Equilibrium Each signaling mechanism induces an equilibrium among the customers: 1. Optimality : Given her prior belief (about queue state) and other 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 ) . q n : probability a customer joins given there are n customers, � q n = σ ( n, s ) f ( s ) s ∈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 ) . q n : probability a customer joins given there are n customers, � q n = σ ( n, s ) f ( s ) s ∈S λq 0 λq 1 λq n . . . . . . n 0 1 2 n +1 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 ) . q n : probability a customer joins given there are n customers, � q n = σ ( n, s ) f ( s ) s ∈S λq 0 λq 1 λq n . . . . . . n 0 1 2 n +1 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 : � if E [ h ( X ∞ , p ) | s ] > 0 ; 1 f ( s ) = if E [ h ( X ∞ , p ) | s ] < 0 . 0
Service provider’s goal The service provider’s revenue is given by ∞ � R ( σ, f, p ) = p · 1 · (1 − π 0 ) = p π i . i =1
Service provider’s goal The service provider’s revenue is given by ∞ � R ( σ, f, p ) = p · 1 · (1 − π 0 ) = p π i . i =1 0 . 30 0 . 25 Full Full No-Info No-Info 0 . 25 0 . 20 0 . 20 0 . 15 Revenue Revenue 0 . 15 0 . 10 0 . 10 0 . 05 0 . 05 0 . 00 0 . 00 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 1 2 3 4 5 λ c (b) λ = 0 . 7 and p = 0 . 3 . (a) c = 0 . 2 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 − π 0 ) = p π i . i =1 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 · · · 0 1 2 3 N ∗ (randomize)
Optimal signaling mechanism Theorem For any fixed-price p > 0 , there exists an optimal signaling mechanism with a threshold structure. Threshold mechanism: join leave · · · 0 1 2 3 N ∗ (randomize) Here, N ∗ ≥ M p (max queue size under full information).
Intuition Suppose the signaling mechanism fully revealed the queue length X : · · · 0 1 2 3 M p
Intuition Suppose the signaling mechanism fully revealed the queue length X : join · · · 0 1 2 3 M p
Intuition Suppose the signaling mechanism fully revealed the queue length X : join · · · 0 1 2 3 M p Joining a queue at state k < M p gets utility u ( k ) − p ≥ 0 .
Intuition Suppose the signaling mechanism fully revealed the queue length X : join · · · 0 1 2 3 M p Joining a queue at state k < M p gets utility u ( k ) − p ≥ 0 . Joining a queue at state M p gets u ( M p ) − p < 0 .
Intuition Instead, suppose the signaling mechanism only revealed whether the queue length satisfies X ≤ M p or not. X ≤ M p · · · 0 1 2 3 M p
Intuition Instead, suppose the signaling mechanism only revealed whether the queue length satisfies X ≤ M p or not. X ≤ M p · · · 0 1 2 3 M p Upon receiving the signal X ≤ M p , the utility for joining is a convex combination of u ( k ) for k ≤ M p .
Intuition Instead, suppose the signaling mechanism only revealed whether the queue length satisfies X ≤ M p or not. X ≤ M p · · · 0 1 2 3 M p Upon receiving the signal X ≤ M p , the utility for joining is a convex combination of u ( k ) for k ≤ M p . If positive, then customer will join, even if the queue length is M p !
Proof sketch max E σ [ R ( σ, f, p )] σ s.t., E σ [ h ( X ∞ , p ) | s = 1] ≥ 0 , E σ [ h ( X ∞ , p ) | s = 0] ≤ 0
Proof sketch max E σ [ R ( σ, f, p )] σ s.t., E σ [ h ( X ∞ , p ) | s = 1] ≥ 0 , E σ [ h ( X ∞ , p ) | s = 0] ≤ 0 We can write the expectations in terms of π , the stationary distribution.
Proof sketch ∞ � max π n σ n =1 ∞ s.t., � h ( n − 1 , p ) π n ≥ 0 max E σ [ R ( σ, f, p )] σ n =1 s.t., E σ [ h ( X ∞ , p ) | s = 1] ≥ 0 , ∞ � h ( n, p ) ( λπ n − π n +1 ) ≤ 0 E σ [ h ( X ∞ , p ) | s = 0] ≤ 0 n =0 λπ n − π n +1 ≥ 0 π T 1 = 1 , π ≥ 0 ∀ n We can write the expectations in terms of π , the stationary distribution.
Proof sketch ∞ � max π n σ n =1 ∞ s.t., � h ( n − 1 , p ) π n ≥ 0 n =1 ∞ � h ( n, p ) ( λπ n − π n +1 ) ≤ 0 n =0 λπ n − π n +1 ≥ 0 π T 1 = 1 , π ≥ 0 ∀ n
Proof sketch ∞ � max π n σ n =1 ∞ s.t., � h ( n − 1 , p ) π n ≥ 0 n =1 ∞ � h ( n, p ) ( λπ n − π n +1 ) ≤ 0 n =0 λπ n − π n +1 ≥ 0 π T 1 = 1 , π ≥ 0 ∀ n Instead of optimizing over the signaling mechanism, we can optimize over the stationary distribution π .
