Post on 18-Aug-2020
transcript
Design of Information Sharing Mechanisms
Krishnamurthy Iyer
ORIE, Cornell University
Oct 2018, IMABased on joint work with David Lingenbrink, Cornell University
Motivation
Many instances in the service economy where users’ payo� fromusing 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 thesystem operator is better informed.
Due to this informational asymmetry, the system operator mayshare information with its users.
Motivation
Many instances in the service economy where users’ payo� fromusing 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 thesystem operator is better informed.
Due to this informational asymmetry, the system operator mayshare information with its users.
Motivation
Many instances in the service economy where users’ payo� fromusing 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 thesystem operator is better informed.
Due to this informational asymmetry, the system operator mayshare information with its users.
Motivation
Many instances in the service economy where users’ payo� fromusing 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 thesystem operator is better informed.
Due to this informational asymmetry, the system operator mayshare information with its users.
Information design: How should an operator share information withpotential users to in�uence their behavior?
Motivation: Uber vs. taxi
Suppose you land at the MSP airport. You don’t want to wait toolong for a ride to your destination, so you decide between usingUber or a taxi service.
• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver isavailable.
Before you make your choice, Uber provides estimates of yourwaiting time.
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Motivation: Uber vs. taxi
Suppose you land at the MSP airport. You don’t want to wait toolong for a ride to your destination, so you decide between usingUber or a taxi service.
• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver isavailable.
Before you make your choice, Uber provides estimates of yourwaiting time.
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Motivation: Uber vs. taxi
Suppose you land at the MSP airport. You don’t want to wait toolong for a ride to your destination, so you decide between usingUber or a taxi service.
• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver isavailable.
Before you make your choice, Uber provides estimates of yourwaiting time.
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Motivation: Uber vs. taxi
Suppose you land at the MSP airport. You don’t want to wait toolong for a ride to your destination, so you decide between usingUber or a taxi service.
• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver isavailable.
Before you make your choice, Uber provides estimates of yourwaiting time.
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Motivation: Uber vs. taxi
Suppose you land at the MSP airport. You don’t want to wait toolong for a ride to your destination, so you decide between usingUber or a taxi service.
• Taxi rides cost more, but are readily available.
• Uber rides cost less, but you have to wait till a driver isavailable.
Before you make your choice, Uber provides estimates of yourwaiting time.
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Motivation: Uber vs. taxi
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Ideas:
• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some �xed probability.
• Partial information?
In this talkHow can a service provider disclose information to increaseparticipation in a queue, thereby increasing revenue?
Motivation: Uber vs. taxi
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Ideas:
• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some �xed probability.
• Partial information?
In this talkHow can a service provider disclose information to increaseparticipation in a queue, thereby increasing revenue?
Motivation: Uber vs. taxi
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Ideas:
• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some �xed probability.
• Partial information?
In this talkHow can a service provider disclose information to increaseparticipation in a queue, thereby increasing revenue?
Motivation: Uber vs. taxi
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Ideas:
• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some �xed probability.
• Partial information?
In this talkHow can a service provider disclose information to increaseparticipation in a queue, thereby increasing revenue?
Motivation: Uber vs. taxi
Uber’s ProblemWhat information can Uber share with you to convince you to waitfor a ride?
Ideas:
• Fully reveal: Customers will only join when wait is short.
• Tell nothing: Customers will join with some �xed probability.
• Partial information?
In this talkHow can a service provider disclose information to increaseparticipation in a queue, thereby increasing revenue?
Literature review
Bayesian persuasion: Optimal information sharing betweenprincipal and a set of uninformed agents.
• Rayo and Segal [2010], Kamenica and Gentzkow [2011]• Mansour et al. [2015], Bergemann and Morris [2017], Dughmiand 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 formore general objectives for service provider.
• Simhon et al. [2016], Guo and Zipkin [2007]: speci�c types ofinformation.
Literature review
Bayesian persuasion: Optimal information sharing betweenprincipal and a set of uninformed agents.
• Rayo and Segal [2010], Kamenica and Gentzkow [2011]• Mansour et al. [2015], Bergemann and Morris [2017], Dughmiand 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 formore general objectives for service provider.
• Simhon et al. [2016], Guo and Zipkin [2007]: speci�c types ofinformation.
Model
Model: Queue
Customers arrive according to a Poisson process with rate λ.
The queue is unobservable to arriving customers who must choosewhether to join the queue upon arrival.
