Design of Information Sharing Mechanisms Krishnamurthy Iyer ORIE, - - PowerPoint PPT Presentation

Design of Information Sharing Mechanisms

Krishnamurthy Iyer

ORIE, Cornell University Oct 2018, IMA Based on joint work with David Lingenbrink, Cornell University

Motivation

Many instances in the service economy where users’ payoff from using the service depends on the state of the system.

• congestion, resource availability, waiting times, etc.

Motivation

Many instances in the service economy where users’ payoff from using the service depends on the state of the system.

• congestion, resource availability, waiting times, etc.

Typically, such system states are unknown to the user, but the system operator is better informed.

Motivation

Many instances in the service economy where users’ payoff from using the service depends on the state of the system.

• congestion, resource availability, waiting times, etc.

Typically, such system states are unknown to the user, but the system operator is better informed. Due to this informational asymmetry, the system operator may share information with its users.

Motivation

Many instances in the service economy where users’ payoff from using the service depends on the state of the system.

• congestion, resource availability, waiting times, etc.

Typically, such system states are unknown to the user, but the system operator is better informed. Due to this informational asymmetry, the system operator may share information with its users. Information design: How should an operator share information with potential users to influence their behavior?

Motivation: Uber vs. taxi

Suppose you land at the MSP airport. You don’t want to wait too long for a ride to your destination, so you decide between using Uber or a taxi service.

Motivation: Uber vs. taxi

Suppose you land at the MSP airport. You don’t want to wait too long for a ride to your destination, so you decide between using Uber or a taxi service.

• Taxi rides cost more, but are readily available.

Motivation: Uber vs. taxi

Suppose you land at the MSP airport. You don’t want to wait too long for a ride to your destination, so you decide between using Uber or a taxi service.

• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver is

available.

Motivation: Uber vs. taxi

Suppose you land at the MSP airport. You don’t want to wait too long for a ride to your destination, so you decide between using Uber or a taxi service.

• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver is

available. Before you make your choice, Uber provides estimates of your waiting time.

Motivation: Uber vs. taxi

Suppose you land at the MSP airport. You don’t want to wait too long for a ride to your destination, so you decide between using Uber or a taxi service.

• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver is

available. Before you make your choice, Uber provides estimates of your waiting time. Uber’s Problem What information can Uber share with you to convince you to wait for a ride?

Motivation: Uber vs. taxi

Uber’s Problem What information can Uber share with you to convince you to wait for a ride? Ideas:

Motivation: Uber vs. taxi

Uber’s Problem What information can Uber share with you to convince you to wait for a ride? Ideas:

• Fully reveal: Customers will only join when wait is short.

Motivation: Uber vs. taxi

Uber’s Problem What information can Uber share with you to convince you to wait for a ride? Ideas:

• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some fixed probability.

Motivation: Uber vs. taxi

Uber’s Problem What information can Uber share with you to convince you to wait for a ride? Ideas:

• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some fixed probability.
• Partial information?

Motivation: Uber vs. taxi

Uber’s Problem What information can Uber share with you to convince you to wait for a ride? Ideas:

• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some fixed probability.
• Partial information?

In this talk How can a service provider disclose information to increase participation in a queue, thereby increasing revenue?

Literature review

Bayesian persuasion: Optimal information sharing between principal and a set of uninformed agents.

• Rayo and Segal [2010], Kamenica and Gentzkow [2011]
• Mansour et al. [2015], Bergemann and Morris [2017], Dughmi

and Xu [2016], Papanastasiou et al. [2017]

Literature review

Bayesian persuasion: Optimal information sharing between principal and a set of uninformed agents.

• Rayo and Segal [2010], Kamenica and Gentzkow [2011]
• Mansour et al. [2015], Bergemann and Morris [2017], Dughmi

and Xu [2016], Papanastasiou et al. [2017] Strategic behavior in queues: Naor [1969], Edelson and Hilderbrand [1975], Chen and Frank [2001], Hassin et al. [2003], Hassin [2016]

• Allon et al. [2011] study cheap talk in unobservable queues for

more general objectives for service provider.

• Simhon et al. [2016], Guo and Zipkin [2007]: specific types of

information.

Model

Model: Queue

Customers arrive according to a Poisson process with rate λ. The queue is unobservable to arriving customers who must choose whether to join the queue upon arrival.

