Design of Information Sharing Mechanisms

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. Due to this informational asymmetry, the system operator may share information with its users.

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. 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 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. 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] 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 obtain 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 1.0 0.9 0.8 u(x) 0.7 0.6 0.5 0.4 0.3 0.2 0 1 2 3 4 5 6 7 8 x Customers...... don t enjoy waiting. u( ) non-increasing.... would join an empty queue. u(0) p 0... would not join long queues. M p s.t. u(m p ) p < 0.

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

Equilibrium

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 = s S σ(n, s)f(s) λq 0 λq 1 λq n 0 1 2... n n+1... 1 1 1 The queue forms a birth-death chain.

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; 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 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 No-Info 0.25 0.20 Full No-Info 0.20 0.15 Revenue 0.15 Revenue 0.10 0.10 0.05 0.05 0.00 0 1 2 3 4 5 λ (a) c = 0.2 and p = 0.3. 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c (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 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.

Proof sketch max σ E σ [R(σ, f, p)] s.t., E σ [h(x, p) s = 1] 0, E σ [h(x, p) s = 0] 0

Proof sketch max σ E σ [R(σ, f, p)] 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 σ E σ [R(σ, f, p)] s.t., E σ [h(x, p) s = 1] 0, E σ [h(x, p) s = 0] 0 max σ s.t., n=1 π n 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 can write the expectations in terms of π, the stationary distribution.

Proof sketch max σ s.t., n=1 π n 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 σ s.t., n=1 π n 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 s.t., n=1 π n 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 s.t., n=1 π n 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 s.t., n=1 π n 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 s.t., n=1 π n 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 0.6 λ n 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 0.6 λ n 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 No-Info 0.25 0.20 Full No-Info 0.20 0.15 Revenue 0.15 Revenue 0.10 0.10 0.05 0.05 0.00 0 1 2 3 4 5 λ (a) c = 0.2 and p = 0.3. 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c (b) λ = 0.7 and p = 0.3.

Revenue comparison u(x) = 1 c (X + 1) 0.30 0.25 Optimal Full No-Info 0.25 0.20 Optimal Full No-Info 0.20 0.15 Revenue 0.15 Revenue 0.10 0.10 0.05 0.05 0.00 0 1 2 3 4 5 λ (a) c = 0.2 and p = 0.3. 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 c (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]. Here, the service provider sets prices p(n) for each queue length n: a cutoff k such that p(n) =, for n k.

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 achieves the same revenue. p = E [u(x ) X < k]

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 Two types: λ 1 = λ 2 = 1. Prices: p 1 = 50, p 2 = 1. 51 n = 0; 2 n = 0; u 1 (n) = 40 n = 1;, u 2 (n) = 2 n = 1; 10000 n 2. 8.5 n 2. Optimal mechanism: 1 n = 0; 0 n = 0; σ(n, 1, 1) = 1/10 n = 1;, σ(n, 2, 1) = 1/10 n = 1; 0 n 2. 0 n 2.

Risk aversion Customers often 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 often 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 where T is the waiting time. E[T ] + β Var[T ] γ,

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 0.95 0.94 Optimal Threshold 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

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 optimal 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 σ x is optimal for x = N + q, where 2(1 p) c 1 if λ = 1; N = if λ 1 c 1 log(λ) (W i ( κe κ ) + κ) otherwise, with κ = when 1 ( 1 p c 1 1 λ c 1 p 1 p ; ) log(λ) and where i = 0 when λ > 1 and i = 1 < λ < 1. For all values of λ <, we have q k<n = λk (1 p c(k + 1)) λ N (c(n. + 1) + p 1)