Dynamic Delegation of Experimentation Yingni Guo Northwestern - PowerPoint PPT Presentation

I. Model Delegation Delegation Principal has full commitment and cannot use transfers. She determines a delegation contract at time 0 . By the Revelation Principle, Principal offers a direct mechanism π : Θ → Π � sup U ρ ( π ( θ ) , θ ) dF ( θ ) , Θ U α ( π ( θ ) , θ ) ≥ U α ( π ( θ ′ ) , θ ) ∀ θ, θ ′ ∈ Θ . subject to Yingni Guo (NU) Delegation of Experimentation 16 / 55

Outline Model 1 Single-player benchmark 2 Characterizing the policy space 3 Main results 4 More general results 5 Yingni Guo (NU) Delegation of Experimentation 17 / 55

II. Single-Player Benchmark Posterior Beliefs Given prior p 0 and the history of events up to time t p t = P t [ ω = 1] . Before the first success, p t satisfies a differential equation p t = − λπ t p t (1 − p t ) . ˙ At the first success, p t jumps to one. Yingni Guo (NU) Delegation of Experimentation 18 / 55

II. Single-Player Benchmark Single Player’s Preferred Policy Player i ’s preferred policy is Markov wrt p t , characterized by a cutoff p ∗ i s.t. � 1 if p t > p ∗ i , π t = if p t ≤ p ∗ 0 i . The cutoff belief is r p ∗ i = . r + ( λ + r ) η i Agent’s cutoff is lower than Principal’s p ∗ α < p ∗ ρ . Yingni Guo (NU) Delegation of Experimentation 19 / 55

II. Single-Player Benchmark Agency Problem Revisited τ i ( θ ) : Player i ’s preferred stopping time given θ . state prob. p 1 θ p ∗ ρ 0 time τ ρ ( θ ) Yingni Guo (NU) Delegation of Experimentation 20 / 55

II. Single-Player Benchmark Agency Problem Revisited For a given prior, Agent prefers to experiment longer than Principal. ( θ ) state prob. p 1 θ p ∗ ρ p ∗ α 0 time τ ρ ( θ ) τ α ( θ ) Yingni Guo (NU) Delegation of Experimentation 20 / 55

II. Single-Player Benchmark Agency Problem Revisited Higher priors warrant longer experimentation. ( θ ) state prob. p 1 θ ′ θ p ∗ ρ 0 time τ ρ ( θ ) τ ρ ( θ ′ ) Yingni Guo (NU) Delegation of Experimentation 20 / 55

II. Single-Player Benchmark Agency Problem Revisited Lower types (those with lower θ ) have incentives to mimic higher types. state prob. p 1 θ ′ θ p ∗ ρ p ∗ α 0 time τ ρ ( θ ) τ α ( θ ) τ ρ ( θ ′ ) τ α ( θ ′ ) Yingni Guo (NU) Delegation of Experimentation 20 / 55

Outline Model 1 Single-player benchmark 2 Characterizing the policy space 3 Main results 4 More general results 5 Yingni Guo (NU) Delegation of Experimentation 21 / 55

III. Characterizing the Policy Space A Policy as a Pair of Numbers (Total Expected Discounted) Resource Pair For a fixed policy π , define w 1 ( π ) and w 0 ( π ) as follows: �� ∞ � � w 1 ( π ) ≡ E re − rt π t dt � π, 1 ∈ [0 , 1] � 0 �� ∞ � � w 0 ( π ) ≡ E re − rt π t dt � π, 0 ∈ [0 , 1] . � 0 w 1 ( π ) : (total expected discounted) resource allocated to R under π in state 1 . w 0 ( π ) : (total expected discounted) resource allocated to R under π in state 0 . Yingni Guo (NU) Delegation of Experimentation 22 / 55

III. Characterizing the Policy Space A Policy as a Pair of Numbers Summary Statistic for the Payoffs Lemma 1 (A Policy as a Pair of Numbers) For a given policy π ∈ Π and prior p 0 ∈ [0 , 1] , player i ’s payoff can be written as � � ( λh i − s i ) w 1 ( π ) � � U i ( π, p 0 ) − s i = p 0 1 − p 0 · . (0 − s i ) w 0 ( π ) Proof Yingni Guo (NU) Delegation of Experimentation 23 / 55

