g ⋅ 𝜈 𝜄 g 23
Revealing Information via Signaling Definition : A signaling scheme is a mapping 𝜌: Θ → Δ P where Σ is the set of all possible signals. 𝜌 is fully described by 𝜌 𝜏, 𝜄 R∈S,T∈P where 𝜌 𝜏, 𝜄 = prob. of sending 𝜏 when observing 𝜄 (so ∑ T∈P 𝜌 𝜏, 𝜄 = 1 for any 𝜄 ) Would such noisy information benefit Receiver? Ø Expected Receiver utility conditioned on 𝜏 : Z T,R ⋅\ R ∑ R∈S 𝑠 𝑗, 𝜄 ⋅ max b∈[d] [ ] 𝑆 𝜏 = ∑ ]^ Z T,R ^ ⋅\ R ^ Ø Pr(𝜏) = ∑ R ^ 𝜌 𝜏, 𝜄 g ⋅ 𝜈 𝜄 g 24
Revealing Information via Signaling Definition : A signaling scheme is a mapping 𝜌: Θ → Δ P where Σ is the set of all possible signals. 𝜌 is fully described by 𝜌 𝜏, 𝜄 R∈S,T∈P where 𝜌 𝜏, 𝜄 = prob. of sending 𝜏 when observing 𝜄 (so ∑ T∈P 𝜌 𝜏, 𝜄 = 1 for any 𝜄 ) Would such noisy information benefit Receiver? Ø Expected Receiver utility conditioned on 𝜏 : Z T,R ⋅\ R ∑ R∈S 𝑠 𝑗, 𝜄 ⋅ max b∈[d] [ ] 𝑆 𝜏 = ∑ ]^ Z T,R ^ ⋅\ R ^ Ø Pr(𝜏) = ∑ R ^ 𝜌 𝜏, 𝜄 g ⋅ 𝜈 𝜄 g ∑ R∈S 𝑠 𝑗, 𝜄 ⋅ 𝜌 𝜏, 𝜄 ⋅ 𝜈 𝜄 Pr(𝜏) ⋅ 𝑆 𝜏 = max • b Ø Expected Receiver utility under 𝜌 : ∑ T max ∑ R∈S 𝑠 𝑗, 𝜄 ⋅ 𝜌 𝜏, 𝜄 ⋅ 𝜈 𝜄 b 25
Revealing Information via Signaling Fact . Receiver’s expected utility (weakly) increases under any signaling scheme 𝜌 . 26
Revealing Information via Signaling Fact . Receiver’s expected utility (weakly) increases under any signaling scheme 𝜌 . Proof: Ø Expected Receiver utility under 𝜌 : ∑ T max ∑ R∈S 𝑠 𝑗, 𝜄 ⋅ 𝜌 𝜏, 𝜄 ⋅ 𝜈 𝜄 b ∑ R∈S 𝑠 𝑗, 𝜄 ⋅ 𝜈 𝜄 Ø Expected Receiver utility without information: max b 27
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