University of Pennsylvania ScholarlyCommons Finance Papers Wharton Faculty Research 1-2014 Equilibrium in Queues Uner Unknown Service Times an Service Value Laurens Debo Senthil K. Veeraraghavan University of Pennsylvania Follow this an aitional works at: http://repository.upenn.eu/fnce_papers Part of the Finance an Financial Management Commons Recommene Citation Debo, L., & Veeraraghavan, S. K. 2014). Equilibrium in Queues Uner Unknown Service Times an Service Value. Operations Research, 62 1), 38-57. http://x.oi.org/10.1287/opre.2013.1232 This paper is poste at ScholarlyCommons. http://repository.upenn.eu/fnce_papers/102 For more information, please contact repository@pobox.upenn.eu.
Equilibrium in Queues Uner Unknown Service Times an Service Value Abstract In the operations research literature, the queue joining probability is monotonic ecreasing in the queue length; the longer the queue, the fewer consumers join. Recent acaemic an empirical evience inicates that queue-joining probabilities may not always be ecreasing in the queue length. We provie a simple explanation for these nonmonotonic queue-joining strategies by relaxing the informational assumptions in Naor's moel. Instea of imposing that the expecte service time an service value are common knowlege, we assume that they are unknown to consumers, but positively correlate. Uner such informational assumptions, the posterior expecte waiting cost an service value increase in the observe queue length. As a consequence, we show that queue-joining equilibria may emerge for which the joining probability increases locally in the queue length. We refer to these as sputtering equilibria. We iscuss when an why such sputtering equilibria exist for iscrete as well as continuously istribute priors on the expecte service time with positively correlate service value). Keywors queueing games, threshol policies, nonmonotone queue joining, ranomization Disciplines Business Finance an Financial Management This journal article is available at ScholarlyCommons: http://repository.upenn.eu/fnce_papers/102
Equilibrium in Queues uner Unknown Service Rates an Service Value Laurens G. Debo Chicago Booth School of Business University of Chicago Laurens.Debo@chicagobooth.eu Senthil K. Veeraraghavan Wharton School University of Pennsylvania senthilv@wharton.upenn.eu January 2011 Abstract We stuy a single queue joining equilibrium when there is uncertainty in the consumers mins about the service rate an value. Without such uncertainty, the joining equilibria are characterize by means of a single threshol queue length above which consumers o not join Naor, 1969). We show that in the presence of such uncertainty, the equilibrium joining strategy is not fully characterize by a single threshol. A sputtering equilibrium might exist. In the sputtering equilibrium, the queue length generally remains within a threshol, but reaches another, strictly higher, threshol, epening on the outcome of the ranomize ecision of the consumer arriving at the lower threshol. We iscuss when an why sputtering equilibria exist. Keywors: Queueing games, Markov Perfect Bayesian Equilibrium, Threshol policies. 1. Introuction This note contributes to the stream of research on consumer queue joining behavior. Equilibrium queue joining behavior have been examine beginning with seminal work by Naor 1969). The reaer is referre to Hassin an Haviv 2003) for an excellent overview. The establishe equilibrium result is as follows: When there are waiting costs, the consumers follow a threshol strategy. In other wors, they join the queue if the length of the queue is below a threshol. Above the threshol, congestion effects ominate any value from service an consumers balk from the service. In the operations research literature, such negative externalities are extensively stuie in a queuing context. In this note, we relax two key assumptions in Naor s classical queue joining game: the observability of the service rates an their service values. We o so by moeling the service rate an value as a ranom variable whose realization is not observe by the consumers. We emonstrate that unobservable service value an service rate may result in a sputtering equilibrium, where the queue length at the server generally resies within a lower threshol, but may probabilistically break through to a strictly higher threshol value, epening on the choice of the consumer who mixes between joining an balking at that particular state. This sputtering equilibrium funamentally expans on the threshol equilibria seen in queues with rational ecisionmaking consumers. A sputtering equilibrium may emerge when service value an service rate are negatively correlate. Thus, we fin that the focus on simple threshol policies in queuing games with uncertainty about the service rate an value might be restrictive. 1
