Designing and Pricing Certificates Nima Haghpanah joint with Nageeb - PowerPoint PPT Presentation

Designing and Pricing Certificates Nima Haghpanah joint with Nageeb Ali, Xiao Lin, Ron Siegel May 8, 2020 1 / 15

Certification Labor markets, Financial markets, Products What certificates would an agent acquire and disclose? How would a profit-maximizing certifier design and price certificates? 2 / 15

A worker, a certifier, a competitive labor market Worker Ability θ ∼ U { 0 , 1 } No φ t Test test ◮ unknown to all A test-fee structure ( T , φ ): s ∼ T ( θ ) 1 Test T : { 0 , 1 } → ∆( S ) Not disclose Disclose WLOG E [ θ | s ] = s “N” φ d 2 Testing fee φ t s Disclosure fee φ d Market observes s or “N” Market offers wage = E [ θ ] 3 / 15

Profit-maximizing test-fee structures? sup sup Profit = Full surplus E [ θ ] = 0 . 5 test-fee structure equilibria Fully reveal, φ t = 0 . 5 , φ d = 0 ◮ Another equilibrium: worker doesn’t take test. Profit = 0 sup inf Profit = 0 . 5 · (1 − 1 / e ) ≈ 0 . 31 equilibria test-fee structure Score distribution 1 “Robustly optimal” test-fee structure: 1 Is unique 2 Zero testing fee 1 / e 3 Not fully revealing: continuum of scores s 0 0 . 5 1 4 / 15

Related Work Profit-maximizing certification: ◮ Lizzeri (1999). Informed worker, mandatory disclosure: ◮ Signaling vs. voluntary disclosure ◮ DeMarzo, Kremer, Skrzypacz (2019). “favorable” selection Adversarial equilibrium selection in information/mechanism design: ◮ Dworczak and Pavan (2020), Halac, Kremer, Winter (2020), Halac, Lipnowski, Rappoport (2020), ... Information design and unit-elastic distributions: ◮ Roesler and Szentes (2017), Ortner and Chassang (2018), Condorelli and Szentes (2020), ... ◮ Indifference condition vs. worst-equilibrium condition 5 / 15

Next Identify optimal test with φ t = 0 and φ d = 0 . 5 sup inf Probability of disclosure equilibria test Exponential distribution maximizes equilibria Probability of disclosure inf Score distribution 1 1 / e s 0 0 . 5 1 6 / 15

Disclosure stage: threshold structure Equilibrium threshold τ : τ − φ d = w N = E [ s | s ≤ τ ] Worst equilibrium τ is largest intersection: τ ′ − φ d � = E [ s | s ≤ τ ′ ] , ∀ τ ′ > τ Claim: Robustly optimal test-fee structure, ◮ Worker participates with probability 1 in all equilibria τ − φ d E [ s | s ≤ τ ] τ 1 τ 2 τ 7 / 15

Fully revealing test Worst equilibrium threshold = φ d ◮ Probability of disclosure = 0 . 5 Ability Score s Prob 1 0 0 2 τ − φ d 0 . 5 + ǫ E [ s | s ≤ τ ] 1 1 1 0 . 5 2 testing fee = 0 τ φ d 0 1 disclosure fee = 0 . 5 − ǫ 8 / 15

Improvement by a noisy test Worst equilibrium threshold = φ d ◮ Probability of disclosure > 0 . 5 Ability Score s Prob 1 − δ 1 − δ 0 0 2 δ τ − φ d 3 2 δ 3 δ 0 . 5 + ǫ 4 E [ s | s ≤ τ ] 0 . 5 1 − 3 δ 1 1 2 1 − 3 δ testing fee = 0 τ φ d 0 3 1 disclosure fee = 0 . 5 − ǫ 4 9 / 15

“Robustly optimal” test subject to φ t = 0, φ d ≃ 0 . 5 Worst equilibrium threshold = φ d ◮ Probability of disclosure 1 − 1 / e ≈ 0 . 63 � τ 0 G ( s ) ds φ d = Ability Score G ( τ ) 2 / e � �� − 1 � � τ 0 0 d = ln( 0 G ( s ) ds ) d τ φ d e τ/φ d c ⇒ G ( τ ) = . . . s ∈ [0 . 5 , 1] τ − φ d . . 0 . 5 + ǫ . E [ s | s ≤ τ ] 1 0 . 5 testing fee = 0 τ φ d 0 disclosure fee = 0 . 5 − ǫ 10 / 15

Robustly optimal test-fee structure Proposition There is a unique robustly optimal test-fee structure. It consists of testing fee φ ∗ t = 0 , disclosure fee φ ∗ d = 0 . 5 , and test T below. Continuum of scores even though abilities are binary. T (0) T (1) Score distribution 1 1 1 2 / e 1 / e s 0 0 . 5 1 s s 0 0 . 5 1 0 0 . 5 1 11 / 15

Arbitrary prior over θ ∈ [0 , 1] with mean µ Proposition − µ 1 − µ ) < µ . Robustly optimal profit ≤ (1 − µ )(1 − e Proposition There exists a robustly optimal test-fee structure with a “step-exponential-step” score distribution. Disclosure fee > 0 Score distribution 1 ◮ Contrast with “maximize value and extract via testing fee” intuition. Testing fee? ◮ Positive for log-concave priors s ◮ May be zero (e.g., for binary prior) 0 1 12 / 15

Precluding no-testing equilibria � 1 µ < max { µ, s − φ d } dG − φ t , 0 � �� � Option Value Rearranging: � 1 φ t < [ s − ( µ + φ d )] dG , (P) µ + φ d Lemma 1 If (P), ∀ equilibria: worker takes test with probability 1 2 If !(P), ∃ equilibrium: worker takes test with probability 0 Proves earlier claim: Robustly optimal test-fee structure, ◮ Worker participates with probability 1 in all equilibria 13 / 15

Optimality of positive disclosure fee profit = φ t profit = φ t + φ d (1 − G ( φ d )) Score distribution G Score distribution G � 1 � 1 φ t = µ (1 − G ( s )) ds φ t = µ + φ d (1 − G ( s )) ds G ( φ d ) s s φ d µ + φ d µ µ 14 / 15

Extensions 1 Small amount of private information ◮ Full surplus extraction remains impossible ◮ Step-exponential-step distributions are approximately optimal 2 Technological constraints: Certifier has a set of feasible tests ◮ Assumption: feasible to garble a feasible test ◮ Step-exponential-step is optimal 3 Score-dependent disclosure fees ◮ Allows for slightly higher profit, still not full surplus Score distribution Score distribution 1 1 1 / e s s 0 0 . 5 1 0 1 Thanks! 15 / 15