Information Design for Congested Social Services: Optimal Need-Based Persuasion
IInformation Design for Congested Social Services:
Optimal Need-Based Persuasion
Jerry Anunrojwong
MIT and Chulalongkorn University, [email protected]
Krishnamurthy Iyer
Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN, [email protected]
Vahideh Manshadi
Yale School of Management, New Haven, CT, [email protected]
We study the effectiveness of information design in reducing congestion in social services catering to userswith varied levels of need. In the absence of price discrimination and centralized admission, the provider relieson sharing information about wait times to improve welfare. We consider a stylized model with heterogeneoususers who differ in their private outside options: low-need users have an acceptable outside option to thesocial service, whereas high-need users have no viable outside option. Upon arrival, a user decides to waitfor the service by joining an unobservable first-come-first-serve queue, or leave and seek her outside option.To reduce congestion and improve social outcomes, the service provider seeks to persuade more low-needusers to avail their outside option, and thus better serve high-need users. We characterize the Pareto-optimalsignaling mechanisms and compare their welfare outcomes against several benchmarks. We show that if eithertype is the overwhelming majority of the population, information design does not provide improvement oversharing full information or no information. On the other hand, when the population is a mixture of thetwo types, information design not only Pareto dominates full-information and no-information mechanisms,in some regimes it also achieves the same welfare as the “first-best”, i.e., the Pareto-optimal centralizedadmission policy with knowledge of users’ types.
Key words : information design; social services; Pareto improvement; congestion
1. Introduction
Social services often face the challenge of congestion due to their limited capacity relative to theirdemand. The congestion partly stems from the inclusionary intent of such services: a toll-free roadis available to all citizens even those who can afford alternative tolled ones. A broad range of low-and middle-income households are eligible to apply for public housing. How can a social serviceprovider reduce congestion and thus the efficiency loss associated with service delay?In this context, the two controls commonly used for managing congestion, i.e., pricing and cen-tralized admission control, are inapplicable due to fairness and implementation considerations.However, the service provider may have control over information about the status of the system a r X i v : . [ c s . G T ] M a y nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services to be shared with users. Local traffic managers and public housing authorities have accurate infor-mation about the level of congestion for their corresponding services. As such, the service providercan leverage this informational advantage to persuade some of those with lower needs to forgo thissocial service and reduce congestion in the system. In this paper, we study how effective such aninformational lever is.To investigate the effectiveness of information design in improving welfare for a congested socialservice, we develop a stylized model that captures the key features of such a system. We considera single server queueing system where users arrive according to a Poisson process and their servicetimes are i.i.d. and exponentially distributed. Upon arrival, each user decides to either wait for theservice by joining an unobservable queue or seek her outside option. To capture the disparity thatusers face with regard to the quality of their outside options, we categorize users into two groups:(1) high-need users that have no feasible outside option and (2) low-need users that have a viablealternative. Both types incur higher waiting costs upon joining a longer queue. A high-need usersalways joins as she does not have any other choice. However, upon arrival, a low-need user makesa join or leave decision to maximize her expected utility. Even though an arriving user does notobserve the queue, her decision relies on her belief about the queue size based on the informationshared by the service provider.We assume the service provider has complete information about the status of the queue which hecan share with the arriving user. However, sharing the information fully may lead to bad welfareoutcomes because a utility-maximizing user does not internalize the negative externality that sheimposes on others (Naor 1969). Instead, the service provider can use the lever of informationsharing to influence users’ beliefs about the queue size and consequently their decisions. We adoptthe framework of Bayesian persuasion or information design (Kamenica and Gentzkow 2011) inwhich the service provider commits to a signaling mechanism in response to which users follow anequilibrium strategy. The welfare of each type is thus determined by the signaling mechanism andthe corresponding equilibrium response of the users. Because high-need users always join the queue,the service provider does not need to know user types to implement a signaling mechanism.Our analysis follows the standard approach (see e.g. Lingenbrink and Iyer (2019), Bergemann andMorris (2016), Candogan and Drakopoulos (2019)) which allows us to only consider obedient binarysignaling mechanisms where upon the arrival of a user, the service provider makes a “ join ” or “ leave ”recommendation and the user finds it incentive compatible to follow that recommendation. Further,it builds on Lingenbrink and Iyer (2019) to establish an equivalence between the class of obedient We assume there is no abandonment. We use the terms Bayesian persuasion and information design interchangeably. nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services binary signaling mechanisms and the set of steady state distributions that satisfy certain linearconstraints. (Lemma 1) To ensure welfare improvement for both types, we focus on Pareto-dominantsignaling mechanisms (Definition 1) and establish structural results for any such mechanisms. Undermild assumptions on utility functions (Assumption 1), we show that any Pareto-dominant signalingmechanism has a threshold structure. (Theorem 1)With these structural results, we compare the optimal signaling mechanism against the bench-marks of full information sharing and no information sharing. We focus on the special settingin which both types have the same linear utility function. Our analysis reveals several intrigu-ing insights into effectiveness of information design. First, there exists a signaling mechanism thatPareto-dominates full information sharing unless full information is itself Pareto-dominant evenwhen the service provider is allowed to disregard user incentives. (Theorem 4) However, if the pop-ulation is mostly comprised of low-need users the welfare gain due to information design is fairlylimited. (Proposition 1) This dichotomy stems from the intuition that in the absence of high-needusers, a low-need user cannot be persuaded to leave if the queue length falls below the threshold upto which she would have joined under full information. On the other hand, when high-need usersare present it is possible to persuade more low-need users to leave. Second, there exists a signalingmechanism that Pareto-dominates no information sharing only if the arrival rate of high-need usersdoes not exceed a threshold. However, if high-need users constitutes the overwhelming majority, theninterestingly, no information is Pareto dominant even when the provider is allowed to disregard userincentives (Theorem 5). The main intuition behind this result is that with abundance of demandfrom high-need users, the system is so congested that the type- L user does not need much persuasionto choose her outside option over the social service. Conversely, if the system is not overcrowdedby high-need users, information design proves effective over sharing no information. Putting theseinsights together, we conclude that signaling is particularly effective if the user population showssufficient heterogeneity.To further study the power of information design, we compare its Pareto frontier with that ofa strong benchmark in which the service provider implements a Pareto dominant admission policy disregarding the user’s incentives. Interestingly, we show that if the arrival rate of high-need users ishigh, the two Pareto frontiers show considerable overlap (Propositions 2 and 3, and Figure 3). Thisfurther illustrates the effectiveness of information design: any Pareto-dominant signaling mechanismthat belongs to the overlapping regions of the frontier achieves the same welfare outcomes as thoseof an admission policy that can not only observe the user types, but also enforce the join or leavedecision without regard to their incentives. Further, in such overlapping cases, no user is indifferentbetween their recommended action and the alternative, implying that signaling mechanism primarily nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services plays the role of a co-ordination device. This is in contrast with usual persuasion settings, wherethe optimal signaling mechanism extracts all user surplus for at least some signals.In summary, our work investigates the effectiveness of information design as a potential approachfor reducing congestion in social services offered to users heterogeneous in their needs. Using astylized model, we show that by implementing a Pareto-dominant signaling mechanism, the serviceprovider can achieve Pareto improvement in the welfare by persuading more low-need users to seekoutside option, thereby reducing congestion. We also identify conditions under which informationdesign not only outperforms the simple mechanisms of full or no information sharing, but alsoachieves the same welfare outcomes as centralized admission policies that know each user’s need forthe service. Our work relates to and contributes to several streams of literature.
Information Design:
Like ours, in many other settings service providers and platforms have accessto more information than their customers. As such, informational aspects of service and platformoperations have been studied in many applications. Adopting the framework of Bayesian persuasion(pioneered by Kamenica and Gentzkow (2011)), Lingenbrink and Iyer (2018) and Drakopoulos et al.(2018) study effectiveness of information design for influencing the customers’ time of purchase inorder to maximize the platform’s revenue. Kremer et al. (2014) and Papanastasiou et al. (2017) focuson information design in a sequential learning setting with the goal of maximizing social welfare.In the context of misinformation on social platforms, Candogan and Drakopoulos (2019) study howthe platform can optimally signal the content accuracy while incentivizing desirable levels of userengagement in the presence of positive network externalities. (See also Candogan (2019) that designsoptimal signaling mechanism for incentivizing product adoption on a social network.) Outside theframework of Bayesian persuasion, Kanoria et al. (2017) show that a two-sided matching platformcan significantly improve welfare by hiding information about the quality of a user’s potentialpartners. In another interesting direction, Nahum et al. (2015) show that in two-sided matching,the presence of experts who can reveal information can lead to an inferior outcome for everyone intwo-sided matching even if the use of such experts is optional.Closest to our setting is the recent work of Lingenbrink and Iyer (2019) that study optimal signal-ing for services with unobservable queues. Even though our work builds on the machinery developedin Lingenbrink and Iyer (2019), there are also key differences which we discuss next. Lingenbrinkand Iyer (2019) are concerned with maximizing the service provider’s revenue using informationsharing as well as static pricing. As such, the goal of an optimal signaling mechanism in that set-ting is to persuade more customers to join the queue. However, in our setting the service provider nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services uses information sharing mechanism to improve welfare outcomes by influencing low-need users’decision in favor of leaving. Further, Lingenbrink and Iyer (2019) mainly focus on a setting withhomogeneous users whereas we study a settings with different user types. Relatedly, Anunrojwonget al. (2019) study persuasion of non-expected-utility maximizing agents, and apply it to studythroughput maximization in queues where customers’ disutility depends on the variance of theirwaiting-times.Finally, Das et al. (2017) also study how optimal information sharing mechanisms can reducecongestion in a traffic network when a user chooses a path among the set of paths some of whichhave uncertain states. In particular, the authors consider a static setting where a continuum of userssimultaneously decide on the path they wish to take to minimize their own cost, and show that allpublic signaling mechanisms yield the same outcomes as full information (or no information). Ourpaper complements this work by considering a dynamic setting in which users of different typessequentially arrive over time. Upon arrival of each user, the service provider sends a state-dependentsignal. We show that public signaling can be effective in improving welfare outcomes when comparedto special mechanisms of full information and no information. Strategic Behavior in Queueing Systems:
Following the seminal work of Naor (1969), a streamof literature has focused on analyzing queuing systems where users are strategic. (See the surveyHassin (2016) and references within.) By and large, the main focus in this line of work (Hassinand Koshman 2017, Hassin 1986, Chen and Frank 2001) is on using the levers of pricing (eitherstatic or state-dependent) and admission control rather than information sharing. Focusing on theinformation sharing aspect (for an unobservable queue), Allon et al. (2011) consider a cheap talksetting where the service provider does not have commitment power. Finally, as discussed above,Lingenbrink and Iyer (2019) consider information design in conjunction with pricing in order tomaximize the service provider’s revenue.