Non-threshol joining strategies have been ientifie in queuing games with priorities Hassin an Haviv, 1996). Recent papers incluing Debo et al. 2010) an Veeraraghavan an Debo 2010) have been stuying informational externalities associate with queue joining ecisions uner waiting costs. In both papers, non-threshol queue joining strategies emerge in queuing games with heterogeneously informe consumers. In our note, just as in Naor s moel, consumers are homogenous an there are no priorities. Hence, we show that the non-threshol queue joining behavior emerges even in the canonical Naor 1969) moel when the service rates an service values are not observable an negatively correlate. Thus, our analysis funamentally expans on the queueing games literature with known service rates an values. Our analysis also suggests that in practice queue joining istributions cannot be generate by single threshol consumer joining strategies in the environments that we characterize. This may lea to further theoretical an empirical exploration. 2. Moel Consumers arrive sequentially to the server accoring to a Poisson process with parameter Λ. If the arriving consumers cannot be immeiately serve, they wait an form a queue. The queue iscipline is first-come, first-serve FCFS). All consumers incur a isutility of c > 0 per unit time while waiting to complete the service. The server belongs to one of the two types, ω {h, l}. The utility that a consumer obtains from the server of type ω is v ω, with v h v l. The consumer s prior about the server s type is Prω = h) = p. We aress the server with service value v h as the high-type or the high-quality server. Thus the low-quality server has v = v l.) The service time of both types ω {l, h} is exponentially istribute. There are two types of processes, a slow process an a fast process. The mean service time of the slow fast) process is 1/µ 1/µ) i.e., µ < µ). As we have two server types an two types of service processes, we analyze four situations. For notational ease, we introuce µ =., ). Specifically, we use µ ff = µ, µ), µ ss = µ, µ), µ sf = µ, µ) an µ fs = µ, µ) as shorthan notations for the four situations. For any vector µ σ, with σ {ss, ff, sf, fs}, the first letter refers to the service spee of the high-quality service slow or fast), while the secon letter refers to the spee of the low quality service slow or fast). For instance, µ sf = µ, µ) correspons to the situation when the high-quality server has slow service spee, an the low quality server has fast service spee. The consumer knows σ, but, oes not know the realization of the server s type. The moel parameters are Λ, v h, v l, µ, µ). Throughout this note, we assume that the value generate per unit of time by the high-quality server is always higher than that low-quality server s. That is, v h µ > v l µ. The game procees as follows. First, Nature etermines the server s type with probability p, ω = h). The server s type etermines the service value v ω ) an spee µ ω ). Then, consumers arrive an observe the queue length, n, base on which they ecie whether to join the queue or balk. The consumers maximize their expecte net utility i.e. service value minus the expecte 2
waiting costs), conitional on their available information. The Consumer Strategies an Beliefs: All consumers ecie the probability of joining the service after observing the queue length n 0. We enote this probability by αn). The consumer s upate belief that the type of the server is high after observing a queue of length n is enote by γn). In short, α = α0), α1), α2),...) an γ= γ0), γ1), γ2),...) enote the consumer s joining strategy an the consumer s upate belief respectively. We nee to etermine the equilibrium strategies of all consumers: α an γ. We specify the equilibrium conitions next. The Equilibrium Conitions: Consier a ranomly arriving consumer. Suppose all other consumers are joining accoring to some strategy α. The consumer s expecte utility of joining the queue, enote by u n, γ), is a function of the queue length upon arrival, n, his/her belief about the server s type, γ. We now efine: Definition 1 Equilibrium). The strategies α, an beliefs γ form a Markov Perfect Bayesian Equilibrium Maskin an Tirole 2001) if: i)-the consumers are rational: for each n N, α n) arg max α [0,1] α u n, γ ). 1) ii)-beliefs are consistent: The belief γ n) satisfies Bayes rule on all queue lengths that are reache with strictly positive probability in the long run uner the strategy α. 1 The conition i) of Definition 1 is referre to as the rationality conition for the consumers. The conition ii) of Definition 1 is referre to as the consistency conition of the beliefs. Using the conitions, we can now analyze the queue joining equilibrium. 3. Analysis of Queue Joining Equilibrium A consumer s utility can be written as, u n, γ) = γ n) v h + 1 γ n)) v l cn + 1) γ n) / + 1 γ n))/ ). 2) With Equation 2), we can rewrite conition i) of Definition 1 for consumers in terms of the likelihoo ratio of the upate probability that the service is of high quality when a consumer observes queue length n to upate probability that the service is of low quality). Now, instea of the upate belief, γ n), it will be convenient to introuce the likelihoo ratio, l n), satisfying: γ n) / 1 γ n)) = l 0 l n), where l 0 = p/ 1 p). In other wors, we ecompose γ n) into the consumer s prior likelihoo that the server is of high quality), l 0, an the likelihoo ratio of observing n consumers upon arrival, l n). Then, with Equation 2), we obtain the following 1 When a queue length is not reache with positive probability, the belief is irrelevant, as well as the action at that queue length. 3