Model: Queue

Customers arrive according to a Poisson process with rate λ. The queue is unobservable to arriving customers who must choose whether to join the queue upon arrival. Joining customers pay price p ≥ 0 and wait in a FIFO queue to

• btain service from a single server.

Service time is exponentially distributed with mean 1.

Model: Customers

Customers are homogeneous, delay sensitive, and Bayesian.

Model: Customers

Customers are homogeneous, delay sensitive, and Bayesian. Customers who do not join the queue receive zero utility.

Model: Customers

Customers are homogeneous, delay sensitive, and Bayesian. Customers who do not join the queue receive zero utility. When the queue has length X, joining customers have expected utility h(X, p) = u(X) − p,

• u(X) is the expected utility the customer gets from the service,
• p is the fixed price for the service.

Model: Assumptions

1 2 3 4 5 6 7 8

x

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

u(x) Customers ...

• ...don’t enjoy waiting.

u(·) non-increasing.

• ...would join an empty queue.

u(0) − p ≥ 0

• ...would not join long queues.

∃ Mp s.t. u(Mp) − p < 0.

Model: Customers

Customers are homogeneous, delay sensitive, and Bayesian. Customers who do not join the queue receive zero utility. When the queue has length X, joining customers have expected utility h(X, p) = u(X) − p,

• u(X) is the expected utility the customer gets from the service,
• p is the fixed price for the service.

Model: Customers

Customers are homogeneous, delay sensitive, and Bayesian. Customers who do not join the queue receive zero utility. When the queue has length X, joining customers have expected utility h(X, p) = u(X) − p,

• u(X) is the expected utility the customer gets from the service,
• p is the fixed price for the service.

Queue length X is unknown: customers maintain beliefs, and maximize expected utility.

Model: Service provider

The service provider aims to maximize expected revenue by choosing

• a fixed price p,

Model: Service provider

The service provider aims to maximize expected revenue by choosing

• a fixed price p,
• a signaling mechanism.

Model: Signaling mechanism

Formally, a signaling mechanism is

• a set of possible signals S and

Model: Signaling mechanism

Formally, a signaling mechanism is

• a set of possible signals S and
• a mapping from queue lengths to distributions over S, σ:

σ(n, s) = P(signal s | queue length = n).

Model: Signaling mechanism

Formally, a signaling mechanism is

• a set of possible signals S and
• a mapping from queue lengths to distributions over S, σ:

σ(n, s) = P(signal s | queue length = n). Examples:

• No-info:

S = {∅}, σ(n, ∅) = 1

Model: Signaling mechanism

Formally, a signaling mechanism is

• a set of possible signals S and
• a mapping from queue lengths to distributions over S, σ:

σ(n, s) = P(signal s | queue length = n). Examples:

• No-info:

S = {∅}, σ(n, ∅) = 1

• Full-info:

S = N0, σ(n, n) = 1,

Model: Signaling mechanism

Formally, a signaling mechanism is

• a set of possible signals S and
• a mapping from queue lengths to distributions over S, σ:

σ(n, s) = P(signal s | queue length = n). Examples:

• No-info:

S = {∅}, σ(n, ∅) = 1

• Full-info:

S = N0, σ(n, n) = 1,

• Random full-info:

S = N0 ∪ {∅}, σ(n, n) = σ(n, ∅) = 1

2.

Equilibrium

Equilibrium

Each signaling mechanism induces an equilibrium among the customers:

• 1. Optimality: Given her prior belief (about queue state) and
• ther customers’ strategies, each customer acts optimally.
• 2. Consistency: A customer’s prior belief is consistent with the

queue dynamics induced by the strategies.

Equilibrium

Each signaling mechanism induces an equilibrium among the customers:

• 1. Optimality: Given her prior belief (about queue state) and
• ther customers’ strategies, each customer acts optimally.
• 2. Consistency: A customer’s prior belief is consistent with the

queue dynamics induced by the strategies. Bayesian Persuasion in Dynamic Setting The choice of the signaling mechanism affects not only what information a customer receives, but also her prior belief.

Dynamics

A customer strategy is a function f : S → [0, 1] such that given a signal s, a customer joins with probability f(s).