Joining customers pay price p ≥ 0 and wait in a FIFO queue toobtain service from a single server.
Service time is exponentially distributed with mean 1.
Model: Queue
Customers arrive according to a Poisson process with rate λ.
The queue is unobservable to arriving customers who must choosewhether to join the queue upon arrival.
Joining customers pay price p ≥ 0 and wait in a FIFO queue toobtain service from a single server.
Service time is exponentially distributed with mean 1.
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 expectedutility
h(X, p) = u(X)− p,
• u(X) is the expected utility the customer gets from the service,
• p is the �xed 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 expectedutility
h(X, p) = u(X)− p,
• u(X) is the expected utility the customer gets from the service,
• p is the �xed 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 expectedutility
h(X, p) = u(X)− p,
• u(X) is the expected utility the customer gets from the service,
• p is the �xed price for the service.
Model: Assumptions
0 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 expectedutility
h(X, p) = u(X)− p,
• u(X) is the expected utility the customer gets from the service,
• p is the �xed price for the service.
Queue length X is unknown: customers maintain beliefs, andmaximize expected 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 expectedutility
h(X, p) = u(X)− p,
• u(X) is the expected utility the customer gets from the service,
• p is the �xed price for the service.
Queue length X is unknown: customers maintain beliefs, andmaximize expected utility.
Model: Service provider
The service provider aims to maximize expected revenue bychoosing
• a �xed price p,
• a signaling mechanism.
Model: Service provider
The service provider aims to maximize expected revenue bychoosing
• a �xed price p,
• a signaling mechanism.
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 .
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 .
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 .
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, ∅) = 12 .
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 thecustomers:
1. Optimality: Given her prior belief (about queue state) andother customers’ strategies, each customer acts optimally.
2. Consistency: A customer’s prior belief is consistent with thequeue dynamics induced by the strategies.
Bayesian Persuasion in Dynamic SettingThe choice of the signaling mechanism a�ects not only whatinformation a customer receives, but also her prior belief.
Equilibrium
Each signaling mechanism induces an equilibrium among thecustomers:
1. Optimality: Given her prior belief (about queue state) andother customers’ strategies, each customer acts optimally.
2. Consistency: A customer’s prior belief is consistent with thequeue dynamics induced by the strategies.
Bayesian Persuasion in Dynamic SettingThe choice of the signaling mechanism a�ects not only whatinformation a customer receives, but also her prior belief.
Dynamics
A customer strategy is a function f : S → [0, 1] such that given asignal 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)
0 1 2 . . . n n+1 . . .
λq0 λq1 λqn
111
The queue forms a birth-death chain.
Let π denote its steady state distribution and X∞ ∼ π.
Dynamics
A customer strategy is a function f : S → [0, 1] such that given asignal 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)
0 1 2 . . . n n+1 . . .
λq0 λq1 λqn
111
The queue forms a birth-death chain.
Let π denote its steady state distribution and X∞ ∼ π.
Dynamics
A customer strategy is a function f : S → [0, 1] such that given asignal 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)
0 1 2 . . . n n+1 . . .
λq0 λq1 λqn
111
The queue forms a birth-death chain.
Let π denote its steady state distribution and X∞ ∼ π.
Dynamics
A customer strategy is a function f : S → [0, 1] such that given asignal 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)
0 1 2 . . . n n+1 . . .
λq0 λq1 λqn
111
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 thatmaximizes a customer’s expected utility (under steady state)assuming all other customers follow f :
f(s) =
{1 if E[h(X∞, p)|s] > 0;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.
0 1 2 3 4 5
λ
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Rev
enu
e
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
Rev
enu
e
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 PricingHow should a service provider choose a price p and a signalingmechanism (S, σ) to maximize her expected revenue R(σ, f, p) inthe resulting equilibrium?
Signaling under Fixed Price
Characterizing the optimal mechanism
LemmaIt su�ces to consider signaling mechanisms (S, σ) whereS = {0, 1} and the customer equilibrium f is obedient: f(s) = s.
Proof: Standard revelation principle argument.
Characterizing the optimal mechanism
LemmaIt su�ces to consider signaling mechanisms (S, σ) whereS = {0, 1} and the customer equilibrium f is obedient: f(s) = s.
Proof: Standard revelation principle argument.
Optimal signaling mechanism
TheoremFor any �xed-price p > 0, there exists an optimal signalingmechanism with a threshold structure.