conition to make a consumer join a queue of length n: u n, γ) > 0 l n) > L n), where L n). = 1 l 0 v l + cn + 1) 1 v h cn + 1) 1. 3) L can be interprete as the minimum likelihoo ratio that is require to make a consumer join a queue of length n. The rationality conition i) of Definition 1 etermines α n) for any n, which is 1 0) when l n) > L n) l n) < L n)). Now, we iscuss conition ii) of Definition 1. This conition nees to be impose on ln). To that en, let πn, α, µ) be the long run probability that n consumers are in the system when the consumer strategy is α an the server s service rate is µ. For a given service rate an consumer joining strategy, the stochastic process that escribes the queue length is a birth an eath process, which allows us to characterize the long-run probability istribution. See Ross 1996, pp. 254.) Suppose consumers follow the strategy profile α. Then, for both server types ω {h, l}, the stationary probability of observing a queue of length n, π n, α, µ), is: n 1 π n, α, µ) = π 0, α, µ) j=0 αj)λ µ where π 0, α, µ) = 1 + n 1 n=1 j=0 αj)λ µ 1. 4) With the PASTA property Wolff, 1982), π n, α, µ), is also the probability that a ranomly arriving consumer observes n consumers in the system. The posterior probability that the server s pπn,α,µ quality is high after observing a queue length of n is h ) pπn,α, )+1 p)πn,α, ). The posterior likelihoo is then l 0 πn,α,) πn,α, ). The consistency conition of Definition 1ii) becomes: l n) = πn,α,) πn,α, ) for all queue lengths that are reache with positive probability in equilibrium. 2 With Equation 4), notice that πn,α,) πn,α, ) becomes equal to π0,α,) π0,α, ) ) n. As the first factor is equal to l0), the consistency conition Definition 1ii) becomes: l 0) = π 0, α, ) π 0, α, ) an l n) = l 0) ) n for n 1. 5) We exploit this structure to obtain the conitions for the equilibrium: instea of fining a fixe point α an γ that satisfies Definition 1i) an ii), we reuce the equilibrium characterization of Definition 1i-ii) to a single-imensional fixe point characterization in Proposition 1. is because the likelihoo ratio at the empty queue, l0), along with Equation 5) immeiately etermines ln) for any n 1. This With Equation 3), the rational actions can be immeiately etermine by comparing ln) with Ln). Thus, the only unknown is l0). In the following Proposition, we characterize the equilibrium by letting ϕ be a conjecture of the equilibrium l0). For any conjecture ϕ of l0), Equation 5) provies a conjecture of ln), which etermines a rational strategy i.e. join iff l n) > L n), or ranomize when l n) = L n), see Equation 3)). This rational structure is an equilibrium if the likelihoo ratio it inuces at the 2 ln) is ineterminate whenever πn, α, ) an πn, α, ) are both zero. 4
empty queue obtaine from Equation 4) is equal to ϕ. We refer to this fixe point as ˆϕ. In that case, with ˆϕ, a strategy profile, α, that is rational Definition 1i) is satisfie) an a corresponing belief, γ, that is consistent Definition 1ii) is satisfie for all n 0) is constructe as escribe below: Proposition 1. Let ˆn = ˆn 0, ˆα 0, ˆn 1 ) where ˆn 0 N, ˆα 0 [0, 1) an ˆn 1 N ˆn 1 > ˆn 0 ) an Ψ ˆn) = [ ˆn0 Λ k=0 ˆn0 k=0 [ Λ ] k + ˆα0 ˆn1 k=ˆn 0 +1 [ Λ ] k + ˆα0 ˆn1 k=ˆn 0 +1 [ Λ ] k ] k, 6) then, i) there exists a classical threshol equilibrium pure or mixe if) for some ˆϕ R +, an ˆn 1 = ˆn 0 + 1 the following conitions are satisfie: an ˆϕ = Ψ ˆn 0, ˆα 0, ˆn 0 + 1) 7) ˆϕ ) n L n), for 0 n < ˆn0, ˆϕ ) n < L n), for n ˆn0. where the first inequality is bining when α 0 > 0. ii) there exists a non-threshol equilibrium if for some ˆϕ R +, an ˆn 1 > ˆn 0 + 1 the following conitions are satisfie: the conition of Equation 7) an ˆϕ ) n > L n), for 0 n < ˆn 0 if 0 < ˆn 0 ), ˆϕ )ˆn0 = L ˆn 0 ), for n = ˆn 0, ˆϕ ) n > L n), for ˆn 0 < n < ˆn 1, ˆϕ ) n < L n), for n ˆn 1. Proposition 1i) provies conitions for a pure or mixe strategy consumer equilibria: When ˆα 0 = 0, a pure strategy equilibrium exists in which consumers join at any queue length that is strictly less than the threshol ˆn 0, an balk from queues equal to or longer than ˆn 0. When ˆα 0 > 0, an ˆn 1 = ˆn 0 + 1, consumers join at any queue length that is strictly less than the threshol ˆn 0, join with probability ˆα 0 at the queue length ˆn 0, otherwise, they balk. In this case, α = 1, 1,..., ˆα 0, 0, 0,...), where ˆα 0 is on the ˆn 0 + 1st position. γ follows immeiately from the long run probability istributions that are characterize by Equation 4). This is an extension of a pure strategy equilibrium, as iscusse in Hassin an Haviv 2003), p. 7-8. We refer to this equilibrium simply as the classical threshol equilibrium. Proposition 1ii) inicates that this classical equilibrium strategy is not always an equilibrium. When ˆn 1 > ˆn 0 + 1, Proposition 1ii) ientifies a nonthreshol equilibrium with mixing with probability ˆα 0 ) at some queue length ˆn 0 ), while joining 8) 9) 5