Dynamics

A customer strategy is a function f : S → [0, 1] such that given a signal s, a customer joins with probability f(s). qn: probability a customer joins given there are n customers, qn =

• s∈S

σ(n, s)f(s)

Dynamics

A customer strategy is a function f : S → [0, 1] such that given a signal s, a customer joins with probability f(s). qn: probability a customer joins given there are n customers, qn =

• s∈S

σ(n, s)f(s) 1 2 . . . n n+1 . . . λq0 λq1 λqn 1 1 1 The queue forms a birth-death chain.

Dynamics

A customer strategy is a function f : S → [0, 1] such that given a signal s, a customer joins with probability f(s). qn: probability a customer joins given there are n customers, qn =

• s∈S

σ(n, s)f(s) 1 2 . . . n n+1 . . . λq0 λq1 λqn 1 1 1 The queue forms a birth-death chain. Let π denote its steady state distribution and X∞ ∼ π.

Customer equilibrium

A symmetric equilibrium among customers is a strategy f that maximizes a customer’s expected utility (under steady state) assuming all other customers follow f: f(s) =

• 1

if E[h(X∞, p)|s] > 0; if E[h(X∞, p)|s] < 0.

Service provider’s goal

The service provider’s revenue is given by R(σ, f, p) = p · 1 · (1 − π0) = p

∞

• i=1

πi.

Service provider’s goal

The service provider’s revenue is given by R(σ, f, p) = p · 1 · (1 − π0) = p

∞

• i=1

πi.

1 2 3 4 5 λ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Revenue

Full No-Info

(a) c = 0.2 and p = 0.3.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c 0.00 0.05 0.10 0.15 0.20 0.25 Revenue

Full No-Info

(b) λ = 0.7 and p = 0.3.

u(X) = 1 − c · (X + 1)

Service provider’s goal

The service provider’s revenue is given by R(σ, f, p) = p · 1 · (1 − π0) = p

∞

• i=1

πi. Problem: Optimal Signaling and Pricing How should a service provider choose a price p and a signaling mechanism (S, σ) to maximize her expected revenue R(σ, f, p) in the resulting equilibrium?

Signaling under Fixed Price

Characterizing the optimal mechanism

Lemma It suffices to consider signaling mechanisms (S, σ) where S = {0, 1} and the customer equilibrium f is obedient: f(s) = s.

Characterizing the optimal mechanism

Lemma It suffices to consider signaling mechanisms (S, σ) where S = {0, 1} and the customer equilibrium f is obedient: f(s) = s. Proof: Standard revelation principle argument.

Optimal signaling mechanism

Theorem For any fixed-price p > 0, there exists an optimal signaling mechanism with a threshold structure.

Optimal signaling mechanism

Theorem For any fixed-price p > 0, there exists an optimal signaling mechanism with a threshold structure. Threshold mechanism: · · ·

join leave

N ∗

(randomize)

1 2 3

Optimal signaling mechanism

Theorem For any fixed-price p > 0, there exists an optimal signaling mechanism with a threshold structure. Threshold mechanism: · · ·

join leave

N ∗

(randomize)

1 2 3 Here, N ∗ ≥ Mp (max queue size under full information).

Intuition

Suppose the signaling mechanism fully revealed the queue length X: · · · Mp 1 2 3

Intuition

Suppose the signaling mechanism fully revealed the queue length X: · · ·

join

Mp 1 2 3

Intuition

Suppose the signaling mechanism fully revealed the queue length X: · · ·

join

Mp 1 2 3 Joining a queue at state k < Mp gets utility u(k) − p ≥ 0.

Intuition

Suppose the signaling mechanism fully revealed the queue length X: · · ·

join

Mp 1 2 3 Joining a queue at state k < Mp gets utility u(k) − p ≥ 0. Joining a queue at state Mp gets u(Mp) − p < 0.

Intuition

Instead, suppose the signaling mechanism only revealed whether the queue length satisfies X ≤ Mp or not. · · ·

X ≤ Mp

Mp 1 2 3

Intuition

Instead, suppose the signaling mechanism only revealed whether the queue length satisfies X ≤ Mp or not. · · ·

X ≤ Mp

Mp 1 2 3 Upon receiving the signal X ≤ Mp, the utility for joining is a convex combination of u(k) for k ≤ Mp.