Dynamic Allocation of Social Goods:
Our paper is also related the literature on dynamic alloca-tion of social goods such as public housing (Kaplan 1984) and donated organs (Agarwal et al. 2019).Recently, Leshno (2017) and Arnosti and Shi (2017) consider settings where the user has a hetero-geneous preference over arriving goods and thus she faces a trade-off between waiting longer andaccepting a less preferred good. (Similar trade-off exists in dynamic matching as studied in Dovaland Szentes (2018) and Baccara et al. (2018).) These papers focus on designing efficient allocationmechanisms such as waitlist mechanisms. We complement this literature by studying the role thatinformation sharing can play in improving welfare for social services.While our contribution is theoretical, there is also an extensive literature on practical aspects ofprovision and prioritization of social services. For example, Brown and Watson (2018) examines the nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services validity and reliability of a widely-used homelessness vulnerability assessment. Segall et al. (2016)outlines criteria for kidney transplantation in elderly patients.Finally, our work investigates the power of information design to reduce congestion in social ser-vices, where the usage of monetary payments to shape agents’ incentives is either impractical orunpalatable. As such, it is broadly related to the growing literature on mechanism design withoutmoney. Motivated by wide-ranging applications, this stream of literature studies resource allocationwithout relying on monetary payments. For examples of static settings, see Procaccia and Tennen-holtz (2009), Prendergast (2017), Arnosti and Randolph (2020); dynamic settings are studied inBalseiro et al. (2019), Gorokh et al. (2019), Feigenbaum et al. (2020).
2. Model
In the following, we describe a model of information design for improving welfare outcomes in aqueueing setting with heterogeneous users. Our model builds upon that of Lingenbrink and Iyer(2019), who study revenue maximization in a related queueing setting.Consider a service provider who provides a social service to a stream of users arriving over time.Due to capacity constraints, the arriving users possibly wait in an unobservable queue for service,where they are served on a first-come-first-serve (FCFS) basis by a single server. Each user’s servicetime is independently and identically distributed as an exponential distribution with rate one. Arriving users must decide whether to join the queue and wait for the service or to leave for anoutside option. Upon joining, we assume there is no abandonment: if a user joins the queue shewill stay until service completion. To describe users’ utility, we start with discussing their outsideoptions. We model the users as belonging to one of two groups which differ in the quality of theiroutside options. Specifically, we assume that each user is either a (1) high-need user, who has noviable outside option, which we model by letting their utility for taking the outside option be −∞ ;or a (2) low-need user who has a viable outside option whose utility we normalize to . We denotea user’s type as H if they are high-need, and by L if they are low-need. We assume that users oftype i ∈ { H , L } arrive according to an independent Poisson process with rate λ i , with λ = λ L + λ H denoting the total arrival rate. To avoid trivialities, we assume λ L > . In our analysis, we alsoassume that λ ≤ , to capture the setting where the social service is not under-capacitated.On joining the queue to obtain service, each user receives a net utility composed of the benefitfrom the social service and a cost of waiting until service completion. Formally, the utility functionof a type i ∈ { L , H } user is given by u i : N → R , where u i ( n ) denotes her utility on joining a queuewith n users already in system, either in queue or being served. We make the natural assumptionsthat u i (0) > , and lim k →∞ u i ( k ) < . Further we make the following assumption: Normalizing the service rate to one is without loss of generality. Here, N = N ∪ { } denotes the set of non-negative integers. nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services Assumption 1.
The utility functions satisfy the following monotonicity assumptions: For each type i ∈ { H , L } , the utility function u i ( n ) is strictly decreasing in n . The difference ∆ u L ( n ) (cid:44) u L ( n ) − u L ( n + 1) is non-increasing in n . We remark that the monotonicity assumption on the utility of both types is natural and it reflectsthe fact that waiting for service completion imposes a waiting cost on the users. The second conditionrequires that while each additional user ahead in queue imposes greater waiting costs on a type- L user, the incremental cost decreases with more users ahead in queue. We note that the linear utilityfunction, i.e., u L ( n ) = 1 − c ( n + 1) for some c > , satisfies both the conditions. We assume that the users are strategic and Bayesian in their joining decisions. Because high-needusers have no viable outside option, any such arriving user always joins the queue for service. Onthe other hand, the low-need users may decide to leave for the outside option, based on their beliefsabout the queue state. Since the queue is unobservable to the users, the service provider seeks toleverage his informational advantage to influence the low-need users’ decision, with the goal towardsimproving welfare outcomes. To that end, the service provider commits to a signaling mechanism as follows: the service provider selects a set of possible signals S , and a mapping σ : N × S → [0 , ,such that, if there are n users already in queue upon the arrival of a user, he sends a signal s ∈ S to the user with probability σ ( n, s ) ∈ [0 , . (We require (cid:80) s ∈ S σ ( n, s ) = 1 for all n .) Note that sincehigh-need users in our model have no viable outside option and hence always join the queue, theservice provider can implement a signaling mechanism without the knowledge of user types.Given the signaling mechanism, we require the low-need users’ choices to constitute an equilib-rium. Informally, the equilibrium requires that in the steady state that arises from the users’ actions,each low-need user is acting optimally. To elaborate further, given the steady state distribution π ,we require that a low-need user joins the queue upon receiving a signal s ∈ S if and only if herexpected utility from joining E π [ u L ( n ) | s ] is greater than zero, the utility of her outside option. (Weassume that ties are broken in favor of joining; we note that due to negative externalities users inthe queue impose on each other, the welfare under other tie-breaking rules can only be better.)Note that the steady state distribution π itself is determined endogenously in equilibrium fromthe users’ actions. To avoid unnecessary notational burden, we refrain from formally defining theequilibrium for general signaling mechanisms, and point the reader to Lingenbrink and Iyer (2019).Instead, using standard revelation principle style arguments (see e.g. Lingenbrink and Iyer (2019),Bergemann and Morris (2016), Candogan and Drakopoulos (2019)) , one can show that it sufficesto consider obedient binary signaling mechanisms. These are the mechanisms where the signals are While our structural results rely only on Assumption 1, to derive further insights on the welfare outcomes underdifferent information sharing mechanisms, in Section 4, we assume that both user types’ utility function for joiningthe queue is the same linear function. nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services limited to “ join ” and “ leave ” — which we represent as and respectively — and for which in theresulting user equilibrium, a high-need user always joins, and a low-need user joins upon receivingsignal and leaves otherwise. We describe such mechanisms more formally next.First note that a binary signaling mechanism can be described by { p n : n ≥ } , where p n denotesthe probability that a type- L user receives the signal s = 1 (" join "), when the queue length is n upon her arrival. Assuming that all users follow their recommendation, let π = { π n : n ≥ } denotethe resulting steady-state distribution. By elementary queueing theory, the steady state distributionsatisfies the following detailed-balance conditions (Gross et al. 2018): π n +1 = ( λ L p n + λ H ) π n , for all n ≥ . (1)Given the steady-state distribution and using Bayes’ rule, an arriving type- L user receiving thesignal s = 1 (" join ") believes the queue-length is n ≥ with probability π n p n / (cid:80) k ∈ N π k p k . Similarly,an arriving type- L user receiving the signal s = 0 (" leave ") believes the queue-length is n ≥ withprobability π n (1 − p n ) / (cid:80) k π k ∈ N (1 − p k ) .For a type- L user, let U L ( s, a ) denote her expected utility upon receiving a signal s ∈ { , } andchoosing an action a ∈ { join , leave } . Note that we have U L ( s, leave ) = 0 . (Recall that type- L ’s outsideoption is normalized to zero.) On the other hand, we have U L (1 , join ) = (cid:88) n ∈ N π n p n (cid:80) k ∈ N π k p k u L ( n ) = (cid:80) n ∈ N ( π n +1 − λ H π n ) u L ( n ) (cid:80) n ∈ N ( π n +1 − λ H π n ) ,U L (0 , join ) = (cid:88) n ∈ N π n (1 − p n ) (cid:80) k ∈ N π k (1 − p k ) u L ( n ) = (cid:80) n ∈ N ( λπ n − π n +1 ) u L ( n ) (cid:80) n ∈ N ( λπ n − π n +1 ) . Here, the second equality in each line follows from the fact that λ L π n p n = π n +1 − λ H π n and λ L π n (1 − p n ) = λπ n − π n +1 , which follow from the detailed-balance condition (1).In an obedient binary signaling mechanism, a type- L user must find it incentive compatible tofollow the service provider’s recommendations. Thus, in such a mechanism, we must have the fol-lowing obedience constraints: U L (1 , join ) ≥ U L (1 , leave ) = 0 and U L (0 , join ) ≤ U L (0 , leave ) = 0 . This inturn yields the following constraints on the steady-state distribution π : J ( π ) (cid:44) ∞ (cid:88) n =0 ( π n +1 − λ H π n ) u L ( n ) ≥ , ( JOIN ) L ( π ) (cid:44) ∞ (cid:88) n =0 ( λπ n − π n +1 ) u L ( n ) ≤ ( LEAVE )Using the preceding constraints, the following result, from Lingenbrink and Iyer (2019), establishesa correspondence between obedient binary signaling mechanisms and a set of all distributions sat-isfying obedience constraints. We omit the proof for brevity. nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services Lemma 1 (Lingenbrink and Iyer (2019)) . For any obedient binary signaling mechanism, thesteady-state distribution π satisfies the following conditions: Distributional constraints: (cid:80) n ∈ N π n = 1 and π n ≥ for all n ≥ ; Detailed-balance constraints: λ H π n ≤ π n +1 ≤ ( λ H + λ L ) π n for all n ∈ N ; and Obedience constraints ( JOIN ) and ( LEAVE ) as defined above.Conversely, for any distribution π satisfying the preceding sets of constraints, there exists an obedi-ent binary signaling mechanism { p n : n ≥ } , with p n = π n +1 − λ H π n λ L π n whenever π n > (and arbitraryotherwise). We let Π SM denote the set of all distributions that satisfy the three sets of constraints mentionedabove. (Here, SM stands for signaling mechanism.) Here, the second constraints arise from thedetailed-balance conditions (1) and the fact that p n ∈ [0 , for all n .In addition to simplifying notation, the preceding result enables us to describe the user welfarein an obedient binary signaling mechanism purely in terms of the resulting distribution π ∈ Π SM . Inparticular, for any π ∈ Π SM , the welfare of type i users, denoted by W i ( π ) , is given by W L ( π ) = λ L ∞ (cid:88) n =0 π n (cid:18) π n +1 − λ H π n λ L π n (cid:19) u L ( n ) = ∞ (cid:88) n =0 ( π n +1 − λ H π n ) u L ( n ) = J ( π ) (2) W H ( π ) = λ H ∞ (cid:88) n =0 π n u H ( n ) . (3)Here, the first line follows from the fact that the arrival rate of type- L users is λ L and that if thequeue-length is n , which occurs with probability π n in steady state, an arriving type- L user joinsthe queue with probability ( π n +1 − λ H π n ) /λ L π n and receives utility u L ( n ) . Similarly, the second linefollows from the fact that a type- H user always joins upon arrival.Since we focus on a social service setting, we seek to understand the effectiveness of informationdesign in improving the welfare outcomes for both types. In this context, the following notion ofPareto dominance is natural: Definition 1.