at longer queues an balking at queue length ˆn 1. In this case, α = 1, 1,..., ˆα 0, 1,..., 1, 0, 0,...), where ˆα 0 is on the ˆn 0 +1)-st position an the last 1 is at the ˆn 1 +1)-st position. Again, γ follows immeiately from the long run probability istributions. In the following subsections, we elaborate further the intuition behin Proposition 1. To that en, we introuce ν ϕ), which is the set of solutions to the ϕ / ) ν = L ν). This is the continuous version of Equation 8), etermining ˆn 0. Define now ˆn ϕ) =. min ν ϕ). ˆn ϕ) is the lowest queue length at which the consumer has negative expecte utility, an hence balks when the conjecture of the likelihoo ratio at the empty queue is ϕ. Keeping all else equal, a larger ϕ means a higher likelihoo ratio at the empty queue. Of course only the equilibrium value of ϕ will be the equilibrium likelihoo ratio. By means of Equation 8), a higher likelihoo ratio at every queue length. Therefore, the balking threshol ˆn ϕ) is non-ecreasing in ϕ. Now, efine: Φˆn) =. Ψ ˆn, 0, )), 10) i.e. the likelihoo ratio at the empty queue when the consumer joining strategy is a pure threshol strategy etermine by balking at ˆn. As ˆn ϕ) N, the function Φˆn ϕ)) is a iscontinuous) staircase function in ϕ. In the following subsections, Φˆn ϕ)) will be iscusse further. We also provie intuition for the equilibrium structure of Proposition 1. 3.1 Classical threshol equilibrium at ˆn 0 with ˆα 0 = 0 A pure strategy threshol joining equilibrium with ˆα 0 = 0) exists if there is a solution, ˆϕ, to: ϕ = Φˆn ϕ)). 11) In wors, if the conjecture about the likelihoo ratio at the empty queue, ˆϕ, leas to a rational joining strategy with threshol ˆn 0 = ˆn ˆϕ), which in turn leas to a likelihoo ratio at the empty queue, Φˆn 0 ) that is consistent with i.e. equal to) ˆϕ, then the two equilibrium conitions of Definition 1i) an ii) are satisfie. ˆn 1 is irrelevant as it is reache with zero probability. Assume now that at one of the iscontinuous points of ˆn ϕ), say, ˆϕ, the consumers threshol changes from ˆn 0 to ˆn 1 an causes the likelihoo ratio inuce by the threshol to change from a level strictly below or above) ˆϕ to a level strictly above or below) ˆϕ. Formally, there exists a ˆϕ such that ˆn ˆϕ ɛ) = ˆn 0 an ˆn ˆϕ + ɛ) = ˆn 1 for which Φ ˆn 0 ) < Φ ˆn 1 ) or: Φ ˆn 0 ) > Φ ˆn 1 )). When ˆϕ Φ ˆn 0 ), Φ ˆn 1 )) or: ˆϕ Φ ˆn 1 ), Φ ˆn 0 ))), ˆϕ cannot characterize an pure strategy equilibrium. In that case, we can construct a consumer joining equilibrium involving mixing at ˆn 0, i.e. α 0 > 0. We consier two cases involving mixing in the following two subsections. In the first case, we analyze ˆn 1 = ˆn 0 + 1. This correspons with a classical threshol joining equilibrium with ranomization. In the secon case, ˆn 1 > ˆn 0 +1. We will enote the latter as sputtering equilibrium, because at ˆn 0 the queue sputters before increasing to ˆn 1 ue to the mixing probability ˆα 0 at ˆn 0. 6
^n=0 Φnφ)) ^ ^n=1 ^n=2 ^n=3 ^n=4 ^n=5 7 νt ^n=6 Upate Service Value ^n=40 Expecte Waiting Cost φ^ Queue Length Figure 1: Determination of equilibrium for µ = µ sf ; Φ ˆn ϕ)) = φ left panel) an the Upate Service Value, ˆγn)v h +1 ˆγn))v l, vs. the Upate Waiting Cost, ˆγn)cn+1)/ +1 ˆγn))cn+ 1)/ right panel). Demonstration of the sputtering equilibrium with ranomization at n = 6 an a balking threshol at n = 40. 3.2 Classical threshol equilibrium at ˆn 0 with ranomization ˆα 0 > 0 an ˆn 1 = ˆn 0 + 1 When ˆn 1 = ˆn 0 + 1, it is easy to see from the efinitions of Equations 6) an 10) that: Ψ ˆn 0, 0, ˆn 0 + 1)) = Φ ˆn 0 ) an Ψ ˆn 0, 1, ˆn 0 + 1)) = Φ ˆn 0 + 1). Then, by continuity of Ψ ˆn 0, α, ˆn 0 + 1)) in α, any likelihoo ratio in [Φ ˆn 0 ), Φ ˆn 0 + 1)] is achieve with a mixe strategy, α [0, 1] at ˆn 0. Thus, there exists an equilibrium with mixing at ˆn 0 such that the likelihoo ratio of that strategy is equal to ˆϕ. Both conitions i) an ii) of Definition 1 are satisfie. It follows that a classical mixe strategy extension of a pure strategy threshol Hassin an Haviv, 2003) is an equilibrium; i.e. all consumers join queues that are strictly shorter than ˆn 0, ranomize with probability ˆα 0 at a queue of length ˆn 0, an balk at queues that are longer than or equal to ˆn 0 + 1. 3.3 Non-threshol or Sputtering equilibrium with ˆα 0 > 0 an ˆn 1 > ˆn 0 + 1 When ˆn 1 > ˆn 0 + 1, an extension of a pure threshol strategy with mixing with probability α 0 at the threshol ˆn 0 an balking at ˆn 0 + 1 cannot cover any likelihoo ratio in [Φ ˆn 0 ), Φ ˆn 1 )] when Φ ˆn 0 + 1) < Φ ˆn 1 ) an ˆϕ Φ ˆn 0 + 1), Φ ˆn 1 )]. In that case, mixing at ˆn 0 only covers the likelihoo ratio range of [Φ ˆn 0 ), Φ ˆn 0 + 1)] an leaves the range Φ ˆn 0 + 1), Φ ˆn 1 )] uncovere. If ˆϕ lies in the latter range, the classical mixe strategy extension cannot etermine the equilibrium. Proposition 1ii) ientifies a non-threshol strategy. Recall that Φ Equation 10)) is a special case of Ψ Equation 6)). By continuity of Ψ ˆn 0, α, ˆn 1 )) in α, any likelihoo ratio in [Φ ˆn 0 ), Φ ˆn 1 )] is achieve with a mixe strategy, α [0, 1] at ˆn 0 as can be seen from the efinitions). Hence, there exists a ranomization probability ˆα 0 that ˆϕ is reache if it lies in Φ ˆn 0 + 1), Φ ˆn 1 )]. An Example of Sputtering Equilibrium: Now, we provie an illustrative example of a case 7
