Motivational Ratings orner 1 , Nicolas Lambert 2 Johannes H 1 Yale - - PowerPoint PPT Presentation

Motivational Ratings

Johannes H¨

• rner1, Nicolas Lambert2

1Yale and CEPR 2Stanford and Microsoft Research

Frontiers of Economic Theory and Computer Science Becker Friedman Institute, August 2016

Focus: Ratings that incentivize effort (moral hazard).

Hospitals, physicians; schools, teachers; companies, executives, etc.

Focus: Ratings that incentivize effort (moral hazard).

Hospitals, physicians; schools, teachers; companies, executives, etc.

Goal: What is the best rating system?

Unknown skill θ Private effort A Forward-looking

Agent

Market Agent dXt

Competitive Rational expectations

At, θt dWt

Market Agent dXt At, θt dπt

Pays upfront

dWt

Rater Market Agent At, θt dXt, dSt dπt {dWk,t}k

Rater Market Agent At, θt dXt, dSt dπt

Ancillary statistics

{dWk,t}k

Rater Market Agent At, θt dXt, dSt dπt Yt = ft({Xs, Ss}s≤t) {dWk,t}k

Rater Market Agent At, θt dXt, dSt dπt Yt = ft({Xs, Ss}s≤t)

Non-strategic Committed Maximizes efficiency

{dWk,t}k

Rater Market Agent At, θt dXt, dSt dπt Yt = ft({Xs, Ss}s≤t)

Non-strategic Committed Maximizes effort

{dWk,t}k

Rater Market Agent At, θt dXt, dSt dπt Yt = ft({Xs, Ss}s≤t) {dWk,t}k {Ys}s≤t: public rating

Rater Market Agent At, θt dXt, dSt dπt Yt = ft({Xs, Ss}s≤t) {dWk,t}k dXt, dSk,t, . . .

Non-exclusive rating

Confidential Public Exclusive Yt {Ys}s≤t Non-exclusive Yt, {Xs, Sk,s}s≤t,k≤K0 {Ys, Xs, Sk,s}s≤t,k≤K0

Confidential Public Exclusive Yt {Ys}s≤t Non-exclusive Yt, {Xs, Sk,s}s≤t,k≤K0 {Ys, Xs, Sk,s}s≤t,k≤K0

Model

Continuous time t ≥ 0, infinite horizon.

Continuous time t ≥ 0, infinite horizon. The relevant processes are: Effort: At ∈ R+; privately known by the agent.

Continuous time t ≥ 0, infinite horizon. The relevant processes are: Effort: At ∈ R+; privately known by the agent. Ability: θt ∈ R; unknown to all.

Continuous time t ≥ 0, infinite horizon. The relevant processes are: Effort: At ∈ R+; privately known by the agent. Ability: θt ∈ R; unknown to all. Output: Xt ∈ R. Ancillary Statistics: St ∈ RK−1.

Ability Process: dθt = −θt dt

Mean-reversion

+

Innovation

γ dW θ

t ,

with θ0 ∼ N(0, γ2/2), γ > 0, and W θ a standard B.M. Rate of mean-reversion: 1.

Ability Process: dθt = −θt dt

Mean-reversion

+

Innovation

γ dW θ

t ,

with θ0 ∼ N(0, γ2/2), γ > 0, and W θ a standard B.M. Rate of mean-reversion: 1.

Ability Process: dθt = −θt dt

Mean-reversion

+

Innovation

γ dW θ

t ,

with θ0 ∼ N(0, γ2/2), γ > 0, and W θ a standard B.M. Rate of mean-reversion: 1.

Ability Process: dθt = −θt dt

Mean-reversion

+

Innovation

γ dW θ

t ,

with θ0 ∼ N(0, γ2/2), γ > 0, and W θ a standard B.M. Rate of mean-reversion: 1.

Output Process: dXt = (At + θt)dt + σ1dW1,t, with X0 = 0, σ1 > 0, and W1 a standard B.M. (W1 ⊥ W θ).

Signal Processes, k = 2, . . . , K: dSk,t = (αkAt + βkθt) dt + σk dWk,t, with Sk,0 = 0, σk > 0, αk, βk ∈ R and Wk a standard B.M.

Signal Processes, k = 2, . . . , K: dSk,t = (αkAt + βkθt) dt + σk dWk,t, with Sk,0 = 0, σk > 0, αk, βk ∈ R and Wk a standard B.M. If αk = βk = 0, the signal is “white noise.”