For any two π, ˆ π ∈ Π SM , we say ˆ π Pareto-dominates π , if W i (ˆ π ) ≥ W i ( π ) for i ∈ { L , H } with a strict inequality for at least one i . Further, we say a distribution π ∈ Π SM isPareto-dominant within the class Π SM if and only if there exists no ˆ π ∈ Π SM that Pareto-dominates π .Hereafter, we frequently abuse the terminology to say an obedient binary signaling mechanism isPareto-dominant (within the class of such mechanisms), if the corresponding steady state distribu-tion (as per Lemma 1) is Pareto-dominant within the class Π SM .For our comparative analysis, we look at two specific signaling mechanisms that capture thetwo extremes of information sharing: (1) the full-information mechanism, denoted by fi , where the nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services service provider always reveals the queue-length to an arriving user — the corresponding steadystate distribution π ∈ Π SM satisfies π n +1 = λπ n for all n such that u L ( n ) ≥ , and π n +1 = λ H π n otherwise; and (2) the no-information mechanism, denoted by ni , where the service provider revealsno information to the users. We characterize the corresponding steady state distribution in Lemma4 in Section 4.In the following, we also consider admission policies , where the service provider can enforce thejoining or leaving of any user, regardless of their type or their incentives. While such enforcementis clearly practically infeasible, it serves as a benchmark against which the welfare outcomes ofsignaling mechanisms can be compared. Formally, an admission policy can be described by theclass of distributions Π AP that satisfy the distributional and the detailed-balance constraints fromLemma 1, but need not satisfy the obedience constraints ( JOIN ) and (
LEAVE ). (Here, AP standsfor admission policy.) Analogous to Definition 1, we define Pareto-dominance within the class Π AP .Observe that Π SM ⊆ Π AP , i.e., any signaling mechanism is also an admission policy (one that alsorespects user incentives), and hence any signaling mechanism π ∈ Π SM that is Pareto-dominantwithin the class of admission policies Π AP is also Pareto-dominant within the class of signalingmechanisms Π SM , but the converse may not hold. This observation motivates our choice of Π AP asa welfare benchmark.Before we end this section, we note that both Π AP and Π SM are closed and convex, and the welfarefunctions as defined in (2) and (3) are linear in π . Thus, the sets { ( W L ( π ) , W H ( π )) , π ∈ Π SM } and { ( W L ( π ) , W H ( π )) , π ∈ Π AP } are also convex. As a consequence, it follows that any ˆ π that is Pareto-dominant within the class of signaling mechanisms Π SM (or admission policies Π AP ), is the solutionto the (linear) optimization problem that maximizes the convex combination of the two user types’welfare over the Π SM (respectively, Π AP ). In particular, let W ( π, θ ) = θW L ( π ) + (1 − θ ) W H ( π ) for all π ∈ Π AP . Then, any Pareto-dominant signaling mechanism is the solution to max π ∈ Π SM W ( π, θ ) forsome θ ∈ [0 , . Similarly, any Pareto-dominant admission policy is the solution to max π ∈ Π AP W ( π, θ ) for some θ ∈ [0 , . Furthermore, for any Pareto-dominant π , the specific θ ∈ [0 , for which π maximizes W ( · , θ ) captures the relative importance the service provider ascribes to improving thewelfare of the two types. In this context, for a given θ , we refer to the admission policy that achievesthe maximum as the “first-best”, and the signaling mechanism that achieves the maximum as the“second-best”.
3. Structural characterization
In this section, we provide structural characterizations of the Pareto-dominant signaling mechanismsand admission policies. We use these structural characterization in Sections 4 and 5 to evaluate theeffectiveness of signaling mechanisms in improving welfare outcomes, and compare its performanceagainst admission policies and simple signaling mechanisms. nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services Before we begin, for the sake of completeness, we state the following technical result that estab-lishes the existence of Pareto-dominant signaling mechanisms and admission policies. The prooffollows from the observation that the sets Π AP and Π SM (or some relevant subsets) are compact, andhence maximizers of W ( π, θ ) over these sets exist for all θ ∈ [0 , . The formal proof is provided inAppendix A. Lemma 2.
There exist Pareto-dominant signaling mechanisms and admission policies.
Next, we define the following threshold structure among distributions π ∈ Π AP . Definition 2.
We say a given π ∈ Π AP has a threshold structure, if there exists an m ∈ N ∪ {∞} ,such that π k +1 = λπ k for all k < m , and π k +1 = λ H π k for all k > m . In such a setting, we say thedistribution π has a threshold x = m + a ∈ R + , where a = ( π m +1 − λ H π m ) /λ L π m ∈ [0 , .Informally, a distribution π ∈ Π AP has a threshold structure with threshold equal to x = m + a ∈ [ m, m + 1] , if an arriving type- L user is asked to join the queue with probability for all queue-lengthstrictly less than m , asked to leave with probability for all queue-lengths strictly greater than m ,and asked to join the queue with probability a ∈ [0 , if the queue length is exactly m . (Note thata threshold ∞ corresponds to the case where an arriving type- L user is asked to join regardless ofthe queue-length.)Our first result states that any Pareto-dominant signaling mechanism has a threshold structure.The proof follows from a perturbation analysis similar to that in Lingenbrink and Iyer (2019): weshow that given any π ∈ Π SM that does not have a threshold structure, one can perturb it to obtaina ˆ π ∈ Π SM that Pareto-dominates it. Theorem 1.
Any signaling mechanism π ∈ Π SM that is Pareto-dominant within the class Π SM has a threshold structure.Proof of Theorem 1. Let π ∈ Π AP be such that there exists an m ≥ with π m +1 < λπ m and λ H π m +1 < π m +2 . In words, this implies that under π , an arriving type- L user is asked to leave withpositive probability if the queue length is m , and asked to join with positive probability if the queuelength is m + 1 . We now show that such a π cannot be Pareto-dominant within Π SM . We do this byconstructing an ˆ π ∈ Π SM that Pareto-dominates π .Towards that end, consider the following perturbation of π for small enough δ > : ˆ π k = π k if k < m + 1 ; π m +1 + δ (cid:80) n>m +1 π n if k = m + 1 ; π k (1 − δ ) if k > m + 1 . nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services First, it is straightforward to verify that ˆ π satisfies the detailed balance constraints in Lemma 1 forall small δ > . In addition, we have J (ˆ π ) = ∞ (cid:88) k =0 ( π k +1 − λ H π k ) u L ( k ) + δ (cid:32) (cid:88) k>m +1 π k (cid:33) ( u L ( m ) − λ H u L ( m + 1)) − δπ m +2 u L ( m + 1) − δ (cid:88) k>m +1 ( π k +1 − λ H π k ) u L ( k )= J ( π ) + δ · (cid:88) k>m +1 π k · ( u L ( m ) − u L ( k − − λ H ( u L ( m + 1) − u L ( k )) ) . Now, as λ H < , for any k > m + 1 , we have u L ( m ) − u L ( k − − λ H ( u L ( m + 1) − u L ( k )) > u L ( m ) − u L ( k − − u L ( m + 1) − u L ( k )= ( u L ( m ) − u L ( m + 1)) − ( u L ( k − − u L ( k ))= ∆ u L ( m ) − ∆ u L ( k − ≥ , where we have used Assumption 1 in both inequalities. Specifically, the first inequality follows fromthe fact that u L ( k ) is strictly decreasing in k and hence u L ( m + 1) − u L ( k ) > , and the secondinequality follows from the fact that ∆ u L ( k ) = u L ( k ) − u L ( k + 1) is non-increasing in k . Using this andthe fact that π m +2 > λ H π m +1 ≥ , we obtain that J (ˆ π ) > J ( π ) ≥ . Hence the obedience constraint( JOIN ) holds for ˆ π .By similar algebraic steps, we have L (ˆ π ) = L ( π ) − δ · (cid:88) k>m +1 π k · ( u L ( m ) − u L ( k − − λ ( u L ( m + 1) − u L ( k )) ) . Using the fact that λ ≤ , by a similar argument as before, we obtain that the parenthetical termis non-negative, and hence L (ˆ π ) ≤ L ( π ) ≤ . Thus, the obedience constraint ( LEAVE ) also holds for ˆ π . Taken together, this implies we have ˆ π ∈ Π SM .Next, note that W H (ˆ π ) = λ H ∞ (cid:88) n =0 ˆ π n u H ( n )= W H ( π ) + λ H δ · (cid:32) (cid:88) k>m +1 π k · ( u H ( m + 1) − u H ( k )) (cid:33) . Since u H ( k ) is non-increasing in k , we obtain W H (ˆ π ) ≥ W H ( π ) . Finally, we have W L (ˆ π ) = J (ˆ π ) >J ( π ) = W L ( π ) . Thus, we obtain that ˆ π Pareto-dominates π .From the above, we conclude that for any Pareto-dominant signaling mechanism π ∈ Π SM , it mustbe the case that whenever there exists an m ≥ with π m +1 < λπ m , we have π m +2 = λ H π m +1 . Thisimplies that π must have one of the following two structures: nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services
1. for all m ≥ , we have π m +1 = λπ m ; OR2. there exists an m ≥ such that π k +1 = λπ k for k < m , π m +1 < λπ m and π k +1 = λ H π k for k > m .In the first case, we have type L users being asked to join the queue for all queue length, implyingthat π trivially has a threshold structure (with threshold equal to ∞ ). In the second case, the type L users are asked to join with probability for queue-lengths strictly less than m and asked toleave with probability for queue-lengths strictly greater than m . Again, this implies a thresholdstructure for π , with threshold in the interval [ m, m + 1] . (cid:3) Furthermore, using the same argument as above, we obtain that the Pareto-dominant admissionpolicies also have a threshold structure. We omit the proof for brevity.
Theorem 2.
Any admission policy π ∈ Π AP that is Pareto-dominant within the class Π AP has athreshold structure. Having established the threshold structure of any Pareto-dominant distribution (either within Π AP or Π SM ), we next state another key structural property of Pareto-dominant distributions within Π SM . In particular, we show that in any Pareto-dominant π ∈ Π SM that is not Pareto-dominantwithin Π AP , the obedience constraint that binds is the constraint ( LEAVE ). The intuition behindthis result lies in the observation, common in many congested service systems, that type- L usersdo not internalize the negative externalities they impose on other users (both type- L and type- H )by joining the queue. Hence, the type- L users are naturally more inclined to join the queue thanleave, and the challenge in information sharing is in ensuring that when the type- L users are askedto leave, they find it incentive compatible to do so. Theorem 3.
Given any Pareto-dominant signaling mechanism π ∈ Π SM , either it is (1) Pareto-dominant within the class Π AP of admission policies or (2) the obedience constraint ( LEAVE ) binds,i.e., L ( π ) = 0 . The proof of the preceding theorem relies on the following lemma which states that for any Pareto-dominant distribution (either within Π AP or Π SM ), the threshold must be less than that under thefull-information mechanism ( fi ) . Note that the full-information threshold, which we denote by m fi ,is the smallest integer k for which u L ( k ) < . Lemma 3.
For any Pareto-dominant π either within the class Π SM or within Π AP , the thresholdmust be less than or equal to the full-information threshold m fi . The proof of this lemma is provided in Appendix A. The above lemma also proves useful in Section 4where we study the effectiveness of signaling mechanisms by comparing their welfare outcomesagainst those of the full-information mechanism. With the lemma in place, we are now ready toprove the theorem. nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services Proof of Theorem 3.