Intuition

Instead, suppose the signaling mechanism only revealed whether the queue length satisfies X ≤ Mp or not. · · ·

X ≤ Mp

Mp 1 2 3 Upon receiving the signal X ≤ Mp, the utility for joining is a convex combination of u(k) for k ≤ Mp. If positive, then customer will join, even if the queue length is Mp!

Proof sketch

max

σ

Eσ [R(σ, f, p)] s.t., Eσ[h(X∞, p)|s = 1] ≥ 0, Eσ[h(X∞, p)|s = 0] ≤ 0

Proof sketch

max

σ

Eσ [R(σ, f, p)] s.t., Eσ[h(X∞, p)|s = 1] ≥ 0, Eσ[h(X∞, p)|s = 0] ≤ 0 We can write the expectations in terms of π, the stationary distribution.

Proof sketch

max

σ

Eσ [R(σ, f, p)] s.t., Eσ[h(X∞, p)|s = 1] ≥ 0, Eσ[h(X∞, p)|s = 0] ≤ 0 max

σ ∞

• n=1

πn s.t.,

∞

• n=1

h(n − 1, p)πn ≥ 0

∞

• n=0

h(n, p) (λπn − πn+1) ≤ 0 λπn − πn+1 ≥ 0 πT1 = 1, π ≥ 0 ∀n We can write the expectations in terms of π, the stationary distribution.

Proof sketch

max

σ ∞

• n=1

πn s.t.,

∞

• n=1

h(n − 1, p)πn ≥ 0

∞

• n=0

h(n, p) (λπn − πn+1) ≤ 0 λπn − πn+1 ≥ 0 πT1 = 1, π ≥ 0 ∀n

Proof sketch

max

σ ∞

• n=1

πn s.t.,

∞

• n=1

h(n − 1, p)πn ≥ 0

∞

• n=0

h(n, p) (λπn − πn+1) ≤ 0 λπn − πn+1 ≥ 0 πT1 = 1, π ≥ 0 ∀n Instead of optimizing over the signaling mechanism, we can

• ptimize over the stationary distribution π.

Proof sketch

max

π ∞

• n=1

πn s.t.,

∞

• n=1

h(n − 1, p)πn ≥ 0

∞

• n=0

h(n, p) (λπn − πn+1) ≤ 0 λπn − πn+1 ≥ 0 πT1 = 1, π ≥ 0 ∀n Instead of optimizing over the signaling mechanism, we can

• ptimize over the stationary distribution π.

Proof sketch

max

π ∞

• n=1

πn s.t.,

∞

• n=1

h(n − 1, p)πn ≥ 0

∞

• n=0

h(n, p) (λπn − πn+1) ≤ 0 λπn − πn+1 ≥ 0 πT1 = 1, π ≥ 0 ∀n We have an (infinite) LP in {πn : n ≥ 0}.

Proof sketch

max

π ∞

• n=1

πn s.t.,

∞

• n=1

h(n − 1, p)πn ≥ 0

∞

• n=0

h(n, p) (λπn − πn+1) ≤ 0 λπn − πn+1 ≥ 0 πT1 = 1, π ≥ 0 ∀n We have an (infinite) LP in {πn : n ≥ 0}. We can perturb any feasible solution to this LP to a threshold mechanism without decreasing the revenue.

Proof sketch

max

π ∞

• n=1

πn s.t.,

∞

• n=1

h(n − 1, p)πn ≥ 0

∞

• n=0

h(n, p) (λπn − πn+1) ≤ 0 λπn − πn+1 ≥ 0 πT1 = 1, π ≥ 0 ∀n We first show that for queue lengths less than Mp, the optimal mechanism must tell customers to join.

Proof sketch

Next, we consider an feasible solution π where πn = λnπ0 for n ≤ N, 0 < πN+1 ≤ πN < λπN−1. λn

5 10 15 0.2 0.4 0.6 0.8 1.0

Proof sketch

Next, we consider an feasible solution π where πn = λnπ0 for n ≤ N, 0 < πN+1 ≤ πN < λπN−1. λn

5 10 15 0.2 0.4 0.6 0.8 1.0

We construct a better solution by increasing πN+1 by β

n>N+1 πn

and scaling down πn by (1 − β) for n > N + 1.

