Designing and Selling Hard Information Nima Haghpanah r S. Nageeb - PowerPoint PPT Presentation

Designing and Selling Hard Information Nima Haghpanah r � S. Nageeb Ali r � Xiao Lin r � Ron Siegel Penn State September 23, 2020 1 / 18

Market for Hard Information 2 / 18

Market for Hard Information asset Agent Market 2 / 18

Market for Hard Information Intermediary Hard info asset Agent Market Hard info 2 / 18

Market for Hard Information ◮ Sellers of physical or financial products - Buyers ◮ Workers - Employers ◮ Entrepreneurs - Investors Intermediary Hard info asset Agent Market Hard info 2 / 18

Market for Hard Information ◮ Sellers of physical or financial products - Buyers ◮ Workers - Employers ◮ Entrepreneurs - Investors Intermediary’s profit-maximization problem ◮ Accurate tests or noisy tests? ◮ Upfront fees or disclosure fees? ◮ How much of the surplus can she appropriate? Intermediary Hard info asset Agent Market Hard info 2 / 18

Agent (with asset), competitive market, intermediary 3 / 18

Agent (with asset), competitive market, intermediary Value θ ∈ { 0 , 1 } , prob. 0 . 5 each ◮ Unknown to all 3 / 18

Agent (with asset), competitive market, intermediary Value θ ∈ { 0 , 1 } , prob. 0 . 5 each ◮ Unknown to all A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) ◮ Unbiased s = E [ θ | s ] 2 Testing fee φ t Disclosure fee φ d 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each ◮ Unknown to all A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) ◮ Unbiased s = E [ θ | s ] 2 Testing fee φ t Disclosure fee φ d 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each Not φ t Test ◮ Unknown to all s ∼ T ( θ ) A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) ◮ Unbiased s = E [ θ | s ] 2 Testing fee φ t Disclosure fee φ d 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each Not φ t Test ◮ Unknown to all s ∼ T ( θ ) A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) Not Disclose ◮ Unbiased s = E [ θ | s ] φ d 2 Testing fee φ t Disclosure fee φ d 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each Not φ t Test ◮ Unknown to all s ∼ T ( θ ) A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) Not Disclose ◮ Unbiased s = E [ θ | s ] φ d 2 Testing fee φ t s Disclosure fee φ d Market observes s 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each Not φ t Test ◮ Unknown to all s ∼ T ( θ ) A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) “N” Not Disclose ◮ Unbiased s = E [ θ | s ] φ d 2 Testing fee φ t s Disclosure fee φ d Market observes s or “N” 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each Not φ t Test ◮ Unknown to all s ∼ T ( θ ) A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) “N” Not Disclose ◮ Unbiased s = E [ θ | s ] φ d 2 Testing fee φ t s Disclosure fee φ d Market observes s or “N” Market offers price = E [ θ | observation] 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each Not φ t Test ◮ Unknown to all s ∼ T ( θ ) A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) “N” Not Disclose ◮ Unbiased s = E [ θ | s ] φ d 2 Testing fee φ t s Disclosure fee φ d Market observes s or “N” Market offers price = E [ θ | observation] – Observe s : price( s ) = E [ θ | s ] = s – Observe “N”: price(N) endogeneous 3 / 18

Agent (with asset), competitive market, intermediary Agent Value θ ∈ { 0 , 1 } , prob. 0 . 5 each Not φ t Test ◮ Unknown to all s ∼ T ( θ ) A test-fee structure ( T , φ ): 1 Test T : { 0 , 1 } → ∆( S ) “N” Not Disclose ◮ Unbiased s = E [ θ | s ] φ d 2 Testing fee φ t s Disclosure fee φ d ( T , φ ) may have multiple equilibria Market observes s or “N” Market offers price = E [ θ | observation] – Observe s : price( s ) = E [ θ | s ] = s – Observe “N”: price(N) endogeneous 3 / 18

Revenue-maximizing test-fee structures? 4 / 18

Revenue-maximizing test-fee structures? max max Revenue test-fee structure equilibria 4 / 18

Revenue-maximizing test-fee structures? max max Revenue = 0 . 5 = E [ θ ] “full surplus” test-fee structure equilibria 4 / 18

Revenue-maximizing test-fee structures? max max Revenue = 0 . 5 = E [ θ ] “full surplus” test-fee structure equilibria Fully reveal, φ t = 0 , φ d = 1 ◮ Equilibrium with revenue = 0 . 5 ◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0 4 / 18

Revenue-maximizing test-fee structures? max max Revenue = 0 . 5 = E [ θ ] “full surplus” test-fee structure equilibria Fully reveal, φ t = 0 , φ d = 1 ◮ Equilibrium with revenue = 0 . 5 ◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0 ◮ Another equilibrium with revenue = 0 ◮ Agent: don’t take test ◮ Market: price(N) = 0 . 5 4 / 18

Revenue-maximizing test-fee structures? max max Revenue = 0 . 5 = E [ θ ] “full surplus” test-fee structure equilibria Fully reveal, φ t = 0 , φ d = 1 ◮ Equilibrium with revenue = 0 . 5 ◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0 ◮ Another equilibrium with revenue = 0 ◮ Agent: don’t take test ◮ Market: price(N) = 0 . 5 ∀ ( T , φ ) with an equilibrium where revenue ≥ 0 . 5 − ǫ ◮ ∃ another equilibrium of ( T , φ ) where revenue ≤ ǫ 4 / 18