Let π be Pareto-dominant within the class Π SM . By Lemma 3, π has athreshold structure with threshold smaller or equal to m fi . This in turn implies that J ( π ) > , asa type- L user always receives non-negative utility upon joining the queue, and receives a positiveutility if the queue is empty (which occurs with positive probability).Now, assume for the sake of contradiction that L ( π ) < and π is not Pareto-dominant within theclass Π AP . This implies that there exists an admission policy ˆ π ∈ Π AP that Pareto-dominates π . Inparticular, we have W i (ˆ π ) ≥ W i ( π ) for i ∈ { H , L } , with at least one inequality strict.Next, let ˜ π = (1 − (cid:15) ) π + (cid:15) ˆ π for some (cid:15) ∈ (0 , to be chosen later. By convexity of Π AP , we have ˜ π ∈ Π AP . Furthermore, by linearity, we have J (˜ π ) = (1 − (cid:15) ) J ( π ) + (cid:15)J (ˆ π ) and L (˜ π ) = (1 − (cid:15) ) L ( π ) + (cid:15)L (ˆ π ) .Since J ( π ) > and L ( π ) < , for all small enough (cid:15) > we have J (˜ π ) ≥ and L (˜ π ) ≤ . Thus, theobedience constraints ( JOIN ) and (
LEAVE ) hold for ˜ π , and hence ˜ π ∈ Π SM . Finally, again by linearity,we have W L (˜ π ) = (1 − (cid:15) ) W L ( π ) + (cid:15)W L (ˆ π ) ≥ W L ( π ) W H (˜ π ) = (1 − (cid:15) ) W H ( π ) + (cid:15)W H (ˆ π ) ≥ W H ( π ) , with at least one inequality strict. Thus, we obtain that ˜ π Pareto-dominates π , which contradictsour assumption that π is Pareto-dominant within the class Π SM . (cid:3) In concluding this section, we note that preceding result raises the intriguing possibility of theexistence of a signaling mechanism that is Pareto-dominant not only within the class Π SM of signalingmechanisms, but also within the broader class Π AP of admission policies. For any such mechanism,it follows that under a practically infeasible setting where the service provider observes the types ofusers and is allowed to enforce the joining and leaving of users without regard to their incentives, hecannot jointly improve both types’ welfare. Put differently, the existence of such mechanisms alsoimplies the existence of admission policies where the type- L users’ incentive constraints are satisfiedfor “free”.A trivial instance of such a scenario can arise, e.g., in cases where λ H is large enough, and theadmission policy always bars type- L users from joining the queue. First, such an admission policymust be Pareto-dominant, as any other policy that lets some type- L users in would necessarilyreduce the welfare of the type- H users. Furthermore, such an admission policy can be implementedas a no-information mechanism, which satisfies obedience constraints as congestion in the queuewith just the type- H users makes joining undesirable for the type- L users. Excluding such trivialscenarios, a natural question is whether there exist signaling mechanisms that do not exclude anytypes, but are still Pareto-dominant within the class of admission policies Π AP . In Section 5, undera linear utility assumption, we show that indeed such mechanisms exist. nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services
4. Mechanism Comparisons
Having characterized the structure of Pareto-dominant signaling mechanisms, in this section, wecompare such mechanisms against various benchmarks in two different settings. First, in Section 4.1,we consider the homogeneous setting where all users have type L , i.e., λ H = 0 . Then, in Section 4.2,we consider the heterogeneous setting with both types of users present. As we discuss below, the twosettings exhibit striking contrast in the effectiveness of signaling mechanisms for welfare improve-ment.In light of Theorems 1 and 2, for ease of presentation, we use the following simplified notationsfor a threshold policy: For x ∈ R + , the threshold policy x is an admission policy that gives rise toa steady state distribution π x ∈ Π AP that has a threshold structure, as defined in Definition 2, withthreshold x . For such a policy, with a slight abuse of notation, for i ∈ { L , H } we denote W i ( π x ) simply by W i ( x ) . We do the same for J ( π x ) and L ( π x ) . We start our comparative studies by analyzing the special case of homogeneous users ( λ H = 0 ).Observe that in this single-type setting, Pareto-dominance is equivalent to optimality, in termsof maximizing the welfare of type- L users. Consequently, we let sm denote the optimal signalingmechanism, the one that maximizes the welfare of type- L users.In the following proposition we compare the optimal signaling mechanism sm with the full-information extreme ( fi ). With a slight abuse of notation, for µ ∈ { fi , sm } , we denote the W L ( µ ) the type- L welfare under mechanism µ . We have the following result, whose proof is provided inAppendix B. Proposition 1.
In the homogeneous setting, we have W L ( fi ) ≤ W L ( sm ) ≤ α fi · W L ( fi ) , where α fi (cid:44) (cid:0) (cid:80) m fi n =0 λ n L (cid:1)(cid:14)(cid:0) (cid:80) m fi − n =0 λ n L (cid:1) ≤ /m fi , and m fi is the full-information threshold. The preceding theorem states that in the homogeneous setting, signaling mechanisms are notvery effective in improving the welfare beyond that achieved by the full-information mechanism. Togain some intuition, observe that in general Bayesian persuasion settings, the performance gains aretypically achieved by pooling, in the persuaded agents’ beliefs, the “good” and the “bad” states ofthe system. However, in a queueing setting, the linear nature of the underlying Markovian systemprecludes any such simple pooling of states in the agents’ belief: the only way for the system to The homogeneous setting where all users have type H is uninteresting from the point of design, as all users join thequeue regardless of their information due to no viable outside option. nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services reach a bad state (one with long queue-length) is by progressing through all intermediate queue-lengths. Because of this, agents are not easily persuaded. Formally, the proof proceeds by showingthat a threshold mechanism with threshold x ≤ m fi − will not be incentive compatible: for sucha mechanism, a type- L user on receiving the signal “ leave ” will realize the queue length is below m fi , and hence will join the queue, implying the ( LEAVE ) constraint is violated. Thus, any smallimprovement in welfare over fi stems from the structure of sm , which may sometimes ask type- L users to leave when queue-length equals m fi − . Next we proceed to a setting where the population is a mixture of the two types. Note that in thiscase, we have two objectives, namely, the welfare of both types. As discussed before, to examine theeffectiveness of information design in improving the welfare of both types, we focus on the notionof Pareto dominance. In the following, we draw comparisons between Pareto-dominant optimalsignaling mechanisms with the extreme forms of sharing information, i.e., full information and noinformation.To gain comparative insights, we make the following assumption:
Assumption 2.
For each user type i ∈ { L , H } , the utility upon joining the queue, with n usersalready present, is given by u i ( n ) = 1 − c ( n + 1) , where c ∈ (0 , is the waiting cost per each additionaluser ahead in the queue. The assumption of linear utility is common in the literature, and is made primarily for technicalconvenience. On the other hand, the assumption that the utility functions of the two user typesare the same needs some justification. First, note that since we focus on the notion on Pareto-dominance, each users’ utility can be scaled by an arbitrary positive number without any effect onour analysis. Thus, the main import of the assumption is that the two user types place the samerelative weights on the value of service and the cost of waiting. We believe this assumption enablesus to neatly isolate the effects of heterogeneity of the users’ outside options.With this assumption in place, our first result compares the power of information design againstthe full-information benchmark.
Theorem 4.
Suppose Assumption 2 holds. Then, we have either the full-information mechanism fi is Pareto-dominant within the class of admission policies Π AP ; OR there exists a signaling mechanism that Pareto-dominates the full-information mechanism.Further, the first case occurs if and only if W L ( m fi ) > W L ( m fi − nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services To understand the implications of the preceding result, consider an admission policy with threshold x ≤ m fi . As x increases, the more type- L users are served by the service provided, increasing theirutility. At the same time, the negative externality each type- L user imposes on other type- L usersincreases as x increases. (This is in addition to the negative externalities imposed on type- H users.)The preceding result states that for the full-information mechanism to be Pareto-dominant, thegains from serving more type- L users must dominate the negative externalities they impose on others(which is succinctly captured by the condition W L ( m fi ) > W L ( m fi − ). This occurs for instance whenthe arrival rate of type- L users is small. Conversely, if serving more type- L users imposes greaternegative externality on other type- L users, our result states that information design can leveragethis to improve the welfare of both types over the full-information mechanism. Proof of Theorem 4.
The proof proceeds in two steps. First, we show that the full-informationmechanism, with threshold m fi is Pareto-dominant within the class Π AP of admission policies if andonly if W L ( m fi ) > W L ( m fi − . Subsequently, we show that when this condition fails, there existsa signaling mechanism π ∈ Π SM that Pareto-dominates the full-information mechanism. We beginwith the first step.Suppose W L ( m fi ) > W L ( m fi − . From Lemma 6, W L ( k ) is unimodal in k ∈ N , with W L ( k ) increas-ing in k until reaching a maximum, and then decreasing. This implies that W L ( k ) < W L ( m fi ) forall k < m fi . Furthermore, from Lemma 6, we also obtain that W L ( · ) is monotone between integers.Hence, we conclude that for all ≤ ˆ x < m fi , we have W L (ˆ x ) < W L ( m fi ) . As Lemma 3 implies thatany Pareto-dominant admission policy must have a threshold less than or equal to m fi , we obtainthat the full-information mechanism, with threshold m fi , is Pareto-dominant within the class Π AP of admission policies.For the converse, suppose the full-information mechanism is Pareto-dominant within the class ofadmission policies Π AP . Consider the admission policy with threshold m fi − . From Lemma 6, wehave W H ( m fi − > W H ( m fi ) . Since the full-information mechanism is Pareto-dominant within Π AP ,we must have W L ( m fi ) > W L ( m fi − .To complete the proof, suppose W L ( m fi ) ≤ W L ( m fi − . Consider a threshold policy x ∈ ( m fi − , m fi ) . We have the following expression for L ( x ) : L ( x ) = λ L (cid:32) π m fi − ( m fi − x ) u L ( m fi −
1) + ∞ (cid:88) k =0 π m fi + k u L ( m fi + k ) (cid:33) . Since u L ( m fi − ≥ and u L ( m fi + k ) < for all k ≥ , we obtain for all large enough x ∈ ( m fi − , m fi ) , we have L ( x ) < . Moreover, it is straightforward to verify that J ( x ) ≥ . Thus, for largeenough x ∈ ( m fi − , m fi ) , the threshold policy x is a signaling mechanism. Using Lemma 6 again,we have W L ( x ) ≥ min { W L ( m fi − , W L ( m fi ) } = W L ( m fi ) . By a stochastic dominance argument, we nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services have W H ( x ) > W H ( m fi ) for x ∈ ( m fi − , m fi ) . Thus, for large enough x ∈ ( m fi − , m fi ) , the signalingmechanism with threshold x Pareto-dominates the full-information mechanism. (cid:3)
Next, we discuss our second benchmark, i.e., sharing no information. Before proceeding, we firstcharacterize the steady-state distribution under the no-information mechanism ( ni ) . Lemma 4.
For p ∈ [0 , , let π ( p ) ∈ Π AP be given by π n ( p ) = (1 − λ L p − λ H )( λ L p + λ H ) n . Then,the steady state distribution under the no-information mechanism ni is given by π ( p ni ) ∈ Π SM , for p ni ∈ [0 , that satisfies the following conditions: if W L ( π (1)) > then p ni = 1 ; if W L ( π (0)) < then p ni = 0 ; otherwise, p ni ∈ (0 , satisfies W L ( π ( p ni )) = 0 .Here, p ni ∈ [0 , denotes the probability under the no-information mechanism that a type- L user joinsthe queue upon arrival. The proof of the preceding lemma is presented in Appendix B. Using this characterization, wecompare the effectiveness of information design against the no-information benchmark.
Theorem 5.