Proof sketch ∞ � max π n π n =1 ∞ s.t., � h ( n − 1 , p ) π n ≥ 0 n =1 ∞ � h ( n, p ) ( λπ n − π n +1 ) ≤ 0 n =0 λπ n − π n +1 ≥ 0 π T 1 = 1 , π ≥ 0 ∀ n Instead of optimizing over the signaling mechanism, we can optimize over the stationary distribution π .
Proof sketch ∞ � max π n π n =1 ∞ s.t., � h ( n − 1 , p ) π n ≥ 0 n =1 ∞ � h ( n, p ) ( λπ n − π n +1 ) ≤ 0 n =0 λπ n − π n +1 ≥ 0 π T 1 = 1 , π ≥ 0 ∀ n We have an (infinite) LP in { π n : n ≥ 0 } .
Proof sketch ∞ � max π n π n =1 ∞ s.t., � h ( n − 1 , p ) π n ≥ 0 n =1 ∞ � h ( n, p ) ( λπ n − π n +1 ) ≤ 0 n =0 λπ n − π n +1 ≥ 0 π T 1 = 1 , π ≥ 0 ∀ n We have an (infinite) LP in { π n : n ≥ 0 } . We can perturb any feasible solution to this LP to a threshold mechanism without decreasing the revenue.
Proof sketch ∞ � max π n π n =1 ∞ s.t., � h ( n − 1 , p ) π n ≥ 0 n =1 ∞ � h ( n, p ) ( λπ n − π n +1 ) ≤ 0 n =0 λπ n − π n +1 ≥ 0 π T 1 = 1 , π ≥ 0 ∀ n We first show that for queue lengths less than M p , the optimal mechanism must tell customers to join.
Proof sketch Next, we consider an feasible solution π where π n = λ n π 0 for n ≤ N , 0 < π N +1 ≤ π N < λπ N − 1 . 1.0 0.8 λ n 0.6 0.4 0.2 0 5 10 15
Proof sketch Next, we consider an feasible solution π where π n = λ n π 0 for n ≤ N , 0 < π N +1 ≤ π N < λπ N − 1 . 1.0 0.8 λ n 0.6 0.4 0.2 0 5 10 15 We construct a better solution by increasing π N +1 by β � n>N +1 π n and scaling down π n 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) 0 . 30 0 . 25 Full Full No-Info No-Info 0 . 25 0 . 20 0 . 20 0 . 15 Revenue Revenue 0 . 15 0 . 10 0 . 10 0 . 05 0 . 05 0 . 00 0 . 00 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 1 2 3 4 5 λ c (a) c = 0 . 2 and p = 0 . 3 . (b) λ = 0 . 7 and p = 0 . 3 .
Revenue comparison u ( X ) = 1 − c · ( X + 1) 0 . 30 0 . 25 Optimal Optimal Full Full 0 . 25 0 . 20 No-Info No-Info 0 . 20 0 . 15 Revenue Revenue 0 . 15 0 . 10 0 . 10 0 . 05 0 . 05 0 . 00 0 . 00 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 1 2 3 4 5 λ c (a) c = 0 . 2 and p = 0 . 3 . (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, optimal 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 λ i , are charged p i , and have utility h i ( n, p i ) = u i ( n ) − p i .
Heterogeneous customers Suppose customers come from one of K types. Type i customers arrive at rate λ i , are charged p i , and have utility h i ( n, p i ) = u i ( n ) − p i . 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 λ i , are charged p i , and have utility h i ( n, p i ) = u i ( n ) − p i . 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: λ 1 = λ 2 = 1 . Prices: p 1 = 50 , p 2 = 1 . n = 0 ; n = 0 ; 51 2 n = 1 ; n = 1 ; u 1 ( n ) = , u 2 ( n ) = 40 2 n ≥ 2 . n ≥ 2 . − 10000 − 8 . 5 Optimal mechanism: 1 n = 0; 0 n = 0; σ ( n, 1 , 1) = , σ ( n, 2 , 1) = 1 / 10 n = 1; 1 / 10 n = 1; n ≥ 2 . n ≥ 2 . 0 0
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’ optimal 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’ optimal strategy involves not joining for at most one signal, and joining for all others. ⇒ an iterative approach to optimize information sharing =
Risk aversion 0 . 95 Optimal Threshold 0 . 94 Sandwich 0 . 93 Throughput 0 . 92 0 . 91 0 . 90 0 . 89 0 . 88 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 5 7 . 0 Risk-aversion, β Threshold = join | leave Sandwich = risky-join | safe-join | risky-join | leave
Conclusion
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