Signal Processes, k = 2 , . . . , K: dSk,t = (αkAt + βkθt) dt + σk dWk,t, with Sk,0 = 0, σk > 0, αk, βk ∈ R and Wk a standard B.M. If αk = βk = 0, the signal is “white noise.” We also write S1 := X (α1 := 1, β1 := 1).

Payoffs

Given a (cumulative) transfer process π, realized payoffs are: Market: ∞ e−rt(

Revenue

• n [t, t + dt)
• dXt

−

Transfer

• n [t, t + dt)
• dπt ),

Agent: ∞ e−rt( dπt

• Transfer
• n [t, t + dt)

− c(At) dt

• Cost
• n [t, t + dt)

), where the discount rate is r > 0, and c(0) = c′(0) = 0 and c′′ > 0.

Payoffs

Given a (cumulative) transfer process π, realized payoffs are: Market: ∞ e−rt(

Revenue

• n [t, t + dt)
• dXt

−

Transfer

• n [t, t + dt)
• dπt ),

Agent: ∞ e−rt( dπt

• Transfer
• n [t, t + dt)

− c(At) dt

• Cost
• n [t, t + dt)

), where the discount rate is r > 0, and c(0) = c′(0) = 0 and c′′ > 0.

Payoffs

Given a (cumulative) transfer process π, realized payoffs are: Market: ∞ e−rt(

Revenue

• n [t, t + dt)
• dXt

−

Transfer

• n [t, t + dt)
• dπt ),

Agent: ∞ e−rt( dπt

• Transfer
• n [t, t + dt)

− c(At) dt

• Cost
• n [t, t + dt)

), where the discount rate is r > 0, and c(0) = c′(0) = 0 and c′′ > 0.

Payoffs

Given a (cumulative) transfer process π, realized payoffs are: Market: ∞ e−rt(

Revenue

• n [t, t + dt)
• dXt

−

Transfer

• n [t, t + dt)
• dπt ),

Agent: ∞ e−rt( dπt

• Transfer
• n [t, t + dt)

− c(At) dt

• Cost
• n [t, t + dt)

), where the discount rate is r > 0, and c(0) = c′(0) = 0 and c′′ > 0.

Payoffs

Given a (cumulative) transfer process π, realized payoffs are: Market: ∞ e−rt(

Revenue

• n [t, t + dt)
• dXt

−

Transfer

• n [t, t + dt)
• dπt ),

Agent: ∞ e−rt( dπt

• Transfer
• n [t, t + dt)

− c(At) dt

• Cost
• n [t, t + dt)

), where the discount rate is r > 0, and c(0) = c′(0) = 0 and c′′ > 0.

Payoffs

Given a (cumulative) transfer process π, realized payoffs are: Market: ∞ e−rt(

Revenue

• n [t, t + dt)
• dXt

−

Transfer

• n [t, t + dt)
• dπt ),

Agent: ∞ e−rt( dπt

• Transfer
• n [t, t + dt)

− c(At) dt

• Cost
• n [t, t + dt)

), where the discount rate is r > 0, and c(0) = c′(0) = 0 and c′′ > 0. Recall that E[dXt] = At dt. Hence, efficiency requires c′(At) = 1 ∀t.

The Optimization Program

The market has some expectation A∗ about the agent’s effort.

The Optimization Program

The market has some expectation A∗ about the agent’s effort. Given its information Mt, the market forms a belief µt = E∗[θt | Mt].

The Optimization Program

The market has some expectation A∗ about the agent’s effort. Given its information Mt, the market forms a belief µt = E∗[θt | Mt]. The agent maximizes over all processes (At): E ∞ e−rt(µt − c(At)) dt

• .

The Optimization Program

The market has some expectation A∗ about the agent’s effort. Given its information Mt, the market forms a belief µt = E∗[θt | Mt]. The agent maximizes over all processes (At): E ∞ e−rt(µt − c(At))dt

• .

The Optimization Program

The market has some expectation A∗ about the agent’s effort. Given its information Mt, the market forms a belief µt = E∗[θt | Mt]. The agent maximizes over all processes (At): E ∞ e−rt(µt − c(At)) dt

• .

The rater’s goal: to maximize the argmax (At) over M.