Suppose Assumption 2 holds. For λ H ∈ [1 − c, , the no-information mechanism ni is Pareto-dominant within the class Π SM of signaling mechanisms, as well as within the class Π AP of admission policies. For λ H ∈ (0 , − c ) , there exists a signaling mechanism that Pareto-dominates the no-information mechanism. The preceding result neatly breaks the analysis into two cases, depending on the magnitude of thearrival rate of type- H users. In the first case, when the arrival rate of type- H users is high, even ifno other type- L users join the queue, the outside option is more desirable for a type- L user. In otherwords, the type- L user does not need much persuasion to forego the social service and avail theoutside option. In such instances, any information shared by the service provider would only inducesome type- L user to join the queue and hence reduce type- H users’ welfare. On the other hand,when the system is not already overwhelmed by type- H users, information design proves effectivein improving welfare of both types over the no-information mechanism. Proof of Theorem 5.
First, suppose λ H ∈ [1 − c, , and consider the admission policy A whereno type- L users are admitted into the queue. Under such policy, the steady state distribution π A isthat of the M/M/ queue with arrival rate λ H , and hence, we have π An = (1 − λ H ) λ n H for all n ∈ N .From this, we obtain than W L ( π A ) = 0 (since all type- L users are made to avail the outside option),and W H ( π A ) = 1 − c − λ H ≤ .Now, consider any other admission policy, where at least some fraction of type- H users are admit-ted into the queue. Using a coupling argument, it is straightforward to show that the resulting nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services steady-state distribution ˆ π stochastically dominates π . Since u H ( n ) is strictly decreasing in n , thisfurther implies that W H (ˆ π ) < W H ( π A ) . Hence, it follows that the admission policy A is Pareto-dominant within the class Π AP .Next, we show that the admission policy A can be implement as the no-information mecha-nism, under which type- L users never join the queue. This follows immediately from the fact that L ( π A ) = (cid:80) n ∈ N ( λπ An − π An +1 ) u L ( n ) = λ L (cid:80) n ∈ N π An u H ( n ) = ( λ L /λ H ) W H ( π A ) < . Here, the secondequality follows from the fact that for π A , π An +1 = λ H π An , and that u L ( n ) = u H ( n ) for all n ∈ N fromAssumption 2. This proves the first statement of the theorem.For the second part of the theorem, suppose λ H ∈ [0 , − c ) . It is straightforward to verify that,1. if λ = λ H + λ L ≤ − c , then under no-information, type- L users join with probability . Thesteady-state distribution is π n = (1 − λ ) λ n for all n ≥ ;2. if λ H < − c < λ , then under no information, type- L users join with probability p ni = − c − λ H λ L ∈ (0 , . The effective arrival rate into the queue is λ p ni + λ = 1 − c . The steady state distribution is π n = c (1 − c ) n for n ≥ .In the first case, the steady-state distribution π has a threshold structure with threshold equal toinfinity. Since the threshold is greater than m fi , Lemma 3 implies that π is not Pareto-dominantwithin Π SM . In the second case, the steady-state distribution π does not have a threshold struc-ture, and by Theorem 1, we conclude that π is not Pareto-dominant within Π SM . Thus, in bothcases, we obtain that there exists a signaling mechanism that Pareto-dominates the no-informationmechanism. (cid:3)
5. Achieving First Best
Having compared the effectiveness of information design against those of the two extreme signalingmechanisms, we now investigate its limitations. Specifically, in this section, we compare signalingmechanisms against Pareto-dominant admission policies, and ask how limiting the requirement ofensuring obedience constraints is in terms of welfare improvement.To better study this question, we consider the problem of maximizing the weighted welfare W ( π, θ ) = θW L ( π ) + (1 − θ ) W H ( π ) , both over the class of admission policies and the class of signalingmechanisms. Note that, due to convexity of Π AP and Π SM , all Pareto-dominant admission policiesand signaling mechanisms can be obtained as the maximizer for some θ ∈ [0 , . As we mention inSection 2, the specific θ for which a Pareto-dominant mechanism (or an admission policy) maximizes W ( π, θ ) captures the relative weight placed by the service provider in improving either types welfare.For notational convenience, for any θ ∈ [0 , , we let sm ( θ ) denote the Pareto-dominant mechanismthat maximizes W ( π, θ ) over π ∈ Π SM , and ap ( θ ) denote the admission policy that does the sameover Π AP . nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services As we discussed in the conclusion of Section 3, Theorem 3 raises the possibility that there exists θ ∈ [0 , such that sm ( θ ) = ap ( θ ) . The main point we make in this section is that, remarkably, for a widerange of θ , sm ( θ ) = ap ( θ ) , i.e., the signaling mechanism sm ( θ ) is Pareto-dominant within the classof admission policies. We have the following proposition, whose proof is provided in Appendix C. Proposition 2.
Suppose Assumption 2 holds. For λ H ∈ [1 − c, , the set of Pareto-dominant signaling mechanisms { sm ( θ ) : θ ∈ [0 , } is thesame as the set of Pareto-dominant admission policies { ap ( θ ) : θ ∈ [0 , } . For each λ H < − c , there exists a θ ( λ H ) > such that the set of θ for which sm ( θ ) = ap ( θ ) hasthe form [ θ ( λ H ) , . (We allow θ ( λ H ) > , in which case the interval is empty.) The preceding result has an interesting implication about the role of information design whensignaling mechanisms achieve Pareto-dominance over Π AP . Note that in such cases, neither obedienceconstraint binds, since sm ( θ ) = ap ( θ ) . Thus, the obedience constraints impose no limitations on theservice provider. In such cases, information design plays solely the role of a co-ordination device,directing some type- L users away from the queue and others to join the queue. In neither instance theuser is indifferent between the recommended action and the alternative. This is unlike what happensin typical persuasion settings, where optimality requires indifference for at least some signals.We also note that the two cases of the proposition are exactly the same as that of Theorem 5.In particular, when no-information mechanism is Pareto-dominant, the set of Pareto-dominant sig-naling mechanisms is same as the set of Pareto-dominant admission policies. Put differently, ininstances where signaling mechanisms lack the power of admission policies, no-information is Pareto-dominated by some signaling mechanisms.Finally, the equivalence of sm ( θ ) and ap ( θ ) is also appealing from an implementation point ofview: the service provider can implement a signaling mechanism without the knowledge of usertypes. However, under an admission policy, the service provider observes the type of each arrivinguser and makes join and leave decision on her behalf.We illustrate the qualitative insights of the preceding proposition along with those of Theorems 4and 5 via numerical examples presented in Figure 1. Here we plot the welfare of Pareto-dominantsignaling mechanisms and admission policies for different values of λ L ∈ { . , . , . } , and c =0 . . We fix λ = 1 in each case, to study the extreme setting where the service capacity exactlymatches the total arrival rate. For each value of λ L , we also plot the full-information mechanism( fi ) and the no-information mechanism ( ni ). First, observe that for λ L = 0 . , we have λ H > − c ,and hence the first case of Theorem 5 applies. Thus, we observe that the no-information mechanism( ni , green square) is Pareto-dominant. On the other hand, we observe that the full-informationmechanism ( fi , green star) is Pareto-dominated by a signaling mechanism (green cross). Further, nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services − . − . − . . . . . W H . . . . . . W L Welfare comparison, c = 0 . , λ H = 1 − λ L λ L = 0 . λ L = 0 . λ L = 0 . apsmfini Figure 1 Welfare of Pareto-dominant signaling mechanisms and admission policies for λ L ∈ { . , . , . } , λ H =1 − λ L and c = 0 . . Here, green (dashes) represents λ L = 0 . , red (dots-dashes) represents λ L = 0 . , andblue (dots) represents λ L = 0 . . Further, circles ( ◦ ) represent dominant admission policies ( ap ), crosses ( × )represent dominant signaling mechanisms ( sm ), star ( (cid:63) ) represents the full-information mechanism ( fi ), andsquare ( (cid:3) ) represents the no-information mechanism ( ni ). (The no-information points for λ L ∈ { . , . } overlap, and so do those corresponding to signaling mechanisms and admission policies for each fixed λ L .) note that as established in the first case of Proposition 2, ap ( θ ) = sm ( θ ) for θ ∈ [0 , . On the otherhand, for the other two values of λ L , we see that the no-information mechanism achieves zero welfarefor both types, and is Pareto-dominated in the class of signaling mechanisms. Finally, we observethat as the proportion λ L of users with viable outside option increases, the welfare of both usertypes increases.We complement the findings of Proposition 2 with numerical computations presented in Figure 2.Here, we plot the welfare of the signaling mechanism sm ( θ ) , the admission policy ap ( θ ) and thefull information mechanism fi for θ ∈ { , . , } , c = 0 . , and λ = 1 . Note that θ = 0 and θ = 1 correspond to the extreme cases where the service provider seeks to maximize the welfare of the onetype, perhaps at the expense of the other. The case θ = 0 . corresponds to the case where the serviceprovider values the two types equally. Together, these three cases provide a representative accountof the service provider’s potential objectives for welfare improvement. In these figures, in the region nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services H w e l f a r e Welfare with =0, c =0.15 apsmfi (a) θ = 0 H w e l f a r e Welfare with =0.5, c =0.15 apsmfi (b) θ = 0 . H w e l f a r e Welfare with =1, c =0.15 apsmfi (c) θ = 1 Figure 2 Welfare of the Pareto-dominant signaling mechanism sm ( θ ) , the Pareto-dominant admission policy ap ( θ ) ,and the full-information mechanism fi for θ ∈ { , . , } . Here λ L = 1 − λ H , and c = 0 . . right of the green line, we have λ H > − c . Thus, as shown in Proposition 2, we have that sm ( θ ) = ap ( θ ) . For θ ∈ { . , } , we see that even for some values of λ H < − c , the two are equal. Finally,we note that as λ H → , we approach the homogeneous setting, and as shown in Proposition 1, weobserve the performance of the signaling mechanism sm ( θ ) approach that of the full-informationmechanism in each case. We formalize these observations in the following proposition, whose proofis provided in Appendix C. Proposition 3.
Suppose λ = λ H + λ L = 1 . Under Assumption 2, we have: sm (0) = ar (0) if and only if λ L ∈ [0 , c ] . sm (1 /
2) = ar (1 / if and only if λ L ∈ (cid:104) , min (cid:16) c − c , (cid:17)(cid:105) . sm (1) = ar (1) if and only if λ L ∈ (cid:104) , min (cid:16) − ck ∗ ck ∗ ( k ∗ − , c − ck ∗ (cid:17)(cid:105) where k ∗ = (cid:24) − c − √ c c (cid:25) is thesmallest positive integer such that c − ck ≥ − c ( k +1) c ( k +1) k . Finally, in Figure 3, we plot for each c ∈ { . , . } , the values of ( θ, λ H ) for which the Pareto-dominant admission ap ( θ ) is the same as the Pareto-dominant signaling mechanism sm ( θ ) . In otherwords, for these values, information design plays mainly the role of a co-ordination device, inducingusers to co-ordinate towards a better welfare outcome. In particular, neither obedience constraintsbind for such values of ( θ, λ H ) . Observe that, as shown in Proposition 2 and 3, for any fixed λ H , thevalues of θ for which this holds is an interval of the form [ θ ( λ H ) , . In particular, for λ H > − c ,this is the entire interval [0 , . Conversely, for small enough values of λ H , i.e., as we approach thehomogeneous setting, we observe that this set is empty. For any fixed intermediate value of λ H ,we note that as θ increases, the threshold value in the Pareto-dominant mechanism sm ( θ ) = ap ( θ ) increases; as more weight is placed on type- L users’ welfare, the Pareto-optimal signaling mechanismasks type- L users to join the queue for a larger range of queue-length values. We also note that thefor any fixed θ , the values of λ H for which sm ( θ ) = ap ( θ ) is fairly complex, with it being an union oftwo intervals for some values of θ . nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services θ . . . . . . . . . . λ H Regions where ap ( θ ) = sm ( θ ) , c = 0 . . . . . . . t h r e s h o l d (a) c = 0 . θ . . . . . . . . . . λ H Regions where ap ( θ ) = sm ( θ ) , c = 0 . . . . . . . t h r e s h o l d (b) c = 0 . Figure 3 Regions of the ( θ, λ H ) plane for which sm ( θ ) = ap ( θ ) , i.e., the signaling mechanism sm ( θ ) is Pareto-dominantwithin Π AP . The colors represent the value of the threshold in ap ( θ ) , i.e., m ap ( θ ) .