Threshold mechanism:
· · ·
join leave
N∗
(randomize)0 1 2 3
Here, N∗ ≥Mp (max queue size under full information).
Optimal signaling mechanism
TheoremFor any �xed-price p > 0, there exists an optimal signalingmechanism with a threshold structure.
Threshold mechanism:
· · ·
join leave
N∗
(randomize)0 1 2 3
Here, N∗ ≥Mp (max queue size under full information).
Optimal signaling mechanism
TheoremFor any �xed-price p > 0, there exists an optimal signalingmechanism with a threshold structure.
Threshold mechanism:
· · ·
join leave
N∗
(randomize)0 1 2 3
Here, N∗ ≥Mp (max queue size under full information).
Intuition
Suppose the signaling mechanism fully revealed the queue lengthX :
· · ·
joinX ≤ Mp
Mp0 1 2 3
Intuition
Suppose the signaling mechanism fully revealed the queue lengthX :
· · ·
join
X ≤ Mp
Mp0 1 2 3
Intuition
Suppose the signaling mechanism fully revealed the queue lengthX :
· · ·
join
X ≤ Mp
Mp0 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 lengthX :
· · ·
join
X ≤ Mp
Mp0 1 2 3
Joining a queue at state k < Mp gets utility u(k)− p ≥ 0.
Joining a queue at stateMp gets u(Mp)− p < 0.
Intuition
Instead, suppose the signaling mechanism only revealed whetherthe queue length satis�es X ≤Mp or not.
· · ·
join
X ≤ Mp
Mp0 1 2 3
Intuition
Instead, suppose the signaling mechanism only revealed whetherthe queue length satis�es X ≤Mp or not.
· · ·
join
X ≤ Mp
Mp0 1 2 3
Upon receiving the signal X ≤Mp, the utility for joining is a convexcombination of u(k) for k ≤Mp.
Intuition
Instead, suppose the signaling mechanism only revealed whetherthe queue length satis�es X ≤Mp or not.
· · ·
join
X ≤ Mp
Mp0 1 2 3
Upon receiving the signal X ≤Mp, the utility for joining is a convexcombination of u(k) for k ≤Mp.
If positive, then customer will join, even if the queue length isMp!
Proof sketch
maxσ
Eσ [R(σ, f, p)]
s.t., Eσ[h(X∞, p)|s = 1] ≥ 0,
Eσ[h(X∞, p)|s = 0] ≤ 0
maxσ
∞∑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
maxσ
Eσ [R(σ, f, p)]
s.t., Eσ[h(X∞, p)|s = 1] ≥ 0,
Eσ[h(X∞, p)|s = 0] ≤ 0
maxσ
∞∑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 stationarydistribution.
Proof sketch
maxσ
Eσ [R(σ, f, p)]
s.t., Eσ[h(X∞, p)|s = 1] ≥ 0,
Eσ[h(X∞, p)|s = 0] ≤ 0
maxσ
∞∑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 stationarydistribution.
Proof sketch
maxσ
∞∑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
maxσ
∞∑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 canoptimize over the stationary distribution π.
Proof sketch
maxπ
∞∑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 canoptimize over the stationary distribution π.
Proof sketch
maxπ
∞∑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 (in�nite) LP in {πn : n ≥ 0}.
Proof sketch
maxπ
∞∑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 (in�nite) LP in {πn : n ≥ 0}. We can perturb any feasiblesolution to this LP to a threshold mechanism without decreasingthe revenue.
Proof sketch
maxπ
∞∑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 �rst show that for queue lengths less thanMp, the optimalmechanism must tell customers to join.
Proof sketch
Next, we consider an feasible solution π where πn = λnπ0 forn ≤ N , 0 < πN+1 ≤ πN < λπN−1.
λn
0 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 forn ≤ N , 0 < πN+1 ≤ πN < λπN−1.
λn
0 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
TheoremFor any �xed-price p > 0, there exists an optimal signalingmechanism with a threshold structure.
For linear waiting costs: closed-form for the threshold.
Optimal signaling mechanism
TheoremFor any �xed-price p > 0, there exists an optimal signalingmechanism with a threshold structure.
For linear waiting costs: closed-form for the threshold.
Revenue comparison
u(X) = 1− c · (X + 1)
0 1 2 3 4 5
λ
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Rev
enu
e
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
Rev
enu
e
Full
No-Info
(b) λ = 0.7 and p = 0.3.