Why is Transparency Suboptimal?

Consider: dXt = At dt + dW1,t, dS2,t = θt dt + dW2,t.

Why is Transparency Suboptimal?

Consider: dXt = At dt + dW1,t, dS2,t = θt dt + dW2,t. If {Xs, S2,s}s≤t is disclosed, then {As}s≤t doesn’t affect µt: At = 0.

Why is Transparency Suboptimal?

Consider: dXt = At dt + dW1,t, dS2,t = θt dt + dW2,t. If {Xs, S2,s}s≤t is disclosed, then {As}s≤t doesn’t affect µt: At = 0. If instead the market only observes: d(Xt + S2,t) = (At + θt)dt + dW1,t + dW2,t, career concerns arise: At > 0.

Why is Transparency Suboptimal?

Consider: dXt = At dt + dW1,t, dS2,t = θt dt + dW2,t. If {Xs, S2,s}s≤t is disclosed, then {As}s≤t doesn’t affect µt: At = 0. If instead the market only observes: d(Xt + S2,t) = (At + θt)dt + dW1,t + dW2,t, career concerns arise: At > 0. One can do better than disclosing the sum of the signals.

Rating Processes

A rating process is a I-adapted Y, where It = σ({Ss}s≤t).

Rating Processes

A rating process is a I-adapted Y, where It = σ({Ss}s≤t). A rating process defines an information structure via Mt = σ(Yt)

Confidential

.

Rating Processes

A rating process is a I-adapted Y, where It = σ({Ss}s≤t). A rating process defines an information structure via Mt = σ(Yt)

Confidential

.

Rating Processes

A rating process is a I-adapted Y, where It = σ({Ss}s≤t). A rating process defines an information structure via Mt = σ(Yt)

Confidential

. Throughout, we impose:

• 1. For all ∆, (Yt, St − St−∆) is normal and stationary.
• 2. The map ∆ → Cov[Yt, St−∆] is absolutely cts., with

integrable and square integrable Radon-Nikodym derivative.

• 3. The mean rating is zero: E∗[Yt] = 0.

Rating Processes

A rating process is a I-adapted Y, where It = σ({Ss}s≤t). A rating process defines an information structure via Mt = σ(Yt)

Confidential

. Throughout, we impose:

• 1. For all ∆, (Yt, St − St−∆) is normal and stationary.
• 2. The map ∆ → Cov[Yt, St−∆] is absolutely cts., with

integrable and square integrable Radon-Nikodym derivative.

• 3. The mean rating is zero: E∗[Yt] = 0.

We usually work with scalar Yt = E∗[θt | Mt] (“Direct ratings”).

Deterministic Information Quality Implies Normality

Lemma (Normal Representation) Let Y be a progressively measurable process on I such that:

• 1. ∀T > t + τ > t, Cov[YT, St+τ | It] is a function of (t, T, τ),

differentiable in τ, with uniformly Lipschitz cts. derivative in t.

• 2. ∀T > t, Cov[YT, θt | It] is a function of (t, T).
• 3. ∀t, E[Y2

t ] < ∞ and E[Yt] = 0.

Then, for all ∆ ≥ 0, (Yt, St − St−∆) is normally distributed.

Methods that Qualify: Exponential smoothing. (Business Week’s b-school ranking.) Yt =

• s≤t

e−a(t−s) dXs.

Methods that Qualify: Exponential smoothing. (Business Week’s b-school ranking.) Yt =

• s≤t

e−a(t−s) dXs. Special case: Transparency (a = κ :=

• 1 + γ2/σ2

1).

Methods that Qualify: Exponential smoothing. (Business Week’s b-school ranking.) Yt =

• s≤t

e−a(t−s) dXs. Special case: Transparency (a = κ :=

• 1 + γ2/σ2

1).

Moving window. (Consumer credit ratings, BBB grades.) Yt = t

t−∆

dXs.

Methods that Qualify: Exponential smoothing. (Business Week’s b-school ranking.) Yt =

• s≤t

e−a(t−s) dXs. Special case: Transparency (a = κ :=

• 1 + γ2/σ2

1).

Moving window. (Consumer credit ratings, BBB grades.) Yt = t

t−∆

dXs. Methods that Don’t: Coarse ratings.

What can the rater Do? More Generally:

· · · · · · · · · dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . .

What can the rater Do? More Generally:

· · · · · · · · · dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . .

What can the rater Do? More Generally:

· · · · · · · · · dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . .

What can the rater Do? More Generally:

· · · · · · · · · dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . .

What can the rater Do? More Generally:

· · · · · · · · · dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . . e−δdt e−δ(t−s) 1 + +

What can the rater Do? More Generally:

· · · · · · · · · u1(·) dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . .

What can the rater Do? More Generally:

· · · · · · · · · u1(·) uk(·) uK(·) dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . .

What can the rater Do? More Generally:

· · · · · · · · · dSK,t dXt dSk,t dSK,t−dt dXt−dt dSk,t−dt dSK,s dXs dSk,s · · · · · · · · · . . . . . . . . . . . . . . . . . . uk(t − s) u1(0)

Lemma (Analytic Representation) Fix a rating process Y . Given a conjectured A∗, there exist unique vector-valued functions uk, k = 1, . . . , K, such that, for all t, Yt =

• k
• s≤t

uk(t − s)(dSk,s − αkA∗ ds).

Main Results for the Confidential/Exclusive Case

The unique optimal confidential rating system is uk(t) = dk √r λ e−rt + βk σ2

k

e−κt. dk := (κ2 − r2)mβ αk σ2

k

− (κ2 − 1)mαβ βk σ2

k

, with

λ := (κ − 1)√r(1 + r)mαβ + (κ − r) √ ∆, mβ :=

• k

β2

k

σ2

k

, mαβ :=

• k

αkβk σ2

k

, mα :=

• k

α2

k

σ2

k

, ∆ := (κ + r)2(mαmβ − m2

αβ) + (1 + r)2m2 αβ,

κ :=

• 1 + γ2

k

β2

k

σ2

k

.

That is, Yt =

• s≤t
• k
• incentive term
• dk

√r λ e−r(t−s) +

belief term

• βk

σ2

k

e−κ(t−s) (dSk,s − αkA∗ ds).

That is, Yt =

• s≤t
• k
• incentive term
• dk

√r λ e−r(t−s) +

belief term

• βk

σ2

k

e−κ(t−s) (dSk,s − αkA∗ ds). The system is a (two-state) mixture Markov rating system.

That is, Yt =

• s≤t
• k
• incentive term
• dk

√r λ e−r(t−s) +

belief term

• βk

σ2

k

e−κ(t−s) (dSk,s − αkA∗ ds). The system is a (two-state) mixture Markov rating system. The rating can be written as the sum of two Markov processes.

That is, Yt =

• s≤t
• k
• incentive term
• dk

√r λ e−r(t−s) +

belief term

• βk

σ2

k

e−κ(t−s) (dSk,s − αkA∗ ds). The system is a (two-state) mixture Markov rating system. The rating can be written as the sum of two Markov processes. One state is the rater’s belief νt := E∗[θt | It]. dνt = − κνt dt + γ2 κ + 1

• k

βk σ2

k

(dSk,t − αkA∗ dt)

That is, Yt =

• s≤t
• k
• incentive term
• dk

√r λ e−r(t−s) + βk σ2

k

e−κ(t−s) (dSk,s − αkA∗ ds). The system is a (two-state) mixture Markov rating system. The rating can be written as the sum of two Markov processes. The other is some incentive state It. dIt = − rIt dt + √r λ

• k dk (dSk,t − αkA∗ dt) .

Two states are needed: keeping track of νt isn’t enough.

Two states are needed: keeping track of νt isn’t enough. νt It Yt

Two states are needed: keeping track of νt isn’t enough. νt It Yt

Two states are needed: keeping track of νt isn’t enough. It Yt (It, νt)

Two states are needed: keeping track of νt isn’t enough. It Yt (It, νt) The rating process Y = I + ν isn’t Markov.

Reality Check

Ratings are not Markov: widely documented for credit rating.

Altman and Kao (1992), Carty and Fons (1993), Altman (1998), Nickell et al. (2000), Bangia et al. (2002), Lando and Skødeberg (2002), Hamilton and Cantor (2004), etc.

Reality Check

Ratings are not Markov: widely documented for credit rating.

Altman and Kao (1992), Carty and Fons (1993), Altman (1998), Nickell et al. (2000), Bangia et al. (2002), Lando and Skødeberg (2002), Hamilton and Cantor (2004), etc.