6. Conclusion
Social services often share two common features: they have limited capacity relative to their demand,and they aim to serve users with varied levels of needs. Reducing congestion for such services usingprice discrimination or admission control is not feasible in this setting. However, the service providercan use its informational advantage, about service availability and wait times, to influence usersdecisions in seeking the service by choosing what information to reveal. How effective will such alever be? Our work seeks to answer this question. Adopting the framework of Bayesian persuasion,we study information design in a queuing system that serves users who are heterogeneous in theirneed for the service. We show that, by and large, information design provides a Pareto-improvementin welfare of all user types when compared to simple mechanisms of sharing full information or noinformation. Further, we show that information design can go beyond and achieve the “first-best”: itcan achieve the same welfare outcomes as those of centralized admission policies that observe eachuser’s type, and disregard user incentives. Thus our work indicates that information design can playa promising role in improving welfare for congested social services.
Acknowledgments
The second author gratefully acknowledges partial support from the National Science Foundation undergrants CMMI-2002155 and CMMI-2002156. nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services References
Agarwal N, Ashlagi I, Rees MA, Somaini PJ, Waldinger DC (2019) An empirical framework for sequentialassignment: The allocation of deceased donor kidneys. Technical report, National Bureau of EconomicResearch.Allon G, Bassamboo A, Gurvich I (2011) We will be right with you: Managing customer expectations withvague promises and cheap talk.
Operations Research
SSRN Electronic Journal
URL http://dx.doi.org/10.2139/ssrn.3386273 .Arnosti N, Randolph T (2020) Alaskan hunting license lotteries are flexible and approximately efficient.Technical report, URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3370605 .Arnosti N, Shi P (2017) Design of lotteries and waitlists for affordable housing allocation.
Columbia BusinessSchool Research Paper (17-52).Baccara M, Lee S, Yariv L (2018) Optimal dynamic matching .Balseiro SR, Gurkan H, Sun P (2019) Multiagent mechanism design without money.
Operations Research http://dx.doi.org/10.1287/opre.2018.1820 .Bergemann D, Morris S (2016) Bayes correlated equilibrium and the comparison of information structuresin games.
Theoretical Economics
Journal of Social Distress and the Homeless .Candogan O (2019) Persuasion in networks: Public signals and k-cores.
Proceedings of the 2019 ACM Con-ference on Economics and Computation , 133–134.Candogan O, Drakopoulos K (2019) Optimal signaling of content accuracy: Engagement vs. misinformation.
Operations Research .Chen H, Frank MZ (2001) State dependent pricing with a queue.
IIE Transactions , 1279–1284 (IEEE).Doval L, Szentes B (2018) On the efficiency of queueing in dynamic matching markets. Technical report,Working paper.Drakopoulos K, Jain S, Randhawa RS (2018) Persuading customers to buy early: The value of personalizedinformation provisioning.
Available at SSRN 3191629 .Feigenbaum I, Kanoria Y, Lo I, Sethuraman J (2020) Dynamic matching in school choice: Efficient seatreassignment after late cancellations.
Management Science .Gorokh A, Banerjee S, Iyer K (2019) From monetary to non-monetary mechanism design via artificial curren-cies. Technical report, URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2964082 . nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services
Fundamentals of Queueing Theory 5th edition (Wiley).Hassin R (1986) Consumer information in markets with random product quality: The case of queues andbalking.
Econometrica
Rational Queueing (CRC Press, Taylor & Francis Group).Hassin R, Koshman A (2017) Profit maximization in the M/M/1 queue.
Operations Research Letters
American Economic Review EC , 117.Kaplan EH (1984) Managing the Demand for Public Housing . Ph.D. thesis, Massachusetts Institute of Tech-nology.Kremer I, Mansour Y, Perry M (2014) Implementing the “wisdom of the crowd”.
Journal of Political Economy https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2967011 .Lingenbrink D, Iyer K (2018) Signaling in online retail: Efficacy of public signals.
Proceedings of the 13thWorkshop on Economics of Networks, Systems and Computation , 1–1.Lingenbrink D, Iyer K (2019) Optimal signaling mechanisms in unobservable queues.
Operations Research
Autonomous Agents andMulti-Agent Systems .Naor P (1969) The regulation of queue size by levying tolls.
Econometrica
Management Science (forthcom-ing).Prendergast C (2017) How food banks use markets to feed the poor.
Journal of Economic Perspectives
Proceedings of the10th ACM conference on Electronic commerce , 177–186.Segall L, Nistor I, Pascual J, Mucsi I, Guirado L, Higgins R, Laecke SV, Oberbauer R, Biesen WV, Abramow-icz D, Cavrilovici C, Farrington K, Covic A (2016) Criteria for and appropriateness of renal transplan-tation in elderly patients with end-stage renal disease.
Transplantation . Appendix nunrojwong, Iyer, and Manshadi:
Information Design for Congested Social Services A. Proofs from Section 3
Proof of Lemma 2.
The proof is immediate for λ < , since the sets Π AP and Π SM are compact (under thetopology of weak convergence). To see this, note that for any π ∈ Π AP , by Lemma 5 (stated at the end of thissection), we have π k ≤ λ k π for all k . Hence, for λ < , Prohorov’s theorem directly implies compactness of Π AP and Π SM .Suppose λ = 1 . Let Π fiAP and Π fiSM denote the set of admission policies and signaling mechanisms that arenot Pareto-dominated by the full-information mechanism. We again use Prohorov’s theorem to show thatthese sets are relatively compact, from which the existence will follow by noticing that any Pareto-dominantdistribution is the maximizer of W ( π, θ ) over these sets.Fix an (cid:15) > . Let W L ( fi ) and W H ( fi ) denote the welfare of each type under the full information mechanism.Next, fix some large enough N to be chosen later. Consider a steady-state distribution π ∈ Π AP . We have thefollowing expressions: W H ( π ) = λ H ∞ (cid:88) n =0 π n u H ( n ) ≤ λ H u H (0) (cid:32) (cid:88) n ≤ N π n (cid:33) + λ H u H ( N + 1) (cid:32) (cid:88) n>N π n (cid:33) ≤ λ H u H (0) + λ H u H ( N + 1) (cid:32) (cid:88) n>N π n (cid:33) . where the first inequality follows from Assumption 1 and the second follows because u H (0) > . Similarly, wehave W L ( π ) = ∞ (cid:88) n =0 ( π n +1 − λ H π n ) u L ( n )= − λ H π u L (0) + ∞ (cid:88) n =1 π n ( u L ( n − − λ H u L ( n )) ≤ ∞ (cid:88) n =1 π n ( u L ( n − − λ H u L ( n )) . Now, by Assumption 1, we have that u L ( n − − λ H u L ( n ) = ∆ u L ( n −
1) + λ L u L ( n ) is strictly decreasing in n (as λ L > ). Thus, by a similar argument, we obtain W L ( π ) ≤ ( u L (0) − λ H u L (1)) + ( u L ( N ) − λ H u L ( N + 1)) (cid:32) (cid:88) n>N π n (cid:33) . Now, choose N large enough that min { u H ( N + 1) , u L ( N ) − λ H u L ( N + 1)) } < − /(cid:15) . Then, we have W H ( π ) ≤ λ H u H (0) − (cid:15) (cid:32) (cid:88) n>N π n (cid:33) W L ( π ) ≤ ( u L (0) − λ H u L (1)) − (cid:15) (cid:32) (cid:88) n>N π n (cid:33) . This implies that if (cid:80) n>N π n > (cid:15) , then for small enough (cid:15) > , W L ( π ) < W L ( fi ) and W H ( π ) < W H ( fi ) , andhence the distribution π is Pareto-dominated by the full information mechanism fi . nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services Π fiAP has the property that for any (cid:15) > , there exists an N (cid:15) such that (cid:80) n>N (cid:15) π n ≤ (cid:15) for any π ∈ Π fiAP . Using Prohorov’s theorem, we then obtain that this set ofdistributions is relatively compact (under weak topology). Finally, the set Π fiSM , being a subset of Π fiAP , is alsorelatively compact. The proof then follows by observing that a maximizer of W ( π, θ ) over the closure of Π fiAP (and, separately, Π fiSM ) exists and is Pareto-dominant within Π AP (resp., Π SM ). (cid:3) Proof of Lemma 3.
We prove the statement for π ∈ Π SM that is Pareto-dominant with the class Π SM . Theproof of the corresponding statement for π ∈ Π AP is analogous and is omitted.The proof proceeds by showing the contrapositive. Let π ∈ Π SM have a threshold structure, with a threshold x > m fi , where x = m + a with m ∈ N and a ∈ [0 , . Thus, we have π k +1 = λπ k for all k < m , and π k +1 = λ H π k for all k > m . Note, we allow m = ∞ , which captures the case where π k +1 = λπ k for all k ∈ N . Observe that x > m fi implies that m ≥ m fi , and hence the threshold structure of π implies π m fi > .Consider ˆ π defined as follows: ˆ π k = (cid:40) Z π k if k ≤ m fi ; Z λ k − m fi H π m fi if k > m fi ,where Z = (cid:80) k ≤ m fi π k + π m fi (cid:80) k>m fi λ k − m fi H . Using the detailed balance constraints in Lemma 1, it follows that π k ≥ λ k − m fi H π m fi for all k > m fi . Thus, as (cid:80) k π k = 1 , we have Z ≤ .Next, consider J (ˆ π ) = ∞ (cid:88) k =1 (ˆ π k +1 − λ H ˆ π k ) u L ( k ) = 1 Z (cid:88) k
LEAVE ) also holds for ˆ π . Hence, we obtain that ˆ π ∈ Π SM .Furthermore, for (cid:96) ≤ m fi , we have (cid:80) k ≤ (cid:96) ˆ π k = Z · (cid:80) k ≤ (cid:96) π k . Since, Z ≤ , this implies (cid:80) k ≤ (cid:96) ˆ π k ≥ (cid:80) k ≤ (cid:96) π k forall (cid:96) ≤ m fi . For (cid:96) > m fi , after some algebra, we obtain (cid:88) k ≤ (cid:96) ˆ π k − (cid:88) k ≤ (cid:96) π k = 1 Z (cid:32) (cid:88) q ≤ m fi (cid:88) k>(cid:96) π q (cid:0) π k − π m fi λ k − m fi H (cid:1) + (cid:96) (cid:88) q = m fi +1 (cid:88) k>(cid:96) π m fi λ q − m fi H (cid:0) π k − λ k − q H π q (cid:1)(cid:33) . Using Lemma 5, we have π k ≥ π q λ k − q H for all k > q . Thus, the right-hand side is non-negative, and hence, (cid:80) k ≤ (cid:96) ˆ π k ≥ (cid:80) k ≤ (cid:96) π k for (cid:96) > m fi as well. Together, this implies that ˆ π is stochastically dominated by π . Since u H ( k ) is strictly decreasing in k , we have W H (ˆ π ) = λ H ∞ (cid:88) k =0 ˆ π k u H ( k ) ≥ λ H ∞ (cid:88) k =0 π k u H ( k ) = W H ( π ) . Finally, since W L (ˆ π ) = J (ˆ π ) > J ( π ) = W L ( π ) , we conclude that ˆ π ∈ Π SM Pareto-dominates π , and hence π cannot be Pareto-dominant within Π SM . (cid:3) nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services Lemma 5.