Optimal signaling mechanism

Theorem For any fixed-price p > 0, there exists an optimal signaling mechanism with a threshold structure.

Optimal signaling mechanism

Theorem For any fixed-price p > 0, there exists an optimal signaling mechanism with a threshold structure. For linear waiting costs: closed-form for the threshold.

Revenue comparison

u(X) = 1 − c · (X + 1)

1 2 3 4 5 λ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Revenue

Full No-Info

(a) c = 0.2 and p = 0.3.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c 0.00 0.05 0.10 0.15 0.20 0.25 Revenue

Full No-Info

(b) λ = 0.7 and p = 0.3.

Revenue comparison

u(X) = 1 − c · (X + 1)

1 2 3 4 5 λ 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Revenue

Optimal Full No-Info

(a) c = 0.2 and p = 0.3.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c 0.00 0.05 0.10 0.15 0.20 0.25 Revenue

Optimal Full No-Info

(b) λ = 0.7 and p = 0.3.

Optimal Signaling and Pricing

Optimal price

We consider the setting where we can choose the optimal fixed price, in addition to subsequent optimal signaling.

Optimal price

We consider the setting where we can choose the optimal fixed price, in addition to subsequent optimal signaling. Benchmark: Optimal state-dependent prices in a fully-observable

• queue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,

2001].

Optimal price

We consider the setting where we can choose the optimal fixed price, in addition to subsequent optimal signaling. Benchmark: Optimal state-dependent prices in a fully-observable

• queue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,

2001]. Here, the service provider sets prices p(n) for each queue length n:

• a cutoff k such that p(n) = ∞, for n ≥ k.

Optimal price

We consider the setting where we can choose the optimal fixed price, in addition to subsequent optimal signaling. Benchmark: Optimal state-dependent prices in a fully-observable

• queue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,

2001]. Here, the service provider sets prices p(n) for each queue length n:

• a cutoff k such that p(n) = ∞, for n ≥ k.
• For n < k, extract entire customer surplus: p(n) = u(n).

Optimal price

We consider the setting where we can choose the optimal fixed price, in addition to subsequent optimal signaling. Benchmark: Optimal state-dependent prices in a fully-observable

• queue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,

2001]. Here, the service provider sets prices p(n) for each queue length n:

• a cutoff k such that p(n) = ∞, for n ≥ k.
• For n < k, extract entire customer surplus: p(n) = u(n).

Question How does our revenue compare with that of the optimal state-dependent pricing mechanism?

Optimal revenue

Theorem Revenue under optimal signaling = Revenue under optimal state dependent prices.

Optimal revenue

Theorem Revenue under optimal signaling = Revenue under optimal state dependent prices. Proof: Under optimal state-dependent prices, the expected revenue is E [I{X∞ < k}u(X∞)] .

Optimal revenue

Theorem Revenue under optimal signaling = Revenue under optimal state dependent prices. Proof: Under optimal state-dependent prices, the expected revenue is E [I{X∞ < k}u(X∞)] . We show that the signaling mechanism with threshold equal to cutoff k, and with fixed price p∗ = E [u(X∞)|X∞ < k] achieves the same revenue.

Optimal revenue

Theorem Revenue under optimal signaling = Revenue under optimal state dependent prices.

Optimal revenue

Theorem Revenue under optimal signaling = Revenue under optimal state dependent prices. In settings where it is infeasible to charge state-dependent prices,

• ptimal signaling can be effective in raising revenue.

Extensions

Extensions

State-dependent service/arrival rates

Extensions

State-dependent service/arrival rates Other service disciplines

Extensions

State-dependent service/arrival rates Other service disciplines Exogeneous abandonment

Extensions

State-dependent service/arrival rates Other service disciplines Exogeneous abandonment Rational abandonment

• Threshold mechanisms do as well; unknown whether optimal.

Heterogeneous customers

Suppose customers come from one of K types. Type i customers arrive at rate λi, are charged pi, and have utility hi(n, pi) = ui(n) − pi.

Heterogeneous customers

Suppose customers come from one of K types. Type i customers arrive at rate λi, are charged pi, and have utility hi(n, pi) = ui(n) − pi. Theorem If all customers types pay the same price, there exists an optimal signaling mechanism with a threshold structure.