Mixture rating models: shown to explain economic differences.

Two-state: Frydman and Schuerman (2008); HMM: Giampieri et al. (2005); Rating momentum: Stefanescu et al. (2006).

Implication: Benchmarking

As an example, suppose there is one signal (output): αk = α, βk = β, σk = σ.

Implication: Benchmarking

As an example, suppose there is one signal (output): αk = α, βk = β, σk = σ. Then, the optimal confidential rating simplifies to u(t) = β σ2

• 1 − √r

κ − √r √re−rt + e−κt

• .

Implication: Benchmarking

As an example, suppose there is one signal (output): αk = α, βk = β, σk = σ. Then, the optimal confidential rating simplifies to u(t) = β σ2

• 1 − √r

κ − √r √re−rt + e−κt

• .

So the incentive state isn’t always “added.” It may be subtracted.

0.02 0.04 0.06 0.08 0.10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t u(t) (r, κ) = (14, 15)

Reality Check

Benchmarking: Prior-year performance widely used for incentives. When standards are based on prior-year performance, man- agers might avoid unusually positive performance outcomes, since good current performance is penalized in the next period through an increased standard. —Murphy, 2001.

Implementation

What about other effort levels, given alternative goals of the rater? Maximum effort A induced by two-state mixture Markov rating Y .

Implementation

What about other effort levels, given alternative goals of the rater? Maximum effort A induced by two-state mixture Markov rating Y . Minimum effort 0 induced by “pure noise” W , B.M.

Implementation

What about other effort levels, given alternative goals of the rater? Maximum effort A induced by two-state mixture Markov rating Y . Minimum effort 0 induced by “pure noise” W , B.M. Can induce any A ∈ [0, A] by λY + (1 − λ)W , some λ ∈ [0, 1]. Two-state mixture Markov ratings (plus noise) are wlog.

In Conclusion. . .

In Conclusion

Our analysis shows why: Insisting on transparency or even publicness isn’t optimal.

In Conclusion

Our analysis shows why: Insisting on transparency or even publicness isn’t optimal. And, more surprisingly: Two-state mixture Markov models are “robust.” Ratings aren’t Markovian. Benchmarking can be optimal.

Technical Aspects

Focus on scalar ratings (wlog). Lemma. The effort A∗ induced by a confidential process Y solves c′(A∗) ∝ Corr[Y , θ]

• Belief term

·

Incentive term

• k
• t≥0 αkuk(t)e−rt dt
• Var[Y ]
• Normalization

Focus on scalar ratings (wlog). Lemma. The effort A∗ induced by a confidential process Y solves c′(A∗) ∝ Corr[Y , θ]

• Belief term

·

Incentive term

• k
• t≥0 αkuk(t)e−rt dt
• Var[Y ]
• Normalization

Focus on scalar ratings (wlog). Lemma. The effort A∗ induced by a confidential process Y solves c′(A∗) ∝ Corr[Y , θ]

• Belief term

·

Incentive term

• k
• t≥0 αkuk(t)e−rt dt
• Var[Y ]
• Normalization

Focus on scalar ratings (wlog). Lemma. The effort A∗ induced by a confidential process Y solves c′(A∗) ∝ Corr[Y , θ]

• Belief term

·

Incentive term

• k
• t≥0 αkuk(t)e−rt dt
• Var[Y ]
• Normalization

Focus on scalar ratings (wlog). Lemma. The effort A∗ induced by a confidential process Y solves c′(A∗) ∝ Corr[Y , θ]

• Belief term

·

Incentive term

• k
• t≥0 αkuk(t)e−rt dt
• Var[Y ]
• Normalization

→ max

{uk}k

Optimal Ratings: Proof Overview

We first guess what optimal ratings look like.

• 1. Write the agent’s marginal cost as a function of {uk}k.

Optimal Ratings: Proof Overview

We first guess what optimal ratings look like.

• 1. Write the agent’s marginal cost as a function of {uk}k.
• 2. Get a set of FOC by adding small perturbations.

(=Calculus of variation.)

Optimal Ratings: Proof Overview

We first guess what optimal ratings look like.

• 1. Write the agent’s marginal cost as a function of {uk}k.
• 2. Get a set of FOC by adding small perturbations.

(=Calculus of variation.)

• 3. Derive systems of differential equations that the uk’s must

satisfy.