For any π ∈ Π AP , and for any k > q ∈ N , we have π k ≥ λ k − q H π q and π k ≤ λ k − q π q . In particular,when λ H > , for any π ∈ Π AP , we have π k ∈ (0 , for all k ∈ N .Proof. The proof follows immediately from the detailed balance constraints in Lemma 1. (cid:3)
B. Proofs from Section 4
Proof of Proposition 1.
First note that < W L ( fi ) ≤ W L ( sm ) simply follows from the observation that fi isa feasible signaling mechanism. Thus its welfare is a lower bound on that achieved by the optimal signalingmechanism sm .Next, we prove W L ( sm ) ≤ α fi W L ( fi ) . The proof consists of two steps. In the first step, we show x sm ≥ m fi − ,where x sm ∈ R + is the threshold of the sm mechanism. We prove this by showing that the second obedienceconstraint, ( LEAVE ), will not be satisfied if the threshold is below m fi − . More precisely, let π sm denote thesteady-state distribution corresponding to sm mechanism, and let x sm = m + a where m ∈ N and a ∈ [0 , .Then L ( π sm ) = (cid:40) λ L π m (1 − a ) · u L ( m ) + λ L π m +1 · u L ( m + 1) , if a > ; λ L π m · u L ( m ) , if a = 0 .Here, the first case follows from the fact that, under the optimal signaling mechanism sm , a user is asked toleave with probability − a if the queue length equals m , which occurs with probability π m , and asked toleave with probability if the queue-length equals m + 1 , which occurs with probability π m +1 . The secondcase follows analogously.Since π sm ∈ Π SM , we have L ( π sm ) ≤ . This condition, along with the fact that u L ( · ) is strictly decreasing,forces u L ( m + 1) < if a > , and u L ( m ) ≤ and u L ( m + 1) < if a = 0 . In both cases, we have m + 1 ≥ m fi ,and hence x sm = m + a ≥ m fi − a ≥ m fi − . Further, from Lemma 3, we have x sm ≤ m fi . Putting these twotogether, we obtain x sm ∈ [ m fi − , m fi ] . Since W L ( x ) is monotone between integers, we thus obtain that W L ( sm ) ≤ max { W L ( m fi − , W L ( m fi ) } . Now, W L ( m fi −
1) = (cid:80) m fi − n =0 λ n L u L ( n ) (cid:80) m fi − n =0 λ n L ≤ (cid:80) m fi − n =0 λ n L u L ( n ) (cid:80) m fi − n =0 λ n L = (cid:32) (cid:80) m fi n =0 λ n L (cid:80) m fi − n =0 λ n L (cid:33) · (cid:80) m fi − n =0 λ n L u L ( n ) (cid:80) m fi n =0 λ n L = α fi · W L ( fi ) . Here, in the inequality follows from the fact that u L ( m fi − ≥ , and the first and the second equalitiesfollow from the definition of a threshold mechanism. In the final equality, we have used the definition of α fi .Thus, taken together, we obtain W L ( sm ) ≤ α fi W L ( fi ) . The statement of the proposition follows after notingthat α fi ≤ m fi for all λ L ≤ . (cid:3) nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services Proof of Lemma 4.
First note that an arriving type- L user has no information about the queue length.Therefore, a symmetric equilibrium strategy consists of a probability p with which she joins the queue. Let π n ( p ) be the steady-state distribution corresponding to such a strategy. By detailed balance constraint, wehave: π n +1 ( p ) = ( λ H + pλ L ) π n ( p ) , n ∈ N This implies π n ( p ) = (1 − λ L p − λ H )( λ L p + λ H ) n , n ∈ N . A type- L users chooses p that maximizes her utility.This gives rise to the three cases listed in the statement of the lemma. (cid:3) The following lemma is repeatedly used in the proof of Theorem 4.
Lemma 6. The welfare function W H ( x ) is strictly decreasing in x ∈ R + . The welfare function W L ( x ) is unimodal over integers, initially increasing up to a maximum, and thendecreasing. Furthermore, W L ( x ) is monotone between consecutive integers.Proof. The proof of the first statement follows from the fact the steady-state distribution under thethreshold policy x is stochastically dominated by that under the threshold policy ˆ x > x . Since u H ( k ) is strictlydecreasing in k , we thus obtain that W H ( x ) > W H (ˆ x ) .For the second statement, note that for x = m + a , where m ∈ N and a ∈ [0 , , we have W L ( x ) = λ L · (cid:80) mk =0 λ k + λ m ( λ H + aλ L )1 − λ H · (cid:32) m − (cid:88) k =0 λ k u L ( k ) + aλ m u L ( m ) (cid:33) . Since this is of the form α + βaγ + δa , where α, β, γ, δ are independent of a , we obtain that W L ( m + a ) is monotonein a ∈ [0 , . Thus, we conclude that W L ( x ) is monotone between consecutive integers. It is straightforwardto verify that W L ( x ) is continuous, and hence the maximum of W L ( x ) is attained at an integer.Next, to show unimodality, we split the analysis into two cases corresponding to λ = 1 and λ < . For λ = 1 , we obtain W L ( m ) = λ L (cid:80) m − k =0 u L ( k ) m + 1 + λ H − λ H = λ L m − cm ( m + 1) / m + 1 + λ H − λ H . Now, considering for a moment that m takes values over the reals, we obtain ∂ W L ( m ) ∂m = − λ L (1 − λ H ) ( cλ H + 2(1 − λ H ))(1 + m (1 − λ H )) ≤ . Hence, this implies that ∂ ( W L ( m +1) − W L ( m )) ∂m ≤ , which further implies that W L ( m + 1) − W L ( m ) ≤ W L ( m ) − W L ( m − . Thus, we obtain that W L ( m ) has decreasing differences. Further, we have W L (1) − W L (0) = (1 − c )(1 − λ H ) λ L − λ H > , and W L (cid:0) c (cid:1) − W L (cid:0) c − (cid:1) = − c (1 − λ H ) c +2(1 − λ H ) < . Thus, we obtain W L ( m ) is unimodular, initiallyincreasing up to a maximum and then decreasing.For λ < , note that we have W L ( m ) = (1 − λ H ) ( c (1 − λ ) λ m m − c (1 − λ m ) + (1 − λ )(1 − λ m ))(1 − λ ) (1 − λ m ( λ − λ H ) − λ H ) Again, considering for a moment that m takes values over the reals, we obtain ∂W L ( m ) ∂m = λ m (1 − λ H )(1 − λ m ( λ − λ H ) − λ H ) ( c (1 − λ H ) m log ( λ ) − cλ m ( λ − λ H ) − cλ H + log( λ )( c + λ −
1) + c ) , nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services A ( m ) denote the term inside the paren-thesis. That is, we have A ( m ) = c (1 − λ H ) m log ( λ ) − cλ m ( λ − λ H ) − cλ H + log( λ )( c + λ −
1) + c, and hence, ∂A ( m ) ∂m = c (1 − λ m ( λ − λ H ) − λ H ) log ( λ ) ≤ . Thus, A ( m ) is decreasing in m . Further, A (0) = c (1 − λ + log( λ )) + log( λ )( λ − ≥ − λ + λ log( λ ) ≥ , and lim m →∞ A ( m ) m = c (1 − λ H ) log( λ ) < . Thus, we obtain that ∂W L ( m ) ∂m is decreasing in m , with it being initiallypositive, and eventually negative. This implies that W L ( m ) is unimodal, increasing up to a maximum andthen decreasing. This concludes the proof. (cid:3) C. Proofs from Section 5
In this section, we provide the proofs for results in Section 5. The following lemma is used in the proof ofProposition 2.
Lemma 7.
For x ∈ R + , the function L ( x ) is decreasing as long as it is non-negative, subsequent to whichit stays negative. Formally, we have L ( x ) ≤ max { inf ≤ u ≤ x L ( u ) , } .Proof. Consider a threshold policy x = m + a , where m ∈ N and a ∈ [0 , . We have L ( x ) = ∞ (cid:88) k =0 ( λπ k − π k +1 ) u L ( k )= λ L π m (1 − a ) u L ( m ) + λ L ∞ (cid:88) k =1 π m + k u L ( m + k )= λ L · λ m (1 − a ) u L ( m ) + λ m ( λ H + λ L a ) (cid:80) ∞ k =1 λ k − H u L ( m + k ) (cid:80) mk =0 λ k + λ m ( λ H + aλ L )1 − λ H . Since this is a ratio of two linear functions of a , we obtain that it monotone in a , and hence, it suffices toanalyze L ( x ) as a function over integers. After some algebra, we have λ L L ( m ) = λ m (cid:80) ∞ k =0 λ k H u L ( m + k ) (cid:80) mk =0 λ k + λ m λ H − λ H = λ m (cid:80) ∞ k =0 λ k H (cid:80) mk =0 λ k + λ m λ H − λ H · (cid:80) ∞ k =0 λ k H u L ( m + k ) (cid:80) ∞ k =0 λ k H . Now, both factors on the right-hand side are decreasing in m . Note that the first factor is positive. If L ( m ) is non-negative, then the second factor is non-negative, and hence L ( m + 1) − L ( m ) ≤ . On the other hand,if L ( m ) < , then the second factor is negative, and since it is decreasing, we obtain L ( m + 1) < as well.Thus, we conclude that L ( x ) is decreasing as long as it is non-negative, subsequent to which it stays negative.Formally, we have L ( x ) ≤ max { inf ≤ u ≤ x L ( y ) , } . (cid:3) Proof of Proposition 2.