III. Characterizing the Policy Space A Policy as a Pair of Numbers Summary Statistic for the Payoffs Lemma 1 (A Policy as a Pair of Numbers) For a given policy π ∈ Π and prior p 0 ∈ [0 , 1] , player i ’s payoff can be written as � � ( λh i − s i ) w 1 ( π ) � � U i ( π, p 0 ) − s i = p 0 1 − p 0 · . (0 − s i ) w 0 ( π ) Proof ( w 1 ( π ) , w 0 ( π )) is a summary statistic of π for the payoffs. Yingni Guo (NU) Delegation of Experimentation 23 / 55

III. Characterizing the Policy Space Feasible Set Feasible Set Feasible set Γ : the set of feasible resource pairs ( w 1 , w 0 ) | ( w 1 , w 0 ) = ( w 1 ( π ) , w 0 ( π )) , π ∈ Π � � Γ = . Yingni Guo (NU) Delegation of Experimentation 24 / 55

III. Characterizing the Policy Space Feasible Set Characterizing the Feasible Set ⇒ ∃ p ∈ R 2 , � p � = 1 , ˆ w ∈ bd (Γ) ⇐ ˆ w ∈ argmax w ∈ Γ p · w. Yingni Guo (NU) Delegation of Experimentation 25 / 55

III. Characterizing the Policy Space Feasible Set Characterizing the Feasible Set ⇒ ∃ p ∈ R 2 , � p � = 1 , ˆ w ∈ bd (Γ) ⇐ ˆ w ∈ argmax w ∈ Γ p · w. Yingni Guo (NU) Delegation of Experimentation 25 / 55

III. Characterizing the Policy Space Feasible Set Characterizing the Feasible Set ⇒ ∃ p ∈ R 2 , � p � = 1 , ˆ w ∈ bd (Γ) ⇐ ˆ w ∈ argmax w ∈ Γ p · w. Yingni Guo (NU) Delegation of Experimentation 25 / 55

III. Characterizing the Policy Space Feasible Set Characterizing the Feasible Set ⇒ ∃ p ∈ R 2 , � p � = 1 , ˆ w ∈ bd (Γ) ⇐ ˆ w ∈ argmax w ∈ Γ p · w. Yingni Guo (NU) Delegation of Experimentation 25 / 55

III. Characterizing the Policy Space Feasible Set Characterizing the Feasible Set ⇒ ∃ p ∈ R 2 , � p � = 1 , ˆ w ∈ bd (Γ) ⇐ ˆ w ∈ argmax w ∈ Γ p · w. Yingni Guo (NU) Delegation of Experimentation 25 / 55

III. Characterizing the Policy Space Feasible Set Characterizing the Feasible Set ⇒ ∃ p ∈ R 2 , � p � = 1 , ˆ w ∈ bd (Γ) ⇐ ˆ w ∈ argmax w ∈ Γ p · w. Lemma 2 (Feasible Set) , where Π M are Markov policies (wrt p ). � ( w 1 ( π ) , w 0 ( π )) , π ∈ Π M � Γ = co Proof Yingni Guo (NU) Delegation of Experimentation 25 / 55

III. Characterizing the Policy Space Feasible Set Canonical Markov Policies: Poisson Conclusive News Stopping-time policies (lower-cutoff Markov policies) allocate all resource to R until a fixed time; if at least one success occurs by then, allocate all resource to R forever; otherwise, switch to S . Slack-after-success policies (upper-cutoff Markov policies) allocate all resource to R until the first success; then allocate a fixed fraction to R . Yingni Guo (NU) Delegation of Experimentation 26 / 55

III. Characterizing the Policy Space Feasible Set Canonical Markov Policies: Poisson Conclusive News w 0 ( π ) slack-after-success policies 1 s e i c i l o p e m i t - g n i p p o t s w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 27 / 55

b III. Characterizing the Policy Space Feasible Set Canonical Markov Policies: Poisson Conclusive News w 0 ( π ) slack-after-success policies 1 A: allocate all resource to S s e i c i l o p e m i t - g n i p p o t s A w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 27 / 55

b III. Characterizing the Policy Space Feasible Set Canonical Markov Policies: Poisson Conclusive News w 0 ( π ) slack-after-success policies 1 B: switch to S at some fixed time if no success occurs s e i c i l o p e m i t - B g n i p p o t s w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 27 / 55