Revenue-maximizing test-fee structures? max max Revenue = 0 . 5 = E [ θ ] “full surplus” test-fee structure equilibria Fully reveal, φ t = 0 , φ d = 1 ◮ Equilibrium with revenue = 0 . 5 ◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0 ◮ Another equilibrium with revenue = 0 ◮ Agent: don’t take test ◮ Market: price(N) = 0 . 5 ∀ ( T , φ ) with an equilibrium where revenue ≥ 0 . 5 − ǫ ◮ ∃ another equilibrium of ( T , φ ) where revenue ≤ ǫ max min Revenue test-fee structure equilibria 4 / 18

Revenue-maximizing test-fee structures? max max Revenue = 0 . 5 = E [ θ ] “full surplus” test-fee structure equilibria Fully reveal, φ t = 0 , φ d = 1 ◮ Equilibrium with revenue = 0 . 5 ◮ Agent: take test, reveal if and only if score = 1 ◮ Market: price(N) = 0 ◮ Another equilibrium with revenue = 0 ◮ Agent: don’t take test ◮ Market: price(N) = 0 . 5 ∀ ( T , φ ) with an equilibrium where revenue ≥ 0 . 5 − ǫ ◮ ∃ another equilibrium of ( T , φ ) where revenue ≤ ǫ max min Revenue = 0 . 5 · (1 − 1 / e ) ≈ 0 . 31 test-fee structure equilibria 4 / 18

“Robustly optimal” test-fee structure 5 / 18

“Robustly optimal” test-fee structure 1 Is unique (and has unique equilibrium) 2 Testing is free φ t = 0 (disclosure is costly φ d > 0) 3 Not fully revealing: continuum of scores 5 / 18

“Robustly optimal” test-fee structure 1 Is unique (and has unique equilibrium) 2 Testing is free φ t = 0 (disclosure is costly φ d > 0) 3 Not fully revealing: continuum of scores c.d.f Score distribution 1 exponential 1 / e score 0 0 . 5 1 5 / 18

Fully reveal, free testing ( φ t = 0) (Robustly) optimal disclosure fee φ d ? Pr [ s | θ ] s = 0 s = 1 θ = 0 1 0 θ = 1 0 1 6 / 18

Fully reveal, free testing ( φ t = 0) (Robustly) optimal disclosure fee φ d ? Pr [ s | θ ] s = 0 s = 1 θ = 0 1 0 θ = 1 0 1 Fee φ d Prob. of disclosure 0 6 / 18

Fully reveal, free testing ( φ t = 0) (Robustly) optimal disclosure fee φ d ? Pr [ s | θ ] s = 0 s = 1 θ = 0 1 0 ◮ φ d ≥ 0 . 5: conceal any s , price(N) = 0 . 5 θ = 1 0 1 Fee φ d 1 conceal 0 . 5 Prob. of disclosure 0 6 / 18

Fully reveal, free testing ( φ t = 0) (Robustly) optimal disclosure fee φ d ? Pr [ s | θ ] s = 0 s = 1 θ = 0 1 0 ◮ φ d ≥ 0 . 5: conceal any s , price(N) = 0 . 5 θ = 1 0 1 ◮ φ d ∈ [0 , 1]: disclose iff s = 1, price(N) = 0 Fee φ d 1 conceal disclose iff s = 1 0 . 5 Prob. of disclosure 0 0 . 5 6 / 18

Fully reveal, free testing ( φ t = 0) (Robustly) optimal disclosure fee φ d ? Pr [ s | θ ] s = 0 s = 1 θ = 0 1 0 ◮ φ d ≥ 0 . 5: conceal any s , price(N) = 0 . 5 θ = 1 0 1 ◮ φ d ∈ [0 , 1]: disclose iff s = 1, price(N) = 0 Fee φ d 1 1 conceal disclose iff s = 1 0 . 5 Prob. of disclosure 0 0 . 5 0 . 5 1 6 / 18

Fully reveal, free testing ( φ t = 0) (Robustly) optimal disclosure fee φ d ? Pr [ s | θ ] s = 0 s = 1 θ = 0 1 0 ◮ φ d ≥ 0 . 5: conceal any s , price(N) = 0 . 5 θ = 1 0 1 ◮ φ d ∈ [0 , 1]: disclose iff s = 1, price(N) = 0 Fee φ d 1 1 conceal disclose iff s = 1 0 . 5 “robust demand curve” Prob. of disclosure 0 0 . 5 0 . 5 1 6 / 18

Fully reveal, free testing ( φ t = 0) (Robustly) optimal disclosure fee φ d ? Pr [ s | θ ] s = 0 s = 1 θ = 0 1 0 ◮ φ d ≥ 0 . 5: conceal any s , price(N) = 0 . 5 θ = 1 0 1 ◮ φ d ∈ [0 , 1]: disclose iff s = 1, price(N) = 0 Fee φ d 1 1 conceal disclose iff s = 1 0 . 5 “robust demand curve” φ d Prob. of Revenue disclosure 0 0 . 5 0 . 5 1 6 / 18