when no threshol consumer joining equilibrium exists. Let v h = 2, v l = 0, p = 0.1, c = 0.051 an µ = 1.05 an µ = 1.25. Let µ = µ sf, that is: the high-quality server is slow while the low-quality server is fast. In Figure 1, left panel, we illustrate Φˆnϕ)). We inicate again the value of ˆnϕ) on each horizontal segment where ˆnϕ) is constant. Notice that at ˆϕ = 0.5439, ˆn increases from ˆn 0 = 6 to ˆn 1 = 40. We obtain a non-threshol equilibrium with ˆϕ = 0.5439, ˆn 0 = 6 an ˆα 0 = 0.1170 an ˆn 1 = 40. This means that consumers always join the queue as long as its length is less than 6, they join with probability 0.1170 when the queue length is 6, they always join when the queue length is between 7 an 39, an they balk from any queue that is 40 or longer. Now, we explain the intuition behin this result: Recall that νϕ) is the set of real roots of ϕ / ) ν = Lν). νϕ) etermines the queue length at which rational consumers o not join, assuming that ϕ is the likelihoo ratio at the empty queue. On the left panel, we plot in ashe lines the corresponence Φ νϕ) ), where for certain values of ϕ, ν ϕ) can take three ifferent values. The values of νϕ) over the ashe branch are between 7 an 40. Recall also that ˆnϕ) = min νϕ), i.e. the lowest queue length at which a consumer balks. Hence, whenever there are multiple solutions in νϕ), the solution on the soli, top branch is selecte. As a result, the range of queue lengths between 7 an 39 is exclue from ˆn ϕ), which jumps from 6 to 40 see Figure 1, left panel). As ˆϕ Φ7), Φ40)], no classical threshol equilibrium can exist. This can also be observe from the left panel of Figure 1 as the 45 egree line has no intersection with Φˆnϕ)). Thus, with Proposition 1ii), a non-threshol equilibrium is ientifie with ranomization at a queue of length 6 an balking at a queue of length 40. The right panel of Figure 1 illustrates the equilibrium consumer upate utility value an waiting cost). Inee, there is one queue length 6), at which the consumer is inifferent between joining an balking, while at queue length 40, the consumer balks. Queue Dynamics: It is interesting to observe that a non-threshol equilibrium causes the following queue ynamics. As at ˆn 0, consumers join the queue with a probability of less than one, thus it will be ifficult for the queue to grow beyon ˆn 0. But, once the queue is larger than ˆn 0, all consumers join again with probability one. Hence, the server will observe that the queue stalls from time to time at ˆn 0. This is why we label this equilibrium as a sputtering equilibrium. These queue ynamics are an immeiate result of the non-threshol consumer strategy. In the following subsection, we iscuss when this phenomenon occurs. 3.4 Equilibrium Conitions In this section, we iscuss some properties of the moel. First, we erive a sufficient conition for the classical threshol consumer joining equilibrium Lemma 2). Next, we iscuss the number of possible equilibria Lemma 3). From the iscussion in the previous section, it follows that it is important to assess when ˆn ϕ) increases in steps of +1, an when it has larger jumps. When / ) ν L ν) is monotone in ν an ν ϕ) is a singleton, then, obviously, ˆn ϕ) increases in steps of +1. In general, however, / ) ν L ν) is not always monotone in ν. 8
ν ϕ) can contain three solutions, as illustrate in Figure 1, left panel, where for a range of values of ϕ two solutions are inicate by means of a ashe line an one solution the lowest value) by means of the soli line. The reason why there are three possible solutions coul be unerstoo from the right panel of Figure 1: As the service value is boune between v l an v h, the upate service value after observing the queue length follows an S-shape for this particular example: the upate value is boune by v h an v l an at very low an very high queue lengths, fining one extra consumer in the queue provies low aitional evience that the server is of high quality. The upate waiting cost, on the other han, is monotone increasing. As a result, there may be three intersection points between the S-shape upate value an the monotone increasing upate cost. Hence, ˆn ϕ) can increase in steps that are larger than +1 in Figure 1, from 6 to 40 at ˆϕ = 0.5439). The following Lemma provies a sufficient conition for ˆn ϕ) to be increasing in steps of +1: Lemma 2. When C µ) > 0, where C µ) = c + 1 4 v h v l ) ln then, ˆn ϕ) increases in steps of +1, otherwise, ˆn ϕ) may increase in steps that are larger than +1. It follows from Lemma 2 that Cµ fs ) = c + 1 ) 4 µvh µv l ln µ µ) is strictly positive as µv h > µv l an µ/µ > 1) an, trivially, Cµ ss ) = Cµ ff ) = c is strictly positive. This implies that when the server s service rate is inepenent of the server s quality, or, when the highest quality server is also