III. Characterizing the Policy Space Feasible Set Canonical Markov Policies: Poisson Conclusive News w 0 ( π ) slack-after-success policies b C 1 s e i c C: allocate all resource to R i l o p e m i t - g n i p p o t s w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 27 / 55

b III. Characterizing the Policy Space Feasible Set Canonical Markov Policies: Poisson Conclusive News w 0 ( π ) slack-after-success policies 1 D s e i c i l o p D: allocate all resource to R e m until 1 st success; then allocate i t - g some fixed fraction to R n i p p o t s w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 27 / 55

b III. Characterizing the Policy Space Feasible Set Canonical Markov Policies: Poisson Conclusive News w 0 ( π ) slack-after-success policies 1 E s e i c i l o p e m i t - g n i p p o E: allocate all resource to R t s until 1 st success; then switch to S w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 27 / 55

III. Characterizing the Policy Space Feasible Set Feasible Set: Poisson Conclusive News w 0 ( π ) slack-after-success policies 1 s e i c feasible set: Γ i l o p e m i t - g n i p p o t s w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 28 / 55

III. Characterizing the Policy Space Feasible Set Feasible Set: Poisson Conclusive News Lemma 3 (Feasible Set: Poisson Conclusive News) The feasible set is the convex hull of the image of stopping-time and slack-after-success policies. Yingni Guo (NU) Delegation of Experimentation 29 / 55

III. Characterizing the Policy Space Preferences over Feasible Pairs Preferences over Feasible Pairs Player i ’s payoff given π and θ is θη i w 1 ( π ) − (1 − θ ) w 0 ( π ) � � U i ( π, θ ) − s i = · s i . Yingni Guo (NU) Delegation of Experimentation 30 / 55

III. Characterizing the Policy Space Preferences over Feasible Pairs Preferences over Feasible Pairs Player i ’s payoff given π and θ is θη i w 1 ( π ) − (1 − θ ) w 0 ( π ) � � U i ( π, θ ) − s i = · s i . Yingni Guo (NU) Delegation of Experimentation 30 / 55

III. Characterizing the Policy Space Preferences over Feasible Pairs Preferences over Feasible Pairs Player i ’s payoff given π and θ is θη i w 1 ( π ) − (1 − θ ) w 0 ( π ) � � U i ( π, θ ) − s i = · s i . Player i ’s preferences over ( w 1 , w 0 ) ∈ Γ are determined by θ : the prior belief that the state is 1 ; η i : the benefit-cost ratio from the experimentation. Yingni Guo (NU) Delegation of Experimentation 30 / 55

b III. Characterizing the Policy Space Preferences over Feasible Pairs Preferences over Feasible Pairs: Indifference Curves w 0 ( π ) 1 θ slope= 1 − θ η ρ P b P Principal’s indifference curve given θ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 31 / 55

b III. Characterizing the Policy Space Preferences over Feasible Pairs Preferences over Feasible Pairs: Indifference Curves w 0 ( π ) 1 θ slope= 1 − θ η ρ P b P Principal’s indifference curve given θ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 31 / 55

b b b b III. Characterizing the Policy Space Preferences over Feasible Pairs Preferences over Feasible Pairs: Indifference Curves w 0 ( π ) 1 Agent’s indifference curve given θ θ slope= 1 − θ η α A A θ slope= 1 − θ η ρ P P Principal’s indifference curve given θ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 31 / 55

b b b b III. Characterizing the Policy Space Preferences over Feasible Pairs Preferences over Feasible Pairs: Indifference Curves w 0 ( π ) 1 Agent’s indifference curve given θ θ slope= 1 − θ η α A A A: ( w 1 α ( θ ) , w 0 α ( θ )) θ P: ( w 1 ρ ( θ ) , w 0 slope= ρ ( θ )) 1 − θ η ρ P P Principal’s indifference curve given θ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 31 / 55

III. Characterizing the Policy Space Delegation Problem Reformulated Delegation Problem Reformulated Replace policy space Π with feasible set Γ : ⇒ θη i w 1 − (1 − θ ) w 0 . ( w 1 , w 0 ) ∈ Γ = Yingni Guo (NU) Delegation of Experimentation 32 / 55