Revenue comparison
u(X) = 1− c · (X + 1)
0 1 2 3 4 5
λ
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Rev
enu
e
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
Rev
enu
e
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 �xedprice, in addition to subsequent optimal signaling.
Benchmark: Optimal state-dependent prices in a fully-observablequeue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,2001].
Here, the service provider sets prices p(n) for each queue length n:
• a cuto� k such that p(n) =∞, for n ≥ k.• For n < k, extract entire customer surplus: p(n) = u(n).
QuestionHow does our revenue compare with that of the optimalstate-dependent pricing mechanism?
Optimal price
We consider the setting where we can choose the optimal �xedprice, in addition to subsequent optimal signaling.
Benchmark: Optimal state-dependent prices in a fully-observablequeue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,2001].
Here, the service provider sets prices p(n) for each queue length n:
• a cuto� k such that p(n) =∞, for n ≥ k.• For n < k, extract entire customer surplus: p(n) = u(n).
QuestionHow does our revenue compare with that of the optimalstate-dependent pricing mechanism?
Optimal price
We consider the setting where we can choose the optimal �xedprice, in addition to subsequent optimal signaling.
Benchmark: Optimal state-dependent prices in a fully-observablequeue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,2001].
Here, the service provider sets prices p(n) for each queue length n:
• a cuto� k such that p(n) =∞, for n ≥ k.
• For n < k, extract entire customer surplus: p(n) = u(n).
QuestionHow does our revenue compare with that of the optimalstate-dependent pricing mechanism?
Optimal price
We consider the setting where we can choose the optimal �xedprice, in addition to subsequent optimal signaling.
Benchmark: Optimal state-dependent prices in a fully-observablequeue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,2001].
Here, the service provider sets prices p(n) for each queue length n:
• a cuto� k such that p(n) =∞, for n ≥ k.• For n < k, extract entire customer surplus: p(n) = u(n).
QuestionHow does our revenue compare with that of the optimalstate-dependent pricing mechanism?
Optimal price
We consider the setting where we can choose the optimal �xedprice, in addition to subsequent optimal signaling.
Benchmark: Optimal state-dependent prices in a fully-observablequeue. [Naor, 1969, Edelson and Hilderbrand, 1975, Chen and Frank,2001].
Here, the service provider sets prices p(n) for each queue length n:
• a cuto� k such that p(n) =∞, for n ≥ k.• For n < k, extract entire customer surplus: p(n) = u(n).
QuestionHow does our revenue compare with that of the optimalstate-dependent pricing mechanism?
Optimal revenue
TheoremRevenue under optimal signaling = Revenue under optimal statedependent prices.
Optimal revenue
TheoremRevenue under optimal signaling = Revenue under optimal statedependent prices.
Proof: Under optimal state-dependent prices, the expected revenueis
E [I{X∞ < k}u(X∞)] .
Optimal revenue
TheoremRevenue under optimal signaling = Revenue under optimal statedependent prices.
Proof: Under optimal state-dependent prices, the expected revenueis
E [I{X∞ < k}u(X∞)] .
We show that the signaling mechanism with threshold equal tocuto� k, and with �xed price
p∗ = E [u(X∞)|X∞ < k]
achieves the same revenue.
Optimal revenue
TheoremRevenue under optimal signaling = Revenue under optimal statedependent prices.
Optimal revenue
TheoremRevenue under optimal signaling = Revenue under optimal statedependent prices.
In settings where it is infeasible to charge state-dependent prices,optimal signaling can be e�ective in raising revenue.
Extensions
Extensions
State-dependent service/arrival rates
Other service disciplines
Exogeneous abandonment
Rational abandonment
• Threshold mechanisms do as well; unknown whether optimal.
Extensions
State-dependent service/arrival rates
Other service disciplines
Exogeneous abandonment
Rational abandonment
• Threshold mechanisms do as well; unknown whether optimal.
Extensions
State-dependent service/arrival rates
Other service disciplines
Exogeneous abandonment
Rational abandonment
• Threshold mechanisms do as well; unknown whether optimal.
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 utilityhi(n, pi) = ui(n)− pi.
TheoremIf all customers types pay the same price, there exists an optimalsignaling mechanism with a threshold structure.
Remark: When prices are di�erent for di�erent types, the optimalsignaling mechanism need not have threshold structure.
Heterogeneous customers
Suppose customers come from one of K types.
Type i customers arrive at rate λi, are charged pi, and have utilityhi(n, pi) = ui(n)− pi.