(Yields exponential ratings.)

Optimal Ratings: Proof Overview

We first guess what optimal ratings look like.

• 1. Write the agent’s marginal cost as a function of {uk}k.
• 2. Get a set of FOC by adding small perturbations.

(=Calculus of variation.)

• 3. Derive systems of differential equations that the uk’s must

satisfy.

(Yields exponential ratings.)

Main difficulties:

◮ Non-standard calculus of variation.

(Multidimensional objective with single-dimensional input, time-delayed controls.)

◮ Set of FOC is a continuum.

We then verify that the guess is correct. To do so, we define an auxiliary principal-agent problem. The agent is as in the main model. The principal pays the agent, as the market does in the main model, but her payoffs includes the objective of the intermediary in the main model.

As in the main model, the agent maximizes E

• s≥t

e−r(s−t)(µs − c(As)) ds | Mt

• ,

but now µ is an arbitrary transfer rate.

As in the main model, the agent maximizes E

• s≥t

e−r(s−t)(µs − c(As)) ds | Mt

• ,

but now µ is an arbitrary transfer rate. The principal maximizes E    ∞ ρe−ρt(c′(At)

Objective term

− φµt(µt − νt)

• Penalty term

) dt    , where ρ, φ > 0 and νt is the belief under “full transparency”.

As in the main model, the agent maximizes E

• s≥t

e−r(s−t)(µs − c(As)) ds | Mt

• ,

but now µ is an arbitrary transfer rate. The principal maximizes E    ∞ ρe−ρt(c′(At)

Objective term

− φµt(µt − νt)

• Penalty term

) dt    , where ρ, φ > 0 and νt is the belief under “full transparency”.

As in the main model, the agent maximizes E

• s≥t

e−r(s−t)(µs − c(As)) ds | Mt

• ,

but now µ is an arbitrary transfer rate. The principal maximizes E    ∞ ρe−ρt(c′(At)

Objective term

− φµt(µt − νt)

• Penalty term

) dt    , where ρ, φ > 0 and νt is the belief under “full transparency”.

As in the main model, the agent maximizes E

• s≥t

e−r(s−t)(µs − c(As)) ds | Mt

• ,

but now µ is an arbitrary transfer rate. The principal maximizes E    ∞ ρe−ρt(c′(At)

Objective term

− φµt(µt − νt)

• Penalty term

) dt    , where ρ, φ > 0 and νt is the belief under “full transparency”. Note: if E[µ] = 0, E [µt(µt − νt)] = 0 ⇔ Var[µt] = Cov[µt, νt].

Why the penalty? We want µ to be a belief process.

Why the penalty? We want µ to be a belief process. It is a belief process iff µt = E[θt | µt] (= E[νt | µt]).

Why the penalty? We want µ to be a belief process. It is a belief process iff µt = E[θt | µt] (= E[νt | µt]). For any Gaussian (µ, ν): E[νt | µt] = E[νt] + Cov[µt, νt] Var[µt] (µt − E[µt]) . = ⇒ A (mean-normalized) process is a market belief iff Var[µt] = Cov[µt, νt].

With a carefully chosen φ, as the principal becomes increasingly patient, E [µt(µt − νt)] → 0 and c′(A) → c′(A∗) where A∗ is the conjectured optimal effort level derived from the FOC, and µ converges to the conjectured optimal rating.

Overview of the Other Cases

Public Ratings

The unique optimal public rating system is uk(t) = dk √r λ e−√rt + βk σ2

k

e−κt. Here,

• dk := κ − √r

κ − r dk + λ √r − 1 κ − r βk σ2

k

.

Public Ratings

The unique optimal public rating system is uk(t) = dk √r λ e−√rt + βk σ2

k

e−κt. So: uk(t) = dk √r λ e−√rt + βk σ2

k

e−κt, Here,

• dk := κ − √r

κ − r dk + λ √r − 1 κ − r βk σ2

k

.

Public Ratings

The unique optimal public rating system is uk(t) = dk √r λ e−√rt + βk σ2

k

e−κt. So: uk(t) = dk √r λ e−√rt + βk σ2

k

e−κt, = dk √r λ exp(−

rate of mean reversion

• 11/2 r1/2
• discount rate

t) + βk σ2

k

e−κt, Here,

• dk := κ − √r

κ − r dk + λ √r − 1 κ − r βk σ2

k

.