the fastest server, threshol joining strategies always exist as the upate service value is either inepenent of the queue length, or ecreases in queue length, while the upate service cost increases in the queue length. Only when the highest quality server is also the slowest an both service value an cost increase in queue length that is: ), service value an rate are negatively correlate), a sputtering equilibrium may exist. As a consequence, if Cµ sf ) = c + ) µ µvh µv l ln µ) > 0, a classical threshol strategy is a consumer joining equilibrium for any 1 4 pure service rate strategy, µ σ with σ {ss, ff, sf, fs}. In terms of the primitives of our moel: If the ifference between the slow µ) an the fast spee µ) is not high, or, that the waiting costs, c, are high enough, provie that the high quality server operating at a slow spee creates more value per unit of service time than the low quality server operating at a fast service spee µv h > µv l ), the consumer joining equilibrium is a classical threshol strategy for any service rate strategy. Otherwise, a sputtering queue joining equilibrium may exist. An Illustration of Cµ): The conition C µ) = 0 is illustrate in Figure 2. The ashe lines are v h = v l an =. Notice that C µ) < 0 only when v h > v l an <, i.e., for a given service rate of the low quality server; when the service rate of the high quality server is lower, but, not too low, it is possible that C µ) < 0. C µ) > 0 is a sufficient conition for a threshol equilibrium strategy. When C µ) > 0, we can analyze the properties of threshol strategy equilibria, i.e. when ˆα 0 = 0 or ˆα 0 > 0 an ˆn 1 = ˆn 0 + 1. In Lemma 3, we characterize the number of pure strategy joining equilibria: 9
Non-threshol Equilibrium may exist Cµ)<0 Cµ)>0 Threshol Equilibrium always exists Figure 2: C µ) = 0 for v h = 1.75, v l = 0.25 an c = 0.1051. The ashe lines are v h = v l an =. Lemma 3. When C µ) > 0, then: i) When <, there exists at most one pure strategy threshol equilibrium. If no pure strategy equilibrium exists, a mixe strategy equilibrium exists. ii) When, there exists at least one pure strategy threshol equilibrium. If more than one pure strategy equilibrium exists, a mixe strategy equilibrium exists. The intuition behin Lemma 3 is the following: Consier the situation when the high quality server is the slowest server σ = sf or < ); it is thus easily unerstoo that the likelihoo ratio at the empty queue, Φˆn), ecreases in the joining threshol, ˆn. Assume that ˆn is very low. The queue is mostly short. Hence, the recurrent state space queue lengths) is small such that the probability of observing an empty queue when the prouct quality is high is not much ifferent from the probability of observing an empty queue when the service quality is low. Hence, Φ is comparable to one. When ˆn is high, there is enough variation in the queue lengths such that there is a ifference in probability of observing an empty queue. In fact, the probability of observing an empty queue when the quality is high will be lower than the probability of observing an empty queue when the quality is low. Hence, Φ is less than one, an is thus ecreasing in ˆn. As ˆn ϕ) is always increasing in ϕ, it follows that Φˆn ϕ)) ecreases in ϕ. As a result, there can be at most one point of intersection for Φˆn ϕ)) an the 45 egree line, an, therefore, at most one pure strategy equilibrium. For such pure strategy equilibrium, ˆϕ < 1. When the low quality server is slow σ = fs), the opposite is true: Φˆn) increases. As a result, there can be multiple points of intersection for Φˆn ϕ)) an the 45 egree line, an thus multiple pure strategy equilibria with ˆϕ > 1 may exist. Recall from Section 3 that ˆϕ is the likelihoo ratio at the empty queue, l 0), an we ecompose the posterior belief that the quality is high via γ n) / 1 γ n)) = l 0 l n). Hence, in any 10
equilibrium in which the high-quality server is the slowest σ = sf), as the equilibrium l 0) is less than one, an empty queue is ba news about the service quality that is: γ 0) < p 0 ). Longer queues, however, make the consumer more confient that the quality is high see Equation 9)). When the low quality server is slow σ = fs), the opposite is true: empty queues carry goo news about the quality γ 0) > p 0 ) an longer queues make the consumer less confient that the quality is high. 4. Conclusions In summary, combining Proposition 1 an Lemmas 2 an 3 yiel the following conclusions. 1. When the high quality server is slow an the low quality server is fast, the upate value from the service increases in the queue length. However, long queues also imply long waiting times. Hence, both costs an value increase in queue length. When the ifference between the service rates of the high- an the low-quality server is small, there will exist a threshol, at which the aitional upate value oes not compensate for the increase in the waiting costs. Consumers balk from the queue at this threshol. 