III. Characterizing the Policy Space Delegation Problem Reformulated Delegation Problem Reformulated Replace policy space Π with feasible set Γ : ⇒ θη i w 1 − (1 − θ ) w 0 . ( w 1 , w 0 ) ∈ Γ = Principal offers a direct mechanism ( w 1 , w 0 ) : Θ → Γ � θη ρ w 1 ( θ ) − (1 − θ ) w 0 ( θ ) � � max dF ( θ ) , Θ 1 − θw 1 ( θ ) − w 0 ( θ ) ≥ θη α θη α 1 − θw 1 ( θ ′ ) − w 0 ( θ ′ ) ∀ θ, θ ′ ∈ Θ . subject to Yingni Guo (NU) Delegation of Experimentation 32 / 55

III. Characterizing the Policy Space Delegation Problem Reformulated Delegation Problem Reformulated Replace policy space Π with feasible set Γ : ⇒ θη i w 1 − (1 − θ ) w 0 . ( w 1 , w 0 ) ∈ Γ = Principal offers a direct mechanism ( w 1 , w 0 ) : Θ → Γ � θη ρ w 1 ( θ ) − (1 − θ ) w 0 ( θ ) � � max dF ( θ ) , Θ 1 − θw 1 ( θ ) − w 0 ( θ ) ≥ θη α θη α 1 − θw 1 ( θ ′ ) − w 0 ( θ ′ ) ∀ θ, θ ′ ∈ Θ . subject to Payoff parameters η α > η ρ ; feasible set Γ ; type distribution F . Yingni Guo (NU) Delegation of Experimentation 32 / 55

III. Characterizing the Policy Space Delegation Problem Reformulated Delegation Problem Reformulated Replace policy space Π with feasible set Γ : ⇒ θη i w 1 − (1 − θ ) w 0 . ( w 1 , w 0 ) ∈ Γ = Principal offers a direct mechanism ( w 1 , w 0 ) : Θ → Γ � θη ρ w 1 ( θ ) − (1 − θ ) w 0 ( θ ) � � max dF ( θ ) , Θ 1 − θw 1 ( θ ) − w 0 ( θ ) ≥ θη α θη α 1 − θw 1 ( θ ′ ) − w 0 ( θ ′ ) ∀ θ, θ ′ ∈ Θ . subject to Payoff parameters η α > η ρ ; feasible set Γ ; type distribution F . Yingni Guo (NU) Delegation of Experimentation 33 / 55

Outline Model 1 Single-player benchmark 2 Characterizing the policy space 3 Main results 4 More general results 5 Yingni Guo (NU) Delegation of Experimentation 34 / 55

IV. Main Results The Cutoff Rule The Cutoff Rule Definition 1 The cutoff rule is the contract ( w 1 , w 0 ) s.t. � ( w 1 α ( θ ) , w 0 if θ ≤ θ ∗ , α ( θ )) ( w 1 ( θ ) , w 0 ( θ )) = ( w 1 α ( θ ∗ ) , w 0 α ( θ ∗ )) if θ > θ ∗ . Yingni Guo (NU) Delegation of Experimentation 35 / 55

b b IV. Main Results The Cutoff Rule Delegation Set under Cutoff Rule w 0 ( π ) 1 θη ρ feasible set: Γ Principal’s preferred policies θη ρ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 36 / 55

b b b IV. Main Results The Cutoff Rule Delegation Set under Cutoff Rule w 0 ( π ) 1 b θη α θη ρ Agent’s preferred policies feasible set: Γ Principal’s preferred policies θη α θη ρ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 36 / 55

b b b b IV. Main Results The Cutoff Rule Delegation Set under Cutoff Rule w 0 ( π ) 1 b θη α θη ρ Agent’s preferred θ ∗ η α policies feasible set: Γ Principal’s preferred policies θη α θη ρ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 36 / 55

b b b b IV. Main Results The Cutoff Rule Delegation Set under Cutoff Rule w 0 ( π ) 1 b θη α θη ρ Agent’s preferred θ ∗ η α policies feasible set: Γ Principal’s preferred policies θη α θη ρ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 36 / 55