Public Ratings

The unique optimal public rating system is uk(t) = dk √r λ e−√rt + βk σ2

k

e−κt. So: uk(t) = dk √r λ e−√rt + βk σ2

k

e−κt, = dk √r λ exp(−

rate of mean reversion

• 11/2 r1/2
• discount rate

t) + βk σ2

k

e−κt, Here,

• dk := κ − √r

κ − r dk + λ √r − 1 κ − r βk σ2

k

.

In common: Differences:

A two-state rating system. One state is the belief. No signal gets discarded. Benchmarking can arise. Impulse response is the harmonic mean between the discount rate and the rate of mean-reversion.

In common: Differences:

A two-state rating system. One state is the belief. No signal gets discarded. Benchmarking can arise. Impulse response is the harmonic mean between the discount rate and the rate of mean-reversion. With homogeneous signals, ˜ dk = 0: transparency is best.

Exclusive vs. Non-Exclusive Information

Suppose some (not all) signals are openly available to the market.

In common: New features:

(With homogeneous signals) A two-state rating system. Private: uk = ˆ dke−rt + βk σ2

k

e−κt Public: uk = ˇ dke−δt + βk σ2

k

e−κt Better informed market. Public information and ratings can be substitutes.

Multi-Dimensional Actions

The analysis extends to multi-dimensional actions (separable cost).

Multi-Dimensional Actions

The analysis extends to multi-dimensional actions (separable cost). As an example: dS1,t = a1,tdt + σ1dW1,t, dS2,t = (a2,t + θt) dt + σ2dW2,t, with cost c(a1, a2) = c · (a2

1 + a2 2).

Multi-Dimensional Actions

The analysis extends to multi-dimensional actions (separable cost). As an example: dS1,t = a1,tdt + σ1dW1,t, dS2,t = (a2,t + θt) dt + σ2dW2,t, with cost c(a1, a2) = c · (a2

1 + a2 2). The best confidential system is

u1(t) = √r σ1 e−rt, u2(t) = e−κt σ2

2

, and effort c′(a1) = κ − 1 4√rσ1 , c′(a2) = κ − 1 2(r + κ)σ2

2

.

How do Different Signals get Weighted?

The confidential process can be rewritten as uk(t) = βk σ2

k

• (κ2 − r2)αk

βk − (κ2 − 1)mαβ mβ √rmβ λ e−rt + e−κt

• .

Fixing the SNR βk

σ2

k , signals are ordered according to the ratio αk

βk :

the higher the ratio, the larger the weight (whether positive or not).

Precision

How well-informed is the market? (as measured by Var[µ].)

Precision

How well-informed is the market? (as measured by Var[µ].) Always better informed with public ratings.

Precision

How well-informed is the market? (as measured by Var[µ].) Always better informed with public ratings. But it isn’t simply a trade-off between effort and information: fixing precision, higher effort under the best confidential rating system.

Precision

How well-informed is the market? (as measured by Var[µ].) Always better informed with public ratings. But it isn’t simply a trade-off between effort and information: fixing precision, higher effort under the best confidential rating system. Variance non-monotone in r.

Limits in the Confidential Case

Recall that uk(t) = dk √r λ e−rt + βk σ2

k

e−κt.

Limits in the Confidential Case

Recall that uk(t) = dk √r λ e−rt + βk σ2

k

e−κt. When r tends to 0: The coefficient dk/λ tends to a nonzero limit. Effort diverges, and market is less informed than under transparency.

Limits in the Confidential Case

Recall that uk(t) = dk √r λ e−rt + βk σ2

k

e−κt. When r tends to 0: The coefficient dk/λ tends to a nonzero limit. Effort diverges, and market is less informed than under transparency. When the rater’s signals become arbitrarily informative: Unless αk/βk is independent of k, effort diverges, and limit rating process is non-degenerate. No transparency.

Limits in the Confidential Case

Recall that uk(t) = dk √r λ e−rt + βk σ2

k

e−κt. When r tends to 0: The coefficient dk/λ tends to a nonzero limit. Effort diverges, and market is less informed than under transparency. When the rater’s signals become arbitrarily informative: Unless αk/βk is independent of k, effort diverges, and limit rating process is non-degenerate. No transparency. When mean-reversion tends to 0: Effort converges to a finite limit; no transparency.