2. When the ifference in service rates between ifferent services is large enough, a sputtering equilibrium involving ranomization at some lower threshol emerges. Consumers are inifferent between joining an balking at that threshol, while, at strictly longer an shorter queue lengths, the consumers may strictly prefer joining the queue. Of course, as waiting costs grow without boun in the queue length an the service value is finite, there will exist a secon, higher threshol above which no consumer ever joins. As a result, the queue generally resies at low lengths, an with some probability epening on the ranom ecision of one consumer), grows to a larger queue, since every one else joins at higher queue lengths. To summarize, we fin that the presence of informational uncertainty on service value an service rate can introuce a sputtering queue joining equilibrium. Thus, a focus on simple threshol policies in queuing games with uncertainty about the service rate an value might be restrictive. References Debo, L., C. Parlour an U. Rajan, 2010. Inferring Quality from a Queue. Chicago Booth Working paper. Hassin, R. an M. Haviv. 2003. To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems. Kluwer Acaemic Publishers. Hassin, R. an M. Haviv. 1997. Equilibrium Threshol Strategies: The Case of Queues with Priorities. Operations Research. 456), pp. 966-973. Maskin, E an J. Tirole. 2001. Markov Perfect Equilibrium. J. of Econ. Theory. 100, pp. 191 219. Naor, P. 1969. The Regulation of Queue Size by Levying Tolls. Econometrica, 37, pp. 15-34. Ross, S. 1996. Stochastic Processes. Wiley an Sons, New York. 11
Veeraraghavan, S. an L. Debo. 2010. Hering in Queues uner Waiting Costs: Rationality an Regret. Forthcoming in M&SOM. Wolff, R. 1982. Poisson arrivals see time averages. Operations Research, 30, pp. 223-231. Proofs Proof of Proposition 1: We impose conitions i) an ii) of Definition 1 for a given joining profile α. We consier only strategy profiles of the following special form, parameterize by some n = n 0, α 0, n 1 ): αj) = 1 for 0 j < n 0 an αn 0 ) = α 0 an αj) = 1 for n 0 + 1 j < n 1 an αj) = 0 for j n 1 + 1. Then, we can rewrite with Equation 4) l n) as: l 0) = Ψ n) an l n) = l 0) ) n for all n that are reache with positive probability on the long run). With Equations 3) an 5), we can rewrite conitions i) an ii) of Definition 1 as: i)-consumers are rational when for each for all n that are reache with positive probability on the long run, they join when ) n ) n Ψ n) > <)L n) ˆαn) = 10) an Ψ n) = L n) ˆαn) [0, 1]. 12) ii)-beliefs are consistent: The belief ˆγ n) satisfies Bayes rule when: ˆγ n) = Ψ n) ) n Ψ n) ) n + 1 for n 0. Hence, 1) the belief on the queue lengths that are reache with strictly positive probability on the long run, ˆγ n) is completely specifie by n an 2) for a given n, the rationality conitions are completely etermine. Equation 12) etermines thus the equilibrium conitions ˆn, which etermine then ˆα. Now, we introuce the variable ϕ > 0. In equilibrium, ˆϕ, will be equal to Ψ ˆn). We replace Ψ n) in Equation 12) by ϕ an for any ϕ > 0, we efine n 0 is the lowest queue length at which the consumer balks: ) n ) n ˆnϕ) = min{n N : ϕ L n), for 0 n < n 0 an ϕ < L n), for n n 0 }. Recall that we introuce Φ ˆnϕ)) = Ψ ˆnϕ), 0, )). The inicates that Ψ ˆn 0, 0, )) oes not epen on n 1. As ˆnϕ) N, ˆnϕ) is iscontinuously increasing in ϕ. Hence, Φ ˆnϕ)) is a iscontinuous function in ϕ. We can exten Φ ˆnϕ)) to a corresponence, ˆΦ ˆnϕ)), where at any iscontinuous point ϕ of ˆnϕ), where ˆnϕ ɛ) = n 0 an ˆnϕ + ɛ) = n 1 for an arbitrary small, but strictly positive ɛ, the image of the corresponence is the set [Φ n 0 ), Φ n 1 )] if Φ n 0 ) < Φ n 1 ) or [Φ n 1 ), Φ n 0 )] otherwise. Existence of a fixe point of ˆΦ ˆnϕ)) = ϕ. Notice that ϕ = 0: ˆn0) 0 for which ˆΦ ˆn0)) > 0 an for ϕ : ˆn ) v h c < for which ˆΦ ˆn )) <. It follows that the corresponence has at least one fixe point, ˆϕ, in 0, + ). Next, we characterize a fixe point. 12
Characterization of a fixe point of ˆΦ ˆnϕ)) = ϕ. We consier three cases: Case 1. Assume that for some continuous point of ˆnϕ), ˆϕ, the corresponing ˆn 0 = ˆn ˆϕ) satisfies ˆϕ = Φ ˆn ˆϕ)) Then, it is easy to see that the strategy profile ˆα, efine by ˆαj) = 1 for 0 j < ˆn 0 an ˆαˆn 0 ) = 0 satisfies i) of Definition 1, by construction of ˆnϕ) as ˆn 0 is the lowest queue length at which the consumer balks, the consumer joins for all queue lengths strictly lower than ˆn 0, as is assume in the special structure of α) an also ii) of Definition 1 because ˆϕ = l0). Hence, ˆα satisfies conitions i) an ii) of Definition 1. Case 2. Assume that no continuous point of ˆnϕ) exists for which the above fixe point equality is satisfie. As the corresponence ˆΦ ˆnϕ)), there must exist at least one iscontinuous point, ˆϕ, that satisfies ˆΦ ˆn ˆϕ)) = ˆϕ. Now, let ˆn 0 = ˆn ˆϕ ɛ) an assume that ˆn ˆϕ + ɛ) = ˆn 0 + 1 then: either Φ ˆn 0 ) > ˆϕ > Φ ˆn 0 + 1) or Φ ˆn 0 ) < ˆϕ < Φ ˆn 0 + 1) 13) Now consier αj) = 1 for 0 j < n 0 an αn 0 ) = α 0 an αj) = 0 for j n 0 +1. Then, conition ii) for consistent beliefs becomes: ˆϕ = Ψ ˆn 0 + 1, α 0, ˆn 0 + 1)). From continuity of the right han sie of the above expression in α 0, which ranges from Φ ˆn 0 ) for α 0 = 0) to Φ ˆn 0 + 1) for α 0 = 1) an by the inequalities 13) it follows that there must exist a ˆα 0 such that the rationality conition at the empty queue nees to be satisfie: ˆϕ = Ψ ˆn 0 + 1, ˆα 0, ˆn 0 + 1)). Then, it is easy to see that the strategy profile ˆα, efine by ˆαj) = 1 for 0 j < ˆn 0, ˆαˆn 0 ) = ˆα 0 an ˆαj) = 0 for j ˆn 0 + 1 satisfies i) of Definition 1, by construction of ˆnϕ) an also ii) of Definition 1 because ˆϕ = l0). Hence, ˆα satisfies conitions i) an ii) of Definition 1. Case 3. Assume again that no continuous point of ˆnϕ) exists for which the above fixe point equality is satisfie. Now, assume that ˆn ˆϕ + ɛ) = ˆn 1 > ˆn 0 + 1. Then, either: Φ ˆn 0 ) > ˆϕ > Φ ˆn 1 ) or Φ ˆn 0 ) < ˆϕ < Φ ˆn 1 ). 