First, suppose λ H ∈ [1 − c, , and fix a θ ∈ [0 , . Consider the admission policy ap ( θ ) . Since the arrival rate of type- H users is so high, even if no type- L users join the queue, the welfare oftype- H users is non-positive. If ap ( θ ) makes some type- L users to join the queue, then the welfare of type- H users can only be lower. Thus, we obtain that W H ( ap ( θ )) ≤ . Furthermore, under ap ( θ ) , the welfare of type- L nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services ap ( θ ) is Pareto-dominated by the admission policy that keeps alltype- L users from joining the queue. Thus, we obtain W L ( ap ( θ )) ≥ . Thus, we have J ( ap ( θ )) = W L ( ap ( θ )) ≥ .Furthermore, we have W H ( ap ( θ )) = λ H λ L ( J ( ap ( θ )) + L ( ap ( θ ))) . Since W H ( ap ( θ )) ≤ and J ( ap ( θ )) ≥ , we obtain that L ( ap ( θ )) ≤ . This implies that ap ( θ ) satisfies boththe obedience constraints ( JOIN ) and (
LEAVE ). Thus, we obtain that ap ( θ ) ∈ Π SM , and hence ap ( θ ) = sm ( θ ) .Next, suppose λ H < − c . Suppose there exists a θ ∈ (0 , with ap ( θ ) = sm ( θ ) , and consider θ > θ .First, note that the threshold of ap ( θ ) is always an integer for all θ , which we denote by m ap ( θ ) . Now, wehave θ W L ( ap ( θ )) + (1 − θ ) W H ( ap ( θ )) ≥ θ W L ( ap ( θ )) + (1 − θ ) W H ( ap ( θ )) θ W L ( ap ( θ )) + (1 − θ ) W H ( ap ( θ )) ≥ θ W L ( ap ( θ )) + (1 − θ ) W H ( ap ( θ )) , After some algebra, we obtain W L ( ap ( θ )) − W L ( ap ( θ )) ≥ W H ( ap ( θ )) − W H ( ap ( θ )) . Suppose m ap ( θ ) > m ap ( θ ) . This implies that W H ( ap ( θ )) > W H ( ap ( θ )) , by Lemma 6. The preceding inequal-ity would then imply W L ( ap ( θ )) − W L ( ap ( θ )) > . However, this would imply that ap ( θ ) Pareto-dominates ap ( θ ) , leading to a contradiction. Thus, we obtain that m ap ( θ ) ≤ m ap ( θ ) , i.e., m ap ( θ ) is weakly increasingin θ .Now, since sm ( θ ) = ap ( θ ) , we have L ( m ap ( θ )) ≤ . Since m ap ( θ ) ≥ m ap ( θ ) , by Lemma 7, we then obtainthat L ( m ap ( θ )) ≤ . Also, by Lemma 3, we obtain that m ap ( θ ) ≤ m fi , and hence J ( m ap ( θ )) ≥ . Together,we obtain that ap ( θ ) ∈ Π SM , and hence sm ( θ ) = ap ( θ ) .Thus, we obtain that if ap ( θ ) = sm ( θ ) for some θ , then the same holds for all θ > θ . Thus, using thecontinuity of W ( π, θ ) with respect to θ , we obtain the set of all θ for which ap ( θ ) = sm ( θ ) takes the form [ θ ( λ H ) , for some θ ( λ H ) ≥ . Finally, we note that θ ( λ H ) > , since for θ = 0 , the admission policy makes alltype- L users take the outside option, which cannot be implemented as a signaling mechanism because, fromTheorem 5, the leave condition does not hold for λ H < − c . (cid:3) The proof of Proposition 3 uses the following lemma.
Lemma 8.
Consider a threshold mechanism with threshold x = m + a where m ∈ N and a ∈ [0 , . Suppose λ = λ H + λ L = 1 . Then, under Assumption 2, we have W L ( x ) = λ L m + a + 1 /λ L (cid:18) m − c m ( m + 1)2 + a (1 − c ( m + 1)) (cid:19) W H ( x ) = (1 − λ L ) m + a + 1 /λ L (cid:18) m + 1 − c ( m + 1)( m + 2)2 + ( λ L a + 1 − λ L )( − c + λ L − cλ L − cmλ L ) λ L (cid:19) L ( x ) = − c + λ L − cmλ L − cλ L aλ L ( m + a + 1 /λ L ) . nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services Proof.
For a threshold mechanism with threshold x = m + a as in the lemma statement, the steady statedistribution is given by π n = (cid:40) π for n ≤ m ; ( λ L a + λ H ) λ n − ( m +1) H π for n ≥ m + 1 .The normalization constraint (cid:80) ∞ n =0 π n = 1 gives π = (cid:32) m + 1 + ∞ (cid:88) n = m +1 ( λ L a + λ H ) n − m (cid:33) − = (cid:18) m + 1 + λ L a + 1 − λ L − λ H (cid:19) − = 1 m + a + 1 /λ L . We therefore have W L ( x ) = ∞ (cid:88) n =0 ( π n +1 − λ H π n ) u L ( n )= m − (cid:88) n =0 (1 − λ H ) π (1 − c ( n + 1)) + ( λ L a + λ H − λ H ) π (1 − c ( m + 1))= λ L π (cid:32) m − (cid:88) n =0 − c ( n + 1) + a (1 − c ( m + 1)) (cid:33) = λ L m + a + 1 /λ L (cid:18) m − c m ( m + 1)2 + a (1 − c ( m + 1)) (cid:19) , and W H ( x ) = ∞ (cid:88) n =0 λ H π n u H ( n )= λ H π m (cid:88) n =0 (1 − c ( n + 1)) + λ H ( λ L a + λ H ) π ∞ (cid:88) n = m +1 (1 − λ L ) n − ( m +1) (1 − c ( n + 1))= λ H π (cid:18) m + 1 − c ( m + 1)( m + 2)2 (cid:19) + λ H ( λ L a + λ H ) π − c + λ L − cλ L − cmλ L λ L = (1 − λ L ) m + a + 1 /λ L (cid:18) m + 1 − c ( m + 1)( m + 2)2 + ( λ L a + 1 − λ L )( − c + λ L − cλ L − cmλ L ) λ L (cid:19) . Furthermore, L ( x ) = ∞ (cid:88) n =0 ( π n − π n +1 ) u L ( n )= (1 − ( λ L a + λ H )) π (1 − c ( m + 1)) + ∞ (cid:88) n = m +1 ( λ L a + λ H ) π λ n − ( m +1) H (1 − λ H )(1 − c ( n + 1))= λ L (1 − a ) π (1 − c ( m + 1)) + λ L ( λ L a + 1 − λ L ) π ( − c + λ L − cλ L − cmλ L ) λ L = λ L m + a + 1 /λ L (cid:18) (1 − a )(1 − c ( m + 1)) + ( λ L a + 1 − λ L )( − c + λ L − cλ L − cmλ L ) λ L (cid:19) = − c + λ L − cmλ L − cλ L aλ L ( m + a + 1 /λ L ) . This completes the proof. (cid:3)
Proof of Proposition 3.
For each θ ∈ { , / , } and nonnegative integer k , we will derive a condition on λ L such that x ap ( θ ) = x sm ( θ ) = k . Since x ap ( θ ) is always an integer, the condition on λ L such that x ap ( θ ) = x ap ( θ ) is the union on such conditions over all k ∈ N . nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services k ≥ , the necessary and sufficient condition for x ap ( θ ) = k is W ( k, θ ) ≥ W ( k − , θ ) and W ( k, θ ) ≥ W ( k + 1 , θ ) . For k = 0 , the necessary and sufficient condition is W (0 , θ ) ≥ W (1 , θ ) . ap ( θ ) and sm ( θ ) are the same if and only if the leave condition holds: L ( k ) ≤ . Together, these are the necessary and sufficientconditions for x ap ( θ ) = x sm ( θ ) = k .Using expressions from Lemma 8, L ( k ) ≤ is equivalent to − c + λ L − ckλ L ≤ or (1 − ck ) λ L ≤ c . We have W L ( k ) = λ L k + 1 /λ L (cid:20) − c k ( k + 1)2 (cid:21) W L ( k ) − W L ( k −
1) = λ L (2 − ck + ckλ L − ck λ L )2(1 + kλ L )(1 − λ L + kλ L ) W H ( k ) − W H ( k −
1) = ckλ L (1 − λ L )( λ L − − kλ L )2(1 + kλ L )(1 − λ L + kλ L ) In particular, W L (1) − W L (0) = λ L (1 − c ) / (1 + λ L ) and W H (1) − W H (0) = − cλ L (1 − λ L ) / (1 + λ L ) .1. We consider the case θ = 0 here. For k ≥ , we note that W ( k, − W ( k − ,
1) = W H ( k ) − W H ( k − < always, so the condition cannot be satisfied. For k = 0 , the condition W (0 , − W (1 , ≥ is always satisfied,and the condition L ( k ) = L (0) ≤ gives λ L ≤ c . We conclude that x ap (0) = x sm (0) if and only if λ L ∈ [0 , c ] .2. We consider the case θ = 1 / here. For k = 0 , we have W (0 , / − W (1 , /
2) = ( W L (1) − W L (0) + W H (1) − W H (0)) ≥ if and only if cλ L (1 − ) λ L − λ L (1 − c ) ≥ if and only if λ L ≤ c . The condition L ( k ) = L (0) ≤ also implies λ L ≤ c .For k ≥ , W ( k, / − W ( k − , / ≥ if and only if λ L (2 − ck + ckλ L − ck λ L ) + ck (1 − λ L )( λ L − − kλ L ) ≥ if and only if (2 − ck ( k − λ L ≥ ck . Therefore, the ap (1 / conditions become (2 − ck ( k − λ L ≥ ck and (2 − c ( k + 1) k ) λ L ≤ c ( k + 1) , and the leave obedient condition is (1 − ck ) λ L ≤ c .For k = 1 , we get λ L ≥ c , (2 − c ) λ L ≤ c , (1 − c ) λ L ≤ c . Together this gives c ≤ λ L ≤ c/ (1 − c ) . For c ≥ / ,this gives λ L ∈ [ c, . Henceforth we assume c < / .We will show that k ≥ is not possible. First we show that c < /k . This is true by assumption for k = 2 ,and for k ≥ , the condition (2 − ck ( k − λ L ≥ ck implies that c < /k ( k − ≤ /k . For k ≥ , we thereforehave ck − ck ( k − ≤ λ L ≤ c − ck . The inequality c − ck ≥ ck − ck ( k − gives c ≥ k − k ( k +1) ≥ k ( k − , a contradiction. For k = 2 , if c ≥ / , we get c ≥ − c , so c > (1 − c ) λ L , a contradiction, and if c < / , we have c − c ≤ λ L ≤ c − c ,but c − c < c − c , a contradiction.To conclude, we have the characterization k = 0 , λ L ∈ [0 , c ] k = 1 , λ L ∈ (cid:20) c, min (cid:18) c − c , (cid:19)(cid:21)
3. We consider the case θ = 1 here. For k ≥ , the conditions are ck ( k − λ L ≤ − ck , c ( k + 1) kλ L ≥ − c ( k + 1) , (1 − ck ) λ L ≤ c . We have − ck ≥ , so c < /k . For k = 0 , we have W (0 , − W (1 ,
1) = W L (0) − W L (1) < always, so k = 0 is not possible. Let n = (cid:98) /c (cid:99) , then ≤ k ≤ n .For k = n , the condition c ( k + 1) λ L ≥ − c ( k + 1) is true automatically since the right hand side is negative,so the conditions reduce to ≤ c ≤ min (cid:16) c − cn , − cncn ( n − (cid:17) .For ≤ k ≤ n , we have − c ( k + 1) c ( k + 1) k ≤ λ L ≤ min (cid:18) − ckck ( k − , c − ck (cid:19) nunrojwong, Iyer, and Manshadi: Information Design for Congested Social Services k < k ∗ , the inequality c − ck ≥ − c ( k +1) c ( k +1) k is false, so it is not possible. For k = n , the lower bound isnegative, so the resulting lower bound for λ L at k = n is in fact zero. For k = n, n − , . . . , k ∗ + 1 , So k ∗ ≤ k ≤ n . For k ∗ + 1 ≤ k ≤ n , − ckck ( k − ≤ c − c ( k − < c − ck , where the first inequality comes from the fact that k − ≥ k ∗ satisfies the inequality defining k ∗ . Therefore, the upper bound is − ckck ( k − , which exactly matchesthe lower bound for λ L at k − . Therefore, the union of the valid ranges of λ L for k = n, n − , . . . , k ∗ gives (cid:104) , min (cid:16) − ck ∗ ck ∗ ( k ∗ − , c − ck ∗ (cid:17)(cid:105) ..