TheoremIf all customers types pay the same price, there exists an optimalsignaling mechanism with a threshold structure.
Remark: When prices are di�erent for di�erent types, the optimalsignaling mechanism need not have threshold structure.
Heterogeneous customers
Suppose customers come from one of K types.
Type i customers arrive at rate λi, are charged pi, and have utilityhi(n, pi) = ui(n)− pi.
TheoremIf all customers types pay the same price, there exists an optimalsignaling mechanism with a threshold structure.
Remark: When prices are di�erent for di�erent types, the optimalsignaling 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;
0 n ≥ 2.
, σ(n, 2, 1) =
0 n = 0;
1/10 n = 1;
0 n ≥ 2.
Risk aversion
Customers o�en perceive uncertain wait-times to be longer thande�nite wait-times (Maister 2005).
• Variance of the waiting time plays a role in a customer’sdecision to join.
Mean-Variance model: Suppose customers will join only if
E[T ] + β ·√Var[T ] ≤ γ,
where T is the waiting time.
QuestionWhat is the optimal signaling mechanism under themean-variance model?
Risk aversion
Customers o�en perceive uncertain wait-times to be longer thande�nite wait-times (Maister 2005).
• Variance of the waiting time plays a role in a customer’sdecision to join.
Mean-Variance model: Suppose customers will join only if
E[T ] + β ·√
Var[T ] ≤ γ,
where T is the waiting time.
QuestionWhat is the optimal signaling mechanism under themean-variance model?
Risk aversion
Customers o�en perceive uncertain wait-times to be longer thande�nite wait-times (Maister 2005).
• Variance of the waiting time plays a role in a customer’sdecision to join.
Mean-Variance model: Suppose customers will join only if
E[T ] + β ·√
Var[T ] ≤ γ,
where T is the waiting time.
QuestionWhat is the optimal signaling mechanism under themean-variance model?
Risk aversion
Main di�culty: Revelation principle no longer holds. Cannot reducethe space of signaling mechanisms to those with binary signals.
Restricted Revelation PrincipleIt su�ces to consider signaling mechanisms where customers’optimal strategy involves not joining for at most one signal, andjoining for all others.
=⇒ an iterative approach to optimize information sharing
Risk aversion
Main di�culty: Revelation principle no longer holds. Cannot reducethe space of signaling mechanisms to those with binary signals.
Restricted Revelation PrincipleIt su�ces to consider signaling mechanisms where customers’optimal strategy involves not joining for at most one signal, andjoining for all others.
=⇒ an iterative approach to optimize information sharing
Risk aversion
Main di�culty: Revelation principle no longer holds. Cannot reducethe space of signaling mechanisms to those with binary signals.
Restricted Revelation PrincipleIt su�ces to consider signaling mechanisms where customers’optimal strategy involves not joining for at most one signal, andjoining 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
Th
rou
ghp
ut
Optimal Threshold
Sandwich
Threshold = join | leaveSandwich = risky-join | safe-join | risky-join | leave
Conclusion
Conclusion
We study Bayesian persuasion in a dynamic queueing setting.
• The optimal signaling mechanism under a �xed price has athreshold structure.
• Under optimal �xed price, optimal signaling achieves theoptimal revenue under state-dependent prices.
Information Design exploits the information asymmetry between aplatform and its users to improve design objectives.
• An important tool in a platform’s arsenal.
Conclusion
We study Bayesian persuasion in a dynamic queueing setting.
• The optimal signaling mechanism under a �xed price has athreshold structure.
• Under optimal �xed price, optimal signaling achieves theoptimal revenue under state-dependent prices.
Information Design exploits the information asymmetry between aplatform and its users to improve design objectives.
• An important tool in a platform’s arsenal.
Conclusion
We study Bayesian persuasion in a dynamic queueing setting.
• The optimal signaling mechanism under a �xed price has athreshold structure.
• Under optimal �xed price, optimal signaling achieves theoptimal revenue under state-dependent prices.
Information Design exploits the information asymmetry between aplatform 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 eachp ∈ [0, 1− c], the threshold mechanism σx is optimal forx = N∗ + q∗, where
N∗ =
⌊2(1−p)c − 1
⌋if λ = 1;
∞ if λ ≤ 1− c1−p ;⌊
1log(λ) (Wi (−κe−κ) + κ)
⌋otherwise,
with κ =(
1−pc −
11−λ
)log(λ) and where i = 0 when λ > 1 and i = −1
when 1− c1−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).