14) Similarly as in Case 2., consier αj) = 1 for 0 j < n 0 an αn 0 ) = α 0 an introuce αj) = 1 for n 0 + 1 j < n 1 an αj) = 0 for j n 1 + 1. Notice that by efinition of Ψ Equation 6) for ˆn 1 = ˆnϕ + ɛ) > ˆn 0 + 1: Φ ˆn 0 ) = Ψ ˆn 0, 0, ˆn 1 )) an Φ ˆn 1 ) = Ψ ˆn 0, 1, ˆn 1 )). From continuity of Ψ ˆn 0, α 0, ˆn 1 )) in α 0, an by the inequalities 14) it follows that there must exist a ˆα 0 such that the rationality conition at the empty queue nees to be satisfie: ˆϕ = Ψ ˆn 0, ˆα 0, ˆn 1 )). Again, it is easy to see that the strategy profile ˆα, efine by ˆαj) = 1 for 0 j < n 0 an ˆαn 0 ) = ˆα 0 an ˆαj) = 1 for n 0 + 1 j < n 1 an ˆαj) = 0 for j n 1 + 1 satisfies i) of Definition 1 by construction of ˆnϕ)) an ii) of Definition 1 because ˆϕ = l0)). Hence, ˆα satisfies conitions i) an ii) of Definition 1. Case 1, Case 2 an Case 3 result in Equations 6), 7), 8) an 9). Proof of Lemma 2: Define: { ) ν ϕ ν) =. L ν) for ν [νl, ν h 1) + for ν ν h 1, 13
Where ν ω = v ω µ ω /c. ϕ n) is the likelihoo ratio upon observing an empty queue that will make the consumer inifferent between joining a queue of length n an balking, assuming that at queue lengths 0 to n, all consumers join an informe consumers only join when the server is of high quality, i.e. when ˆn ϕ ) = n, then ϕ n ) = ϕ. When ϕ n) is increasing in n, this means that as ϕ increases, ˆn ϕ) increases in increments of 1. When ϕ n) is not monotone, this means that ˆn ϕ) increases in increments of potentially more than 1. In orer to establish this, we consier ν as a continuous variable an erive ϕ ν). We fin that over [ν l, ν h 1), the erivative can have zero or two zero points. We ignore the knife-ege case of one egenerate zero point. Let νan νbe the zero points of ν ϕ ν). It can easily be establishe that ν ϕ 0) > 0, hence, ϕ ν) increases first, reaches a local maximum an then ecreases to a local minimum, after which it increases again. It can also be establishe easily that lim ν νh 1 ϕ ν) = +. As a result, over [ν, ν], where ϕ ν) ecreases, ˆn ϕ) may increase in jumps that are larger than 1. Now, we take the erivative: ν ϕ ν) = Hence, it follows that c + v l v h ) 2 v h c ν+1 c + v l + c ν+1 v h c ν+1 ln ) ) ) c 1 v h c ν+1 + v l + c ν+1 c 1 ) 2 + v h c ν+1 ) v l + c ν+1 v h c ν+1 c v h v l + ln ) c + c ν + 1 v l ) c ν + 1 ) v l µ } l {{} 0 v l + c ν+1 v h c ν+1 ) 1 l 0 v l + c ν + 1 ) ln ) v h c ν + 1 v h c ν + 1 ) µ }{{ h } 0 ) ν ν ϕ ν) > 0 )) ln > 0 ) ln ) ) > 0 > 0 < c v h v l. 15) This is a quaratic equation in ν. Hence, ν ϕ ν) = 0 has zero, or two real solutions, epening on the parameters. Notice that the left han sie reaches a maximum when v l + c ν + 1 ) v h c ν + 1 )) = 0 ν = 1 + v h + v l ν 2c v l c ν + 1 < 0 < v h c ν + 1 1 v h v l < 0 < 1 v h v l 2 2 an as a result, assuming that v h v l > 0 is always satisfie because in the worst case, µv h > µv l all other cases are trivially satisfie), ν lies in between ν l an ν h 1. Thus, we can write: v l + c ν + 1 ) v h c ν + 1 ) 1 v h v l ) 2 16) 4 14
Plugging Equation 16) in Equation 15), we obtain that: or c v h v l > 1 v h v l ) 2 ln 4 C µ) = c + 1 4 v h v l ) ln ) ) ν ϕ ν) > 0, ν [ν l, ν h 1] > 0 ν ϕ ν) > 0, ν [ν l, ν h 1]. We have thus proven that when C µ) > 0, ϕ ν) is increasing. Hence ˆn ϕ) increases in jumps of one. When C µ) < 0, there exist two roots of Equation 15), νan ν. ϕ ν) is ecreasing in between the two roots, which implies that ˆn ϕ) will never take values over ν, ν). Hence ˆn ϕ) may increase in jumps that are larger than one. Notice that when > 1, C µ) > 0. Only when < 1, we may obtain that C µ) < 0. Proof of Lemma 3: First, we write Φ ν) as follows: Φ ν) = ν n=0 ν n=0 [ Λ ] n [ Λ ] n = [ ] ν+1 [ ] ν+1 1 Λ 1 Λ 1 Λ )/ 1 Λ ) We show that Φ ν) increases in ν iff <. Now, we consier ν as a continuous variable an etermine the conition when Φ ν) is increasing: [ ] ) ν+1 [ ] ) ν+1 1 Λ ν 1 Λ 1 Λ [ ] ) ν+1 > Φ ν) 1 Λ Λ ν µ h ν 1 Λ 1 Λ [ ] ) ν+1 > Φ ν) Λ ν as ν a ν+1 ) = a ν+1 ln a, we obtain that Now, consier ν Φ ν) > 0 ) Λ ν+1 ln ν Φ ν) > 0 an let z = x ν+1, then notice an as z lnz) 1 z ) [ ] ) ν+1 ν+1 1 Λ ln Λ µ h 1 Λ ) > 1 Λ 1 Λ µ h ln Λ 1 Λ [ ] ν+1 or: 1 Λ ) ) Λ ) Λ ν+1 ln Λ [ ] ν+1 > 1 Λ x ν+1 ln x) 1 x ν+1 ) ln z 1 ν+1 z = 1 ln z) z 1 z ν + 1 1 z [ ] ν+1 17) 1 Λ < 0 an continous ecreasing for z > 0. Hence, the conition of Equation 17) is satisfie iff Λ > Λ or >, then ν Φ ν) > 0, otherwise ν Φ ν) < 0. 15