Heterogeneous customers

Suppose customers come from one of K types. Type i customers arrive at rate λi, are charged pi, and have utility hi(n, pi) = ui(n) − pi. Theorem If all customers types pay the same price, there exists an optimal signaling mechanism with a threshold structure. Remark: When prices are different for different types, the optimal signaling mechanism need not have threshold structure.

Heterogeneous customers

Two types: λ1 = λ2 = 1. Prices: p1 = 50, p2 = 1. u1(n) =        51 n = 0; 40 n = 1; −10000 n ≥ 2. , u2(n) =        2 n = 0; 2 n = 1; −8.5 n ≥ 2. Optimal mechanism: σ(n, 1, 1) =        1 n = 0; 1/10 n = 1; n ≥ 2. , σ(n, 2, 1) =        n = 0; 1/10 n = 1; n ≥ 2.

Risk aversion

Customers ofen perceive uncertain wait-times to be longer than definite wait-times (Maister 2005).

• Variance of the waiting time plays a role in a customer’s

decision to join.

Risk aversion

Customers ofen perceive uncertain wait-times to be longer than definite wait-times (Maister 2005).

• Variance of the waiting time plays a role in a customer’s

decision to join. Mean-Variance model: Suppose customers will join only if E[T] + β ·

• Var[T] ≤ γ,

where T is the waiting time.

Risk aversion

Customers ofen perceive uncertain wait-times to be longer than definite wait-times (Maister 2005).

• Variance of the waiting time plays a role in a customer’s

decision to join. Mean-Variance model: Suppose customers will join only if E[T] + β ·

• Var[T] ≤ γ,

where T is the waiting time. Question What is the optimal signaling mechanism under the mean-variance model?

Risk aversion

Main difficulty: Revelation principle no longer holds. Cannot reduce the space of signaling mechanisms to those with binary signals.

Risk aversion

Main difficulty: Revelation principle no longer holds. Cannot reduce the space of signaling mechanisms to those with binary signals. Restricted Revelation Principle It suffices to consider signaling mechanisms where customers’

• ptimal strategy involves not joining for at most one signal, and

joining for all others.

Risk aversion

Main difficulty: Revelation principle no longer holds. Cannot reduce the space of signaling mechanisms to those with binary signals. Restricted Revelation Principle It suffices to consider signaling mechanisms where customers’

• ptimal strategy involves not joining for at most one signal, and

joining for all others. = ⇒ an iterative approach to optimize information sharing

Risk aversion

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Risk-aversion, β 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 Throughput

Optimal Threshold Sandwich

Threshold = join | leave Sandwich = risky-join | safe-join | risky-join | leave

Conclusion

Conclusion

We study Bayesian persuasion in a dynamic queueing setting.

• The optimal signaling mechanism under a fixed price has a

threshold structure.

Conclusion

We study Bayesian persuasion in a dynamic queueing setting.

• The optimal signaling mechanism under a fixed price has a

threshold structure.

• Under optimal fixed price, optimal signaling achieves the
• ptimal revenue under state-dependent prices.

Conclusion

We study Bayesian persuasion in a dynamic queueing setting.

• The optimal signaling mechanism under a fixed price has a

threshold structure.

• Under optimal fixed price, optimal signaling achieves the
• ptimal revenue under state-dependent prices.

Information Design exploits the information asymmetry between a platform and its users to improve design objectives.

• An important tool in a platform’s arsenal.

Thank you!

(paper available at: https://ssrn.com/abstract=2964093)

References

Exact Thresholds

Suppose u(n) = 1 − c(n + 1) with c ∈ (0, 1). Then, for each p ∈ [0, 1 − c], the threshold mechanism σx is optimal for x = N ∗ + q∗, where N ∗ =       

• 2(1−p)

c

− 1

• if λ = 1;

∞ if λ ≤ 1 −

c 1−p;

• 1

log(λ) (Wi (−κe−κ) + κ)

• therwise,

with κ =

• 1−p

c

−

1 1−λ

• log(λ) and where i = 0 when λ > 1 and i = −1

when 1 −

c 1−p < λ < 1. For all values of λ < ∞, we have

q∗ =

• k<N ∗ λk(1 − p − c(k + 1))

λN ∗(c(N ∗ + 1) + p − 1) .