IV. Main Results The Cutoff Rule Optimality Main assumption For all θ ≤ θ ∗ , the following condition is satisfied: ≥ (3 θ − 1) − f ′ ( θ ) η α f ( θ ) θ (1 − θ ) . η α − η ρ Proposition 1 The cutoff rule is optimal if the main assumption holds. Yingni Guo (NU) Delegation of Experimentation 37 / 55

IV. Main Results Implementation Implementing the Cutoff Rule Calibrated belief ( p t ) prior belief p 0 = θ ∗ ; without any success, it drifts down according to ˙ p t = − λπ t p t (1 − p t ) ; upon the first success, it jumps to one. Behavior a cutoff imposed at p ∗ α ; Agent has full flexibility if the belief stays above the cutoff; Agent is required to stop once the cutoff is reached. Yingni Guo (NU) Delegation of Experimentation 38 / 55

b IV. Main Results Implementation Implementing the Cutoff Rule (cont.) calibrated belief p t 1 p 0 = θ ∗ cutoff: p ∗ α time t 0 Yingni Guo (NU) Delegation of Experimentation 39 / 55

b IV. Main Results Implementation Implementing the Cutoff Rule (cont.) calibrated belief p t 1 p 0 = θ ∗ cutoff: p ∗ α time t 0 Yingni Guo (NU) Delegation of Experimentation 39 / 55

b IV. Main Results Implementation Implementing the Cutoff Rule (cont.) calibrated belief p t 1 1st success p 0 = θ ∗ cutoff: p ∗ α time t 0 Yingni Guo (NU) Delegation of Experimentation 39 / 55

b IV. Main Results Implementation Implementing the Cutoff Rule (cont.) calibrated belief p t 1 p 0 = θ ∗ cutoff: p ∗ α time t 0 Type θ stops. Yingni Guo (NU) Delegation of Experimentation 39 / 55

b IV. Main Results Implementation Implementing the Cutoff Rule (cont.) calibrated belief p t 1 p 0 = θ ∗ cutoff: p ∗ α time t 0 Type θ stops. Yingni Guo (NU) Delegation of Experimentation 39 / 55

b b IV. Main Results Implementation Implementing the Cutoff Rule (cont.) calibrated belief p t 1 p 0 = θ ∗ cutoff: p ∗ α time t 0 Types with θ ≥ θ ∗ stop. Type θ stops. Yingni Guo (NU) Delegation of Experimentation 39 / 55

IV. Main Results Time Consistency Time Consistency Definition 2 Fix a (direct or indirect) mechanism. It is time-consistent if Principal finds it optimal to fulfill the mechanism after any history on path. Formal definition Yingni Guo (NU) Delegation of Experimentation 40 / 55

IV. Main Results Time Consistency Time Consistency Definition 2 Fix a (direct or indirect) mechanism. It is time-consistent if Principal finds it optimal to fulfill the mechanism after any history on path. Formal definition Proposition 2 The cutoff rule is time-consistent if the main assumption holds. Yingni Guo (NU) Delegation of Experimentation 40 / 55

b IV. Main Results Time Consistency Time Consistency: Principal’s Posterior Belief Calibrated belief p t 1 p 0 ( θ ∗ ) b Cutoff: p ∗ α time t 0 Types with θ ≥ θ ∗ stop. Type θ stops. More general results Yingni Guo (NU) Delegation of Experimentation 41 / 55

b IV. Main Results Time Consistency Time Consistency: Principal’s Posterior Belief Calibrated belief p t 1 p 0 ( θ ∗ ) b Cutoff: p ∗ ρ Cutoff: p ∗ α time t 0 Types with θ ≥ θ ∗ stop. Type θ stops. More general results Yingni Guo (NU) Delegation of Experimentation 41 / 55

b IV. Main Results Time Consistency Time Consistency: Principal’s Posterior Belief Calibrated belief p t 1 p 0 ( θ ∗ ) b Cutoff: p ∗ ρ Cutoff: p ∗ α time t 0 Types with θ ≥ θ ∗ stop. Type θ stops. More general results Yingni Guo (NU) Delegation of Experimentation 41 / 55

b IV. Main Results Time Consistency Time Consistency: Principal’s Posterior Belief Calibrated belief p t 1 p 0 ( θ ∗ ) b Cutoff: p ∗ ρ Cutoff: p ∗ α time t 0 Types with θ ≥ θ ∗ stop. Type θ stops. More general results Yingni Guo (NU) Delegation of Experimentation 41 / 55

b IV. Main Results Time Consistency Time Consistency: Principal’s Posterior Belief Calibrated belief p t 1 p 0 ( θ ∗ ) b Cutoff: p ∗ ρ Cutoff: p ∗ α time t 0 Types with θ ≥ θ ∗ stop. Type θ stops. More general results Yingni Guo (NU) Delegation of Experimentation 41 / 55

b IV. Main Results Time Consistency Time Consistency: Principal’s Posterior Belief Calibrated belief p t 1 p 0 ( θ ∗ ) b Cutoff: p ∗ ρ Cutoff: p ∗ α time t 0 Types with θ ≥ θ ∗ stop. Type θ stops. More general results Yingni Guo (NU) Delegation of Experimentation 41 / 55

IV. Main Results Cutoff Type The Cutoff Type The cutoff type θ ∗ : the lowest value in Θ s.t. Agent’s preferred policy given θ ∗ equals Principal’s preferred policy if she believes that θ ≥ θ ∗ . For any ˆ θ > θ ∗ , Agent’s preferred policy given ˆ θ is above Principal’s preferred policy if she believes that θ ≥ ˆ θ . Yingni Guo (NU) Delegation of Experimentation 42 / 55

b b b b IV. Main Results Cutoff Type The Cutoff Type (cont.) w 0 ( π ) 1 b θη α θη ρ Agent’s preferred θ ∗ η α policies feasible set: Γ Principal’s preferred policies θη α θη ρ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 43 / 55

b b b b b IV. Main Results Cutoff Type The Cutoff Type (cont.) w 0 ( π ) 1 b θη α θη ρ Agent’s preferred θ ∗ η α policies θ ∗ η ρ feasible set: Γ Principal’s preferred policies θη α θη ρ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 43 / 55

b b b b b b IV. Main Results Cutoff Type The Cutoff Type (cont.) w 0 ( π ) 1 b θη α θη ρ Agent’s preferred θ ∗ η α policies θ ∗ η ρ feasible set: Γ Principal’s preferred policies θη α θη ρ w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 43 / 55

IV. Main Results Cutoff Type Over- and Under-Experimentation stopping time τ Agent’s preferred stopping time type θ θ θ More general results Yingni Guo (NU) Delegation of Experimentation 44 / 55

IV. Main Results Cutoff Type Over- and Under-Experimentation stopping time τ Agent’s preferred stopping time Principal’s preferred stopping time type θ θ θ More general results Yingni Guo (NU) Delegation of Experimentation 44 / 55

IV. Main Results Cutoff Type Over- and Under-Experimentation stopping time τ Agent’s preferred stopping time τ α ( θ ∗ ) delegation rule Principal’s preferred stopping time type θ θ θ ∗ θ More general results Yingni Guo (NU) Delegation of Experimentation 44 / 55

IV. Main Results Cutoff Type Over- and Under-Experimentation stopping time τ τ α ( θ ∗ ) delegation rule Principal’s preferred stopping time type θ θ θ ∗ θ More general results Yingni Guo (NU) Delegation of Experimentation 44 / 55

IV. Main Results Cutoff Type Over- and Under-Experimentation stopping time τ τ α ( θ ∗ ) delegation rule Principal’s preferred stopping time type θ θ θ ∗ θ over-experimentation More general results Yingni Guo (NU) Delegation of Experimentation 44 / 55

IV. Main Results Cutoff Type Over- and Under-Experimentation stopping time τ τ α ( θ ∗ ) delegation rule Principal’s preferred stopping time type θ θ θ ∗ θ over-experimentation under-experimentation More general results Yingni Guo (NU) Delegation of Experimentation 44 / 55

Outline Model 1 Single-player benchmark 2 Characterizing the policy space 3 Main results 4 More general results 5 Yingni Guo (NU) Delegation of Experimentation 45 / 55

V. More general results Poisson Inconclusive News Feasible Set: Poisson Inconclusive News w 0 ( π ) 1 s e i c i l o p v o k r a M ff to u c s feasible set: Γ e - i r c e i l pp o p u v o k r a M ff o t u c - r e w o l w 1 ( π ) 0 1 Yingni Guo (NU) Delegation of Experimentation 46 / 55